The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P₀(X) of the nonempty subsets of X is bicomplete.Indeed we present an explicit method to construct the bicompletion of the T₀-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T₀-spaces (X,U) for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete.     

Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X and a field K, let .It is shown that if X,Y are complexes on the same vertex set, then for k?0

     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that

     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

Two results on spaces with a sharp base

Example. There exists a space X with a sharp base and a perfect mapping onto a space Y which does not have a sharp base.     

Nilpotent orbits and some small unitary representations of indefinite orthogonal groups

For 2?m?l/2, let G be a simply connected Lie group with as Lie algebra, let be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra , and let be the universal enveloping algebra of . This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of .The roots with respect to the usual compact Cartan subalgebra are all ±e_i±e_j with 1?i<j?l. In the usual positive system of roots, the simple root e_m−e_m+1 is noncompact and the other simple roots are compact. Let be the parabolic subalgebra of for which e_m−e_m+1 contributes to and the other simple roots contribute to , let L be the analytic subgroup of G with Lie algebra , let , let be the sum of the roots contributing to , and let be the parabolic subalgebra opposite to .The members of are nilpotent members of . The group acts on with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on that carries a fully reducible representation of .For , let . Then λ_s defines a one-dimensional module . Extend this to a module by having act by 0, and define . Let be the unique irreducible quotient of . The representations under study are and , where and Π_S is the Sth derived Bernstein functor.For s>2l−2, it is known that π_s=π_s′ and that π_s′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m?s?2l−2 that π_s=π_s′ and that π_s′ is still unitary. The present paper shows that π_s′ is unitary for 0?s?m−1 even though π_s≠π_s′, and it relates the K spectrum of the representations π_s′ to the representation of on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in π_s′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on π_s′ does not make π_s′ unitary for s<0 and that the K spectrum of π_s′ in these cases is not related in the above way to the representation of on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra , 2?m?l/2.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Ratio asymptotic of Hermite-Padé orthogonal polynomials for Nikishin systems. II

We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures N(σ₁,…,σm) such that for each k, σk has constant sign on its support consisting on an interval , on which almost everywhere, and a set without accumulation points in .     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.