Examples from Jerzy Trzeciak's "Mathematical English Usage -- a Glossary" Collected into one ASCII-file on August 3rd, 2005 ...... as <was> noted in Section 2. [Not: as it was noted''] ......, as claimed<desired/required>. ......, as desired <claimed/required>. ......, as is easily verified. ......, as noted <as was noted> in Section 2. [Not: as it was noted''] ......, as required <claimed/desired>. ......, contrary to assumption. ......, in contradiction with Lemma 2. ......, that is,...... ......, the constant C being independent of n. ......, the constant C being independent of...... ......, the last equality arising from (8). ......, the last inequality coming from (5). ......, the limit being assumed to exist for every real x. ......, the prime denoting the omission of the zero term. ......, where C can be made arbitrarily small by taking...... ......, where each function g is as specified <described> above. ......, where G is uniquely determined up to unitary equivalence <up to an additive constant>. ......, where the prime indicates that only terms with p>0 may appear. ......, which completes the proof. ......, which contradicts F being countable. ......, which is a contradiction. ......, which is impossible. [Not: what is impossible''] ......, which is the desired conclusion. ......, which proves the assertion. [Not: the thesis''] ......, which we can write as Df=...... ......, which, together with (2), shows...... ......, with equality if a=1. ......; examples are c0 and l1. ......(a fact which was critical to our arguments in ). ......where C can be chosen independent of n. ......where C can be made arbitrarily small by taking...... ......where C is so chosen <chosen so> that...... ......where E runs over <runs through> the family B. ......where E runs over the family B. ......where F will be defined shortly. [= in a moment] ......where none of the sums is of the form...... ......where P(d) denotes the space of (not necessarily monic) polynomial functions of degree d. ......where the Pk are polynomials. ......where the prime means that...... ......where the subscript r denotes radial differentiation. ......where the sup is taken over all intervals I. ......where zk=z... z (k times) ......with e0 denoting multiplication by f. ......with the same symbols as are used in...... (possibly with a different C) (see the last paragraph but one of page 24) (The precise definitions follow.) (This is where the local convexity of E is needed.) (with the usual modification for p={\infty}) [Not: Denote f=......'', but: Define <Write/Set/Let> f=......] [see also: basis] the base p expansion <representation> of x  contains an extension of Proposition 2 to the setting of finitely additive set functions. [Do not overuse equation'' for expressions without equality sign; use estimate, inequality, formula, relation etc. instead.] [Do not use not to negate an adjective placed before a noun. Write: a non-Lipschitz continuous function, and not: a not Lipschitz continuous function''.] [Do not write the function f being the solution of (1)'' when you mean the function f which is the solution of (1)''.] [Do not write: Every subspace is not of the form......'' if you mean: No subspace is of the form......; every has to be followed by an affirmative statement.] [Note that decreasing function is preferable to monotone decreasing function''.] [Note that downward and downwards can be used without distinction as adverbs, but the only form of the adjective is downward (e.g., in a downward direction).] [Note that where is sometimes ambiguous: where i\in I'' after a formula can mean either for all i\in I'' or for some i\in I''.] [Note the difference between besides, except and apart from}: besides usually adds'' something, except subtracts'', and apart from can be used in both senses; after no, nothing etc., all three can be used.] [Note the difference between what and which in sentences similar to the last two examples: what refers to what follows it, while which refers to what precedes it.] [Note the difference: absence = non-presence; lack = shortage of something desirable.] [Use neither when there are two alternatives; if there are more, use none.] a 180o rotation a 3 by 4 matrix = a 3× 4 matrix a 3-element set a ball of radius r a ball of radius r about the origin a ball with centre at x [Or: with centre x, centred at x] a Blaschke product with at least one zero off the origin a circle with centre a and radius r A close inspection of the proof reveals that...... A combination of (3.1) and (3.2) yields...... a complete set of representatives of the isomorphism classes of A-modules A completely different method was used to establish Theorem 2 in full generality. A computational restraint is the algebraic number theory involved in finding these ranks, which will typically be more demanding than in our example of Section 1. A concept which has proved useful in the study of measures is tameness. a conformal mapping of Q onto the complex plane minus the nonnegative real axis a connection of some kind a considerable <major/marked/radical/remarkable/significant/minor/slight> improvement A counterexample is afforded by the Klein--Gordon equation. a criterion for membership of C a crystal with hexagonal symmetry a curve parametrized on the interval [0,1] A deformation retract argument completes the proof. A detailed exposition, more suited to the purposes of the present article, is given in . A direct application of Theorem 4 only tells us that...... a direction pointing downward with respect to τ a first category set a fragment of a greater whole a function bounded below <above> by 1 a function continuous from the right a function continuous in space variables A function exhibiting this type of behaviour has been constructed by...... A function exhibiting this type of behaviour has been constructed in . A function f having the form f=...... A function given on G gives rise to an invariant function on G'. a function in two indeterminates a function of moderate growth a function of several variables [Not: of many variables''] a function with compact support a function with poles off K <vanishing off K> A further tool available is the following classical result of Chen. A geodesic which meets bM does so either transversally or...... a highly informative book a key <principal/significant/important> role A key step in obtaining (6) is Hölder's inequality. a laborious <tedious/complicated/routine/straightforward/ingenious> proof A linear transformation brings us back to the case in which...... A little reflection on the definitions makes it clear that...... a long-standing conjecture a lower bound a lucid <clear, plausible, likely, straightforward> explanation a manifold all of whose geodesics are closed [= a manifold whose geodesics are all closed] a map of period p in t a marked difference [= obvious, noticeable] a matter of primary importance a method for recognizing pure injective modules A moment's consideration will show that...... A more complete theory may be obtained by...... A natural question is how sharp the bounds given in Theorem 6 are. A necessary and sufficient condition for A to be open is that C be closed. a nowhere vanishing vector field A number of authors have considered, in varying degrees of generality, the problem of determining...... a one-to-one map a p-integrable function a partition of unity subordinate to the covering {Ui} a path obtained by going from A to B along the lower half of the circle a path starting <terminating> at x a path terminating at x a polynomial with integer coefficients A positive percentage of summands occur in all the k partitions. A principal ideal is one that is generated by a single element. a progression each of whose terms can be written as...... a rapidly decreasing function a real n× n matrix A rephrasing of the definition is that...... a result of independent interest a selection of open problems A semilattice A has breadth n iff whenever E< A and |E|>n, there is an x such that...... a sequence of distinct complex numbers a sequence of smooth domains that approximates D from within a sequence with only finitely many terms nonzero A set D is called a phase diagram if...... [No comma before if here.] a set of cardinality ω a set of no more than k elements a set whose complement is first category a set with <of> diameter 1 A short calculation shows that...... A short computation shows that...... A shortcoming of our method is the inability to compare three or more progressions. a shorter such proof A similar argument holds for the other cases. A similar reformulation can be made for...... A similar result holds for...... A similar result holds with compact'' replacing convex''. A simple argument shows that we cannot hope to have Df=0. a slight <considerable, radical, substantial> change a slight improvement a slight strengthening of Theorem 1 a slowly varying function A solution will be found by a process of trial and error. a somewhat better result A standard verification shows that...... A stronger result is in fact true. A stronger topology makes it easier for a given function to be continuous. A substantial no-man's-land persists in which neither alternative has been proved. a succession of more and more refined discrete models a succinct account of the theory A survey of the research on fn(x,y) up to 1970 (most of it dealing with the case n=1) was given in . a two-variable characterization a vector of norm 1 A very readable account of the theory has been given by Zagier . A weighted graph is one in which each vertex is assigned an integer (called its weight). a word starting with a and ending with b About 40 percent of the solar energy is concentrated in the region...... Actual construction of...... may be accomplished in a variety of ways. Actually, [3, Theorem 2] does not apply exactly as stated, but its proof does. Actually, S has the much stronger property of being convex. Actually, the proof gives an even more precise conclusion:...... Actually, Theorem 3 gives more, namely,...... Adding (7) and (8), we find that...... Adding E to both sides of (1), we can call upon (2) to obtain (3). Adding equations (2) and (3), we obtain...... [Not: Adding by sides''] Adding over n=0,1,... leads to this result:...... Addition of (2) and (3) gives...... Addressing this issue requires using the convergence properties of Fourier series. After each comparison the index is decreased. After having finished proving (2), we will return to...... [Not: finished to prove (2)''] After making a linear transformation, (9) becomes...... After making a linear transformation, we can assume...... After receiving his PhD he took a position at <he came to> the University of Texas. After receiving his PhD he took a position at the University of Texas. After the change of variable z=...... After, perhaps, passing to a subsequence, we have...... All but a finite number of the Gs are empty. All inputs of size n are equally likely to occur. All our estimates hold without this restriction. All our results can be extended in this way, but we shall stick to considering P rather than P'. All our results hold independently of whether the underlying field is R or C. All possible types are listed in Table 4. All sides were increased by the same proportion. All the computation will be done by one processor in one step. All the evidence points to the validity of the conjecture. All the Knaster continua are known to have...... All these methods had severe limitations. All three cases bear a striking resemblance:...... Almost everywhere convergence is the best we can hope for. Along the way, we come across some perhaps unexpected rigidity properties of familiar spaces. Also, F does not increase distances. Also, wherever possible, we work with integer coefficients, enabling us to obtain information about torsion. Altering finitely many terms of the sequence un does not affect the validity of (9). Alternatively, it is straightforward to show directly that...... Although  deals mainly with the unit disc, most proofs are so constructed that they apply to more general situations. Although these proofs run along similar lines, there are subtle adjustments necessary to fit the argument to each new situation. Among all X with fixed L2 norm, the extremal properties are achieved by multiples of U. Among the attempts made in this direction, the most notable ones were due to Jordan and Borel. An affirmative answer is given to the question of . [Not: a positive answer''] an algebra with unit An alternative way to analyze S is to note that...... an arc of length 1 an element of finite order an element of order a power of p an element of prime power order an element of prime power order <of order a power of p> an eventually increasing sequence An examination of the argument just given shows that...... An example is plotted in Figure 1. An example to bear in mind is behaviour in the basin of a periodic point. An explicit example is two planes through the origin. an extension of f off U An extensive treatment of the h-principle can be found in . An important point is that...... An important special case is when L is empty. an impulse acting at time t=0 an increase of 5% <a 5% increase> in the cost of living An ingenious alternative proof, shorter but still complicated, can be found in [MR]. an interval of successive integers an interval of unit length An obvious consequence of Theorem 2 is the following. An obvious question to ask is whether the assertion of Theorem 1 continues to hold for...... An obvious question to ask is whether Theorem 1 continues to hold for...... an orientation preserving homeomorphism An original impulse for this investigation came from the study of...... Analogously to Theorem 2, we may also characterize...... Analysis of the proofs of these previous results shows that...... Analysis similar to that in Section 2 shows that...... Another group of importance in physics is SL2(R). Another kind of modification is illustrated in the next lemma. Another way is to extend the definition of the index to closed curves by setting...... Any algorithm to find max must do at least n comparisons. Any congruence arises this way. Any map either has a fixed point, or sends some point to its antipode. Any other unexplained notation is as found in Fox (1995). Any vector with three or fewer 1's in the last twelve places has at least eight 1's in all. Apart from being very involved, the proof requires the use of...... Apart from these two lemmas, we make no use of the results of . Apply Theorem 3 to get a function...... Apply this to f in place of g to obtain...... Applying this argument k more times, we obtain...... area shaded with slanting lines Arguing by duality we obtain...... arranged in increasing order as <so> long as = on condition that, provided that As a by-product, we obtain an explicit formula for...... As a consequence of Lemma 2, there is...... As a first step we identify the image of Δ. As a first step we will bound A below. As a result, B is isomorphic to C. As an application, consider the Dirichlet problem Lf=0. As an example of the application of Theorem 5, suppose...... As an example, let...... As defined in Section 1, these are structures of the form...... As is customary, we use ln for Cn with the norm...... As n decreases, the class of such covers diminishes. As opposed to the situation considered in , the functions used here are...... As opposed to the situation considered in , the functions used here are...... [Not: Contrary to the situation''] As shown by Faraut,...... As shown in Figure 3, neither curve intersects X. As t runs from 0 to 1, the point f(t) runs through the interval [a,b]. As the proof will show, these properties, with the exception of (c), also hold for complex measures. As usual, we can rephrase the above result as a uniqueness theorem. As usual, we can rephrase the above result as a uniqueness theorem. [Not: As usually''] As we let t vary, f(t) describes a curve in M. As with the digit sums, we can use alternating digit sums to prove...... [= Just as in the case of digit sums] Associated with each Steiner system is its automorphism group, that is,...... Assume that such a g exists. Assume, to derive a contradiction, that...... Asymptotically, more than one-fifth of the polynomials Bn(x) are irreducible. at each step of the construction At first glance Lemma 2 seems to yield four possible outcomes. at first sight at stage n At the core of our proof of Theorem 1 is a simple counting argument. at the end of Section 2 At the expense of replacing b by b2 we may remove the condition...... At the fourth comparison we have a mismatch. at the same time = yet, still, nevertheless At the time of writing , I was not aware of this reference. at the top of page 4 At this point, the reader is urged to review the definitions of...... At this stage we appeal to Theorem 2 to deduce that...... At times [= Occasionally] it will be useful to consider...... Basic material on semigroups of operators can be found in . Before going to the proof, it is worth noting that...... Before passing to 4(b), we observe that...... Before proceeding we record an inequality for the size of an admissible X. Before we go on, we need a few facts about the spaces Lp. Besides being very involved, the proof gives no information on...... Besides being very involved, the proof requires the use...... Both cases can occur. Both conditions <Both these conditions/Both the conditions> are restrictions only on...... [ Note: the after both] Both F and G are connected, but the latter is in addition compact. Both f and g are obtained by...... [ Or: f and g are both obtained] Both its sides are convex. [ Or: Its sides are both convex.] Both of these conditions are satisfied if f is bounded (the second in view of Assumption 3). Both proofs are easy, so we give neither. Both X and Y are countable, but neither is finite. But (1) can be interpreted to mean that...... But (3) is merely an abbreviation for the statement that...... But (9) needs handling with greater care. But A has three times as many elements as B has. But Anzn is much larger than the sum of all the other terms in the series ∑ Akzk. But B is not divisible, hence C cannot be divisible either. But H itself can equally well be a member of S. But if E is not reflexive or---what is the same---w is weak, then...... But in fact we get the same thing if we consider all maps into S. But M does not consist of 0 alone. But this last assertion follows from Corollary 2, it being plain that...... [= because it is plain] But we have been unable to find any magic squares with seven square entries. But, curiously, the equivalence of (A) and (B) may fail. But...... it being impossible to make A and B intersect. [= since it is impossible to make......] But......, it being impossible to make A and B intersect. [= since it is impossible to make......] But......, it being impossible to make A and B intersect. [= since it is impossible to make] By abuse of notation, we continue to write f for f1. By allowing f to have both positive and negative forms, we obtain...... By an elementary argument,...... By analogy with...... By and large, we shall use the same notation as in...... [= In general] By assumption,...... By changing at least one of the circles to a rectangle, we obtain...... By choice of V,...... By continuity <By the continuity> of f,...... By contrast, T does not have this symmetry. By convention, we set a(x,y)=0 if no such spaces exist. By Corollary 2, distinct 8-sets have either zero, two or four elements in common. By decreasing induction on p,...... By deleting the intervals containing x, if any, we obtain...... By induction, this process produces a sequence (xn) such that...... By induction, we are reduced to proving the following lemma. By its very definition, f is continuous. By modifying the technique set out [= presented] in , we obtain...... By relatively straightforward means one can show that...... By Remark 3 below,...... By reversing the steps above, we see that...... By shrinking A to a point, we obtain...... By suitable translation of variables in (5), we may arrange that k=...... [Not: By adequate translation''] By symmetry considerations, it is sufficient to search over a region in which...... By the above,...... By the induction hypothesis,...... by the preceding lemma [Not: preceeding''] By the same kind of reasoning it suffices to consider...... By the smoothness assumption on f,...... by using a somewhat different method By way of illustration, here is an example of...... By what has been proved, there exists n such that...... By writing out the appropriate equations, we see that this is equivalent to...... Calculating V is an application