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The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.

We improve Pellet's theorem for both scalar and matrix polynomials by using polynomial multipliers.

The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).

In this paper, we establish a general semilocal convergence theorem (with computationally verifiable initial conditions and error estimates) for iterative methods for simultaneous approximation of polynomial zeros. As application of this theorem, we provide new semilocal convergence results for Ehrlich's and Dochev-Byrnev's root-finding methods. These results improve the results of Petkovic et?al. (1998) and Proinov (2006). We also prove that Dochev-Byrnev's method (1964) is identical to Presic-Tanabe's method (1972).

We present a restricted variable generalization of Warning's Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink's restricted variable generalization of Chevalley's Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning's Second Theorem implies Chevalley's Theorem, our result implies Schauz-Brink's Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems. Let q = pl be a power of a prime number p, and let Fq be "the" finite field of order q. For a1,...,an, NźZ+, we denote by m(a1,...,an;N)źZ+ a certain combinatorial quantity defined and computed in Section 2.1.

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.

For actions of a sofic group on probability spaces, the entropy has been defined by Bowen, with an extension by Kerr-Li. In particular, when the action is by homeomorphisms of a compact space preserving a given measure, Kerr-Li show one can compute the measure-theoretic entropy in a manner which uses the topology of the space. We show how to compute the entropy of a an action of homeomorphisms of a Polish space preserving a given measure, also in a manner which uses the topology of the space. We give applications to spectral properties of actions with positive entropy, as well as for actions of completely positive entropy., 26 pages. Removed the section on distal actions as we will prove more general results in "Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups". This is the final version to appear in Journal of Institute of Mathematics of Jussieu

Let L be the Laplace matrix of a weighted digraph. The aim of the paper is to establish a simple way for computing any coefficient of the characteristic polynomial of L as a constant sign sum over the incoming spanning forests. The idea is to express L as the product of generalized (weighted) incidence matrices. It turns out that the minors of them can be studied in terms of the tree-like structure of the digraph. This makes it possible to compute the minors of L.

In 1976, Suffridge proved an intriguing theorem regarding the convolution of polynomials with zeros only on the unit circle. His result generalizes a special case of the fundamental Grace–Szego convolution theorem, but so far it is an open problem whether there is a Suffridge-like extension of the general Grace–Szego convolution theorem. In this paper, we show that Suffridge’s convolution theorem holds for a certain class of polynomials with zeros in the unit disk and thus obtain an extension for one further special case of the Grace–Szego convolution theorem.

We show that a discrete version of Tchakaloff’s theorem on the existence of positive algebraic cubature formulas, entails that the information required for multivariate polynomial approximation can be suitably compressed.

The problem of finding or characterizing the zeros of polynomial functions has a long history. The obtained results varied from theory, algorithms and numerical simulations. Historically, this study began with the fundamental theorem of algebra proved by Gauss. The most known results in this direction is the fact that a real polynomial of degree n has at most n real zeros. There is also the so called Descartes rule of signs concerning the number of positive zeros. Some generalizations of Descartes rule are know such as the BudanFourier theorem that gives an upper bound for the number of zeros of a polynomial. Also, the Sturm’s theorem that gives a method for determining the exact number of zeros in an interval [Hen, chapter 6] and [Hou, chapter 2]. Recent results uses the classical Enestrom-Kakeya theorem to restricts the location of the zeros based on a condition imposed on the coefficients of the polynomial under invistigation. See [Bre] and references therain. In this paper, we will use a dynamical system result concerning periodic points of a continuous function. The result is called Sharkovsky theorem [Sha1, Sha2, Sha3, Sha4, Sha5] that gives a complete description of possible sets of periods for continuous mappings defined on an interval. The interval need not be closed or bounded. The main idea used here is the notion of

In this paper we give interesting generalizations of some well-known Enestrom-Kakeya type results on the location of zeros of a complex polynomial under less restrictive conditions on the coefficients of the polynomial.BIBECHANA 13 (2016) 1-8

This paper focuses on some main properties of the finite partially ordered sets. These properties are furnished in the form of theorems. Here we have presented three such theorems. The first theorem is called as ‘duality theorem’. This fundamental theorem was first obtained by Greene. Few years later it was rediscovered and given an alternative proof by Fomin. The second theorem bestows the functionality property. The proof of this was also done by Greene. However, Gansner gave an alternative proof of the theorem taking advantage of a connection between poset and linear algebra. The proof of the third theorem is fully due to us. This theorem gives rise to a recursive computation of the shape. In the present paper we have discussed the first two properties through suitable illustrations only whereas a complete proof is furnished for the last one. BIBECHANA 13 (2016) 126-130

