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 symmetrized metric ramsey theorem

For every finite metric space A there exists a finite metric space B and a real number r > 0 such that for every coloring of B by two colors there exists a monochromatic A′ ⊆ B such that every isometry between two subsets of A′ extends to a full autoisometry of B and A′ is either isometric to A or is r-homothetic to A.     

A density version of the Hales-Jewett theorem

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

On a metric generalization of ramsey’s theorem

An increasing sequence of realsx=〈x _i :i<ω〉 is simple if all “gaps”x _i +1−x _i are different. Two simple sequencesx andy are distance similar ifx _i +1−x _i <x _j +1−x _j if and only ify _i +1−y _i <y _j +1−y _j for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequencey distance similar tox all of whose pairs are of the same color. We also consider many related problems and generalizations. Partially supported by OTKA-4269. Partially supported by NSF grant STC-91-19999. Partially supported by OTKA-T-020914, NSF grant CCR-9424398 and PSC-CUNY Research Award 663472.     

A geometric Gordan-Stiemke theorem

Let C and K be closed cones in Rⁿ. Denote by φ (K∩C) the face of C generated by K ∩ C, by φ(K ∩ D)^D the dual face of φ(K ∩ C) in C¹, and by φ(-K¹ ∩ C¹) the face of C¹ generated by -K¹ ∩ C¹. It is proved that φ(K ∩ C¹) if and only if -C¹ ∩ [span(K ∩ C)] ⊥ ? C¹ + K¹. In particular, the closedness of C¹ + K¹ is a sufficient condition. Our result contains a generalization of the Gordon-Stiemke theorem which appeared in a recent paper of Saunders and Schneider.     

A new version of a theorem of Kaplansky

Abstract

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the w-version of this theorem, where w is a hereditary torsion theory for modules over a commutative ring.

Communicated by Silvana Bazzoni     

A probabilistic version of a theorem of Kegel

Let G be a finite group and α an automorphism of G of prime order p. It is shown that if the elements x of G satisfy the equation with sufficiently high probability (depending only on p) then G is nilpotent.Received: 8 January 2005     

A geometric proof of a theorem about non-dual renormings

We give a simple geometric proof of a result by Davis and Johnson that every nonreflexive Banach space admits an equivalent norm in which is not isometric to a dual space. Moreover, our renorming keeps unchanged the original norm on a given finite-codimensional subspace and makes this subspace norm-one complemented.

     

An application of ergodic theory to a problem in geometric ramsey theory

LetE be a measurable subset of ℝ ^k ,k>2, with XXX(E)>0. LetV = {0,υ ₁, …,υ _k+1} ε ℝ ^k , whereυ ₁, …,υ _k+1 are affinely independent. We show that forr large enough, we can find an isometric copy ofrV arbitrarily close toE. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss [FKW] showing a similar property for ℝ²,V = {0,υ ₁,υ ₂}.     

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 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 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 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 ramsey theorem for trees,with an application to Banach spaces

LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (x _s) _sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (x _s ) _{s ∈β} is either a weak Cauchy sequence or equivalent to the usuall ¹ basis.     

A characterization theorem for geometric logic

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.     

A nonlinear version of Roth's theorem for sets of positive density in the real line

It is proved that given ε>0, there is δ(ε)>0 such that ifS is a measurable set of [0,N], |S|>εN, then there is a triplex, x+h, x+h ² inS withh satisfyingh>δ(ε)N ^1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h ³,h ⁴ etc.