An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

ON EXISTENCE OF SOLUTIONS OF DIFFERENCE RICCATI EQUATION

Consider the difference Riccati equation f(z+1) =(A(z)f(z)+B(z))/(C(z)f(z)+D(z)),where A,B, C,D are meromorphic functions, we give its solution family with one-parameter H(f(z))={f_0(z),f(z)=((f_1(z)-f_0(z))(f_2(z)-f_0(z)))/(Q(z)(f_2(z)-f_1(z))+(f_2(z)-f_0(z)))}, where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f_0(z),f_1(z),f_2(z) are its three distinct meromorphic solutions.     

Global Solutions for Nonlinear Delay Evolution Inclusions with Nonlocal Initial Conditions

We prove a sufficient condition for the existence of global C ⁰-solutions for a class of nonlinear functional differential evolution equation of the form $ \left\{{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \right. $ \left\{\begin{array}{ll} \displaystyle u'(t)\in Au(t)+f(t),&t\in\mathbb{R}_+, \\[2mm] f(t)\in F(t,u(t),u_t),&t\in\mathbb{R}_+, \\[2mm] u(t)=g(u)(t),& t\in [\,-\tau,0\,], \end{array}\right.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

A symmetry-breaking problem in the annulus

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument     

Integral manifolds and a reduction principle in stability theory. II

In this work we prove reduction theorems, according to which the problem of stability of the zero solution of a system of differential equationsdx/dt=A(t)x+B(t)z+g(t, x, z), dz/dt=C(t)z+h(t, x, z) reduces to the problem of stability of the zero solution of the equationdx/di=A(t)x+B(t) (t,x)+g(t, x, (t, x)), in which the vector function y=(t,x) defines the local or the nonlocal integral manifold that contains the graph of the zero solution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1315–1321, October, 1990.     

Computation of multiple eigenvalues of infinite tridiagonal matrices

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of ; (B) the zeros of ; (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their ``double pairs'.

     

Об одной теореме И. И. П ривалова

Let G be a finite Jordan domain,f(z)=u(z)+iv(z) an analytic function inG. Connections between smoothness of the functionu(z) on dG and smoothness of the functionf(z) on ¯G are obtained. In these results the regionG is assumed to satisfy the condition

Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics

Cauchy problems for a second order linear differential operator equation

On the Existence of Weak Solutions for a Degenerate and Singular Elliptic System in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ ^N , N ≧2 of the form

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

A compact set K ì \mathbbC^N{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

一类无穷下级整函数的Julia集的径向分布

主要研究方程f"(z)+A(z)f'(z)+B(z)f(z)=0(A(z)),B(z)为整函数)的解、解的多项式或微分多项式这些具有无穷下级的整函数的Julia集的径向分布问题.     

On Interpolating Blaschke Products and?Blaschke-Oscillatory Equations

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+A(z)f=0, where A(z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case A(z)=\frac1+4g²(1-z²)²A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}} reveals that an infinite zero sequence is always a union of two exponential sequences.     

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

Generalized equations,variational inequalities and a weak Kantorovich theorem

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ' 0f(u)+g(u)\owns 0 where f is a Fréchet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type Az+g(z) ' bAz+g(z)\owns b where A is a linear operator.     

Hlawka’s functional inequality

The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality) $$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$ for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality: $$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$ which in particular holds true for all x, y, z from a real or complex inner product space.     

Spectral estimates of Cauchy's transform inL 2(Ω)

We prove that the singular numbers of the Cauchy transform onL ²(D) are asymptotically , whiles _n (C _| L _a ² (D))1/n (whereL _a ² (D) is the subspace of analytic functions inL ²(D)). Also, the singular numbers of the logarithmic potential onL ²(D) are asympoticallys _n (L)1/n, whiles _n(L _{|L a} ² (D))1/n ². Our methods yield the asymptotic behavior of the singular numbers of the Cauchy Transform fromL _L ² () intoL ²() where and are rotation-invariant measures on .The author was partly supported by a grant from the national Science Foundation.     

An analysis of approximations for maximizing submodular set functions—I

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)z(ST)+z(ST) for allS, T inN. Such a function is called submodular. We consider the problem max_SN{a(S):|S|K,z(S) submodular}.Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a greedy heuristic always produces a solution whose value is at least 1 –[(K – 1)/K] ^K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e – 1)/e, where e is the base of the natural logarithm.On leave of absence from Cornell University and supported, in part, by NSF Grant ENG 75-00568.Supported, in part, by NSF Grant ENG 76-20274.