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 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.     

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.     

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.     

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 \(\mathbb {F}\!_q(s)\) .     

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.     

Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${?}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${?}_p$-norm.     

A sharp version of Bauer–Fike’s theorem

In this paper, we present a sharp version of Bauer–Fike’s theorem. We replace the matrix norm with its spectral radius or sign-complex spectral radius for diagonalizable matrices; 1-norm and ∞$\infty$-norm for non-diagonalizable matrices. We also give the applications to the pole placement problem and the singular system.     

Graded versions of Goldie’s theorem

We prove graded variants of Goldie’s theorem of existence, structure, and coincidence of right classical quotient ring and right maximal quotient ring of a semiprime (prime) right Goldie’s ring (Theorems 10, 11, and 13). The main difficulty consisting in the problem of existence of a homogeneous regular element in each gr-essential right ideal is solved by posing some additional requirements onto the group grading the ring or onto the homogeneous components of the ring.     

A generalization of Obata’s theorem

In a complete Riemannian manifold (M, g) if the hessian of a real-valued function satisfies some suitable conditions, then it restricts the geometry of (M, g). In this paper we characterize all compact rank-one symmetric spaces as those Riemannian manifolds (M, g) admitting a real-valued functionu such that the hessian ofu has at most two eigenvalues ?u and $ - \frac{{u + 1}}{2}$ under some mild hypotheses on (M, g). This generalizes a well-known result of Obata which characterizes all round spheres.     

A generalization of Baer’s theorem

In this paper, we studied the supersolvability of the product of two subgroups and got a generalization of Baer’s theorem.