F.P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930) 264-286.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....N . The existence of ultraflat unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [Er1] asserting that, for all Pn with n z##D Pn(z) #(1 #) # n 1, where # 0 is an absolute constant (independent of n) Yet, refining a method of Korner [Ko], Kahane [Ka] proved that there exists a sequence (Pn)withP n#K n which is (#n ) ultraflat, where #n = O n 1 17 # log n . Thus the Erdos conjecture was disproved for the classes . in general, not necessarily those produced by Kahane in his paper [Ka] We prove five closely related ....

....of an ultraflat sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [Er1] asserting that, for all Pn n with n (1. 5) max n (z) #(1 #) # n 1, where # 0 is an absolute constant (independent of n) Yet, refining a method of Korner [Ko], Kahane [Ka] proved that there exists a sequence (Pn )withP n#K n which is (# n ) ultraflat, where (1.6) # n = O n 1 17 log n . Thus the Erdos conjecture (1.5) was disproved for the classes . For the more restricted class the analogous Erdos conjecture is unsettled to this date. It is ....

T. Korner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--

....an ultraflat sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [Er] asserting that, for all Pn n with n 1, 1. 2) max n (z) #(1 #) # n 1, where # 0 is an absolute constant (independent of n) Yet, refining a method of Korner [Ko], Kahane [Ka] proved that there exists a sequence (Pn )withP n#K n which is (# n ) ultraflat, where # n = O n 1 17 # log n . Kahane s paper contained though a slight error which was corrected in [QS2] Thus the Erdos conjecture (1.2) was disproved for the classes . For the more ....

T. Korner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--

....an ultraflat sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [Er] asserting that, for all Pn n with n 1, 1. 2) max n (z) #(1 #) # n 1, where # 0 is an absolute constant (independent of n) Yet, refining a method of Korner [Ko], Kahane [Ka] proved that there exists a sequence (Pn )withP n#K n which is (# n ) ultraflat, where # n = O n 1 17 # log n . Kahane s paper contained though a slight error which was corrected in [QS2] Thus the Erdos conjecture (1.2) was disproved for the classes . For the more ....

T. Korner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--

....sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [28] asserting that, for all P n n (z) #(1 #) # n 1, 1. 5) where # 0 is an absolute constant (independent of n) Yet, combining some probabilistic lemmas from Korner s paper [47] with some constructive methods (Gauss polynomials, etc. which are completely unrelated to the deterministic part of Korner s paper) Kahane [43] proved that there exists a sequence (P n ) n which is (# n ) ultraflat, where # n = O n 1 17 . 1.6) Thus the Erdos conjecture (1.5) was ....

....# n 1foreveryz##D, and on the other hand A # n 1 with an absolute constant A 0 for every z #D except for a small arc. In the light of this result he asked how close we can get to satisfying n (z) # n 1foreveryz##D if P n n . The first result in this direction is due to Korner [47]. By using a result of Byrnes, he showed that there are absolute constants 0 A Bsuch that A # n 1# P n (z) #B # n 1 for every z #D. Then Kahane [43] constructed a sequence (P n ) of polynomials n for which # n ) # n 1# P n (z) #(1 # n ) # n 1,z##D , with a sequence (# n ) of ....

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T.W. Korner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--224.

....it may be more convenient to study projectively convex position of hyperplanes, that is if there exists a permissable projective transformation that maps the family of hyperplanes onto a family in convex position. In this formulation the problem is closely related to a question of McMullen [6], 7] In fact the results in these papers provide a lower bound for the possible Carath eodory number. ....

D.G. Larman On sets projectively equivalent to the vertices of a convex polytope Bull. London Math. Soc., 4, 6-12 (1963)

....them which are either pairwise disjoint or pairwise intersecting. Then 0:2 f(n) n Let us remark that we cannot expect any superlogarithmic lower bound to hold for the analogously de ned functions in higher dimensions. This follows from a result of Tietze [12] rediscovered by Besicovitch [2]) which shows that any graph can be represented as the intersection pattern of a family of 3 dimensional convex bodies. If instead of general convex compact sets in the plane, we consider rectangles with sides parallel to the x and y axes, then we can prove the following lower bound. Theorem 2 ....