of Theorem 5. Call a set a phase diagram if...... Call this distance d(f,g) for the moment. Can f(x)>1 be replaced by just x>1? Canonical products are of great interest in the study of entire functions of finite order. Care needs to be exercised in forming the Cauchy kernel of G(x). Certain other classes share this property. Chapter 2 of the classic text  by R. Nevanlinna has a detailed treatment of this construction. Choose δ in accordance with Section 8. Choose one out of these ten. Choose points x in M and y in N, both close to z, such that...... Choose Sk according to the following scheme. Choosing n small enough that na<1 gives the estimate...... Clearly, A is neither symmetric nor positive. Clearly, A is thereby put in one-to-one correspondence with B. Clearly, A{\infty} weights are sharp weights. That there are no others is the main result of Section 2. Clearly, Aj is increasing in j. Clearly, F has size ≤ p. Clearly, F is bounded but it is not necessarily so after division by G. Clearly, F is invertible except possibly on an at most countable set. Clearly, F is less than 1 in absolute value. Clearly, F is r times as long as G. Clearly, F leaves the subspace M invariant. Clearly, M is a Banach algebra having for its identity the unit point mass at 0. Collecting the terms with even powers of x, we obtain...... Compact multipliers, as one would expect, are those elements of A which...... Comparison of (2) and (3) gives...... Comparisons are done in left-to-right order. composition on the right with p Concerning (i) <Regarding (i)/For (i)>, we first prove that...... Condition (a) is so d esigned that a(x)=0. Condition (c) is intended to give us firm control over...... Condition (i) becomes more stringent as n increases. Conditions relating to bounds on the eigenvalues appear to be rare in the literature. Conjecture 2 of [KH], to the effect that [= meaning that] there is no relation P with E(P)=1, still remains open. Consequently, A has all geodesics closed if and only if B does. Consequently, A has two elements too many. [Or: A has two too many elements.] Consequently, A is compact. So also is B. Consequently, F has the Δ2 property. [= F has property Δ2.] Consequently, F is greater by a half. Consequently, H is a free R-module on as many generators as there are path components of X. Consider a pair of points a, b at distance 1. Consider the differences between these integrals and the corresponding ones with f replaced by g. Consider the family of ordered triples of elements from F. Construct an example, like that of Example 9, in which (1) fails but (2) holds. Continuing in this fashion, we get a collection {Vr} of open sets, one for every rational r, with...... Continuing this process, we get...... Continuity then finishes off the argument. convergence in probability <in distribution> Corollary 2 generalizes and strengthens Theorem 3 of . Corresponding to each choice of V there is a function f such that...... curves starting at the origin Define A to be the matrix with 1 in the (i,j) entry and 0 elsewhere. Define f(z) to be that y for which...... Define the category of...... to have as objects pairs of......, and as morphisms...... Definition 3 is motivated in part by certain differential operators to be introduced in Section 3. Denote by θ the angle at x which is common to these triangles. Denote by P the space of...... Denote by q the point of A nearest to p. Denote the largest of those wi by wp. Should there be no such wi, let wp=0. [= If there is no] Differentiation of (5) and (6) gives the respective equations A=B and C=D. Does the limit of f(z) exist as z goes to zero? If so, what is it? Draw the half-line from x at angle φ to θ. Each A(n) corresponds to an element A'(n) in V, and similarly for B(n). Each Ai meets A in a finite set. [= Ai∩ A is finite] Each component that meets X lies entirely within X. Each component which meets X lies entirely within Y. Each f can be expressed in either of the forms (1) and (2). Each f lies in zA for at least one A. Each f lies in zA for some A. Each factor in (4) has absolute value 1 on T. Each of the functions on the right of (9) is one to which our theorem applies. each of the spaces concerned Each of the terms that make up G(t) is well defined. Each of these three integrals is finite. Each of these three integrals is finite. These curves arise from......, and each consists of...... Each row of A has a single ± 1 and the rest of the entries 0. Each set A carries a product measure. Each tree is about two-thirds as deep as it was before. Each vertex is adjacent to q others. Each x here really designates the pair (x,Ax). Either f or g must be bounded. equal up to sign Equality holds <occurs> in (9) if...... Equality is achieved only for a=1. Equality is attained only for a=1. Equality occurs in (9) if...... Equating coefficients we see that...... Equating the coefficient of x2 in V to zero, we get...... Equation (2) now reads Ax=...... Essential to the proof are certain topological properties of G. Even in the case n=2, the application of Theorem 6 gives essentially nothing better than the inequality...... Even though we were able to derive a formula for......, it is not easy to use. Every element of A has f-value 2. [= the value of f at this element is 2] Every F is a sum of irreducible elements. Every invariant subspace is of the form...... Every nonincreasing function occurs as the rearrangement of some C(P) function. Every open set is a union of balls. Every path on G passes through vertices of V and W alternately. Every possible such sequence gives rise to...... Every possible such sequence gives rise to...... [Not: gives raise''] Every region in the plane (other than the plane itself) is conformally equivalent to U. every such map Examination of the left and right members of (1) shows that...... Examining how the Lipschitz constant depends on F, we find that...... Examples 1 and 2 give two operators, the former bounded and the latter not, with...... Examples abound in which P is discontinuous. Examples are given to show that...... Except for these two lemmas, we make no use of the results of . Exercises 2 to 5 [= Exercises 2--5; amer.: Exercises 2 through 5] Expand f in powers of x. expansion in powers of x Explicitly, we have the formula...... Extend this sequence of numbers backwards, defining N-1, N-2 and N-3 by...... Fn tends to zero as n→{\infty}. Few of various existing proofs are constructive. Finally, (2) shows that...... [Not: At last''] Finally, (d) is clear from the very last statement of Theorem 4. Finally, case (E) is completed by again invoking Theorem 1. Finally, multiplication by a permutation matrix will get the exponents in descending order. Find integral formulas by means of which the coefficients cn can be computed from f. First we take up the trivial case h=0. First, some remarks concerning the term closure'' are in order. First, we establish a technical point already alluded to in Section 3. Fix n and let c vary. Fix n for the moment. Following the argument in , set...... Following these preliminary remarks, we now state...... For (ii), consider...... [= To prove (ii), consider] For a counterexample, consider S=...... For a fuller discussion of this topic, see . For a list of relevant references, see . [= references connected with the subject being considered] For a recent account we refer to . For a survey of what is known to date, see [G]. For a thorough treatment of...... we refer the reader to Section 5 of [Ho]. for all f of the type specified in Theorem 4 for all n except a finite number for all sufficiently small h for any family connecting the two flows shown for any two triples [Not: for every two triples''; every'' requires a singular noun.] For background information, see . For binary strings, the algorithm does not do quite as well. For both C&#8734; and analytic categories,...... For brevity, we drop the subscript t on ht. For brevity, we drop the subscript t on ht. [Not: For shortness''] For convenience of exposition, we work with an error term of the form...... For convenience we ignore the dependence of f on g. For convenience we set a(0)=0 <we ignore the dependence of f on g>. For D a smooth domain, the following are equivalent. For direct constructions along more classical lines, see . For direct constructions along more classical lines, see [KL]. For ease of notation, set I=If. For every g in X <not in X> there exists an X...... [ But: for all f and g, for any two maps f and g; every is followed by a singular verb.] For example, F reaches a relative maximum of 5.2 at about x=2.1. For explicit solutions, it may be necessary to have rather precise information about the amplitude φ. For f not in B,...... For f=g, this specializes to the result of . For F=R, no integration over M is needed in (5). For fixed k,...... For fixed n,...... For general linear operators, there is not such an extensive functional calculus as there is for self-adjoint operators. For general rings, Out(R) is not necessarily well-behaved. for greater generality For j=1 the operator is bounded, yet the integral (8) fails to be finite. For j=1 the operator is bounded, yet the integral (8) fails to be finite. [= but the integral] for k in the indicated range For K, this is no longer true. [ Compare: For K, it is no longer true that......] For k=2 the count remains as is. For later use in conjunction with the weighted averages occurring in (2), we next consider...... [Note the double r in occurring.] For later use, we record the following formulas:...... For m not an integer, the norm can be defined by interpolation. For more details we refer the reader to . For most of the proof it suffices to use the rough bound p<1. for n sufficiently large for no x apart from the unique solution of...... [Note the difference between besides, except and apart from}: besides usually indicates adding'' something, except subtracts'', and apart from can be used in both senses; after no, nothing etc., all three can be used.] for no x besides the unique solution of...... For no x does the limit exist. [Note the inversion after the negative clause.] for no x except the unique solution of...... For our application though, we need a stronger statement. for p in the range 1<p<{\infty} for p near enough to q For real x,...... For simplicity of notation <By abuse of notation>, we use the same letter f for...... For simplicity, we confine attention to radial moments. For simplicity, we suppress the explicit dependence on x in the notation. for some set of input data for t close to 0 For the benefit of Pascal programmers, we explain...... For the converse, consider...... For the opposite inclusion, suppose that...... For the other direction, take...... For the sake of clarity, we shall indicate in what follows to which space X belongs. for the sake of simplicity = for simplicity For these considerations it will be convenient to state beforehand, for easy reference, the following variant of...... For this it is sufficient to check that x is in V, or, what is equivalent, that a(x) is bounded. For this purpose, it is necessary to understand the mapping properties of B on as large a function space as possible. For this purpose, we first define...... For which f does equality hold in this inequality? Formula (7) exhibits the same phenomenon. Fortunately, F does not get too close to p. Fortunately, there is a very satisfactory solution to this problem, due to Vermes. From (5) we infer that...... From (ii), with an obvious change in notation, we get...... from line 6 onwards From now on all logarithms are to base two. From now on we confine attention to R2. From now on, F will be fixed. From Proposition 2, and in view of Theorem 3, it suffices to show that...... from stage A up to, but not including, stage B From the viewpoint of the Fox theorem, there is not an exact parallel between the odds and the evens. functions of several variables [Not: of many variables''] Functions which are equal almost everywhere are indistinguishable as far as integration is concerned. Further motivation for looking at ideal class groups comes from the field of cryptography. Further research may yet explain the enigma. Further results along these lines were obtained by Clark . Furthermore, K is an upper bound on <for> f(x) for x in K. Galois correspondences are uninteresting from the dynamical point of view. Generally we add a tilde to distinguish between quantities associated with \tilde G and those associated with G. Generally we add a tilde to distinguish between quantities associated with \widetilde G and those associated with G. Give a proof of Theorem 2 which requires no knowledge of the boundary values of f. Given δ >0, we can find β such that...... H is a free R-module on as many generators as there are path components of X. ha analysis to follow only covers the case d=1. Half of the sets of R miss i and half the remaining also miss j. Half of the sets of the family R miss i and half the remaining also miss j. Having illustrated our method in Section 2, we can afford to be brief in our proof of Theorem 5. He accounts for all the major achievements in topology over the last few years. He accounts for all the major achievements in topology over the last few years. [= He records] He held a Rockefeller Foundation fellowship. He held the Courant Chair at New York University for three years before his retiring. He is currently Professor of Mathematics at Texas State University. He makes the following comments:...... He received his master and Ph.D. degrees from the University of Texas. He received his master and PhD degrees from the University of Texas. He was the first to propose a complete theory of triple intersections. Hence (8) can be rewritten in the form...... Hence A and B are at distance precisely d. Hence A fails to satisfy (3). Hence A has the stated continuity properties. Hence both decay exponentially as x→{\infty}, therefore certainly remain bounded. Hence F is compact, which yields M=N. [Not: what yields''] Hence F is invariant under φ. [= φ-invariant] Hence F is invertible except at countably many points. Hence F is twice as long as G. Hence F vanishes to order 3 <to infinite order> at x. Hence fn also converges to f. Hence M is the unique largest submodule of type (a). Hence there can be no condition on the norms which guarantees (7). Hence V(x) is the maximum value of Jx(v) over all controls v. Hence we would expect the functions fj to behave similarly. Hence we would expect the functions...... to behave similarly. Hence Z enters D without meeting x=0. Hence, although the topology of reducts of A is uniformly controlled, so to speak, by that of A, the model theory of the reducts can be much wilder. Hence, by (7) again, we have...... Here (1) can be interpreted to mean that...... Here (6) merely means that...... Here {x} is the set whose only member is x. Here a and b are chosen to satisfy...... Here a and b are chosen to satisfy...... [Not: to verify''] Here A is a coset of H in G. [H is a subgroup of G.] Here A is small relative to B. [= in comparison with B] Here A, B and C all belong to the same class, which we represent by the symbol P2. Here and subsequently, M denotes...... Here and throughout, P(E) denotes...... Here are some elementary properties of these concepts. Here are some other situations in which we can draw conclusions only almost everywhere. Here B and C are nonnegative numbers, not both 0. Here C behaves covariantly with respect to maps of both X and G. Here c denotes a constant which can vary from line to line. Here D0 and D1 are discs with the same centre, namely b. Here dx stands for Lebesgue measure. [Or: the Lebesgue measure] Here F is assumed to be open. Here F is only defined up to an additive constant. Here F is strictly less than G. Here F is the sum of a collection of...... Here f takes over the role of the time parameter. Here Ff is the previously defined Fourier transform of f. Here G enters essentially in an algebraic way. Here G is discontinuous and, what is more, it does not belong to V. Here is a more explicit statement of what the theorem asserts. Here is a simple direct proof. Here is an alternative phrasing of part (1):...... Here is an example of...... Here is another way of stating (c). Here J is defined to equal Af, the function f being as in (3). [= where the function f is......] Here L can be taken as large as desired. Here one seeks to minimize P(x) over the class of feedback controls. Here p2(r) is the sum of the squares of the divisors of r. Here Q can, for definiteness, be taken as Ff. Here S may be P, but it may also not be P. Here s takes over the role of the time parameter. Here U is considered <viewed/regarded> as acting on M. Here u+ and u- are the positive and negative parts of u, as defined in Section 5. Here we allow a=0. Here we are viewing the coefficients as reduced fractions. Here X assumes values 0,1,...,9, each with probability 1/10. Here Y is a Poisson variable suitably chosen to approximate X in distribution. Here, continuity precludes the existence of singularities. His argument is as follows. His method of proof was to first exhibit a map...... His proof is unnecessarily complicated. His techniques work just as well for general v. Homogeneous spaces are, in a sense, the nicest examples of Riemannian manifolds. How close does Theorem 1 come to this conjecture? How is the result affected if we assume merely that f is bounded? How many entries are there in this section? How many more men attended the meeting than the women? How many multiplications are done on average? How many of them are convex? How many such expressions are there? How many zeros can f have in the disc D? How much of the foregoing can be extended to the noncompact case? How\-ever, particularly in the case of...... However small a neighbourhood of x we take, the image will be...... However, (1) has been proved so far only in the case where...... However, (5) is sufficient to guarantee invertibility in A. However, (5) is superior to (4) only when n is cubefree. However, (9) needs handling with greater care. However, (ii) is nothing but the statement that...... However, A clearly has a topology T on it, namely,...... However, A fails to satisfy (3). However, a few tentative [= cautious, not firm] conclusions can be drawn. However, a few tentative conclusions can be drawn. [= cautious, not firm] However, even for k=1 the problem proved quite difficult. However, F can be as great as 16. However, F is only nonnegative rather than strictly positive, as one may have expected. However, if B were omitted in (1), the case n=0 would imply Nf=1, an undesirable restriction. However, in many applications the existence of such an R is practically impossible to verify. However, it does not seem to have been observed that...... However, it need not be the case that V> W, as we shall see in the following example. However, M is generally not a manifold. However, no extension in this direction has appeared in the literature. However, not every ring enjoys the stronger property of being bounded. However, one needs to perform computations which become time consuming as n increases. However, only 5 of these are distinct. However, the chance of success is very slight. However, the connection with Gromov's work has been obscured in recent years by an emphasis (in the algebraic topology literature) on configuration spaces. However, they now differ by a considerable amount. However, this argument can be easily refuted by showing that...... However, this argument is fallacious, because as remarked after Lemma 3,...... However, this cannot be proved of the cardinal function d(X). However, this equality turned out to be a mere coincidence. However, this still left open the possibility that the converse of Theorem 3 held for m=3. However, to our knowledge this is not fully resolved. However, we immediately encounter the problem of nonregularity of the data. However, we know of no way of deriving one theory directly from the other. However, we shall show in Section 3 that this simply results in Definition 3 again, albeit with complex weight. However, X does have finite uniform dimension. However, X is not finite, nor is Y countable. [Note the inversion.] I am greatly indebted to S. Brown for this example. I shall limit myself to three aspects of the subject. Identical conclusions hold in respect of the condition BN. [= concerning BN] If (ii) is required for finite unions only, then M is called an algebra of sets. If A consists of at most one point, then...... If A is such an operator (appropriately chosen) then...... If a, b, and c are permuted cyclically, the left side of (2) is unaffected. If A=B, how does the situation differ from the preceding one? If any element is removed from an orthonormal set, its span is diminished. If f is an n-simplex in U, then f' is one in V. If f is as in (8), then...... [Not: like in (8)''] If f maps D into itself, then T is commonly called a composition operator. If H is decomposed in the form H=...... If it were true that......, the same argument would apply to f and would show...... If K is empty, part of the hypothesis is vacuously satisfied. If K is now any compact subset of H, there exists...... [Any = whatever you like; write for all x'', for every x'' if you just mean a quantifier.] If n=1, there is nothing to prove. If no confusion can arise, we write K for both the operator and its kernel. If one studies the proof of...... it is apparent that (2) is never used. If one thinks of x, y as space variables and of z as time, then...... If p=0 then there are an additional m arcs. [Note the article an.] If t does not appear in P at all, we can jump forward n places. If the boundary is never hit then xt is a Feller process under reasonable continuity assumptions. If there are to be any nontrivial solutions x then any odd prime must satisfy...... If there is an f...... then...... If there are <is> none, we define...... If this happens on a set of positive measure, then f cannot be continuous. If this is not so, a linear fractional transformation will make it so. If this is so, we may add...... If we adjoin a third congruence to F, say a≡ b, we obtain...... If we adjoin a third congruence to F, say a= b, we obtain...... If we apply induction to (3) we see that...... If we combine this with Theorem 2, we see that...... If we know a covering space E of X then not only do we know that...... but we can also recover X (up to homeomorphism) as E/G. If we know a covering space E of X then not only do we know that......, but we can also recover X (up to homeomorphism) as E/G. If we prove (8), the assertion follows. If we prove that G>0, the assertion follows. If we restrict A to sections coming from G, we obtain...... If we simply mimic the standard proof of...... we are led to...... If we simply mimic the standard proof of......, we are led to...... If y is a solution, then ay also solves (3) for all a in B. If...... then (1) holds but not (2). If...... then R is right Noe\-ther\-ian provided R is semiprime  or commutative  or R/N has zero socle. If......, it is customary to write...... rather than...... Illiterates comprise 20% of the population. [= constitute] Implementation is the task of turning an algorithm into a computer program. Important analytic differences appear when one writes down precisely what is meant by...... Important cases are where S=...... In , this theorem is made the starting point of Gelfand theory. In  we only allowed weight functions that were C1. In  there occur the following formulas. In [K] we raised the question of recovering X from XM. In 1925 Franklin, unaware of Stackel's work, showed...... In 1935 he was promoted to the rank of associate professor at Cornell. In 1987 he went to Delhi University. In 1988, while attempting to generalize this result, the second author noticed that...... In 2000, two important number theory conferences were held at Princeton University. in a broader perspective in a companion paper  In a companion paper  we treat the case...... in a concise manner in a fairly straightforward way In a sequel to the present article, we shall consider...... [= in another paper to be published; do not write In the sequel'' when you mean: In what follows.] in a similar fashion in about 1885; in the year 2000; as early as 1885; in Hilbert's 1905 paper; the revised 1993 edition In addition to a contribution to W1, there may also be one to W2. In addition to f being convex, we require that F be holomorphic. [Note the subjunctive be.] In addition to f being convex, we require that...... In addition to illustrating how our formulas work in practice, it provides a counterexample to Brown's conjecture. In addition to illustrating how our formulas work in practise, it provides a counterexample to Brown's conjecture. In all our analysis, only the order of magnitude of P will be significant. In an effort to generalize Robert's method, we gave in  the following criterion for non-finite generation of kernels. in an exactly similar way In analogy with (1) we have...... In brief outline, here is the main idea of the proof. In case F=R, no integration over M is needed in (5). In case M is empty we write simply f≈ g. In closing this section we take up a result which will play a pivotal role in the characterization of...... In contrast to the previous example, membership of D(A) does impose some restrictions on f. In contrast, Theorem 2 shows that...... In detail the classification is complex, but in essence it is simple. in diagonal form in dimension n In dimensions three or more, this would imply...... In discussing structures, we shall employ the standard terminology of first-order logic. In either case, it is clear that...... [= In both cases] In fact, the computation for A becomes somewhat simpler. In fact, this proof shows that a stronger result holds, namely,...... In fact, we can do even better, and prescribe finitely many derivatives at each point of A. In fact, we can do even better, and prescribe finitely many derivatives at each point of M. In Figure 2, the set A is marked by a square with a small triangle on top. in geometric language In his Stony Brook lectures, he laid great emphasis <stress> on the use of...... In his Stony Brook lectures, he laid great emphasis on the use of...... in infinite dimensions In less precise language, the requirement is that the two angles are the same in size and in orientation. In light of <In the light of> the obvious inequality a≥ b,...... In more detail, the assertion is this: if...... In most cases it turns out that...... In neither case can f be smooth. [Note the inversion after the negative clause.] in nine cases out of ten In order to justify this, we now show that...... In order to make our description of this process precise, we define...... [Not: to precise our description''] In order to state these conditions succinctly, we introduce the following terminology. In other algorithms, this may not be true. in paper  [Or: in the paper ; better: in ] In particular, E contains a copy of l1. In particular, for only finitely many k do we have F(ak)>1. [Note the inversion after only.] In particular, the theorem applies to weakly confluent maps. In representation theory, there can never be a B-map whose domain is finite-dimensional. in reverse order In Section 2 we review the separation of variables formula. In Section 2 we set up notation and terminology. [= prepare] In Section 2, we lay the foundations for a systematic study of...... In Section 3 we proceed with the study of...... [= start or continue the study] In Section 4 we worked out a fairly detailed picture of linear H-systems. In short,...... [= To sum up briefly,......] in some degree in striking contrast to...... In summary, if F is...... In that case Y need only be metrizable (rather than completely metrizable). In that population, women outnumbered men by 2 to 1. in that same paper in the Boolean algebra sense In the case k=0 condition (A) becomes vacuous. In the case of finite additivity, we have...... In the case of n≥ 1 <In case n≥ 1>,...... [Not: In case of n>1''; better: If n≥ 1 then......] In the contrary case,...... In the course of proof, we will encounter...... In the course of writing this paper we learned that P. Fox has simultaneously obtained results similar to ours in certain respects. In the event of a tie, the winner is decided by the toss of a coin. In the final section of the paper, we list some open problems. In the following applications use will be made of...... In the following, all topological notions refer to the weak topology of Y. In the function field case the poles at s=0 and 1 are still present. In the next section we introduce yet another formulation of the problem. In the next theorem, we give fairly minimal conditions that imply...... In the notation above <In the above notation>,...... In the physical context already referred to, K is the density of...... In the physical context already referred to, K is the density of...... [Note the double r in referred.] In the plane, the open sets are those which are unions of open circular discs. in the present context in the previous chapter [= in the immediately preceding one] In the process of replacing each A by a smaller set, we cannot make B so small that m(B)<1. in the ratio of approximately 3:1. In the remainder of this section we will be trying to answer the question:...... In the set-up of condition (H), let...... In the study of infinite series ∑ an it is of significance whether the an approach zero rapidly. In the year 2000 <In 2000>, two important number theory conferences were held at Princeton University. In Theorem 2 we show that the bound is not far from best possible. In this and the other theorems of this section, the Xn are any independent random variables with a common distribution. In this case it is advantageous to transfer the problem to (say) the upper half plane. In this case the method of  breaks down. In this case we need to take into account the difficulty with fitting a ball into a sector. The obvious remedy is to replace the ball by a suitable disk. In this connection, we remark that...... In this paper emphasis is on the case where...... In this section I shall focus attention on...... In this section we discuss in some detail the relationship between...... In this section we discuss in some detail the relationship...... In this section we investigate under what conditions the converse holds. In this section we return to our general consideration of p(R). In this section, however, we will not use it explicitly. In this way, D yields operators D+ and D-. These are formal adjoints of each other. In what follows <In all that follows>, L stands for...... In what follows, we will concern ourselves only with...... Incidentally, our computation shows that...... inequality in the opposite direction infinitely many times [= an infinite number of times] infinitely often Influenced by (2) of Theorem 3, Danes (1989) suggested that...... Informally said, gaps go to gaps. Inserting additional edges destroys no edges that were already present. Inside U, the zeros are also quantitatively restricted. Instead of using the Fourier method we can multiply...... [Not: Instead using''] Integrating both sides over X shows that...... Intuitively, entropy of a partition is a measure of its information content---the larger the entropy, the larger the information content. Is there a relation between A and B? There certainly is not if...... It became clear that the Riemann integral should be replaced by some other type of integral, better suited for dealing with limit processes. It becomes impracticable to compute the zeros of F for degrees greater than 6; in any event, deciding whether the divisors found in this way represent irreducible curves becomes increasingly difficult. It can be inferred from known results that these series at best converge conditionally in Lp. It contributes half of the amount on the right hand side of (1). It does not appear feasible to adapt the methods of this paper to...... It does not matter in the least whether...... It follows from the way f was defined that...... It follows from this representation together with Lemma 3 that...... It follows that a is positive. [= Hence <Consequently,/Therefore,> a is positive.] It follows that G is maximal under <for> the usual partial ordering of B. It follows that the semigroup St is none other than e(t)T. It follows, by interchanging the roles of X and Y, that...... It has been known for some time that...... It has long been known that...... It has properties reminiscent of partition functions. It has some basic properties in common with another most important class of functions, namely, the continuous ones. It has to be assumed that...... It is a consequence of the Hahn--Banach theorem that...... It is a pleasure to thank R. Greenberg for bringing his criterion for...... to our attention, and for generously sharing his ideas about it. It is a pleasure to thank...... It is a routine matter <a matter of routine> to show that...... It is a simple matter <a routine matter/a matter of routine> to show that...... It is a simple matter to remove all type 1 edges. It is almost as easy to find an element...... It is also clear that their are extensions to......, but they do not seem to be worth the effort of formulating them separately. It is also clear that there are extensions to the case of......, but they do not seem to be worth the effort of formulating them separately. It is also not difficult to obtain the complete additivity of μ. It is also tempting to get round this problem by working with...... It is an elementary check that A is a vector space. It is an idea worth carrying out. [Not: worth while carrying out''] It is assumed that...... It is best not to use...... It is beyond the scope of this paper to give a complete treatment of...... It is convenient to view G as a nilpotent group. It is easily seen that...... It is easy to see, by means of an example, that...... It is for this reason that his argument is incorrect. It is generally a highly nontrivial question whether...... It is helpful to keep these similarities in mind. It is here assumed that...... It is hoped that a deeper understanding of these residues will help establish new results about the distribution of modular symbols. It is important that the orders of F and G are comparable, a statement made more precise by the following lemma. It is important to keep in mind that...... It is important to notice some of the weaknesses inherent in the above approach. It is important to pay attention to the ranges of the mappings involved when trying to define...... It is impossible to predict the eventual outcome of the process. It is in all respects similar to matrix multiplication. It is in all respects similar to matrix multiplication. [Not: similar as''] It is interesting that (1) is a necessary condition in a much larger class of functions, which we now describe. It is intuitively clear that the amount by which Sn exceeds zero should follow the exponential distribution. It is not clear to what extent this can be generalized to other varieties of loops. It is not generally possible to restrict f to the class D. It is not immediately obvious what this generalization has to be. It is now apparent what the solution for K will be like:...... It is perhaps worth remarking that...... It is possible for the hull of J to have exactly two points. It is possible that the methods of this paper could be used to......, but there remain considerable obstacles to overcome. It is Proposition 8 that makes this definition allowable. It is smoothness with which we are specifically concerned. It is sufficient to make the computation for T. It is sufficient to prove that this vector field points outwards on ∂ M. It is the connection between...... that will concern us here. It is the freedom of choice of D in this construction that enables us to...... It is therefore enough to show that...... It is therefore of interest to look at the asymptotic behaviour of...... It is therefore of interest to look at...... It is therefore reasonable that the behaviour of p should in some rough sense approximate the behaviour of q. It is therefore unnecessary to specify G on M. It is this idea that underlies some of the results of . It is this point of view which is close to that used in C*-algebras. It is unlikely that the disturbances will eventually disappear. It is written clearly and concisely. It may be difficult to write down an explicit domain of F. It may be worth reminding the reader that...... It may be worth reminding the reader that...... [Not: reminding that''] It may happen that B1 is the only compatible compactification. It may seem strange to define 0.{\infty}=0. It may well be that no optimal time exists, as the following example shows. It meets only countably many of the Yi. It might be argued that the h-principle gives the most natural approach to...... It often does not matter whether...... It produces the same outcome whichever path is taken. [= no matter which] It remains to exclude the case where...... It seems likely that the arguments would be much more involved. It seems likely that their results can be extended to...... It seems plausible that{......} but we have been able to establish this only in certain cases. It seems preferable, for clarity's sake, not to present the construction at the outset in the greatest generality possible. It seems reasonable to expect that......, but we have no proof of this. It seems that the relations between these concepts emerge [= become apparent] most clearly when the setting is quite abstract, and this (rather than a desire for mere generality) motivates our approach to the subject. It seems that the relations between these concepts emerge most clearly when the setting is quite abstract, and this (rather than a desire for mere generality) motivates our approach to the subject. It seems that the solution of Problem 1 is still out of reach. It should be no surprise that a condition like ai≠ bi turns up [= appears, shows up] in this theorem. It should be no surprise that a condition like ai≠ bi turns up in this theorem. It should be noted that...... It should be stressed, however, that...... It simplifies the argument, and causes no loss of generality, to assume...... It suffices to take a suitable translate of U. It supplies the key to the proof of Theorem 2. It turns out that A is not merely symmetric, but actually selfadjoint. It turns out that A, B and C all belong to the same class, which we represent by the symbol P2. It turns out that it suffices to show that...... For if this is proved, the preceding remark shows that...... It turns out that nothing more need be done to obtain...... It turns out that these properties play no role in the proof. It turns out that this problem has a very natural connection with...... It will cause no confusion if we denote...... It will eventually appear that the results are much more satisfactory than one might expect. It will thus be sufficient to...... [= therefore] It will usually be assumed that...... It would clearly have been sufficient to assume...... Iterated correspondences display many of the features of...... Its description is far from complete. Its restriction to M is obviously just f. Keep in mind that we are now using algebraic notation. Keep only those vertices whose coordinates sum to 4. [= add up to 4] Kirk, building on work of Penot, developed a more abstract version of...... Knowing this matrix is equivalent to knowing the multiplicities of the <mbdai. Later, by bringing in the injectivity radius, Fong simplified the argument. Lebesgue discovered that a satisfactory theory of integration results if the sets Ei are allowed to belong to a larger class of subsets of the line. Lemma 3 does not require D to be prime. Lemma 3 suggests that we start by considering A1. Less than 1 in p of its points will result in a quartic with ideal class number p. Let (an) be the sequence of zeros of f arranged so that |a1|≤|a2|≤...... Let <mbda be an eigenvalue of A, and let v be an associated eigenvector. Let 2I denote the interval concentric with I but of twice its length. Let A be the union of the sets f(Q) for f in F. Let A denote the rectangle B rotated through π/6 in a clockwise direction about the vertex (0,1). Let Ai be disjoint members of M whose union is E. Let An be a sequence of positive integers none of which is 1 less than a power of two. Let An be a sequence of positive integers none of which is one less than a power of two. Let ABC be an angle of sixty degrees. Let D be a disc (with centre at a and radius r, say) in C. Let D be the coefficient of xn in the expansion of f. Let D be the plane with three disjoint discs removed. Let E be Cantor's familiar middle thirds set. Let E be Cantor's familiar middle thirds set. Exercises 2 to 5 [= Exercises 2--5; amer.: Exercises 2 through 5] Let f be a map with f|M having the Mittag-Leffler property. Let F be as just described. Let F be of class Tk for some k. Let f be the linear form g→ (m,g). Let F be the unique map such that...... Let f satisfy (2). [Not: Let f satisfies''] Let I be the family of all subalgebras that contain F. [Or: which contain F] Let I be the family of all subalgebras which contain F. [Or: that contain F] Let M be the intersection of the sets Mi. Let M be the manifold to whose boundary f maps K. Let