Conditions for a class of dimensional dual hyperovals to be of polar type

A d-dimensional dual hyperoval with monomial is of polar type if and only if d is even, Gal(GF(2^d+1)/GF(2)) and σ²=id_GF(2^d+1).     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

Closed ideals in with the Duhamel product as multiplication

Let (C^∞,) denote the algebra of infinitely differentiable functions in [0,1] with Duhamel product as multiplication. We describe all the closed ideals in (C^∞,). As a consequence we obtain that the integration operator I, , is unicellular in the space C^∞[0,1], which is the solution of a long-standing problem.     

Existence of positive solutions of nonlinear -point BVP for an increasing homeomorphism and positive homomorphism on time scales

In this paper, we will consider the following multipoint boundary value problem for the following second-order dynamic equations on time scales

where :RR is an increasing homeomorphism and positive homomorphism and (0)=0. By using fixed point theorems, we obtain an existence theorem of positive solutions for the above boundary value problem, which includes and improve some related results in the relevant literature. As an application, an example to demonstrate our results is given.     

Über ein Turánsches problem für radiale, positiv definite Funktionen, II

Turán's problem is to determine the greatest possible value of the integral for positive definite functions f(x), , supported in a given convex centrally symmetric body , . We consider the problem for positive definite functions of the form f(x)=(x₁), , with supported in [0,π], extending results of our first paper from two to arbitrary dimensions.Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, ℓ-1 summability of the inverse Fourier integral on . Their investigations gave rise to a pair of transformations (h_d,m_d) on which they studied using special functions, in particular spherical Bessel functions.To study the d-dimensional Turán problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turán's problem.     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

Widths of weighted Sobolev classes on the ball

We study the Kolmogorov n-widths and the linear n-widths of weighted Sobolev classes on the unit ball B^d in L_q,μ, where L_q,μ, 1≤q≤∞, denotes the weighted L_q space of functions on B^d with respect to weight . Optimal asymptotic orders of and as n→∞ are obtained for all 1≤p,q≤∞ and μ≥0.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

Conditional limiting distribution of beta-independent random vectors

The paper deals with random vectors in , possessing the stochastic representation , where R is a positive random radius independent of the random vector and is a non-singular matrix. If is uniformly distributed on the unit sphere of , then for any integer m<d we have the stochastic representations and , with W≥0, such that W² is a beta distributed random variable with parameters m/2,(d−m)/2 and (U₁,…,U_m),(U_m+1,…,U_d) are independent uniformly distributed on the unit spheres of and , respectively. Assuming a more general stochastic representation for in this paper we introduce the class of beta-independent random vectors. For this new class we derive several conditional limiting results assuming that R has a distribution function in the max-domain of attraction of a univariate extreme value distribution function. We provide two applications concerning the Kotz approximation of the conditional distributions and the tail asymptotic behaviour of beta-independent bivariate random vectors.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

Values of the Pukánszky invariant in free group factors and the hyperfinite factor

Let AMB(L²(M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II₁ factor M in its standard representation. The abelian von Neumann algebra generated by A and JAJ has a type I commutant which contains the projection onto L²(A). Then decomposes into a direct sum of type I_n algebras for n{1,2,…,∞}, and those n's which occur in the direct sum form a set called the Pukánszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S{∞} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of . The only previously known values for masas in free group factors were {∞} and {1,∞}, and some values of the form S{∞} are new also for the hyperfinite factor.We also consider a more refined invariant (that we will call the measure-multiplicity invariant), which was considered recently by Neshveyev and Størmer and has been known to experts for a long time. We use the measure-multiplicity invariant to distinguish two masas in a free group factor, both having Pukánszky invariant {n,∞}, for arbitrary .     

A characterization of convex calibrable sets in with respect to anisotropic norms

A set is called “calibrable” if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the “-calibrability” of bounded convex sets in with respect to a norm (called anisotropy in the sequel) by the anisotropic mean -curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body C satisfying a -ball condition contains a convex -calibrable set K such that, for any V[|K|,|C|], the subset of C of volume V which minimizes the -perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data the characteristic function of a bounded convex set.     

Asymptotic behavior of large solution to elliptic equation of Bieberbach–Rademacher type with convection terms

We analyze the asymptotic behavior of solutions to nonlinear elliptic equation Δu±|u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as , where Ω is a smooth bounded domain in R^N, for some , and . Our approach employs Karamata regular variation theory combined with the method of lower and supper solution.     

Region of variability for close-to-convex functions-II

For a complex number α with let be the class of analytic functions f in the unit disk with f(0)=0 satisfying in , for some convex univalent function in . For any fixed , and we shall determine the region of variability V(z₀,α,λ) for f(z₀) when f ranges over the class

In the final section we graphically illustrate the region of variability for several sets of parameters z₀ and α.     

Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and -minimal polynomials,

Let be a sequence of polynomials with real coefficients such that uniformly for [α-δ,β+δ] with G(eⁱ)≠0 on [α,β], where 0α<βπ and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of p_n+1(cos) on [α,β] for every nn₀. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of p_n(cos) and strictly interlace on [α,β] if on [α,β]. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [-1+,1-], >0, is obtained. Finally it is shown that the results hold for wide classes of weighted L_q-minimal polynomials, q[1,∞], linear combinations and products of orthogonal polynomials, etc.     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Error bounds for the perturbation of the Drazin inverse under some geometrical conditions

This paper studies the Drazin inverse for perturbed matrices. For that, given a square matrix A, we consider and characterize the class of matrices B with index s such that , and , where and denote the null space and the range space of a matrix A, respectively, and A^D denote the Drazin inverse of A. Then, we provide explicit representations for B^D and BB^D, and upper bounds for the relative error B^D-A^D/A^D and the error BB^D-AA^D. A numerical example illustrates that the obtained bounds are better than others given in the literature.