A. S. Besicovitch: On Crum's problem, J. London Math. Soc. 22 (1947), 285-287.

....(t)j p Bateman, Chowla and Erdos [1] showed that the left hand side of (3.2) can be of order p ln ln(p) On the other hand, an upper bound of this order was proved by Montgomery and Vaughan [23] under the assumption of the Riemann Hypothesis for Dirichlet functions. Further, Burgess [4] showed that there are nontrivial upper bounds for b p by proving the following result: For 0 there exists a positive number ffi such that if is a nontrivial character to a sufficiently large p and if b is an integer satisfying b p then the left hand side of (3.2) is less than bp ....

D. A. Burgess: On character sums and primitive roots. -- Proc. London Math. Soc. 12 (1962), 179--192.

....1 c 15 14. He also showed that one may let # be a function of N which tends to zero as N tends to infinity. Subsequently, several authors sharpened Tolev s result improving on the range for c (see [2, 7, 8] The most recent improvement is due to the first author [7] he used Harman s sieve [3, 4] to show that Tolev s theorem holds for 1 c 61 55. Our purpose in this paper is to provide an improvement on a recent result [9] of the second author regarding the solvability of inequality (1.1) for s = 2. In [9] it is proved that, for almost all y # [N, 2N) i.e. the Lebesgue measure of ....

....a common technique for obtaining sharp lower bounds is to combine upper bounds and (not as sharp) known lower bounds by means of combinatorial identities like Buchstab s S(E, z 1 ) S(E, z 2 ) # z2 #p z1 S(E p , p) 2.2) Our proof of (2. 1) uses a version of this idea developed by Harman [3, 4]. Let B be the set of integers in [X, 2X) We use arithmetic information in the form of asymptotic formulas # m#M a(m) S(Am , z(m) # # m#M a(m) S(Bm , z(m) error terms, 2.3) where # is suitably chosen. We expect the error terms here to be always small , but of course, we can prove ....

G. Harman, On the distribution of #p modulo one, J. London Math. Soc. (2) 27 (1983), 9--18.

....#x log p 1 log p 2 = 2x log x O(x) 13) which is known as the Selberg s inequality. Using a modified version of this result Selberg proved the Dirichlet s Theorem 4 in elementary way the same year (see [30] It has been checked and always found that #(n) Li(n) However, Skewes proved (see [32, 33]) that the first crossing of #(n) Li(n) occurs before 10 10 10 34 . This number was known as the largest useful number in mathematics . Since then, the upper bound for the crossing point has subsequently been reduced to 10 371 . John Edensor Littlewood (1885 1977) proved in 1914 (see [23] ....

S. Skewes, On the Di#erence #(x) - Li(x), II, J. London Math. Soc., 8, 1955, p.48-70.

....#x log p 1 log p 2 = 2x log x O(x) 13) which is known as the Selberg s inequality. Using a modified version of this result Selberg proved the Dirichlet s Theorem 4 in elementary way the same year (see [30] It has been checked and always found that #(n) Li(n) However, Skewes proved (see [32, 33]) that the first crossing of #(n) Li(n) occurs before 10 10 10 34 . This number was known as the largest useful number in mathematics . Since then, the upper bound for the crossing point has subsequently been reduced to 10 371 . John Edensor Littlewood (1885 1977) proved in 1914 (see [23] ....

S. Skewes, On the Di#erence #(x) - Li(x), J. London Math. Soc., 8, 1933, p.277-283.

....be an integer lattice of discriminant disc(L) and C R n a set of class m (e.g. a convex set) Suppose that C lies in a ball of radius R around the origin. Then (B. 6) #(L C) vol(C) disc(L) O(R n 1 ) This follows from the Lipschitz principle for the integer lattice proven by Davenport [5], as adapted by W. Schmidt ( 19] Lemma 1) We will apply the Lipschitz principle to certain subsets of convex sets. For this purpose we will need: Lemma 17. Let C R n be a convex set, d 0 and de ne C d : fx 2 C : dist(x; C) dg to be the set of points of C of distance at least d from ....

H. Davenport On a principle of Lipschitz, J. London Math. Soc. 26 (1951), 179-183.

....subgroup of a compact semitopological semigroup. For instance, every Hausdorff locally compact group is Eberlein. Problem 3.2. Ruppert [R1, p.114 115] Find a Hausdorff topological group which is not Eberlein. This problem is open even in the case when G is algebraically isomorphic to Z (cf. [R2]) Every topological group G is a topological subgroup of Is(X) s for a certain Banach space X (take, for example, X = C r (G) the space for all right uniformly continuous functions on G (as in Teleman [T] The natural question is: how good may X be When X may be Asplund or even reflexive ....

