The generalized Levinger transformation

In this paper, we present new results relating the numerical range of a matrix A with the generalized Levinger transformation L(A,α,β)=αH_A+βS_A, where H_A and S_A, are, respectively the Hermitian and skew-Hermitian parts of A. Using these results, we then derive expressions for eigenvalues and eigenvectors of the perturbed matrix A+L(E,α,β), for a fixed matrix E and α, β are real parameters.     

A general vanishing theorem

Let E be a vector bundle and L be a line bundle over a smooth projective variety X. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,{\mathcal S}^{\alpha}E\otimes \wedge^{\beta} E\otimes L)$ when S ^α?+?β E???L is ample. This condition is shown to be invariant under the interchange of p and q. The optimality of this condition is discussed for some parameter values.     

The singularity spectrum of some non-regularity moran fractals

In this paper, we shall study the multifractal decomposition behavior for a family of sets E known as Moran fractals. For each value of the parameter α ∈ (α_min, α_max), we define “multifractal components” E_α of E, and show that they are non-regularity fractals (in the sense of Taylor). By obtaining the new sufficient conditions for the valid multifractal formalisms of non-regularity Moran measures, we give explicit formula for the Hausdorff dimension and Packing dimension of E_α respectively. In particular, we describe a large class of non-regularity Moran measure satisfying the explicit formula.     

Supersingular parameters of the Deuring normal form

It is proved that the supersingular parameters α of the elliptic curve E ₃(α): Y ²+αXY+Y=X ³ in Deuring normal form satisfy α=3+γ ³, where γ lies in the finite field $\mathbb{F}_{p^{2}}$ . This is accomplished by finding explicit generators for the normal closure N of the finite extension k(α)/k(j(α)), where α is an indeterminate over $k=\mathbb{F}_{p^{2}}$ , and j(α) is the j-invariant of E ₃(α). Computing an explicit algebraic form for the elements of the Galois group of the extension N/k(j) leads to some new relationships between supersingular parameters for the Deuring normal form. The function field N, which contains the function field of the cubic Fermat curve, is then used to show how the results of Fleckinger for the Deuring normal form are related to cubic theta functions.     

Constructive almost periodic factorization of some triangular matrix functions

A factorability criterion is obtained constructively, and the respective factorization obtained explicitly, for 2×2 triangular almost periodic matrix functions of the form . Here f=c−1e−α−c₀+c₁eβ, eμ(x):=eiμx, cj are non-zero constants and 0<α,β, α+β<λ?α+β+max{α,β} with α/β being irrational. Note that the factorization problem, even for triangular matrix functions as above with an arbitrary trinomial f, is open. The result obtained is yet another step towards its solution.     

On the convergence of the linear means of Jacobi series at Lebesgue points in the case of half-integer α

We investigate the convergence of the linear means of the Fourier-Jacobi series of functions ?(x) from the weight space L _α,β for x = 1 for the case in which this point is a Lebesgue point for ?. We establish su.cient summability conditions depending on the behavior of the function on the closed interval [?1, 0] and on the properties of the matrix involved in the summation method.     

Some radius problems related to a certain subclass of analytic functions

For real parameters α and β such that 0≤α1β,we denote by S(α,β) the class of normalized analytic functions which satisfy the following two-sided inequality:αR(zf′(z)/f(z))β,z∈U,where U denotes the open unit disk.We find a sufficient condition for functions to be in the class S(α,β) and solve several radius problems related to other well-known function classes.     

Convex relaxation for solving posynomial programs

Convex underestimation techniques for nonlinear functions are an essential part of global optimization. These techniques usually involve the addition of new variables and constraints. In the case of posynomial functions \({x_1^{\alpha _1 } x_2^{\alpha _2 }\ldots x_n^{\alpha _n } ,}\) logarithmic transformations (Maranas and Floudas, Comput. Chem. Eng. 21:351–370, 1997) are typically used. This study develops an effective method for finding a tight relaxation of a posynomial function by introducing variables y _j and positive parameters β _j , for all α _j > 0, such that \({y_j =x_j^{-\beta _j }}\) . By specifying β _j carefully, we can find a tighter underestimation than the current methods.     

The extended gamma class and applications

The gamma class Γ _α (g) consists of positive and measurable functions that satisfy f(x+yg(x))/f(x)→exp(αy). In most cases, the auxiliary function g is Beurling varying, i.e. g(x)/x→0 and g∈Γ₀(g). Taking h=logf, we find that h∈EΓ _α (g,1), where EΓ _α (g,a) is the class of ultimately positive and measurable functions that satisfy (f(x+yg(x))?f(x))/a(x)→αy. In this paper, we discuss local uniform convergence for functions in the classes Γ _α (g) and EΓ _α (g,a). From this we obtain several representation theorems. We also prove some higher order relations for functions in the classes Γ _α (g) and EΓ _α (g,a). Some applications conclude the paper.     

