Interlacing properties and the Schur-Szegő composition

Each degree n polynomial in one variable of the form (x+1)(x ^n?1+c ₁ x ^n?2+???+c _n?1) is representable in a unique way as a Schur-Szeg? composition of n?1 polynomials of the form (x+1) ^n?1(x+a _i ), see Kostov (2003), Alkhatib and Kostov (2008) and Kostov (Mathematica Balkanica 22, 2008). Set $\sigma _{j}:=\sum _{1\leq i_{1}<\cdots <i_{j}\leq n-1}a_{i_{1}}\cdots a_{i_{j}}$ . The eigenvalues of the affine mapping (c ₁,…,c _n?1)?(σ ₁,…,σ _n?1) are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.     

An extension of the Erdős-Stone theorem

LetK _p(u₁, ..., u_p) be the completep-partite graph whoseith vertex class hasu _i vertices (lip). We show that the theorem of Erds and Stone can be extended as follows. There is an absolute constant >0 such that, for allr1, 0<1 and=">1/r, every graphG=G ⁿ of sufficiently large order |G|=n with at least

A generalization of the Erdős–Ko–Rado theorem to t-designs in certain semilattices

The Erd?s–Ko–Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.     

A mapping connected with the Schur–Szegő composition

Every monic polynomial in one variable of the form $(x+1)S$, $\mathrm{deg}S=n?1$, is presentable in a unique way as a Schur–Szeg? composition of $n?1$ polynomials of the form ${(x+1)}^{n?1}(x+a_i)$. We prove geometric properties of the affine mapping associating to the coefficients of S the $(n?1)$-tuple of values of the elementary symmetric functions of the numbers $a_i$. To cite this article: V.P. Kostov, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A new algebraic approach to the Szegő-Widom limit theorem

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if are certain matrix valued functions defined on the unit circle, then

Numerical realization of the conditions of Max NSther＇s residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nother＇s residual intersection theorem. The numerical realization relies on obtaining the inter- section of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determin- ing the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

A survey of the Szegő equation

Science China Mathematics - We survey the main properties of the cubic Szegő equation from the PDE viewpoint, emphasising global existence of smooth solutions, analytic regularity, growth of...     

Numerical realization of the conditions of Max Nther's residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nther's residual intersection theorem. The numerical realization relies on obtaining the intersection of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determining the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic,even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.     

A little theorem of the bigℳ in interior point algorithms

When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constant (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2 ^L ) to in linear cases, whereL denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too large, we may have numerical instability and/or computational inefficiency, while the artificial problem with not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents a little theorem of the big, which will enable us to find whether is not large enough, and to update during the iterations of the algorithm even if we start with a smaller. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal—dual interior point algorithm using Lustig's limiting feasible direction vector.     

A stability version for a theorem of Erdős on nonhamiltonian graphs

Let $n,d$ be integers with $1\le d\le ?\frac{n?1}{2}?$, and set $h(n,d)?\left(\genfrac{}{}{0ex}{}{n?d}{2}\right)+d^2$ and $e(n,d)?max\{h(n,d),h(n,?\frac{n?1}{2}?)\}$. Because $h(n,d)$ is quadratic in $d$, there exists a $d_0\left(n\right)=(n∕6)+O\left(1\right)$ such that

$e(n,1)>e(n,2)>?>e(n,d_0)=e(n,d_0+1)=?=e\left(n,?\frac{n?1}{2}?\right).$

A theorem by Erd?s states that for $d\le ?\frac{n?1}{2}?$, any $n$-vertex nonhamiltonian graph $G$ with minimum degree $\delta \left(G\right)\ge d$ has at most $e(n,d)$ edges, and for $d>d_0\left(n\right)$ the unique sharpness example is simply the graph $K_n?E\left(K_{?(n+1)∕2?}\right)$. Erd?s also presented a sharpness example $H_{n,d}$ for each $1\le d\le d_0\left(n\right)$.We show that if $d<d_0\left(n\right)$ and a $2$-connected, nonhamiltonian $n$-vertex graph $G$ with $\delta \left(G\right)\ge d$ has more than $e(n,d+1)$ edges, then $G$ is a subgraph of $H_{n,d}$. Note that $e(n,d)?e(n,d+1)=n?3d?2\ge n∕2$ whenever $d<d_0\left(n\right)?1$.     

A generalization of the Beurling—Lax theorem

We obtain conditions for the completeness of the system {G(z)e ^τz , τ ≤ 0} in the space H _σ ² (?₊), 0 < σ < + ∞, of functions analytic in the right-hand half-plane for which $$\parallel f\parallel : = \mathop {\sup }\limits_{ - \pi /2 < \varphi < \pi /2} \left\{ {\int_0^{ + \infty } {|f(re^{i\varphi } )|^2 } e^{ - 2r\sigma |\sin \varphi |} dr} \right\}^{1/2} < + \infty $$ .     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

A generalization of the Lindelöf theorem

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ? in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.     

Extensions and consequences of Chvátal-Erdős' theorem

A classical result of Chvátal and Erds states that a graph with independence number smaller or equal to its connectivity contains a Hamilton cycle. In this note we discuss some extensions of this theorem and show how they can be used to proof several other results in hamiltonian graph theory. Although several of the results are known, the proofs in this note are in general essentially shorter than the original proofs, and also give an indication of the relations between the results.Supported by a grant from the Natural Sciences and Engineering Research Council of Canada