A Generalized Fejér’s Theorem for Locally Compact Groups

The classical Fejér’s theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.     

Generalized Browder’s Theorem and SVEP

A bounded operator a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H ₀(λI − T) as λ belongs to certain subsets of . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.     

An Analogue of Beurling’s Theorem for NA Groups

In this paper, we prove Beurling’s theorem for NA groups, from which we derive some other versions of uncertainty principles.     

Engel’s Theorem for Generalized Lie Algebras

We prove analogues of the classical Engel’s Theorem for Lie algebras in the category of comodules over a cotriangular Hopf algebra, generalizing the known result for Lie coloralgebras.     

Generalized Bochner’s Theorem for radial function

A radial function Φ(x) can be expressed by its generator ?(·) through Φ(x)=?(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ? for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ? to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in ICM₀; a positive monotone decreasing function is in ICM₁ and a positive monotone decreasing and convex function is in ICM₂. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as $F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$ , then corresponding radial function Φ(x) is positive definite as a n-variate function iff $\tilde F$ is an incompletely monotone function of order α=(n-1)/2 (or simply $\tilde F \in ICM_{\frac{{n - 1}}{2}} $ .     

A Generalized Form of Whitney＇s Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

Differentiated Generalized Voronovskaja’s Theorem in Compact Disks

In this paper we obtain the differentiated generalized Voronovskaja’s theorem in complex setting with upper and exact quantitative estimates. The results extend that obtained in the real case on [0, 1] in Gonska and Ra?a (Mat Vesnik 61:53–60, 2009) and generalize those obtained in the complex case in Gal (Results Math 53:257–268, 2009).     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

A Generalized Form of Whitney's Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

Combinatorial Generalizations of Jung’s Theorem

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.     

A Generalisation of Tverberg’s Theorem

We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝ ^d of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A ₁,A ₂,…,A _k such that for any C⊂S of at most r points, the convex hulls of A ₁\C,A ₂\C,…,A _k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).     

Symmetric Versions of Laman’s Theorem

Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.     

On Analogues of Heilbronn’s Theorem

Mathematical Notes - Continued fractions with rational partial quotients arise in a natural way in the course of applying any $$k$$ -ary gcd algorithm to the ratio of natural numbers $$a$$ , $$b$$...     

A generalization of Menger’s Theorem

This paper generalizes one of the celebrated results in Graph Theory due to Karl. A. Menger (1927), which plays a crucial role in many areas of flow and network theory. This paper also introduces and characterizes strength reducing sets of nodes and arcs in weighted graphs.     

Extension of Pauli’s theorem to Clifford algebras

Doklady Mathematics -     

Solution of Tikhonov’s Motion-Separation Problem Using the Modified Newton–Kantorovich Theorem

The paper presents a new way to prove the existence of a solution of the well-known Tikhonov’s problem on systems of ordinary differential equations in which one part of the variables performs “fast” motions and the other part, “slow” motions. Tikhonov’s problem has been the subject of a large number of works in connection with its applications to a wide range of mathematical models in natural science and economics. Only a short list of publications, which present the proof of the existence of solutions in this problem, is cited. The aim of the paper is to demonstrate the possibility of applying the modified Newton–Kantorovich theorem to prove the existence of a solution in Tikhonov’s problem. The technique proposed can be used to prove the existence of solutions of other classes of problems with a small parameter.