P be a point of U closest to Q. Let q be the maximum number of variables occurring in...... [Note the double r in occurring.] Let R be a ring and A a right R-module. Let R consist of all z=x+iy such that |x|<1. Let r1, r2,... be an enumeration of the rationals in [0,1]. Let Rn be Euclidean n-space. [Or: the Euclidean n-space] Let S be the set of all solutions of (8) of the form (3). Let Si be the first of the remaining Sj. Let T be an isometric semigroup as above. Let u and v be two distributions neither of which has compact support. Let us continue with the proof of Theorem 2. Let us emphasize that K was not assumed to be connected in Theorem 4. Let us introduce the temporary notation Ff for gfg. Let us introduce the temporary notation Ff for gfg-1. Let us make the following observation <assumption/definition>. Let us note (for later <future> reference) that...... Let us now prove directly (without recourse to ) that...... Let us now take a look at the class N, with the purpose of determining how...... Let us now take a quick look at the class N, with the purpose of determining how much of Theorems 1 and 2 is true here. Let us stress that c is a term and not a subset of C. Let us take B as our range space. Let v be an arbitrary control and xt the corresponding controlled process. [Not: the respective process''; respective'' requires a plural noun.] Let x1, x2 be variable points in the intervals (a,b), (c,d) respectively. Let x1,...,xn be the distinct values assumed by f and A1,...,An the respective sets where these values are assumed. Letting m tend to zero identifies this limit as H. Letting m→{\infty} identifies this limit as H. Likewise, if A does not span C(I), removal of any of its elements will diminish the span. Likewise,...... [= Also,...... = Moreover,......] Littlewood proved that certain subharmonic functions on the unit disc resemble bounded analytic functions in having radial limits almost everywhere. lower-case letters [≠ capital letters = upper-case letters] Making use of a technical estimate, the proof of which is deferred to Section 3, we establish in Section 2 the estimates for...... Many equivalents are known for this property. Many interesting examples are known. We now describe a few of these. Many of the results that follow only require W to satisfy (8). Many of them were already known to Gauss. many such maps Mary Lane deserves our special thanks for her part in bringing this volume to a successful completion. Maurer, in the same year, showed that...... More concisely, we consider the measurable space (X,B). More precisely, f is just separately continuous. More specialized notions from Banach space theory will be introduced as needed. Moreover, {x} is the set whose only member is x. Moreover, H is a free R-module on as many generators as there are path components of X. Moreover, it follows at once from (1) that...... Moreover, M splits as K× L. Most distribution spaces can be characterized through the action of appropriate convolution operators. Most measures that one meets are already complete. Most of the theorems presented here have never been published before. Most of these phenomena can be subsumed under two broad categories. [= included in] Most of this book is devoted to...... Most probably, his method will prove useful in...... Motivated by (4), we obtain the following characterization of...... Much less is known about hyperbolically convex functions. Much of the foregoing can be extended to the noncompact case. Much of the rest of the paper is devoted to a general study of...... Multiplying through by f and integrating shows that...... near the top of the scale Necessarily, one of X and Y is in Z. Neither (1) nor (2) alone is sufficient for (3) to hold. Neither is the problem simplified by assuming f=g. Neither of them is finite. Neumann's characterization of the elements which induce an indecomposable operator Nevertheless, in interpreting this conclusion, caution must be exercised because the number of potential exceptions is huge. Next divide J in half. Next we relabel the collection {An,Bn} as {Cn}. Next, (1) shows that (2) holds whenever g=f(n) for some n. Next, F preserves angles at each point of U. Nitrogen constitutes 78% of the earth's atmosphere. No analogue of such a metric appears to be available for Z. No point of T can be a regular point of f. No two members of A have an element in common. No x has more than one inverse. non-overlapping intervals None of these implications can be reversed. Note in connection with (iv) that...... Note that (2) is a slightly weakened version of the Pólya inequality. Note that (2) serves as the definition of its left side. Note that (3) is merely an abbreviation for the statement that...... Note that (4) covers the other cases. Note that (A) means precisely that condition (B) is not satisfied. Note that An makes only a contribution of at most N2. Note that both sides of the inequality may well be infinite. Note that C behaves covariantly with respect to maps of both X and G. Note that dim E, if it exists, is...... Note that E can be given a complex structure by setting...... Note that F is defined only up to an additive constant. Note that f is determined only to within a set of measure zero. Note that F is independent of the choice of the family S. Note that F is only nonnegative rather than strictly positive, as one may have expected. Note that F is the product over the integers m in B. Note that f=lim fn automatically exists. Note that for semilattices one could not claim that A is algebraically closed in B. Note that M being cyclic implies F is cyclic. Note that m is permitted to vary with the number of inputs. Note that no boundedness assumption is made in this definition; in fact, this would be redundant as shown by Theorem 3 below. Note that O(g) depends on g only through its differential dg. Note that P comes before Q along the arc l. Note that R2 witnesses that A is compact. Note that some of the an may be repeated, in which case B has multiple zeros at those points. Note that some of the Xn may be repeated. Note that the apparently infinite product in the denominator is in fact finite. [= seemingly] Note that the P produced in Theorem 2 need not have dP=0. Note that there is a mistake on p. 3 of , where the condition...... should be deleted. Note that this lemma does not give a simple criterion for deciding whether a given topology is indeed of the form Tf. Note that this too is best possible. Note that we explicitly exclude {\infty} from the values of a simple function. Note that X decomposes into a direct sum of 2-planes. Note that, unlike the maximal function, the Hilbert transform is not...... Note the contrast with Theorem 3. Note the similarity with Lusin's theorem. Now (1) can be interpreted to mean that A=B. Now (1) follows after passage to the limit as n→{\infty}. Now (1) is just (2) and (3) combined. Now (1) splits into the pair of equations...... Now (2) and (3) together are equivalent to (4). Now (2) is clearly equivalent to the assertion that...... Now (3), specialized to our case, becomes...... Now (5) follows from (4) if (2) is applied to the last equation. Now (8) makes it obvious that...... Now (c) asserts only that the overall maximum of f on U is attained at some point of the boundary. Now (d) is clear from the very last statement of Theorem 4. Now A consists of those vectors with eight 2's. Now A, B and C all extend to a small neighbourhood of x. Now choose t appropriately as a function of β. Now E, F and G all extend to U. Now equate the coefficients of x2 at either end of this chain of equalities. Now F has the additional property of being convex. Now F is additionally assumed to satisfy...... Now F is defined by setting F(z)=...... Now F is defined to make G and H match up at the left end of I. Now F is greater than or equal to G. [Not: greater or equal to'' or greater or equal than''] Now F is half as long as G. [Or: as G is] Now f is independent of the choice of γ (although the integral itself is not). Now G can be handled in much the same way. Now it is a simple matter to change the definition of the Fi at the single point zero, still maintaining condition (C), so that F is no longer discontinuous. Now J is defined to equal Af, the function f being as in (3). [= where the function f is......] Now M does not consist of 0 alone. Now M is defined to be the set of all sums of the form...... Now R is the localization of Q at a maximal ideal. Now that we have the above claim, we can select...... Now the reader will have no trouble verifying that...... Now we have the required contradiction since...... Now X can be taken as coordinate variable on M. Now, (3) is merely an abbreviation for the statement that...... Now, F has many points of continuity. Suppose x is one. Now, for arbitrary n, a glance at the derivative shows that...... Now, just the fact that F is a homeomorphism lets us prove that...... Number the successive segments of the boundary line between A and B (marked thickly in the picture) with the numbers 0,1,...,n, starting at the bottom. numbered in Arabic numerals Observe that (1) just uses the fact that m is unary. [Not: uses that m is unary''] Observe that A is thereby put into one-to-one correspondence with B. The elements of C are in one-to-one correspondence with...... Obviously, S may be P, but it may also not be P. Of course, it is tacitly understood that it is this measure that is really under discussion. of degree at most k Of particular interest to us is S(5,3). Of these, (i) and (ii) are almost immediate from the definition. Offsetting the effect of the pole at t=0 requires more work. On average, about half the list will be tested. On dividing through by f, we see that...... On making use of the bound...... we conclude that...... on page 13 [Not: on the page 13''] On the other hand, F fails property P. [Not: On the other side''] on the right of (8) = on the right-hand side of (8) On the way we analyze the relationship between...... On the whole, the solution can hardly be considered satisfactory. On TK we set up the symplectic structure induced by the metric. [= introduce] Once the dissipation relation is in hand, no further work is required. Once this is done, the proof continues thus:...... [= in this way] One can also study these equations on manifolds, but we stick to Rn for simplicity. One can also study these equations on manifolds, but we stick to Rn for simplicity. One can iterate Theorem 2 to conclude that...... One can, for example, take A to be the rationals in X. One cannot in general let A be an arbitrary substructure here. One cannot in general let A be an arbitrary substructure of B here. One delicate issue is that it is not known whether...... One is tempted to reverse the order of integrations but that is illegitimate here. One major advantage of...... is that...... One might hope that this method would work at least for sufficiently regular maps; however,...... One more case merits mentioning here. One more piece of notation: throughout the paper we write...... for...... One must therefore also introduce the class of...... One of the appealing aspects of the spectral set γ is that it readily lends itself to explicit computation. One of the major differences between F and G is that...... One of these lies in the union of the other two. One should take great care with...... One such mapping is the function f given in...... One unusual feature of the solution should be pointed out. Only a few of those results have been published before. Only for x=1 does the limit exist. Only for x=1 does the limit exist. [Note the inversion.] Only in exceptional circumstances is it true that f(x+y)=f(x)+f(y). [Note the inversion.] Only shifts with k=1 need be considered. opposite <reverse> inequality Other types fit into this pattern as well. Our approach provides an alternative derivation of the functional equation. Our basic strategy for proving (1) is different. Our choice of V shows that...... Our definition agrees with the one in . Our definition agrees with the one of . Our development will require a detailed understanding of cozero covers. Our first result generalizes (8) by exploiting some general facts seem\-ing\-ly overlooked by the aforementioned authors. Our first result generalizes (8) by exploiting some general facts seemingly overlooked by the aforementioned authors. Our focus now will be on one-sided averages. Our interest in extending Strang's results comes from the fact that...... Our interest in...... comes from the fact that...... Our last example was kindly provided by B. Johnson. Our main results state in short that MEP characterizes type 2 spaces among reflexive Banach spaces. Our method has the disadvantage of not being intrinsic. Our method of proof will be an adaptation of the reasoning used on pp. 71--72 of . Our notion of nonanalytic integral encompasses such well known examples as...... Our present assumption implies that the last inequality in (8) must actually be an equality. Our presentation is therefore organized in such a way that the analogies between the concepts of topological space and continuous function, on the one hand, and of measurable space and measurable function, on the other, are strongly emphasized. Our principal tools in the proof are...... Our procedure will be to find...... Our proof involves looking at...... Our proof of Theorem 2 is based upon ideas found in [BN]. Our result sheds a new light on...... Our results are closely related to those of Strang . Our results in this section can be refined somewhat by considering...... Our solution is completely different to theirs. Our study grew out of some valuable conversations with Kirk Douglas. Our ultimate objective is to eliminate this assumption completely. Parallel lines never join. Part (b) follows from (a) on noting that A=B under the conditions stated. Part (c) is a frequently used criterion for the measurability of a real-valued function. Passing to a subsequence if necessary, we can assume that...... Pointwise convergence presents a more delicate problem. Precisely r of the intervals Ai are closed. Prewar levels of production were surpassed in 1929. Proceeding as in the proof of......, let...... Proceeding further in this direction, we obtain the following corollary. processes of an entirely new kind Properties (a) to (c) are analytic in character. Proposition 2 presents examples of...... Puiseux's theorem asserts the existence of...... Put this way, the question is not precise enough. Putting these results together, it is seen that...... Quite a few of them are now widely used. Quite a few of them are now widely used. [= A considerable number] Raise both sides to the power p to obtain...... Raising this to the pth power [= to the power p], we obtain...... Rather than discuss this in full generality, let us look at a particular situation of this kind. Rather than using the definition of...... we will use the fact that...... Rearranging terms we obtain the inequality...... Recall the definition of T from Section 3. Recent improvements in the HL-method enable us to do better than this. Recent work of Kirby shows that...... Recently proofs have been constructed which make no appeal to integration. Recently, progress has been made in this case by Barnes (1999). Regarded as the intersection of two quadrics, E represents...... Regarding (8) [= Considering (8)], we have...... Repeated application of (4) shows that...... Repeated application of Lemma 2 enables us to write...... [ Or: enables one to write; not: enables to write''] Replace each occurrence of b by c. Replacement of z by 1/z transforms (3) into (4). Replacement of z by 1/z transforms (4) into (5). Replacing f by log f, we recover the theorem of . Research partially supported by NSF Grant No. 23456. rotation through π/3 <through an angle θ> Roughly 0.7 comparisons were done for each character. Roughly speaking, we shall produce a synthesis of index theory with Fourier analysis. Sales increased almost fourfold in this period. Section 2 presents two L1-type characterizations. Section 6 contains a formula which permits transfer of the results in Section 2 to sums of independent random variables. See [AB] for a proof that is close to the original one. See [KT] for discussion of this technical point. See also , where functions of exponential type are the main subject. See the simplified account in [2, Section 4]. Sets of the form (6) are disjoint and fill out any such orbit. Setting α equal to β in Corollary 3, we obtain...... Setting f=0, we arrive at a contradiction. Show by example that a P-system need not be a field. Similar arguments to those above show that...... Similar results were obtained independently in [AB]. Similarly to , we first consider the nondegenerate case. [Or: Just as in , not: Similarly as in ''] Since f is compact, it follows that Lf=0. [Not: Since f is compact, then Lf=0.''] Since f is convex and g is not,...... Since M is an intersection of closed subspaces, it is a closed subspace of E. Since most of the results presented are quite classical (the novelty lies in the arrangement, and some of the proofs are new), I have not attempted to document the source of every item. Since R is polynomial in x, so also is P. Since the code is linear, showing that it is 3-error-correcting is the same as showing that...... Since the entire argument is based solely upon assumption (6.1), the conclusion of the theorem must hold. Since the integrands vanish at 0, we may as well assume that...... Since...... we see that...... Since......, we have Tf equal to 0 or 2. Slightly refining a result of , Davenport proved...... So all the terms of (2) are accounted for, and the theorem is proved. So far it seems not to be known whether the geometric condition on X can be omitted. So far we have not topologized M(R). So one is naturally led to an investigation of...... So s can be thought of as q with Fq extended but Xq left the same. So we must in particular show that sets like this are not added. Solving it in a general setting is a complicated task. Some members go into more than one Vk. Some of isomorphisms classes above will have a rank of 2. Some of the basic ideas from functional analysis are also included. Some other problems suggest themselves in the topological setting <in the discrete time setting>. Some other problems suggest themselves in the topological setting. Some partial evidence to support this conjecture is discussed in . Some proofs may be simplified by the use <by use> of...... Some restrictions must be placed on the behaviour of f. Some such difficulty is to be expected. Specifically, choose A=B=id/dx. Standard Banach space notation is used throughout. For clarity, however, we record the notation that is used most heavily. statement (ii) of Proposition 7 Strict inequality [= with a < or > sign] can occur <hold> in (8) only if...... Strict inequality can occur <hold> in (8) only if...... Strictly speaking, we should write something like a(l,m,n) to reflect the dependence; we will rely upon context instead. Striving for a contradiction, assume that...... Strong compactness will be addressed in Section 3. Subsidies on these commodities total 25 per cent of the budget. Substitute this value of z into <in> (7) to obtain...... Successive vertices on a path have alternating labels. Such cycles are said to be homologous (written c≈ c'). Such was the case in (8). Suppose A is maximal with respect to having connected preimage. Suppose first that...... Suppose that of all such solutions, (x,y,z) is one with y minimal. Suppose that T* is continuous whenever T is. Suppose that the process continues indefinitely. Suppose that, contrary to our claim,...... Suppose the first three characters of the pattern match three consecutive text characters. Suppose the lemma were false. Then we could find...... Suppose we are given an f of the form f=...... Suppose X is a separable space of cardinality continuum. Suppose x were not in B. Then there would be...... Suppose, to look at a more specific situation, that...... Suppose, towards a contradiction, that...... Take A to be the matrix with all entries zero except for i-j at (i,j). Take as base for a topology on X the sets of the form...... Take for H the set...... Take g1,...,gn without common zero. Take g1,…,gn without common zero. Taking supremum over all g such that...... we get...... Taking y=x we get...... Temporarily assuming that E is positive, we multiply...... term-by-term differentiation That (2) implies (1) is contained in the proof of Theorem 1 in . That (2) implies (1) is contained in the proof of Theorem 1 of . That (2) implies (1) is the content of Corollary 3. That approach was used earlier in . There, however, it was