Compactly supported multiwindow dual Gabor frames of rational sampling density

We consider multiwindow Gabor systems (G _N ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski–Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski–Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a?=?1 and b?=?1. As another application, we show that a multiwindow Gabor system (G _N ; 1, 1) forms an orthonormal basis if and only if it has only one window (N?=?1) which is a sum of characteristic functions whose supports ‘essentially’ form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4)     

Regular bases in products of power series spaces

We prove that Λ_∞(α) × Λ₁(β) has a regular basis iff Λ_∞(α) ? Λ₁(β) has a regular basis iff Λ_∞(α) ? Λ₁(β) is isomorphic to a Cartesian product of two power series spaces. We give a simple condition on α, β which determines when these equivalent statements hold.     

Binary games in constitutional form and collective choice

In this paper, we define the notion of binary game in constitutional form. For this game, we define a core and give a necessary and sufficient condition for a game to be stable.We define a representation of a collective choice rule by a binary game in constitutional form and characterize those collective choice rules which are representable.We finally introduce the notion of c-social decision function and characterize, as an application of our theorem on stability of binary constitutional games, the collective choice rules which are c-social decision functions.Our representation of a collective choice rule by a binary game in constitutional form is an obvious improvement of the classical representation by a simple game.     

Embeddings and separable differential operators in spaces of Sobolev-Lions type

We study embedding theorems for anisotropic spaces of Bessel-Lions type H _p,γ ^l (Ω; E ₀, E), where E ₀ and E are Banach spaces. We obtain the most regular spaces E _α for which mixed differentiation operators D ^α from H _p,γ ^l (Ω; E ₀, E) to L _p,γ(Ω; E _α ) are bounded. The spaces E _α are interpolation spaces between E ₀ and E, depending on α = (α ₁, α ₂, …, α _n ) and l = (l ₁, l ₂, …, l _n ). The results obtained are applied to prove the separability of anisotropic differential operator equations with variable coefficients.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Integral transforms of analytic functions on abstract Wiener spaces

Let (H, B) be an abstract Wiener pair and p_t the Wiener measure with variance t. Let $\text{E}$_a be the class of exponential type analytic functions defined on the complexification [B] of B. For each pair of nonzero complex numbers α, β and f ? $\text{E}$_a, we define

$\text{F}{}_{\mathrm{\alpha ,\beta }}\text{f(y)=}\text{∫}\text{B}\text{f(αx+βy)p}{}_1\text{(dx) (y ?[B]).}$

We show that the inverse $\text{F}$_α,β^?1 exists and there exist two nonzero complex numbers α′,β′ such that

. Clearly, the Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are special cases of $\text{F}$_α,β. Finally, we apply the transform to investigate the existence of solutions for the differential equations associated with the operator $\text{N}$_c, where c is a nonzero complex number and $\text{N}$_c is defined by

$\text{N}{}_{\mathrm{c}}\text{u(x)=?Δu(x)+c(x,Du(x))}$

where Δ is the Laplacian and (·, ·) is the $\text{B-B}{}^1$ pairing. We show that the solutions can be represented as integrals with respect to the Wiener measure.     

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.     

The range of a vector measure

Let X be a completely regular Hausdorff space and E be a locally convex Hausdorff space. Then C_b(X) ? E is dense in (C_b(X, E), β₀), (C_b(X), β) ?_?E = (C_b(X) ? E, β) and (C_b(X), β₁) ?_?E = (C_b(X) ? E, β₁). For a separable space E, (C_b(X, E), β₀) is separable if and only if X is separably submetrizable. As a corollary, for a locally compact paracompact space X, if (C_b(X, E), β₀) is separable, then X is metrizable.     

Limits of translation invariant experiments

In this article we study the behaviour of translation invariant experiments. The main result establishes necessary and sufficient conditions for the local approximation of a sequence of suitably normalized product experiments by Gaussian shifts with respect to the weak convergence. Moreover, it turns out that the local asymptotically normal (LAN) condition with uniform remainders with respect to the local parameter can be characterized in terms of the local behavior of the Hellinger distances. This result depends on the fact that a convergent sequence E_n is always equicontinous if E_n is convergent. The result answers a question which has been posed in the author's article about “the convergence of almost regular statistical experiments to Gaussian shifts” (In Proceedings, 4th Pannonian Sympos. Math. Statist., Bad Tatzmannsdorf, 1983). The proofs rely on a theorem showing that binary experiments can be treated by positive definite functions.     

Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν×H=ν×(|E×ν|^α−1E×ν) for some α∈(0,1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E₀ and H₀. We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.     

Majorizing potentials in strong ratio limit theorems

In this paper, we study the asymptotic properties of the polynomials P _n (z) = P _n (z; f), corresponding to an interpolation table α ? E, where E is a bounded continuum in the complex plane with a connected complement, the table α satisfies the Kakehashi condition, and f is an arbitrary function holomorphic on E. In particular, for zeros of such polynomials, we obtain a generalization of the classical Jentzsch-Szeg? theorem on the distribution of zeros of partial sums of Taylor series.