Counting partitions on the abacus

In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case ℓ=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss. Dedicated to the memory of Dr. Manfred Schocker (1970–2006)     

Smoothness and estimates of sequences in Banach spaces

An upper bound for the order of smoothness of bump functions in Banach spaces without copy ofc ₀ is found in terms of lower and upper estimates of their sequences. It is also shown that everyC ^∞-smooth Banach space with symmetric basis either containsc ₀ or is isomorphic toℓ _2n for some integern. Partially supported by DGICYT grant PB 90-0044.     

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

Hyperspaces of Banach Spaces with the Attouch—Wets Topology

Let X be an infinite-dimensional Banach space with weight τ. By Cld _AW (X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin _AW (X), Comp _AW (X) and Bdd _AW (X), we denote the subspaces of Cld _AW (X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that Fin _AW (X)≈Comp _AW (X)≈ℓ₂(τ)×ℓ₂ ^f ℓandℓBdd _AW (X)≈ℓ₂(2^τ)×ℓ₂ ^f where ≈ means ‘is homeomorphic to’, ℓ₂(τ) is the Hilbert space with weight τ (ℓ₂(ℵ₀)=ℓ₂ the separable Hilbert space) and ℓ₂ ^f ={(x _i ) _iεN εℓ₂∣x _i =0 except for finitely many iεN}.     

Large almost monochromatic subsets in hypergraphs

We show that for all ℓ and ε > 0 there is a constant c = c(ℓ, ε) > 0 such that every ℓ-coloring of the triples of an N-element set contains a subset S of size $ c\sqrt {\log N} $ c\sqrt {\log N} such that at least 1 − ε fraction of the triples of S have the same color. This result is tight up to the constant c and answers an open question of Erdős and Hajnal from 1989 on discrepancy in hypergraphs. For ℓ ≥ 4 colors, it is known that there is an ℓ-coloring of the triples of an N-element set whose largest monochromatic subset has cardinality only Θ(log log N). Thus, our result demonstrates that the maximum almost monochromatic subset that an ℓ-coloring of the triples must contain is much larger than the corresponding monochromatic subset. This is in striking contrast with graphs, where these two quantities have the same order of magnitude. To prove our result, we obtain a new upper bound on the ℓ-color Ramsey numbers of complete multipartite 3-uniform hypergraphs, which answers another open question of Erdős and Hajnal.     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

Uniformly cross intersecting families

Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff |A∩B|=ℓ for all A∈ A and B∈B. Denote by P _e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P _e (n) is Θ(2 ⁿ ), by Frankl and R?dl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with |A||B| = $ \left( {{*{20}c} {2\ell } \\ \ell \\ } \right) $ \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) 2 ^n−2ℓ = Θ(2 ⁿ /$ \sqrt \ell $ \sqrt \ell ), and conjectured that this is best possible. Consequently, Sgall asked whether or not P _e (n) decreases with ℓ.     

An Inequality from Moment Theory

We show how certain simple ℓ^p–inequalities may be proved by “ignoring the p.” An application to moment sequences is considered.     

There Exist Highly Critically Connected Graphs of Diameter Three

Let κ(G) denote the (vertex) connectivity of a graph G. For ℓ≥0, a noncomplete graph of finite connectivity is called ℓ-critical if κ(G−X)=κ(G)−|X| for every X⊆V(G) with |X|≤ℓ. Mader proved that every 3-critical graph has diameter at most 4 and asked for 3-critical graphs having diameter exceeding 2. Here we give an affirmative answer by constructing an ℓ-critical graph of diameter 3 for every ℓ≥3.     

The inclusion of the Schur algebra in B(?2) is not inverse-closed

The Schur algebra is the algebra of operators which are bounded on ℓ ¹ and on ℓ ^∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ ², and whose inverse is not bounded on ℓ ¹ nor on ℓ ^∞. In particular, this shows that the Schur algebra is not inverse-closed.     

ℓ-Divisibility of ℓ-regular partition functions

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b _ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏ _m=1^∞(1−q ^m ) ^ℓ−1.      

On the classes of hereditarily ℓp Banach spaces

Let X denote a specific space of the class of X _α,p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ_p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ_p. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ_p where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c ₀. Here we give a direct proof of the known result that X contains asymptotically isometric copies of ℓ₁.     

Isomorphic embeddings and harmonic behaviour of smooth operators

LetY be a Banach space, 1<p<∞. We give a simple criterion for embedding ℓ _p ⊂Y, namely it suffices that the positive cone ℓ _p ⁺ ⊂Y. This result is applied to the study of highly smooth operators from ℓ _p intoY (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless ℓ _p/K ⊂Y for someK ∈ ℕ. Supported by grants GAUK 277/2001, GAČR 201-01-1198, A1019205.     

Concerning Bourgain’s ℓ1-Index of a Banach space

A well known argument of James yields that if a Banach spaceX contains ℓ ₁ ⁿ ’s uniformly, thenX contains ℓ ₁ ⁿ ’s almost isometrically. In the first half of the paper we extend this idea to the ordinal ℓ₁-indices of Bourgain. In the second half we use our results to calculate the ℓ₁-index of certain Banach spaces. Furthermore we show that the ℓ₁-index of a separable Banach space not containing ℓ₁ must be of the form ω^α for some countable ordinal α. Research supported by the NSF and TARP.     

A propos de la tour localement cyclotomique d’un corps de nombres

We adapt some recent results on Hilbert ℓ-towers of number fields to locally cyclotomic ℓ-towers (i.e. logarithmic towers).Résumé. Nous adaptons des résultats récents sur le problème de la tour de Hilbert d’un corps de nombres au cas de la ℓ-tour localement cyclotomique.      

Biflatness of semigroup algebras

We shall study the biflatness of the convolution algebra ℓ ¹(S) for a semigroup S. We show that for any semigroup S such that ℓ ¹(S) is biflat the canonical partial ordering on the idempotents must be uniformly locally finite. We use this to characterize the biflatness of ℓ ¹(S) for an inverse semigroup S.     

Linear codes with covering radius 3

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by ℓ _q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on ℓ _q (r, 3). General constructions are given and upper bounds on ℓ _q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.     

The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry

The lower dimensional Busemann-Petty problem asks, whether n-dimensional centrally symmetric convex bodies with smaller i-dimensional central sections necessarily have smaller volumes. For i = 1, the affirmative answer is obvious. If i > 3, the answer is negative. For i = 2 or i = 3 (n > 4), the problem is still open, however, when the body with smaller sections is a body of revolution, the answer is affirmative. The paper contains a solution to the problem in the more general situation, when the body with smaller sections is invariant under rotations, preserving mutually orthogonal subspaces of dimensions ℓ and n − ℓ, respectively, so that i + ℓ ≤ n. The answer essentially depends on ℓ. The argument relies on the notion of canonical angles between subspaces, spherical Radon transforms, properties of intersection bodies, and the generalized cosine transforms.     

Congruences for ?-regular partition functions modulo 3

Let b _ℓ (n) denote the number of ℓ-regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b ₄(n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b ₁₃(n). In this paper we prove similar families of congruences for b _ℓ (n) for other values of ℓ.     

A remark on presentations of certain Chevalley groups

Let Φ be a root system of typeA _ℓ, ℓ ≧ 2,D _ℓ, ℓ ≧ 4 orE _ℓ, 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA _r,r∈Φ, satisfying: