A discrete version of the Mishou theorem. II

In 2007, H. Mishou obtained a joint universality theorem for the Riemann zetafunction ζ(s) and the Hurwitz zeta-function ζ(s, α) with transcendental parameter α. The theorem states that a pair of analytic functions can be simultaneously approximated by the shifts ζ(s + iτ ) and ζ(s + iτ, α), τ ∈ R. In 2015, E. Buivydas and the author established a version of this theorem in which the approximation is performed by the discrete shifts ζ(s + ikh) and ζ(s + ikh, α), h > 0, k = 0, 1, 2.... In the present study, we prove joint universality for the functions ζ(s) and ζ(s, α) in the sense of approximation of a pair of analytic functions by the shifts ζ(s + ik ^β h) and ζ(s + ik ^β h, α) with fixed 0 < β < 1.     

The discrete version of a geometric duality theorem

It was shown by Rhodes [1] that a theorem about subsets in the plane specified by the Euclidean metric generalizes to an interesting duality between the absolute and the maximum metrics in the real plane. In this paper the discrete version of this duality is shown to hold between the cityblock (absolute) and the chessboard (maximum) metrics in the quantized space. The characterization of the bisector and the near-bisector under the above metrics is obtained as a by-product.Dedicated to Professor Srinivas Ramanujan on his birth centenary     

A discrete Korovkin theorem

In this paper we give a sufficient condition for the pointwise Korovkin property on B(X), the space of bounded real valued functions on an arbitrary countable set X = {x_l,…, x_j,…}. Our theorem follows from its L_p(X, μ) analogue (and conversely); here 1 p < ∞ and μ is a positive finite measure on X such that μ({x_j}) > 0 for all j.     

A Banach-lattice version of the Josefson-Nissenzweig theorem

Let E, F be two Banach lattices with E order continuous. If F can be mapped positively onto E then the dual F^* contains a weak^* -null sequence of positive and norm-one elements (Theorem 1). This is a Banach-lattice version of the classical Josefson-Nissenzweig theorem. It is an immediate consequence of the dual characterization of order continuity: E is order continuous iff E is Dedekind complete and every norm-one and pairwise disjoint sequence in E^* is weak^*-null (Theorem 2).     

A density version of the Hales-Jewett theorem

Research supported by the National Science Foundation under Grant No. DMS86-05098.     

A new version of the Hahn-Banach theorem

We discuss a new version of the Hahn-Banach theorem, with applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunctions. We show how our result can be used to prove a localized version of the Fenchel-Moreau formula - even when the classical Fenchel-Moreau formula is valid, the proof of it given here avoids the problem of the vertical hyperplane. We give a short proof of Rockafellars fundamental result on dual problems and Lagrangians - obtaining a necessary and sufficient condition instead of the more usual sufficient condition. We show how our result leads to a proof of the (well-known) result that if a monotone multifunction on a normed space has bounded range then it has full domain. We also show how our result leads to generalizations of an existence theorem with no a priori scalar bound that has proved very useful in the investigation of monotone multifunctions, and show how the estimates obtained can be applied to Rockafellars surjectivity theorem for maximal monotone multifunctions in reflexive Banach spaces. Finally, we show how our result leads easily to a result on convex functions that can be used to establish a minimax theorem.     

A generalized version of the Birkhoff-Kellogg theorem

The present note deals with generalizations of the classical Birkhoff-Kellogg theorem. By means of the Leray-Schauder degree and the introduced notion of a flat neighborhood, some results on the existence of eigenvectors for compact operators defined on the boundary of an unbounded set in a separated linear topological space being admissible in the sense of Klee are obtained. Positive operators on ordered spaces are considered, as well.     

A local version of the two-circles theorem

A necessary and sufficient condition is given so that in a domain Ω there are no functions whose average over all balls contained in Ω of radiir ₁,r ₂ vanish except the zero function. Partially supported by NSF grant DMS-8401356 and by NSF grant OJR 85-OV-108 through the Systems Research Center of the University of Maryland.     

A discrete version of the Landau-Lifshits equation

A discrete version of the Landau-Lifshits equation from the theory of ferromagnetism is investigated within the framework of the method of the inverse-scattering problem. Variations of action-angle type are constructed, and the energy spectrum of the model is described. The procedure of dressing is used to obtain the simplest soliton solution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 145, pp. 62–71, 1985.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

A mixed version of Menger's theorem

An(a, b)-n-fan means a union ofn internally disjoint paths. Menger's theorem states that a graphG has an(a, b)-n-fan if and only ifG isn-connected betweena andb. We show thatG contains edge-disjoint(a, b)-n-fans if and only if for anyk withk0min{n–1, |V(G)|–2} and for any subsetX ofV(G)-{a, b} with cardinalityk, G-X is (n-k)-edge-connected betweena andb.     

A quantitative version of Picard's theorem

Letf be an entire function of order at least 1/2,M(r)=max_| z|=r|f(z)|, andn(r, a) the number of zeros off(z)-a in |z|≤r. It is shown that lim sup _r→∞ n(r, a)/logM (r)≥1/2π for all except possibly onea∈C. Supported by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft.     

A discrete four vertex theorem

A discrete four vertex theorem is proved for a general plane polygon using a method of proof that also yields a proof, that appears to be new, for the classical four vertex theorem.     

A constructive version of the Ribes-Zalesski product theorem

For any given finitely generated subgroups H₁,...,H_n of a free group F and any element w of F not contained in the product H₁H_n, a finite quotient of F is explicitly constructed which separates the element w from the set H₁H_n. This provides a constructive version of the product theorem, stating that H₁H_n is closed in the profinite topology of F. The method of proof also applies to other profinite topologies. It is efficient for the profinite topology as well as for the pro-p topology of F. The main tools used are universal p-extensions and inverse automata.The authors gratefully acknowledge support from INTAS project 99–1224. The second author was supported in part by NSERC and by the FCT and POCTI approved projects POCTI/32817/MAT/2000 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER.     

A homoclinic version of the central limit theorem

The definitions of homoclinic partitions and transformations are given in situations that are standard for topological dynamics and ergodic theory. A variant of the central limit theorem is proved, the formulation of which makes use of homoclinic transformations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 80–91, 1990.     

A strong constructive version of engel's theorem

We examine two problems in the computational theory of Lie algebras. First, we prove a constructive version of Engel's theorem: if L is a finite-dimensional Lie algebra that is not nilpotent, we show how to construct an element x in L such that the linear transformation ad x is not nilpotent. No special assumptions about the underlying field are needed. Second, as an important application of the first result, we give an algorithm for the construction of a Cartan subalgebra of a finite-dimensional Lie algebra. This solves the problem of finding a totally constructive proof of the existence of a Cartan subalgebra, posed by Beck, Kolman, and Stewart in the paper "Computing the Structure of a Lie Algebra". Our proofs are ordinary mathematical proofs that do not employ the general law of excluded middle. The advantage of this approach to mathematics is that our proofs, which are not burdened or obscured by the details of a particular programming language, can nevertheless be routinely turned into computer programs