applied in simply connected regions only. That is the least one can expect. That is---apart from the use of relaxed controls---precisely the stochastic Bellman equation. That is---except the use of relaxed controls---precisely the stochastic Bellman equation. The if'' part is straightforward. The above bound on an is close to best possible <to the best possible>. The above bound on an is close to best possible. The above construction has led the author to believe that...... The above construction suggests investigating the solutions of...... [Not: suggests to investigate''] the above-cited paper the above-mentioned element The above-mentioned measure is of course intimately related to the geometry of the real line. The action of G is given by gf=...... The advantage of applying...... lies in the fact that...... The advantage of using...... lies in the fact that...... The aim of this article is to study the relationship between the size of A, as measured by its diameter, and the extent to which A fails to be convex. The aim of this paper is to bring together two areas in which...... The algebraic properties (i) and (ii) required of j are evidently true. The algorithm compares x to each entry in turn until a match is found or the list is exhausted. The algorithm compares x with each entry in turn until a match is found or the list is exhausted. The algorithm examines only roughly one-quarter to one-third of the characters. The algorithm outputs a list of...... The algorithm returns 0 as its answer. the all-one sequence the all-zero vector The analogy with statistical mechanics would suggest...... The analysis is similar to that of . The analysis of PDO involves the geometry of T*M. The answer depends on how broadly or narrowly the term matrix method'' is defined. The answer is not known to us. The argument can easily be modified to yield a proof for the case of k positive. The argument is a variant of one in  and has been used several times since. The argument is now completed by means of techniques originated in the work of Stolz . The argument just given shows that...... The argument works equally well for R being a general ring with unity. The arguments from this point up to Theorem 2 do not depend on...... The as yet unproved conjecture of Bieberbach is that...... The assumed positivity of un is essential for these results. The assumption that Q is not a torsion point is built into (5). The assumptions imposed on f circumvent measurability difficulties. The assumptions of Section 3 are still in force except that for greater generality we do not assume f to be continuous. The author thanks H. Miller for a careful reading of an earlier draft. The author thanks the referee for his helpful suggestions concerning the presentation of this paper. the base p expansion <representation> of x The basic improvement is the smoothness condition on G given in (6). The basis of most of these theorems is Jensen's formula. The best general reference <An excellent recent reference> for...... is . The best known of these is the Knaster continuum. The best one could hope for is K≥ F. The budget has increased by more than a third. The calculation of M(f) is usually no harder than the calculation of N(f). The case a=1 requires a different approach. The case f=1 requires a different argument. The case n=2 is of no interest since...... The case when f is decreasing can be proved similarly, or else can be deduced from...... The cases p=1 and p=2 will be the ones of interest to us. The choice of A is clearly irrelevant, so assume A=0. The choice of A is clearly irrelevant, so assume A=0. With this choice of b,...... The coefficients of A add up to unity. the complement of A relative to B The complex case then follows from (a). The conclusion in (2.3) can be seen by observing that...... The conclusion of Theorem 3 becomes false if this requirement is omitted. The condition (8) holds with strict inequality. The condition A<B excludes quite a few of the standard Young functions. The conference laid the basis for a series of annual gatherings. The conjecture (now known not to be true in general) was that...... The conjecture has not been proved yet. The conjecture will be disproved by exhibiting...... The constant C may be taken as double the constant appearing in (5). the constantly zero function The constants are so adjusted in (6) that (8) holds. The constants implicit in the symbol ≈ depend on r. The context will make it clear whether S denotes a permutation of M or the corresponding symmetry. The continuity of f implies that of g. The continuity of f is readily checked, and we can then apply Morera's theorem. The continuum in question is then called arc-like. The continuum Y is tree-like since it admits a map onto X. The contour Γ surrounds D. The control problem is to choose an investment strategy so as to minimize...... the converse implication The converse is far from obvious. The corollary gives a necessary and sufficient condition on p for g to belong to Ap. The correspondence f→ Af embeds F as a closed ideal in G. The criterion for its existence is Af=0. The crucial set-theoretic properties involved are the following. [Note the position of involved; placed before properties, it would mean complicated''.] The curve C encircles the origin twice. The data were gathered for about a year. The definition is legitimate because...... The definition is stated in terms of local martingales, rather than martingales, for the technical reason that the former are easier to characterize in applications. The definition of generator is designed to make the proof above work for M=Z. The definition of M makes it clear that F is continuous. [Note that the it is necessary here.] The degree of F exceeds that of G by at least 4. The degree of P equals that of Q. The demand that each entry be a perfect square results in nine equations. [Note the subjunctive be.] the density of the zeros of f The derivation D is the same whether we regard E as a derivation on X or on Y. the derivative of f at x in direction v The description of Q makes it evident that (3) holds. [Note that the it is necessary here.] The desingularization of f is X except when Q and Q' have a common factor, in which case...... The detailed analysis of...... is carried out in Section 2. The diagram of L+M is obtained by taking the rows of the diagrams of L and M and reassembling them in order of decreasing length. The diameter of A is three times that of B. The diameter of F is about twice that of G. The difference between these maps is primarily in their kneading sequences. The difference can change only by an even integer. The difficulty consists in generalizing (b). The difficulty disappears entirely if...... The difficulty is that it is by no means clear what one should mean by a normal family. The dimension of a simplex is its cardinality diminished by 1. The distinction between these two realizations lies in the treatment of...... The distributions U and V differ only <merely> by scale factors from the distribution Z. the element so obtained The elements of each array must occupy consecutive memory locations. The elements of F are not in S, as they are in the proof of...... The elements of G, numbering 122 in all, range from 9 to 2000. The empirical data quite clearly indicate that we should expect proportions close to 3/4. [Or: indicates] The endpoint common to A and B will be called the free endpoint of N. The entries in each row are increasing from left to right. The equality f=g, which is part of Theorem 2, implies...... The equation PK=0 then goes over to QK=0. The equations are satisfied with error at most O(n). the especially interesting case The estimate is sharp, as the following example shows. The estimates (5) then follow by slightly increasing the value of m. The eventual aim is to obtain a closed form expression for...... The exact sequence ends on the right with H(X). The example is a variation on the space known as the Warsaw circle''. The existence of a large class of measures, among them that of Lebesgue, will be established in Chapter 2. The explanation for this definition is that...... the expression in parentheses The factor Gf poses no problem because...... The family of 4-sets will be used to generate a symmetry outside N but in M. The final lemma is due to F. Black and is included with his kind permission. The first 15 chapters should be taken up in the order in which they are presented, except for Chapter 9, which may be postponed. The first and third terms are each less than β/3. the first and third terms in (5) The first assertion of the theorem, namely that a is in A, is equivalent to...... the first author = the first-named author The first conclusion is immediate from Theorem 1. The first equality is understood to mean that...... The first of the above equalities is a matter of definition, and the second follows from (3). The first of these was suggested by J. Serrin, who showed how to modify my earlier treatment of...... so as to obtain stronger results with no extra effort. The first two are simpler than the third one. [Or: the third; not: The first two ones''] The first two are simpler than the third. [Or: the third one; not: The first two ones''] The following construction can be carried out. The following easy lemma is surely folklore. The following example shows that condition (2) cannot generally be dispensed with if V is to be continuous. [= cannot be discarded] The following example shows that the degree of smoothness predicted by Theorem 6 cannot be improved on. The following is an explicit criterion for f to be hamiltonian. The following lemma is the key to estimating...... The following lemma, crucial to Theorem 2, is also implicit in . The following proposition relates the two definitions. The following result illustrates the utility of (3). The following theorem is clearly motivated by the classical LP-decomposition. The following three statements are equivalent:...... the following two maps The following variant of Theorem 2 is occasionally useful. The function f (initially defined on C0) determines a functional on S. The function f (initially defined on C{\infty}0) determines a functional on S. the function F is bounded above <below> by 1. The function F is uniquely defined: indeed,...... The function f so defined satisfies...... The function F vanishes to order exactly n at zero. The function g achieves its maximum at x=5. The function g attains <takes/achieves> its maximum at x=5. The function g attains its maximum at x=5. The function g takes its maximum at x=5. The function of Lemma 2 can be easily modified to obey the extra condition. The function of Lemma 2 can be made to satisfy...... The function thus defined is a semigroup morphism. The functions fi (i=1,…,n) have no common zero in Ω. The G is a subgroup of R (relative to addition). the gap condition (3) The general case follows by changing x to x-a. The generally accepted point of view in this domain of science seems to be changing every few years. The geodesics (8) are the only ones that realize the distance between their endpoints. The graph of Fig. 5 shows that...... The greatest element is marked with a dot. the group of invertible elements of E the growth rate of Vn as n→ {\infty} The holomorphic dependence of the integral on <mbda follows from (4). The hypothesis f(0)≠ 0 causes no harm in applications, for if...... the hypothesis of F being projective the hypothesis of positivity [= the positivity hypothesis] The idea behind our use of the σ's is that...... The idea is that C is fixed, but X and Y vary according to circumstances. The idea is to relax the constraint of being a weight function in Theorem 3. The idea of the ensuing computations is the following:...... The ideal is defined by m=......, it being understood that...... the image of A under f = the f-image of A The implication one way follows from Theorem 2. The implied constants <The constants implicit in the symbol ≈> depend on r. The implied constants depend on r. The importance of these examples lay not only in lowering the dimension of known counterexamples, but also in...... [Note that the past tense of lie is lay, not lied''.] The index increases by 1 when the path is crossed from right to left. The induced homomorphism is multiplication by 2. The induced topology is not compact, but we can always get it to be contained in a Bohr topology. The inner sum is zero (and so too is S(a,b)). the innermost sum in (4) the inside diameter The inspiration for Theorem 1 was the paper of Snow. The integral of F is r times the sum of...... the integral of f on A <over A> The integrand is zero outside D. The interchange in the order of integration was legitimate, since...... The interest of the lemma is in the assertion that...... The interested reader is referred to  for further information. [Note the double r in referred.] The interpretation of σ is that for every x,...... The interval J has length 2k. the inverse image = the preimage The iterates eventually reach the value 1. The ki are not values of u and hence also not values of w. The kernel is C0 on X× X off the diagonal. The kernel satisfies good large time bounds if...... The key <fundamental> observation is that if...... The key observation is that if...... The Knaster continua are distinguished by the property that...... The L2 theory has more symmetry than is the case in L1. the largest k value The last but one step is justified by the fact that...... The last term is bounded by a constant multiple of the norm of g. the last two rows the last-mentioned map The latter hypothesis can be removed at the cost of an extra factor on the right hand side of (5). The Laurent expansion of f around <about> zero is...... The least such constant is called the norm of G. The lectures were written up by M. Stong. the left half of the interval The left side of (3) obviously cannot decrease if r increases. The lemma motivates our calling R a generalized Picard group. The lemma raises an interesting question:...... The length of F is thus reduced by half. The letter \chi will be reserved for characteristic functions throughout this book. The level of...... has remained largely unchanged for many years. The level of...... has remained substantially [= largely, essentially] unchanged for many years. The limit always exists (we allow it to take the value {\infty}). The limit in 4(b) is unchanged if g is replaced by f. The line l meets A in a point x <in a subset S>. The line T makes a right angle with the chosen direction. The local homeomorphism property of F does not, however, prevent it from self-intersections. The location of the zeros of a holomorphic function in a region Ω is subject to no restriction except the obvious one concerning the absence of limit points in Ω. The location of the zeros of a holomorphic function in a region Ω is subject to no restriction except the obvious one concerning the absence of limit points in Ω. [Note the difference: absence = non-presence; lack = shortage of something desirable.] The lower limit is defined analogously: simply interchange sup and inf in (1). The m points x1,…,xm are regularly spaced t units apart. The main <crucial/key> ingredient in the proof of...... is...... The main idea of the proof is to take...... The main information conveyed by this formula is that...... The main problems that we address are...... The main results of the paper are found in Section 2. the main source of inspiration The map f assigns to each x the unique solution of...... The map F being continuous, we can assume that...... The map F can be brought into this form by setting...... The map F can be put <brought> into this form by setting...... The map F can be put into this form by setting...... The map f is fixed for the remainder of the proof. The map F is said to be proper if G is dense. [ Note: no comma before if here.] The map f takes a to f(a). The map f takes the value 1 for t=1. The map f, which we know to be bounded, is also right-continuous. The map f, which we know to be bounded, is also right-continuous. [Not: that we know''; do not use that'' in non-defining clauses.] The map G can be handled in much the same way. The map G is also not C1. The map k is no longer continuous. The map U(t) takes values in some compact space G. The mapping f keeps <leaves> the origin fixed. The mapping f leaves <keeps> the origin fixed. The mapping f leaves the origin fixed. The match occurs at position 7 in T. the maximum possible density The measure of A is the supremum of the measures of its compact subsets. The merit of Theorem 5 is that it identifies...... The method applies equally well to certain other error terms. The method falls short of providing an explicit formula for the index. The method is to change variables to y=x+c so that...... The method of proof carries over to domains satisfying...... The method works irrespective of [= regardless of] whether A or B is used. The method works irrespective of whether A or B is used. the minus sign The model as it stands does not satisfy the conditions of...... The monograph comprises three parts. [= contains] the monotone convergence theorem The most frequently used models fall into one of the following two categories. The motivation for the results of this section is the following result of John (paraphrased slightly to suit our purposes). The motivation for writing this paper was twofold. The MR algorithm takes n steps to solve the problem. the multi-index with all entries zero except the kth which is one The name Riesz theorem'' is sometimes given to the theorem which asserts that...... The name of Harald Bohr is attached to bG in recognition of his work on almost periodic functions. the negative of a [= -a] The new X is no more continuous, although it still has norm 1. The next corollary shows among other things that...... [Not: among others''] The next lemma shows how such a semilattice looks when embedded in a larger compact semilattice. The next step is to establish...... The next theorem provides conditions for the existence of...... The next theorem ties together the concepts of...... The next three comparisons are:...... The next two theorems reveal the importance of this concept. the next-to-last inequality the next-to-last inequality = the second last inequality= the penultimate inequality The notation F<G will mean that...... The notion of backward complete is defined analogously by exchanging the roles of f and f-1. The notion of turbulence almost exactly captures the combinatorial content of not being an S-action. The novelty of our approach lies in using...... The number of distinct values that could be in a memory cell is at most s. the number of elements of X the number of solutions (x1,…,xn) in which there are fewer than r distinct values amongst the xi the number of terms in the sequence (an) such that...... the number of zeros of f in D The number of...... has increased substantially [= considerably] in recent years. The objective is to choose a control u so as to minimize...... The observed values of X will on average cluster around points where...... The obstacle we have is that we must control the area of G. The obvious rearrangement reveals the right side to be identical with (8). The only additional feature is the appearance of a factor of 2. The only case requiring further analysis occurs when f=0. The only difficulty is in showing that...... The only edges out of 3 lead to 2 or into B. The only modification to be made is that we define...... The only point remaining concerns the behaviour of...... The only point that requires care is the verification of...... The only points (z,w) at which the continuity of g is possibly in doubt have z=0. The only possibility for G not to be 1-1 is that there exists...... The only property of M needed is local compactness. [Not: the only needed property''] The operator A is not merely symmetric, but actually selfadjoint. The operator H is again homogeneous. The operator P satisfies essentially the same inequality as does F. The operator P satisfies essentially the same inequality as F does. The operators An have still better smoothness properties. the opposite inequality The orbits of H on B are unions of orbits of N on G, which in turn are orbits of N on G1, G2 and G3. The order of G is completely determined by the assumption that...... The ordered pair (a,b) can be chosen in 16 ways so as not to be a multiple of (c,d). the other end of the interval [Not: the second end''---unless of course the ends are ordered in some way] The other inequality is just as easy to prove. The other inequality is just as easy to prove. [= second of the two] The other player is one-third as fast. The other two defining