, On signed a-adic expansions and weakly almost periodic functions, Proc. London Math. Soc. (3) 63 (1991), 620-656.

....G This function will behave in a very erratic fashion. For example, the symmetric group S n is the automorphism group of the null graph on n vertices, but the smallest graph whose automorphism group is the alternating group A n has about 2 n vertices (the exact number was calculated by Liebeck [40]) Clearly this number is not smaller than the degree of the smallest faithful permutation representation of G; this has been investigated in detail by Babai et al. 6] 8 Other measures of the size of the graph could be used; for example, the number of edges, or the number of orbits of G on ....

On graphs whose full automorphism group is an alternating group or a nite classical group, Proc. London Math. Soc. (3) 47 (1983), 337-362.

....can we get a graph G without 4 NZF We know the answer already: The Petersen graph. Petersen graph has a 5 NZF and this is indicated on Fig. 7 3 1 4 1 2 1 2 3 1 2 1 1 2 1 2 Figure 7: 5 NZF Now we can try to go further. Any more examples Tutte did this in the fifties and made (in several papers [57, 58, 59, 60]) the following beautiful conjectures: Conjecture 1 Every bridgeless graph has a 5 NZF. We can notice that the smallest counterexample for this conjecture is a 3 edge connected graph. Assume the contrary, let G be a graph with a 2 cut C = fe 1 ; e 2 g, by contracting the edge e 1 the obtained ....

W.T. Tutte On the embedding of linear graphs in surfaces. Proc. London Math. Soc., II. Ser. 51, 474-483 (1949).

....sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdos (Problem 22 in [Er] asserting that, for all Pn #K n with n # 1, 1. 4) max z##D P n (z) #(1 #) # n 1, where # 0 is an absolute constant (independent of n) Yet, refining a method of Korner [Ko], Kahane [Ka] proved that there exists a sequence (Pn )withP n#K n which is (# n ) ultraflat, where (1.5) # n = O # n 1 17 # log n # . Kahane s paper contained though a slight error which was corrected in [QS2] Thus the Erdos conjecture (1.4) was disproved for the classes Kn . For the ....

T. Korner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--

....problem are quite deep; Heegner s and Stark s proofs appeal to the theory of elliptic modular functions while Baker s proof involves consideration of lower bounds on linear forms of three logarithms. Nice surveys of the solution can be found in [Star2] Star3] or [Gol] From 1969 to 1975 Baker [B2, B3], Stark [Star4, Star5] and Montgomery and Weinberger [MonWe] classi ed all 18 imaginary quadratic number elds of class number two. In the process Hendy [Hen] extended the Frobenius Rabinowitsch result to prove that the only quadratic elds F = Q( p D) with D 0, having class number hF = 2 are ....

, A remark on the class number of quadratic elds, Bull. London Math. Soc. 1 (1969), 98-102.

.... 38 times t 2 [1=2; 2] with constants independent of 2 h0; 1] Combining these facts with the above homogenization result, standard arguments lead to the following rather strong form of semigroup convergence: lim 0 ke tL e t b L k 2 2 = 0 (35) for all t 0 (see, for example, BBJR] [DER2], or the arguments given below) De ne a closed subspace of L 2 (R d ) by M = U 1 Z f 2R d d 0 :j j 1g d L 2 (R d 0 ) 36) where U is given by (23) Clearly M is an invariant subspace of the operators L and b L, and we denote their restrictions to M by (L ) M and b LM . Then ....

, On second-order almost-periodic elliptic operators. J. London Math. Soc. (2000). To appear.

....diagram implies that each A automorphism oe : H H permutes the blocks of KH and induces a symmetry of the Brauer graph. Remark 6.3. Let R = Z and f : A Z, v 7 1, so that RH is the group algebra of W over Z. Since QW is semisimple, Tits Deformation Theorem (see [9] x68A and [12]) implies that the decomposition map d f is a bijection between Irr(KH) and Irr(W ) Let oe : H H be an A algebra automorphism and oe 1 the corresponding automorphism . The commutative diagram in 6.2 shows that the bijection Irr(KH) Irr(W ) is compatible with the induced actions of oe on ....