The central sets theorem originally proven by H. Furstenburg is a powerful result which is applicable to derive many combinatorial conclusions. Furstenburg's original theorem applied to N and finitely many sequences in Z . Some strengthenings of this theorem have been derived first by V. Bergelson and N. Hindman in 1990. Later in 2008, D. De, N. Hindman, and D. Strauss proved a stronger version of the central sets theorem for arbitrary semigroups S which applied to all sequences in S. We provide here a generalization of the stronger version and some applications of this new generalization.

In this paper, we put restrictions on the coefficients of a polynomial in order to improve the bounds for their zeros in a specific region. Our results extend and generalise a number of previously well known theorems including Enestrom–Kakeya theorem.

An alternative derivation for the general condition for delivering maximum power from a 2-terminal network n to another 2-terminal network N is given. This condition is solely dependent upon the Th...

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.

Every nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers.

In a prior paper, the author generalized the classical factor theorem of Sinai to actions of arbitrary countably infinite groups. In the present paper, we use this theorem and the techniques of its proof in order to study connections between positive entropy phenomena and the Koopman representation. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. By combining this result with the generalized factor theorem of the previous paper, we conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley--Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap. We also prove generalizations of these results that apply relative to sub-$\sigma$-algebras and apply to non-free actions.

The Matrix-Tree Theorem is a formula for the number of spanning trees of a graph in terms of the determinant of a certain matrix. We begin with the necessary graph-theoretical background. Let G be a finite graph, allowing multiple edges but not loops. (Loops could be allowed, but they turn out to be completely irrelevant.)

In this paper, we improve the convergence theorem in the paper by Yang (Journal of Industrial and Management Optimization 1, 211---217, 2005), and propose a new modified convergence theorem. The theorem and the proof presented in the present paper are interesting improvements on the convergence theorem of Yang.

In this paper we consider the class of polynomials of the type , , having some zeros at origin and rest of zeros on or outside the boundary of a prescribed disk, and obtain the generalization of well known results.

The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary (i.e., first-order definable) class of Kripke frames to be modally definable. Here we obtain a local analogue of this result, where in place of frames we consider n-frames, which are frames with n distinguished worlds. For talking about n-frames, we generalize customary modal formulas to what we call modal expressions. Unlike a modal formula, which is evaluated at a single world of a Kripke model, a modal expression with n individual variables is evaluated at an n-tuple of worlds, just as a first-order formula with n free variables. We introduce operations on n-frames that preserve validity of modal expressions, and show that closure under these operations is a necessary and sufficient condition for an elementary class of n-frames to be modally definable. We also discuss the relationship between modal expressions and hybrid logic and leave open questions.

A mean-value theorem for the sum of a “saw-tooth” function of a polynomial is proved. The estimate is exact relative to the main parameter which is the length of the interval of summation.

In this note we present a counterexample to Theorem 24 in G.S.F. Frederico and D.F.M. Torres (2010) [3] .

Conditions are given under which a normal closed subset of an association scheme of finite order possesses a complement. Our attempts are in the spirit of the Schur–Zassenhaus Theorem on finite groups and partially generalize that theorem.

We consider a notion of approximation for terms of de Groote–Saurin Λμ-calculus. Then, we introduce an intersection type assignment system for that calculus which is invariant under subject conversion. The type assignment system also induces a filter model, which is an extensional Λμ-model in the sense of Nakazawa and Katsumata. We then establish the approximation theorem, stating that a type can be assigned to a term in the system if and only if it can be assigned to same of its approximations.

We prove an inequality for concave functions and partitioned matrices whose numerical ranges lie in a sector. This complements a theorem by E.Y. Lee concerning the positive semi-definite case.