properties of a σ-algebra are verified in the same manner. The others being obvious, only (iv) needs proof. the paper cited on p. 45 The paper is organized as follows. The paper provides a proof of a combinatorial result that pertains to the characterization of...... The parameter interval was here taken to be (0,1). The parameter of the model is changed from p to 1-p. The parent and child relations often used with trees can be derived once a root is specified. The passage from bounded f to general f is easily effected. The penultimate inequality is justified by the fact that...... The performance of the device has improved considerably. The player has to decide which of the two strategies is better for him and act accordingly. The point A can be reached from B by moving along an edge of G. The point is that (5) can stay small because...... The point is that the operator is now much easier to analyse than is the case in the original setting of the space B. The point of the lemma is that it allows one to...... The point p is within distance d <within a distance d> of X. The point p is within distance d of X. The point x maps to {\infty} under f. The polynomials Ri do not have this stability property, and are therefore of little interest. The possibility A=Ø is not excluded. The possibility A=B is ruled out in the same way. The preceding observation, when looked at from a more general point of view, leads to...... The prerequisite for this book is a good course in advanced calculus. The present paper is motivated by the desire to make the subject as accessible as possible. The present paper owes a great debt to Strang's work. The present proof is so arranged that it applies without change to holomorphic functions of several variables. The pressure increases are significantly below those in Table 2. the primary purpose The prime 2 is anomalous in this respect, in that the only edge from 2 passes through 3. The probability of X being rational equals 1/2. The probability that any particular edge is a bond is 2p. the problem discussed [Not: the discussed problem''] The problem is that, whatever the choice of F, there is always another function f such that...... The problem is to move all the disks to the third peg by moving only one at a time. the problem mentioned [Not: the mentioned problem''] The problem now reduces to establishing that...... The problem one runs into, however, is that f need not be...... The problem with this approach is that V has to be C1 for (3) to be well defined. the process described The process slows down for large x. The product being considered grows like n3. The products FiGi are very close to satisfying (1). The projection technique requires the introduction of an appropriate homomorphism. The proof concludes by observing that...... The proof consists in the construction of...... [= The main part of the proof is] The proof conveniently splits into two cases. The proof follows very closely the proof of (2), except for the appearance of the factor x2. The proof is by induction on n. The proof is completed by invoking Theorem 5. The proof is essentially a repetition of the arguments used to prove...... The proof is given in Section 3. [Not: in the Section 3''] The proof is mainly included to keep the exposition as self-contained as possible. The proof is not direct, but relies on the results of . The proof is rather cumbersome. The proof is similar in spirit to that of . The proof is similar in spirit to that of...... The proof is similar to the proof of Theorem 4, with two principal modifications. The proof is tedious but routine. The proof makes essential use of the Sobolev inequalities. The proof makes use of many of the ideas of the general case, but in a simpler setting. The proof of (2) is a matter of straightforward computation, and depends on the relation ab-cd=1. The proof of (8) will be given after we have proved that...... The proof of the lemma is complete. [Not: completed''; cf. next example] The proof of Theorem 5 is easily adapted to any open set. The proof proceeds along the same lines as the proof of Theorem 4. The proof proceeds along the same lines as the proof of Theorem 5, but the details are more complicated. The proof proper [= The actual proof] will consist of establishing the following statements in sequence. The proof runs parallel to...... The proof shows that if the points are drawn at random from the uniform distribution, most choices satisfy the required bound. The proof we give is based instead on Galois theory. The proof will be divided into a sequence of lemmas. The proof will only be indicated briefly. The proofs are, for the most part, only sketched. The properties of A are preserved in the passage from V to C(V). The proportion of M with hM=1 is between 70% and 80%. The proportion of men to women in the population has changed in recent years. The purpose of this paper is to provide simple exact upper bounds for the case where...... The quantities F and G differ by an arbitrarily small amount. [Not: arbitrary small''] The quantity A was greater by a mere 20%. The question arises whether...... The question as to how often we should expect the class number to be divisible by p is also of some interest. The question of whether B is ever strictly larger than A remains open. The question of whether B is ever strictly larger than A remains open. [Or: The question whether] The question of...... has been explored under a variety of conditions on A. The question we shall be concerned with is whether or not f is...... The quiver Q1 is the same as Q but with x deleted. the ratio between Haar measures on G and H the ratio of P to Q The reader is assumed to be familiar with elementary K-theory. The reader is cautioned that our notation is in conflict with that of . The reader may wonder why we have apparently ignored the possibility of obtaining a better lower bound by considering...... The reader might want to compare this remark with [2, Cor. 3]. The reason for preferring (1) to (2) is simply that (1) is manifestly invariant. [Note the double r in preferring.] The reason this is significant is that...... The reason why...... is that...... The region A has an area of 15 m2. The relation becomes an equality if the wi form an orthonormal basis. The remainder of the estimation is largely as before with B replaced by C. The remainder of the proof is routine. The remaining requirements of Definition 3 are also easily seen to hold in L1. The remarkable feature of this theorem is that...... The remarks following Theorem 2 show that a=0. The rest of the paper concerns the existence of greedy bases (as opposed to basic sequences). The rest of the proof goes through as for Corollary 2, with hardly any changes. The rest of the proof goes through as for Corollary 2, with hardly any changes. [= with almost no changes] The rest of the proof runs as before. The result above seems not to be a consequence of previous results. The result will now be derived computationally. The resulting formula exhibits u as the Laplace transform of x. The results are rather sensitive to the value of the recovery rate σ. The results have been encouraging enough to merit further investigation. the reverse inequality <inclusion> the revolution of the earth about <round> the sun The right hand side is dominated by 2M. The right hand side of (3) is unaffected if we replace H by G. the right side = the right-hand side The right-hand members of (1) and (2) are to be interpreted by continuity when f is in Z. the right-hand side expression the rightmost integral The S-function f is zero unless l=m, in which case it depends only on M. The saddle-point conditions are satisfied up to an error o(n). The same applies to D. The same definition serves for f in F. The same year, Maurer showed that...... The search will succeed provided there is...... the second author = the second-named author The second inequality follows upon considering Ri for i>0. the second largest element the second last row = the last but one row The second term can be absorbed by the first. The second way of constructing K(F) is through the use of...... the segment connecting <joining> a to b The semidirect product of H and G has H× G as its underlying set. The semigroup F can be explicitly determined. The series can easily be shown to converge. The set A is roughly triangular in shape. The set E has the cardinality of the continuum. The set F has the most points when...... the set of all primes not exceeding k the set of points at a distance less than 1 from K the set of points with <at> distance 1 from K The set WF(u) is made up of bicharacteristic strips. the shortest possible way The significance of this fact for our purposes is captured by Corollary 3. The significance of...... is that...... The simplest example of this is furnished by...... The situation is quite different if we replace H(U) by certain subclasses. The situations with domains other than sectors remain to be investigated. The skew diagram L-M is the shaded region in the picture below. The solution is not in L(E) unless a stability condition is imposed, as was demonstrated in Example 3. The solutions are f or g according as t=1 or t=2. The solutions can be carried back to H(V) with the aid of the mapping function φ. the space of all continuous functions on X the space of all functions with the property that...... the space of pth power integrable functions The space X fails the Radon--Nikodym property. The spectral radius formula asserts that...... The spectrum of T was defined in  and identified with the spectrum of a certain algebra AT. The standard proofs proceed via the Cauchy formula. The statement does appear in  but there is a simple gap in the sketch of proof supplied. The statement of Theorem 5 remains valid if we replace f is compact'' by the norm of f is bounded''. The string N (read from right to left) starts with...... The structure of a Banach algebra is frequently reflected in the growth properties of its analytic semigroups. The sum in (2), though formally infinite, is therefore actually finite. The sum is taken over all a dividing p. The sum of the depths is at most two-thirds of what it was before. The support of F has diameter not exceeding l. The supremum in (1) does not change if we restrict ourselves to rational points in Q. The symmetry of G about 0 gives...... The tangent space to N at x is identified with M via left translation. The task is now to obtain...... The Taylor expansion of f about <around> zero is...... The Taylor expansion of f at 0 can be prescribed up to any finite order. The technique is the same as in the proof of...... The temperature has to be maintained at a very high level. The tensor product makes G a module over R. The terms involving a≠ 1 contribute O(n), since...... The terms of the series (1) decrease in absolute value and their signs alternate. The terms with n>N add up to less than 2. The theorem gains in interest if we realize that...... The theorem implies that some finite subcollection of the fi can be removed without altering the span. The theorem indicates that arbitrary multipliers are much harder to handle than those in M(A). The theorem is definitely false without the assumption that......, as an inspection of Example 3 shows. The theorem is false without the assumption that......, as an inspection of Example 3 shows. The theorem to be proved is the following. [= which will be proved] The theory of correspondences may be viewed as bridging the gap between...... The theory of elementary divisors now shows that by pre- and post-multiplying x by suitable elements of K we can reduce x to a diagonal matrix. The theory of...... is entirely <completely> analogous to...... the third last row = the antepenultimate row The three groups have the same number of generators. [ The if only three groups were mentioned before.] the top row The total amount of information lost is...... The total number of vectors of this type is 36. The two cases are not mutually exclusive. The two characteristics are connected, but the relationship is quite a complex one. The two classes coincide if X is compact. In that case we write C(X) for either of them. The two codes differ only in the number of their entries. The two examples, E1 and E2, differ by only a single sequence, e, and they serve to illustrate the delicate nature of Theorem 2. The two figures appear strikingly different. The two functions differ at most on a set of measure zero. The two lines intersect at an angle of ninety degrees. The two notions of rank are independent of each other. The two probabilities remain essentially what they were before. The two questions listed below remain unanswered. The two sets intersect in at most three points. The uniqueness of f is easily proved, since...... the unit mass concentrated at x the upper half of the unit disc The vector field H always points towards the higher A-level. The vector points outwards from M. The vector v has at least n ones in its last m positions. The vertex F has label 1. The wedge denotes that ei has been omitted. The weight satisfies a weak type (1,1) estimate. The word ends in a. The words collection, family and class will be used synonymously with set. The z-component can be expressed in terms of the gauge function. the zero solution The zeros appear at intervals of 2m. The zeros of L-functions are all accurate to within 10-5. Their centers are a distance at least N apart. Their centres are a distance at least N apart. Their result gives no information when k is large, whereas (5) is significant regardless of the size of k. Their study resulted in proving the conjecture for...... Then (5) combined with (6) gives...... Then (5) takes on the form...... [Or: takes the form] Then (6) merely means that...... Then \cal D is an equivalence relation on S. Each \cal D-class of S contains...... Then A equals B. [= A is equal to B] Then A has three times as many elements as B has. Then A is an algebra of type II. Then A=B, as one sees by multiplying out the product on the right. Then A=B, as one sees by multiplying out the product on the right. One unusual feature of the solution should be pointed out. Then a=b, so that f is not injective. Then B contains a unique element of smallest norm. Then B does not have the Radon--Nikodym property. Then C lies on no segment both of whose endpoints lie in K. Then D is the face of the simplex s opposite to A. [Or: opposite A] Then D lies to the right of G. Then either...... or...... In the latter case,...... Then either......, or...... In the former case,...... Then either......, or...... In the latter <former> case,...... Then F and G are homotopic by means of a homotopy H such that...... Then F and G are homotopic via a homotopy H such that...... Then F and G have a factor in common. Then F and G make angle α. Then F becomes inner when extended to B. Then F can be as great as 16. Then F can be decomposed according to the eigenspaces of P. Then F has a unique lift F'. Then F has continuous y-derivatives of all orders <up to order k>. Then F has n or more zeros in Ω. Then F has no holomorphic extension to any larger region. Then F has simple zeros with residue 1 at the integers. Then F has T as its natural boundary. Then F has the property that...... Then F is 2 less than G. Then F is 3 greater than G. Then F is a function of x alone. Then F is bounded but it is not necessarily so after division by G. Then F is bounded in modulus by one. Then F is bounded on each bounded set. Then F is constructed from the Fj. Then F is continuous, hence bounded on D. Then F is functorial in the sense that...... Then F is invertible except possibly on an at most countable set. Then F is less than 1 in absolute value. Then F is Poisson distributed with mean <mbda. Then F is similarly obtained from G. Then F is smooth away <bounded away> from zero. Then F is strictly increasing and yet has zero derivative on a dense set. Then F is strictly less than G. Then F is supported in {|z|≤ 1}. Then F is the homeomorphism X→ Y inverse to G. Then F is within d of the integers. Then F may or may not fix B. Then F need not satisfy (2). [= F does not necessarily satisfy (2).] Then F restricts to a C0 flow on M. Then F satisfies the following replacement for condition (e). Then F varies smoothly in t. Then f(x) and f(y) differ in at least n bits. Then F1,...,Fn vary each in the interval [0,1]. Then Fn converges simply <uniformly/weakly/ weak*/in the norm of Lp/in Lp norm/in norm/in probability> Then Fn tends to F in norm <in the Lp norm>. Then Fn(x,y) converges to F(x,y) uniformly in x. Then f=g, or equivalently a(f)=a(g). Then for such a map to exist, we must have H(M)=0. Then G has 10 normal subgroups and as many non-normal ones. Then G has the structure of a group such that...... Then G is a group with composition as group operation. Then G is bounded away from zero. Then G is differentiable except for a jump at x=0. Then G is simply g with its periodic string read backwards. Then G is uniquely determined up to unitary equivalence. Then H is a free R-module on as many generators as there are path components of X. Then J contains an interval of half its length in which f is positive. Then K appears in the last position in the list. Then Lemma 2 furnishes the bound f≤...... Then M is a Banach algebra having for its identity the unit point mass at 0. Then M is obtained by glueing <gluing> X to Y along Z. Then n(r) is about krn. Then P covers M twofold. Then P is said to be elliptic relative to the action of G if...... [Not: relatively to''] Then P is the product of several integer factors of about xn in size. Then T tends to zero only through positive values. Then the conclusion holds with P-cell'' in place of cell''. Then the sequence (8) breaks off in split exact sequences. Then V has the following subspaces, and no others:...... Then V(x) is the maximum value of Jx(v) over all controls v. Then we have a commutative triangle...... Then we will use (2.3) in order to make the transition from M to a universal Turing machine U. Then X assumes values 0,1,...,9, each with probability 1/10. Then X has no fewer than twenty elements. Then X is a Banach algebra under convolution multiplication <under this norm>. Then X is the Swiss cheese obtained from the family D. Then x=y and y=z implies x=z. [Or: imply] Then, of course, F will be one-to-one. Theorem 1 can be used to bound the number of such T. Theorem 1 gives information on <about>...... Theorem 1 provides an answer in the negative. Theorem 1 relates the quantity a(n) to the quantity b(n). [Not: with the quantity b(n)''] Theorem 2 can be proved a number of different ways. Theorem 2 has a very important converse, the Radon--Nikodym theorem. Theorem 2 is interesting in its own right. Theorem 2 makes it legitimate to apply integration by parts. Theorem 2 of  Theorem 2 still holds for A(x) provided that k is restricted to the range [0,1]. Theorem 2 will form the basis for our subsequent results. Theorem 2, at the end of Section 2, was not originally obtained in the manner indicated there. Theorem 3 becomes false if this requirement is omitted. Theorem 3 below expands on this idea. Theorem 3 can be applied to the estimation of bk. Theorem 3 is remarkable in that considerably fewer conditions than in the previous theorems ensure universality. Theorem 3 may be interpreted as saying that A=B, but it must then be remembered that...... Theorem 7 imposes a quantitative restriction on the location of the zeros of...... Theorems 1 and 2 combine to give...... Theorems 1 and 2 combined give a theorem of Fix on...... Theorems 2 and 3 may be summarized by saying that...... Theorems 3 and 6 of , with the appropriate changes, are also valid. There are 12 indecomposables in total. There are a and b such that...... There are a few exceptions to this rule. [= some] There are a large number of examples showing that...... [Not: There is a large number''] There are a number of equivalent definitions of the regular set. There are at most two such r in (0,1). There are few exceptions to this rule. There are few exceptions to this rule. [= not many] There are few, if any, other significant classes of processes for which such precise information is available. There are many situations where this occurs naturally. There are n continua in X the union of whose images under f is K. There are numerous results in the literature relating spectral conditions to invertibility of f. There are O(1) possible choices for x. There are several cases to consider:...... There are several theorems for a number of other varieties. Among these are the Priestley duality theorem and...... There cannot be two edges between one pair of vertices. There exists a function f and