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 noncommutative version of the John-Nirenberg theorem

We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated space.

     

A density version of the Hales-Jewett theorem

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

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

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 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 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 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 constructive version of Birkhoff's theorem

A version of Birkhoff's theorem is proved by constructive, predicative, methods. The version we prove has two conditions more than the classical one. First, the class considered is assumed to contain a generic family, which is defined to be a set‐indexed family of algebras such that if an identity is valid in every algebra of this family, it is valid in every algebra of the class. Secondly, the class is assumed to be closed under inductive limits. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 weak version of Rolle's theorem

We investigate the fields with the property that any polynomial over the field which splits in the field has a derivative which also splits.

     

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.