Let $$G$$G be a graph and $$f:V(G)\rightarrow \{1,2,3,4,\ldots \}$$f:V(G)?{1,2,3,4,?} be a function. We denote by $$odd(G)$$odd(G) the number of odd components of $$G$$G. We prove that if $$odd(G-X)\le \sum _{x\in X}f(x)$$odd(G-X)≤?x?Xf(x) for all $$ X\subset V(G)$$X?V(G), then $$G$$G has a $$(1,f)$$(1,f)-factor $$F$$F such that, for every vertex $$v$$v of $$G$$G, if $$f(v)$$f(v) is even, then $$\deg _F(v)\in \{1,3,\ldots ,f(v)-1,f(v)\}$$degF(v)?{1,3,?,f(v)-1,f(v)}, and otherwise $$\deg _F(v)\in \{1,3, \ldots , f(v)\}$$degF(v)?{1,3,?,f(v)}. This theorem is a generalization of both the $$(1,f)$$(1,f)-odd factor theorem and a recent result on $$\{1,3, \ldots , 2n-1,2n\}$${1,3,?,2n-1,2n}-factors by Lu and Wang. We actually prove a result stronger than the above theorem.

Sturm's theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semi-open interval, not counting multiplicity. A generalization of Sturm's theorem is known as Tarski's theorem, which provides a linear relationship between functions known as Tarski queries and cardinalities of certain sets. The linear system that results from this relationship is in fact invertible and can be used to explicitly count the number of roots of a univariate polynomial on a set defined by a system of polynomial relations. This paper presents a formalization of these results in the PVS theorem prover, including formal proofs of Sturm's and Tarski's theorems. These theorems are at the basis of two decision procedures, which are implemented as computable functions in PVS. The first, based on Sturm's theorem, determines satisfiability of a single polynomial relation over an interval. The second, based on Tarski's theorem, determines the satisfiability of a system of polynomial relations over the real line. The soundness and completeness properties of these decision procedures are formally verified in PVS. The procedures and their correctness properties enable the implementation of PVS strategies for automatically proving existential and universal statements on polynomial systems. Since the decision procedures are formally verified in PVS, the soundness of the strategies depends solely on the internal logic of PVS rather than on an external oracle.

In this letter we determine the largest dynamic range within which any two nonnegative integers can be uniquely determined from their residues modulo a set of moduli. We also propose an efficient determination algorithm that extends a recent known result.

Inspired by Gale's proof of Farkas's lemma, this expository note aims to give a very simple and intuitive proof of Gordan's theorem and the equivalence of this theorem to Farkas's lemma and some formulations of the separating hyperplane theorems. The tool we have employed is limited to only very simple linear algebra over the eld of real numbers.

A theorem of Boyle and Handelman gives necessary and sufficient conditions for an [math] -tuple of nonzero complex numbers to be the nonzero spectrum of some matrix with nonnegative entries, but is not constructive and puts no bound on the necessary dimension of the matrix. Working with polynomial matrices, we constructively reprove this theorem in a special case, with a bound on the size of the polynomial matrix required to realize a given polynomial.

The present paper is devoted to studying the problem of checking whether a given singular design is L optimal. As a possible solution, a general version of the equivalence theorem for L-optimal designs is formulated. A method for finding coefficients of an extreme polynomial in the case where this polynomial cannot be unambiguously determined by the dispersion matrix of a design is proposed.

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If one adds one extra assumption to the classical Knaster– Kuratowski–Mazurkiewicz (KKM) theorem, namely that the sets Fi are convex, one gets the “Elementary” KKM theorem; the name is due to A. Granas and M. Lassonde (1995) who gave a simple proof of the Elementary KKM theorem and showed that despite being “elementary,” it is powerful and versatile. It is shown here that this Elementary KKM theorem is equivalent to Klee’s theorem, the Elementary Alexandroff– Pasynkov theorem, the Elementary Ky Fan theorem and the Sion–von Neumann minimax theorem, as well as a few other classical results with an extra convexity assumption; hence the adjective “elementary.” The Sion–von Neumann minimax theorem itself can be proved by simple topological arguments using connectedness instead of convexity. This work answers a question of Professor Granas regarding the logical relationship between the Elementary KKM theorem and the Sion–von Neumann minimax theorem.

We obtain a new infinite-dimensional linking theorem. This theorem replaces the upper semi-continuous assumption in the classical infinite-dimensional linking theorem of Kryszewski and Szulkin by a new assumption. As an application of this theorem, we obtain a nontrivial solution for a strongly indefinite periodic Schrodinger equation with sign-changing nonlinearity.

For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds., 31 pages