....= hS i W the corresponding parabolic subgroup and H : hT w j w 2 W i H the corresponding parabolic subalgebra. Denote by Ind S 0 the operation of induction of representations; note that this works on the level of W and of the level of KH, and that we have a commutative diagram (cf. [12], 1.5) N 0 Irr(W ) 0 Irr(W 0 Irr(KH Moreover, the operation Ind S 0 takes projectives characters of KH to projective characters of KH (see [11] 12.2) Now let n : jSj and B = fae j 0 k ng. By Lemma 6.5 we know that B is a union of non trivial blocks of KH. ....

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, On the character values of Iwahori-Hecke algebras of exceptional type, Proc. London Math. Soc. 68 (1994), 51--76.

....; x] with x x 0 , and compares with recent results by Baker and Harman [1] and by Baker, Harman and Pintz [3] in which the lengths of the intervals are x 0:535 and x 0:525 , respectively. Our method is an adaptation to algebraic number elds of the sieve method introduced by the rst author [8, 9] and shares many features with [2] and [3] Our result is not quite as strong as the result in [3] because we do not have an analogue for Hecke L functions of Watt s mean value theorem [18] In order to state the main result we need to establish some notation. We denote by a; b; the integral ....

G. Harman, On the distribution of p modulo one, J. London Math. Soc. (2) 27 (1983), 9-18.

....epimorphism : G 1 Delta Delta Delta G n F G which maps each free factor G i identically onto the subgroup G i of G. By (1) splits globally [in a locally conjugate way] and any such splitting gives the desired embedding. 5] 1] Assume [5] Then, by the subgroup theorem of Haran ([Ha], Theorem 5.1) ae(G) is strongly projective relative to ae(G 1 ) ae(G n ) since G 1 Delta Delta Delta G n F is strongly projective relative to G 1 ; G n . Compare the following remark) 2 10 Remark 1.5 As the proposition shows, our notions of relative [strong] ....

D.Haran: On closed subgroups of free products of profinite groups, Proc.London Math.Soc. 55 (1987), 266-298. 36

....2. y is incomparable with the events of M 0 x;y . Pomsets for Local Trace Languages 27 3. If u 0 is a linear extension of the pre x t 0 x;y of t associated to F 0 x;y , then we have (u 0 ; M 0 x;y [ fx; yg) 62 I. 2 We shall also use here (a restricted version of) Ramsey s Theorem [37]: Let S be a nite set. Then there is a positive integer R(S) such that for any mapping d of the two elements subsets of f1; 2; R(S)g into S, there exist three elements x; y; z 2 f1; 2; R(S)g such that d(B) d(C) for any two elements subsets B and C of fx; y; zg. Lemma 3.16. If L is ....

F.P. Ramsey: On a problem of formal logic. Proc. London Math. Soc. 30 (1930) 264-286

....s 2 S] be a polynomial ring as in Section 3 and H be the Iwahori Hecke algebra over A corresponding to W and with parameters v = v 2 s j s 2 S) Let K be the eld of fractions of A and HK the algebra obtained by scalar extension from A to K. Then it is known that HK is semisimple and split (see [19]) This algebra plays the analogous role as the group algebra of W over Q. Its representations and characters will be called ordinary. Let Irr(H K ) be the set of irreducible characters of HK , and fCg be the set of conjugacy classes of W . For each class C we choose an element wC of minimal ....

....known. For type I 2 (m) they are easily constructed from [12] Theorem (67.14) For type H 3 , the W graphs and explicit matrix representations are given in [30] The table for H 4 has been computed by means of the W graphs in [1] The tables for the remaining exceptional types are computed in [19], 18] 21] algorithms for computing the tables for type An and Bn are contained in [42, 24] 40, 41] for type Dn see [24] For each type, there is a function (e.g. HeckeCharTableG2) which takes as input a list of parameters and returns the character table record of the corresponding generic ....

: On the character values of Iwahori-Hecke algebras of exceptional type. Proc. London Math. Soc. 68, 51-76 (1994)

No context found.

F.P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930) 264-286.

No context found.

F.P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930) 264--286.

No context found.

Korner, T.W., On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219--224.

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