a constant c such that...... [ Or: There exist a function f and a constant c] There exists a function f and a constant c such that...... [Or: There exist a function f and a constant c] There has recently been increasing interest in the theory of...... There is a close analogy between...... There is a family of related examples which includes the space S(M). There is a fourth notion of phantom map which bears the same relation to the third definition as the first does to the second. There is a map such that...... [Not: There is such a map that''] There is a marked difference between X and Y. There is a natural way to glue the associated varieties together along their common boundary. There is a related result concerning primitivity. There is a smallest algebra with this property. There is an extra statement that causes a new character to be read. There is equality if a=1. There is equality if...... There is no condition relating the sections on one side of N to those on the other side. There is no evidence to the contrary. There is no loss of generality in assuming that...... There is no map such that...... There is no need for the assumption that...... There is no recursive or definable R such that...... There is not space to enumerate them all here. There is now an extensive literature dealing with...... There is quite an extensive literature concerning resonance problems, beginning with the work of Lazer. There is r≥ 0 such that...... [Or: There is an r≥ 0] There is, however, a simple condition, satisfied in the vast majority of applications, which ensures...... There ought therefore to be a point x such that...... There remain four intervals of length 1/4 each. There remains the second question. There seems to be no simple formula for ...... Therefore F has a countable spectrum <a finite norm/a compact support>. [Or: F has countable spectrum etc.] Therefore F is at most a multiple of G plus...... Therefore, A has two elements too many. [Or: A has two too many elements.] Therefore, F has the fewest points when the index vanishes. Therefore, F has the unique lift F'=p-1F. Therefore, F is not <no> smaller than G. Therefore, G is uniquely determined up to unitary equivalence. Therefore, whenever it is convenient, we may assume that...... These are conveniently divided into three disjoint sets. These are precisely the linear functionals which also preserve multiplication, i.e.,...... These calculations can be performed entirely in terms of the generators of G. These estimates only require that f have a a certain polynomial rate of decay at infinity. These estimates only require that f have a certain polynomial rate of decay at infinity. These facts shed new light <a new light> on...... These from G number 45. [= there are 45 of them] These idempotents provide a useful tool for analysing the structure of G. These intervals are disjoint from all those used in defining J1. These last two are called the Banach maps. These might be called respectively the regular and the singular parts of B. These models furnish integral formulas for the matrix entries of F. These n disjoint boxes are translates of each other. These patterns are illustrated in Fig. 4. These properties led him to suggest...... These results leave open the basic case k=ω. These results show that an analysis purely at the level of functions cannot be useful for describing...... These slits are located on circles about the origin of radii rk. These spaces are defined and variously characterized in . These theorems allow one to guess the Plancherel formula. [ Or: allow us to guess; not: allow to guess''] These two possibilities cannot arise simultaneously. These volumes bring together all of R. Bing's published mathematical papers. They all have their supports in V. They are all slightly different. They are all zero at p. They are numbered in order of increasing diameter. They established the Hasse principle subject to a rank condition on the coefficients. They form a base of the topology of X. This abstract theory is not in any way more difficult than the special case of the real line. This achieves our objective of describing...... This allows proving the representation formula without having to integrate over X. This approach does not seem to generalize to arbitrary substructures. This approach fails to take advantage of the Gelfand topology on the character space. This approach is standard in homotopy theory. This argument also settles the case of K= Γ. This argument comes from . This argument goes back to Banach. This argument is invalid for several reasons. This assumption is certainly necessary if the distribution of xt is to converge to F. This attempt is doomed because the homogeneity condition fails. [= the attempt is certain to fail] This book is wholly concerned with...... This bound is integrable and on integration one again obtains the same form of...... This can be easily reformulated in purely geometric terms. This can be performed in O(n) time. This can be read off from (8). This can be translated into the language of differential forms. This can result in a significant loss of smoothness. This case arises when...... This case can in fact be treated without recourse to the methods of this paper. This change is unlikely to affect the solution. This choice was prompted by substantial numerical evidence. This claim, however, might be difficult to substantiate. [= prove] This class is strictly larger than...... This class is wide enough to include a number of examples of interest. This comment is relevant in proving Theorem 1. This conclusion extends to the general diffraction problem. This condition also turns out to be necessary. This condition completely characterizes Ap when...... This condition may seem unnatural, but it simplifies some of the technicalities of the proof. This contradicts the fact that...... This contrasts with (but does not contradict) Theorem 2 of . This corollary calls for some explanation and comment. This decrease is offset by the contribution from the poles. This definition best suits our purpose of showing that...... This definition is well adapted for dealing with meromorphic functions. This device makes it possible to replace multivalued functions by functions with...... This easily allows the cases c=1,2,4 to be solved. This effect is largely due to the presence of the logarithmic factor. This enables discontinuous derivations to be built. This enables us to define solution trajectories x(t) for arbitrary t. This enables us to pass from a compact group G to a maximal torus T. [Not: enables to pass''] This equation has a solution in integers provided <providing> that N>7. This equation has a solution in integers provided that N>7. This equation has a unique solution for every p. This equation supplies the key to the proof of Theorem 2. This establishes (i). This example falls within the scope of Cox's theorem. This exhibits a compact set K with...... This explains why we chose 9 rather than, say, 1 for the second coordinate. This extension retains control on...... at the sacrifice of loosing some control on...... This extension retains control on...... at the sacrifice of losing some control on...... [Not: loosing''] This fact alone changed the entire situation. This fact is typical of diffraction. This figure is drawn to a scale of one to ten. This finally yields f=g. [Not: yields that f=g''] This finishes <completes> the proof. This finishes the proof. This follows from Lemma 2 just the way (a) follows from (b). This follows immediately from (8) coupled with the fact that...... This forces f to satisfy (6). This gives (1) and shows that...... This gives some control over the behaviour of...... This graph is depicted in Fig. 1. This group consists of the elements of GL(n) of determinant 1. [= is made up of] This group will be designated by E3 (not to be confounded with Euclidean 3-space). This guarantees that f satisfies all our requirements. This has a maximum value of 4 when x=2. This has already been proved in Section 4. This has deeper significance than one might first realize. This idea goes back at least as far as . This idea is very little different from what can already be found in . This implies that however we choose the points yi, the intersection point will be their limit point. This implies that the local martingale must take a very specific form. This includes all Knaster continua. This inequality admits of a simple interpretation. This inequality persists as r→ 1. [= continues to hold] This inspired us to take a fresh look at all the results in [BG]. This interpretation is confirmed by (4). This interval is much smaller than that suggested by (8). This involves no loss of generality. This is a clear contradiction of the fact that...... This is a condition on how large f is. This is a consequence of Dini's theorem. This is actually a special case of the preceding framework. [Not: preceeding''] This is an exact analogue of Theorem 1 for closed maps. This is an interesting area for future research. This is because the factor M satisfies...... This is certainly reasonable for Algorithm 3, given its simple loop structure. This is clearly sufficient to make the computation for T. This is derived in Section 3 along <together> with a new proof of Morgan's theorem. This is derived in Section 3 along with a new proof of Morgan's theorem. This is discussed more fully in . This is done to simplify the notation. This is easily seen to be an equivalence relation. This is equivalent to requiring <saying> that...... This is equivalent to requiring that...... This is exactly our definition of a weight being regulated. This is immediate from 3.2. This is impossible in consequence of the last corollary. This is in agreement with our previous notation. This is in marked contrast to the behaviour of orthonormal sets in a Hilbert space. This is made more precise by the following definition. [Not: This is precised''] This is most readily shown using the theory developed in Section 6. This is necessary for determining the constants in Theorem 2. This is no coincidence, in the light of the remarks preceding Definition 2. This is not true of F. [Or: for F] This is of course still an implicit characterization of V, since...... This is part of a larger project to study the Galois groups of periodic points of arbitrary polynomial maps. This is sometimes expressed by saying that...... This is somewhat surprising since...... This is the essential tool in proving Theorem 5. This is the hard part of Jones's theorem. This is the lattice packing rotated 45o. This is the least useful of the four theorems. This is the reason for calling f the derivative of g. This is the same as asking which row vectors in R have differing entries at positions i and j. This is the same as saying that...... This is the way Theorem 3 was proved. This is usually called the area theorem, for reasons that will become apparent in the proof. This is where the notion of an upper gradient comes in. This is why no truncation is required here. This isomorphism converts the shift operator to the multiplication operator. This itself does not produce a solution of (1), but an additional hypothesis such as the Palais--Smale condition does provide such a solution. This just amounts to a choice of units. This last means that...... This last result means that...... This lends precision to an old assertion of Dini:...... This makes a total of 170 elements adjacent to A. This makes it possible to show that...... [Note that the it is necessary here.] This makes possible the proof of...... This map carries lines to lines <carries M onto N>. This map extends to all of M. This method can be used for solving...... This method has the disadvantage of not being intrinsic. This method is recently less and less used. This metric produces the usual topology of X. This module is denoted by H(X), or H(X,R) if we want to make explicit the coefficient ring. This motivated the second author to introduce the notion of...... This norm makes X into a Banach space. This normalization determines V uniquely. This notion is closely connected with that of packing dimension. This observation prompted the author to look for a more constructive solution. This paper enlarges the class of continua with this property, namely to include those which...... This paper is in final form. This paper was written in the course of the semester on...... This paper, for the most part, continues this line of investigation. This path stays in B for all time. This permits the extension of boundary estimates to systems...... This phenomenon is called overconvergence. This problem is still far from being solved. This procedure can be extended to take care of any number of terms. This procedure, once implemented, can thereafter be applied with great effectiveness. This proof is due to R. P. Boas. This proof will require substantial changes to cover the case of...... This property is characteristic of holomorphic functions with...... This property is of course already well known in many cases, often with little or no restrictions on V. This property will be used repeatedly hereafter. This proves one half of (2); the other half is a matter of direct computation. This provides an effective means for computing the index. This question is pursued further in Example 3. This question was answered negatively in . However, on the positive side, Davies  proved that...... This raises the following question. [Not: rises''] This realization is particularly convenient for determining...... This result is best possible. [Or: the best possible] This result may be thought of as a sort of regularity theorem. This result will prove extremely useful in Section 2. This says that the real part of g is...... This section makes heavy use of a theorem of Alsen and related results. This series is manifestly convergent. [= evidently] This serves to simplify the construction of...... This set has no fewer elements than K has. This shape bears a striking similarity to that of...... This shows that f could not have n zeros without being identically zero. This shows that the sequence (1) is bounded below, and so the definition of L(f) is meaningful. This shows that...... This simplifies to f=g for x=1. This sort of proof will recur frequently in what follows. This sort of tacit convention is used throughout Gelfand theory. This space is naturally isomorphic to B. This space of curves also shows up in the theorem of Meyer on...... This square has a perimeter equal to the circumference of the circle. This strikingly simple proof was discovered by J. Dixon. This subject has recently been extensively studied. This subject is treated at length in Section 2. This suggests a question: under what conditions is it true that...... ? This suggests a question: under what conditions is it true that......? This suggests that A can exist. This t has the feature we want. This term drops out when f is differentiated. This terminology is used slightly differently in . This theorem accounts for the term subharmonic''. This theorem is a converse <partial converse> of Theorem 2. This theorem removes the restriction to convex regions which was imposed in Theorem 8. This theorem will henceforth be referred to as the minimum principle. [Note the double r in referred.] This theorem will hereafter be referred to as the minimum principle. This theorem will hereafter be referred to as the minimum principle. [Note the double r in referred.] This theory was originated by Gelfand in 1941. This topic has been dealt with by many authors. This topology is compact, but not usually Hausdorff, nor even T1. This underlines the importance of the Bass conjecture. This volume was originally intended as a celebration volume marking the occasion of N. Wiener's seventieth birthday. This was one of the major steps in Wiener's original proof of his Tauberian theorem. This was proved in elementary fashion by P. J. Cohen. This will be proved as soon as we show that...... This will be proved by showing that H has but a single orbit on M. This will help us find what conditions on A are needed for T(A) to be analytic. This will help us find what conditions on A are needed for T(A) to be analytic. [Or: help us to find] This will hold for n>1 if it does for n=1. This will permit us to demonstrate that...... [Not: permit to demonstrate''] This will provide us with a way of getting some information about M without relying on general theory. This will provide us with a way of getting some information about...... without relying on general theory. This works regardless of [= irrespective of] whether B is true or false. This works regardless of whether B is true or false. This, in conjunction with (6), yields...... Thorin discovered the complex-variable proof of Riesz's theorem. Those more than half a square count as whole ones. Throughout integration theory, one inevitably encounters {\infty}. Throughout what follows, we shall freely use without explicit mention the elementary fact that...... Thus πn(X) can be interpreted as the homotopy classes of maps Sn→ X. [= as the set of all homotopy classes] Thus A and B are at distance precisely d. Thus A can be written as a sum of functions built up from B, C, and D. Thus A is compact, and so is B. Thus A is neither symmetric nor positive. Thus A is the smallest algebra with this property. Thus A is the union of all the sets Bx. Thus A is the union of B plus an at most countable set. Thus A-equivalence and B-equivalence are the same thing. Thus C behaves covariantly with respect to maps of both X and G. Thus C can be removed without changing the union. Thus C lies on no segment both of whose endpoints lie in K. Thus E is similar to F with a scale factor 1/3. Thus every subsequence contains a further subsequence converging weakly to some limit. Thus F and G differ by an arbitrarily small amount. Thus F can be equivalently defined as F=...... Thus F can be recovered from XkF by k-fold integration. Thus F has no pole in U (hence none in V). Thus F has x as its unique fixed point. Thus F is at most a multiple of G plus...... Thus f is bounded, and (1) says that f(a)=0. Thus F is G-invariant <invariant under the action of G>. Thus F is greater by a half. Thus F is independent of the choice of the family S. Thus F is integrable for the product measure. Thus F is less than or equal to G. [Not: less or equal to''] Thus F is near 1. [Or: near to 1] Thus F is no less than G. Thus F is not <no> greater than G. Thus f is the function sought. Thus F is very much larger than G. Thus F vanishes to order 3 <to infinite order> at x. Thus Fn(x,y) converges to F(x,y) uniformly in x. Thus G has 10 normal subgroups and as many non-normal ones. Thus in the end F will be homogeneous. [= finally] Thus modules over categories are in many ways like ordinary modules. Thus N separates M into two disjoint parts. Thus R has rank 2 <determinant zero/cardinality c>. Thus the conjecture can be restated as follows. Thus the long exact sequence breaks up into short sequences. Thus the paper is intended to be accessible both to logicians and to topologists. Thus we may count the contribution to the periodic points from each prime separately. Thus X assumes the values 0 and 1 only. Thus X can be taken as coordinate variable on M. Thus X has r times the length of Y. Thus X is not finite; neither <nor> is Y. Thus, F is invertible except where <at> x=0. Thus, whether or not x is in the list, one comparison is done. Tietze's theorem is a fairly direct consequence of Urysohn's lemma. to <in> a lesser degree To assess the quality of this lower bound, we consider the following special case. To avoid trivialities, we shall assume that...... To avoid undue repetition in the statements of our theorems, we adopt the following convention. To be more specific, consider...... To be precise, A is only left-continuous at 0. To be sure we can carry out step n, we need to know that...... To calculate (2), it helps to visualize the Sn as the successive positions in a random walk. To compute how many such solutions there are, observe...... To deal with (3), consider...... To deal with Tf, we note that...... [= Regarding <Concerning> Tf, we note] To deal with the zero characteristic case, let...... To deduce (9) from (5), choose...... To define Pn+1 from Pn, we let...... To establish uniqueness, suppose...... To every f there correspond two functions a and b. To every F there corresponds a unique G such that...... To every F there corresponds a unique...... to fill in the details To get around <get round/overcome> this difficulty, assume...... To get around <overcome> this difficulty, assume...... To go into this in detail would take us too far afield. To grasp the