Ideal version of Ramsey’s theorem

We consider various forms of Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey’s theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems.     

A local version of Gearhart’s theorem

Let {e ^tA: t ≥ 0} be a C₀—semigroup on the Hilbert space ?. If x ₀ ∈ ? is such that the local resolvent R(λ,A) x ₀ admits a bounded holomorphic extension to the open half plane {Reλ > 0}, then lim _t→∞‖e ^tA R(λ₀, A) x ₀‖ = 0 for each λ₀ ∈ ρ(A). This resuit is used to find mild spectral conditions which ensure the decay at infmity to zero of solutions of higher order abstract Cauchy problems.     

The matrix version of Hamburger’s theorem

Mathematical Notes - Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely...     

A difference version of Nori’s theorem

We consider (Frobenius) difference equations over \((\mathbb {F}\!_q(s,t), \phi _q)\) where \(\phi _q\) fixes \(t\) and acts on \(\mathbb {F}\!_q(s)\) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group \(\mathcal {G}\) defined over \(\mathbb {F}\!_q\) can be realized as a difference Galois group over \((\mathbb {F} \! _{q^i} (s,t),\phi _{q^i})\) for some \(i \in \mathbb {N}\) . The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori’s theorem which states that \(\mathcal {G}(\mathbb {F}\!_q)\) occurs as a (finite) Galois group over \(\mathbb {F}\!_q(s)\) .     

A topological version of Liapunov’s theorem

Given a nonatomic finite-dimensional vector measure on a topological space, a criterion is established for obtaining its full range by considering open (or closed) sets only.     

Semialgebraic version of Whitney’s extension theorem

Archiv der Mathematik - In this note we prove a semialgebraic counterpart of Whitney’s extension theorem.     

A Global version of Grozman’s theorem

Let $X$ be a manifold. The classification of all equivariant bilinear maps between tensor density modules over $X$ has been investigated by Grozman (Funct Anal Appl 14(2):58–59, 1980), who has provided a full classification for those which are differential operators. Here we investigate the same question without the hypothesis that the maps are differential operators. In our paper, the geometric context is algebraic geometry and the manifold $X$ is the circle $\text{ Spec}\, \mathbb{C }[z,z^{-1}]$ . Our main motivation comes from the fact that such a classification is required to complete the proof of the main result of Iohara and Mathieu (Proc Lond Math Soc, 2012, in press). Indeed it requires to also include the case of deformations of tensor density modules.     

Two comments on Dvoretzky’s sphericity theorem

For any two positive integersk, l and anyɛ>0 there exists anN(k, l, ɛ) so that given anyl convex bodiesC ₁, …,C _l symmetric about the origin inE ⁿ withn≧N there exists a subspaceE ^k so that eachC _i intersectsE ^k, or has a projection intoE ^k, in a set which is nearly spherical (asphericity <ɛ). The measure of the totality ofE ^k which intersect a given body inE ⁿ in a nearly ellipsoidal set is considered and an affine invariant measure is introduced for that purpose.     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

A right triangulated version of Gentle-Todorov’s theorem

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We prove a right triangulated version of Gentle-Todorov’s theorem by introducing the notion of right homotopy cartesian square.     

On a discrete version of alexandrov’s projection theorem

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.     

A locally convex version of Kadison’s representation theorem

Positivity - In this paper, we study the local ordered $$*$$ -vector spaces and their representations. We prove that each Archimedean local ordered $$*$$ -vector space, can be represented as a...     

A discrete version of Liouville’s theorem on conformal maps

Geometriae Dedicata - Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial...     

Hartogs-stawski’s theorem in discrete valued fields (revised version)

This paper is a revised version of my previous paper [4]. Main purpose is to give a right proof for non-archimedean Hartogs theorem(Stawski’s theorem) in case when the value group of the field K is discrete.     

Random differences in Szemerédi’s theorem and related results

We extend the interior gradient estimate due to Korevaar-Simon for solutions of the mean curvature equation from the case of euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result due to Dajczer, Hinojosa, and Lira. In addition, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity.