A note on the colorful fractional Helly theorem

Helly’s theorem is a classical result concerning the intersection patterns of convex sets in ${\mathbb{R}}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.     

A version of Rolle's theorem and applications

This paper deals with global injectivity of vector fields defined on euclidean spaces. Our main result establishes a version of Rolle's Theorem under generalized Palais-Smale conditions. As a consequence of this, we prove global injectivity for a class of vector fields defined on n-dimensional spaces.Research partially supported by CNPq/Brazil under grant # 307014/89-4Research partially supported by CNPq/Brazil under grant # 301251/78-9     

An explicit version of a theorem of Stickelberger

We give an explicit version of a classical theorem of Stickelberger on the representation of certain integers by binary quadratic forms. This is achieved by generalizing Stickelberger's original congruences via an extension of a recent result of Young.     

An extension of a theorem of Kaplansky

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

A new type of fixed point theorem in metric spaces

We prove a generalization of Edelstein’s fixed point theorem. Though there are thousands of fixed point theorems in metric spaces, our theorem is a new type of theorem.     

Eisenstein series of 3/2 weight and one conjecture of Kaplansky

We shall prove thatf ₁ =x ² +y ⁴ + 7z ² represents all eligible numbers congruent to 2 mod 3 except 14 × 7^2k which was conjectured by Kanplansky. Our method is to use modular forms of weight 3/2. Our method can also be applied to other ternary quadratic forms.     

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 contractive version of a Schur-Horn theorem in II1 factors

We prove a contractive version of the Schur-Horn theorem for submajorization in II₁ factors that complements some previous results on the Schur-Horn theorem within this context. We obtain a reformulation of a conjecture of Arveson and Kadison regarding a strong version of the Schur-Horn theorem in II₁ factors in terms of submajorization and contractive orbits of positive operators.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

On a tauberian theorem of Hardy and Littlewood

In this paper, we give a simple alternative proof of a Tauberian theorem of Hardy and Littlewood (Theorem E stated below, [3]).     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

On a kähler version of cheeger-gromoll-perelman's soul theorem

In this note, we will prove a Kähler version of Cheeger-Gromoll-Perelman's soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

The transporter theorem—a new version of the Dynkin class theorem

Heinz Günther Tillmann zum 65. Geburtstag gewidmet