difference between the two notions, consider...... To guarantee existence and uniqueness of a solution to (3), the function g must be...... To guarantee Q(3), we use MA a second time to control the perfect sets. To handle the three cases in a uniform way, it is convenient to...... To illustrate, let us state the following corollary. To justify the interchange of summation and integration, consider...... To make this argument rigorous, apply (1.12) to...... To obtain a contradiction, we suppose that...... To obtain the required map one must modify the method as explained in . To overcome this problem, we revise our definition of a branch. To place Theorem 1 in context, consider two real vector fields...... To produce such an f, consider...... To prove (8), it only remains to verify that...... To round out the picture presented by Theorem 5, we mention the following consequence of...... To save space later on, all modules are given in the form...... To say that A is totally disconnected means that...... To see an example, set...... To see that A=B we argue as follows. To see that f=g is fairly easy. To show that...... is equivalent to showing that...... To show the greater simplicity of our method over Brown's, let us...... To simplify the writing, we take a=0 and omit the subscripts a. To state these conditions succinctly, we introduce the following terminology. To the best of the author's knowledge, the problem is still open. To this end,...... [= For this purpose; not: To this aim''] To understand why, let us remember that...... Together with Lemma 4, this implies that...... Two consecutive elements do not belong both to A or both to B. Two necessary conditions come to mind immediately. Under <In> what circumstances can this happen? under certain conditions under mild restrictions on f Under those conditions, what does the sum on the left hand side of (8) signify? Under what conditions can f have a local minimum in A? Unfortunately, F is defined only up to an additive constant. Unfortunately, the connection with Gromov's work has been obscured in recent years by an emphasis (in the algebraic topology literature) on configuration spaces. Unfortunately, this is rarely the case. Unless otherwise stated, we assume that...... Unlike Bell's method, Hall's does not use transfinite induction. Up to equivalence it only depends on the pair (f,g). Up to now, we have assumed that...... Upon combining the estimate for B with (5), we ahve now established the first conclusion of Theorem 8. Using (2) and following steps analogous to those above, we obtain...... Using some facts about polynomial convexity, we are able to deduce...... Using the standard inner product we can identify H with H*. Values computed for the right side of (2) were rounded up in the fourth decimal place. vectors in V at maximum distance from v We abbreviate this as f=g a.e. We acknowledge a debt to the paper of Black . We add the word positive'' for emphasis. We adhere to the convention that 0/0=0. We adopt the convention that the first coordinate i increases as one goes downwards, and the second coordinate j increases as one goes from left to right. We adopt throughout the convention that compact spaces are Hausdorff. We aim at fitting this result into the general scheme of...... We apply this to g to obtain...... We are guaranteed only one dense product for each k. We are left with the task of determining...... We are now in a position to prove our main result. We are now ready to proceed to the final stage of our construction. We begin by constructing a function f which has these prescribed zeros. We begin by describing the class of functions f considered, which includes the special cases quoted above. We begin by extending Construction 2.1 to encompass B-algebras. We begin by showing that α-collections and β-collections are quite different things. We begin with a detailed analysis of...... We briefly outline this refinement at the end of Section 2. [Not: in the end''] We call Ex and Ey the x-section and y-section respectively of E. We call F and G equivalent (in symbols F≡ G) if...... We can accomplish both tasks by showing that...... We can arrange the Xi's to include every...... We can assume that p is as close to q as is necessary for the following proof to work. We can assume, by decreasing n if necessary, that...... We can do a heuristic calculation to see what the generator of xt must be. We can do this by restricting attention to...... We can express f in terms of...... We can extend f by zero to the whole Ω. We can extend f by zero to the whole Ω. [Or: the whole of Ω] We can factor g into a product of irreducible elements. We can immediately use one of these poles to cancel out <compensate for> the effect of the pole at s=0. We can immediately use one of these poles to cancel out the effect of the pole at s=0. We can immediately use one of these poles to compensate for the effect of the pole at s=0. We can join a to b by a path π. [ Or: join a and b; not: join a with b''] We can join a to b by a path π. [Not: join a with b''] We can make g Lipschitz at the price of weakening condition (i). We can make this clear with the following example. We can now integrate n times to conclude that...... We can now pose a problem whose solution will afford an illustration of how (5) can be used. We can now restate Theorem 1 in concrete terms:...... We can partition [0,1] into n intervals by taking...... We can regard (8) as an equation for μ. We can represent W by the integral...... We can retrieve H from H' by the formula...... We can solve the resulting equations successively for c1,...,cn. We can substitute a subspace A of X for the set {0,1}. We can suppose R is semilocal (and hence a PID). We can then successively determine F in the other components. We cannot survey this whole subject here. We characterize the Banach spaces X for which n(X)=1. We claim that f(z)>1. Otherwise, the disc D would intersect B. We close this off by characterizing...... We conclude that whether a space X is an RG-space is not determined by the ring structure of C(X). We conclude that, no matter what the class of b is, we have an upper bound on M. We conclude this section with a useful lemma. We conclude this section with one more result. We conclude with a brief mention of free inverse semigroups. We conclude with two simple lemmas to be used mainly in...... We confine ourselves to discussing the case...... We consider every subset of N, whether finite or infinite, to be an increasing sequence. We define a (possibly unbounded) operator A by...... We define a <the> function f by <by setting> f=...... We define the set T to consist of those f for which...... We deliberately write SP(m)---rather than P(m)---in the complex case, because...... We denote it briefly by c. [Not: shortly''] We denote the complement of A by Ac whenever it is clear from the context with respect to which larger set the complement is taken. We denote this, too, by Q. We describe how the notion of positivity relates to the other properties. We did not really have to use the existence of T. We divide N in half. We divide them into three classes depending on the width of k. We divide them into three classes depending on the width of the middle term as compared with the width of the other terms. We divide them into three classes depending on the width of...... We do not exclude the possibility that A consists of precisely the polynomials. We do not expect to get F closed. We do not know whether or not Q(R)=R in this situation. We end this section by stating without proof an analogue of...... We establish our results both unconditionally and on the assumption of the Riemann Hypothesis. We estimate the two terms separately. We extend f to be homogeneous of degree 1. We extract the following from the proof of Lemma 5 of [KL]. We find ourselves forced to introduce an extra assumption. We first consider the M/G/1 queue, where M (for Markov'') means that...... We first do the case n=1. We first prove a reduced form of the theorem. We first replace a(n) by the related and simpler function b(n), where...... We first show that existence of a solution of...... provides a sufficient condition for optimality. We follow Kato  in assuming f to be upper semicontinuous. We follow where possible the argument of Lang . We give a number of results concerning...... [= several] We give an affirmative answer to the question of . [Not: a positive answer''] We give the argument when I=R. We give X the topology of uniform convergence on compact subsets of I. We have A≡ B as right modules. We have been working under the assumption that...... Suppose now that this is no longer so. We have d(f,g)=0 precisely when f=g a.e. We have F=G=H, the last equality being due to the fact that...... We have made modifications in the interest of readability. We have not required f to be compact, and we will not do so except when explicitly stated. We have not required f to be......, and we will not do so except when explicitly stated. We have shown that......, whence it is readily inferred that...... We have thus found another three solutions of (5). [= three more] We have thus proved that (a) implies (b). Conversely, define...... We have thus proved that B is the union of a collection of balls. We have thus proved the theorem without any reference to integration. We have to change the proof of Lemma 3 only slightly. We have to keep track of how the constant K depends on the domain D. We have to show that M itself is an algebra. [Or: M is itself an algebra.] We identify A and B whenever convenient. We include the proof for completeness. We leave it to the reader to verify that...... We leave it to the reader to verify that...... [Note that the it is necessary here.] We leave the details to the reader. We leave this as an exercise. We leave to the reader the proof that f is indeed self-adjoint, and not merely symmetric. We let T denote the set of...... We make G act trivially on Y. We make the convention that f(Q)=i(Q). We make the following definition. We make two standing assumptions on the maps under consideration. We may (and do) assume that...... We may assume, upon replacing F by F1, that...... We must have Lf=0, for otherwise we can replace f by f-Lf. We must then have...... We need only consider 3 cases. [Or: We only need to consider] We need only establish (8) for intervals of the form...... We need to make the smallest possible jump. We next show how the continuity requirement on f in Theorem 2 can be weakened. We next show that...... We note in passing that Fox has subsequently improved Barnes's result by showing that...... We note that H is in fact not Lipschitz continuous if this condition is violated. We now apply (2) to get Nf=0. We now apply (3) to both sides of (6). We now apply the previous observation to estimate F. We now bound v on <under> the hypothesis Hm. We now come to a theorem which was first proved by...... We now come to the theorem which was alluded to in the introduction of the present chapter. We now construct a group that will be of aid in determining the order of G. We now construct f as in the proof of Theorem 5, with V replaced by W. We now define three more groups of interest. We now exploit the relation (15) to see what else we can say about G. We now give an estimate for...... in terms of...... We now give some applications of Theorem 3. We now indicate how that difficulty can be circumvented. We now indicate some of the inherent difficulties. We now integrate around the circle <over |x|<1 /on |x|<1> to obtain...... We now introduce the algebras with which we shall be concerned. We now move on to the question of local normal forms. We now multiply (7) by p and take nth roots to obtain...... We now proceed to matrix rings. We now prove a lemma which is interesting in its own right. We now prove necessity. [Or: the necessity] We now prove sufficiency. [Or: the sufficiency] We now prove that f cannot have compact support unless f=0 a.e. We now prove that the positivity assumption is in fact superfluous. We now prove Theorem 4. [Not: the Theorem 4''] We now prove...... Indeed, suppose otherwise. Then...... We now provide a bound applicable to systems of...... We now show that A is closed. Suppose that, on the contrary, there is an x...... We now show that BL-algebras exist in great profusion. We now show that G is in the symbol class indicated. We now show the following improvement on (2). We now solve this equation for x. We now specialize to the situation of Lemma 8. We now start piling the pieces on top of each other. We now state a result that will be of use later. We now turn our attention to...... We now turn to a brief discussion of another concept which is relevant to John's theorem. We now turn to estimating Tf. We now turn to Gromov's original result. We now wish to discuss in some depth the problem of...... We now work toward the establishment of properties (A) to (D). We obtain (using the fact that Q is a probability measure) Q(A)=...... [Not: using that''] We omit further details. We partially order M by declaring X<Y to mean that...... We pause to record a generalization of Theorem 2 in a different direction. We postpone the proof until Section 2. We proceed by induction. We produce an evolution equation which differs from (2.3) only in the replacement of the F2 term by F3. We prove both lemmas at the same time. We prove, under mild conditions on f, that...... We put b in R unless a is already in. We put off discussing this problem to Section 5. We quote for future reference another result of Fox: there exists...... We recall what this means. [Not: We remind''] We refer to these as homogeneous Sobolev spaces. We regard (1) as a mapping of S2 into S2, with the obvious conventions concerning the point {\infty}. We remark at the outset that this formula makes sense, because...... We remark that Weyl's inequalities were proved for positive functions. We represent A as a quotient space of X by sending the point...... to...... [Not: We present''] We saw this in Section 2. We see from (2.3) that...... We see with the aid of an integration by parts that...... We select a system for which the power of b dividing n is least. We shall also refer to a point as backward nonsingular, with the obvious analogous meaning. We shall assume that this is the case. We shall be interested in seeing whether...... We shall discuss this again at somewhat greater length in Section 2.1. We shall do this by showing that...... We shall draw heavily on ideas from . We shall find it convenient not to distinguish between two such sequences which differ only by a string of zeros at the end. We shall only use (2), but have included (1) for completeness. We shall restrict the discussion to plane regions. We shall return to this central theme in Chapter 7. We shall return to this central theme in Section 4. We shall split up K as follows. We shall suppose that Assumptions 2.1 hold mutatis mutandis. We shall then show that this f can be represented in the form (5). We shall touch only a few aspects of the theory. We shall write a.e.[ω]'' whenever clarity requires that the measure be indicated. We shall write a.e.[ω]'' whenever clarity requires that the measure be indicated. [Note the subjunctive be.] We shall, by convenient abuse of notation, generally denote it by xt whatever probability space it is defined on. We should avoid using (2) here, since...... [Not: avoid to use''] We show how this method works in two cases. In both, C is...... We show that A is negligible compared with B. We show that nevertheless a positive proportion of the polynomials Bn(x) satisfy Eisenstein's criterion. We show that one can drop an important hypothesis of the saddle point theorem without affecting the result. We sketch below one possible approach to obtaining such refinements. We sketch the proof of the easy half of the theorem. We sketch the rest of the proof, leaving routine details to the reader. We start with a brief overview of our strategy. We start with the observation that...... We suc\-cee\-ded in proving (4) for amenable groups. [Not: succeeded to prove''] We tacitly assume that...... We take advantage of this fact on several occasions, by not actually specifying the topology under consideration. We take the same approach as in . We take this opportunity to correct a minor error in Lemma 2 of [PS]. We temporarily fix n in N. We then show how this leads to stronger results in applications. [Do not use application'' when you mean map'': a map f:X→ Y ( not: an application f'').] We thus get f=g. [Not: We thus get that f=g.''] We thus obtain f=g. [Not: We obtain that f=g'', We receive f=g''.] We turn the set of...... into a category by defining the morphisms to be...... We use the letter m for...... We use the same trick as Boas <as Boas does/that Boas does>. We were surprised to find out that...... <at finding out that......> We will also leave to the reader the proof of (5). We will be considering L on various function spaces. We will frequently write w.w. for weakly wandering''. We will not use this fact in any essential way. We will proceed without making explicit distinctions between the two types of convergence. We will prove this theorem shortly, but first we need a key lemma. We will pursue our investigation of conservation laws in Section 5. We will see later that the values of h(n) for large n are irrelevant to the problem. We will show that V is the required open cover. We will thus avoid doubly counting the same contribution. We will try to give it the simplest representation possible. We will use the symbol ∩ to represent intersection. We will write H(x,t) and Ht(x) interchangeably. We wish to arrange that f be as smooth as possible. We would like to know...... but that is beyond our reach at this point. We would like to know......, but that is beyond our reach at this point. We write G=FHF-1 for short. [= for brevity; not: We write shortly''] We write H for the value of G at zero. We write K(Q) for K to make the dependence on Q explicit. What happens if the word nonnegative'' is omitted? What is F(c) if c is a positively oriented circle? What is left is to show that...... What is still lacking is an explicit description of ker C. What is the answer if a=0? What is the relevance of this example to Fatou's lemma? What most interests us is whether...... What relations exist between A and B? What the theorem is saying in substance is that...... What would this imply about the original series? When A is commutative, the answer to both questions is yes''. When clarity requires it, we shall write...... When h is in H, the integral formula becomes Af=...... When is it the case that......? When it is necessary to emphasize one particular coordinate, we write...... When n=0, (7) just amounts to saying that...... When operated on by a rotation, each of these vectors is mapped to...... When r>3 things become much more difficult. When the rank condition fails we use (3) instead. When the target space is clear from context, we just write...... When we talk of a complex measure, it is understood that μ(E) is a complex number. Whenever the dimension drops by 1, the rank drops by at most Z. Where it is important to distinguish different norms on E, we will use the notation...... Where there is a choice of several acceptable forms, that form is selected which...... Where we could, we have chosen these examples from naturally occurring mathematical structures. [Note the double r in occurring.] While topological measures resemble Borel measures, they in general need not be subadditive. Why not increase the precision of these statements? With a little more work we can prove...... With a view to bounding I in (8) by the right side of (6), we first...... With each D there is associated a region VD. with minor modifications With the customary abuse of notation, the same symbol is used for both...... With the customary abuse of notation, the same symbol is used for...... With this agreement, it is clear that...... With this definition of a tree, no vertex is singled out as the root. With this definition, the set of equivalence classes is a metric space. With this example in mind, let...... With this in hand, we can finally define E to be equal to P(m)/H. Within I, the function f varies <oscillates> by less than 1. Without losing any generality, we could have restricted our definition of integration to integrals over all of X. Without losing any generality, we could have restricted our definition of integration to integrals over all of X. [Not: Without loosing''] Without loss of generality we can assume that...... Write out the integers from 1 to n. Pair up the first and the last, the second and next to last, etc.