Hua’s theorem with nine almost equal prime variables

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

Clarkson-Erdős theorem for many variables

In the present paper, the Clarkson-Erdős, which complements the well-known Müntz theorem, is extended to the case of many variables. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 594–598, April, 1999.     

Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

A generalization of Fan��s matching theorem

We introduce a generalized coercivity type condition for setvalued maps defined on topological spaces endowed with a generalized convex structure and we extend Fan’s matching theorem.     

Hardy’s theorem and rotations of several complex variables

In this paper, Hardy’s theorem and rotations characterized by complex Gaussians in the complex plane due to Hogan and Lakey are extended to complex spaces of several variables. We point out that conditions under which a function on the n-dimensional real Euclidean space has an analytic extension to the complex space. Moreover, we prove that the function is a rotation of a multiple of real Gaussians through some angle if the extension satisfies certain assumptions.     

Permutation-invariance in Komlós type theorem for non-negative random variables

We provide a permutation-invariant version of Komlós’ type convergence for non-negative random variables.     

Projection pencils of quadrics and Ivory��s theorem

Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785?C794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib Math J 45(4):785?C794, 2004). We use this definition to prove a more general form of Ivory??s theorem.     

A simple generalization of Ger?gorin��s theorem

It is well known that the spectrum of a given matrix A belongs to the Ger?gorin set ??(A), as well as to the Ger?gorin set applied to the transpose of A, ??(A ^T ). So, the spectrum belongs to their intersection. But, if we first intersect i-th Ger?gorin disk ?? _i (A) with the corresponding disk $\Gamma_i(A^T)$ , and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set ??(A)??????(A ^T ). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.     

Chevalley–Weil theorem and subgroups of class groups

We prove, under some mild hypothesis, that an ´etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an “absolute” version of the Chevalley–Weil theorem. Using this result, we are able to generalise the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.     

Second Noether’s theorem with time delay

In this paper, we consider variational problems with delayed argument. Simple proofs of the first and second Noether theorems are given. Applying our results to some concrete problems we obtain conservation laws for systems with delay.     

Warning’s second theorem with relaxed outputs

We present a generalization of Warning’s second theorem to polynomial systems over a finite local principal ring with restricted input and relaxed output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.     

Springer��s theorem for tame quadratic forms over Henselian fields

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.     

Liouville’s theorem and the restricted biharmonic mean value property on the plane

In [3, 4], under some conditions we proved that a bounded Lebesgue measurable function satisfying the restricted biharmonic mean value property in ? ⁿ , where n ≥ 3 or n = 1, is constant. In the present paper, we study the case n = 2.     

Generalization of Kummer’s second theorem with applications

The aim of this research paper is to obtain single series expression of

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

Khinchin��s theorem for approximation by integral points on quadratic varieties

We prove an analogue of the Khinchin??s theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often.     

Berkson’s theorem

We give a new proof of the theorem that Amitsur’s complex for purely inseparable field extensions has vanishing homology in dimensions higher than 2. This is accomplished by computing the kernel and cokernel of the logarithmic derivativet →Dt/t mapping the multiplicative Amitsur complex to the acyclic additive one (D is a derivation of the extension field). This research was supported by National Science Foundation grant NSF GP 1649.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).