Data-sparse approximation to a class of operator-valued functions

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator .

     

Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

All maps of type are boundary maps

Let be a continuous map of an interval into itself having periodic points of period for all and no other periods. It is shown that every neighborhood of contains a map such that the set of periods of the periodic points of is finite. This answers a question posed by L. S. Block and W. A. Coppel.

     

Commuting Toeplitz operators with pluriharmonic symbols

By making use of -harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in -dimensional complex space.

     

On the correlations of directions in the Euclidean plane

Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Irregular hypergeometric systems associated with a singular monomial curve

In this paper we study irregular hypergeometric systems defined by one row. Specifically, we calculate slopes of such systems. In the case of reduced semigroups, we generalize the case studied by Castro and Takayama. In all the cases we find that there always exists a slope with respect to a hyperplane of this system. Only in the case of an irregular system defined by a integer matrix we might need a change of coordinates to study slopes at infinity. In the other cases slopes are always at the origin, defined with respect to a hyperplane. We also compute all the -characteristic varieties of the system, so we have a section of the Gröbner fan of the module defined by the hypergeometric system.

     

On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

We are concerned with the derivation of Poincaré-Friedrichs type inequalities in the broken Sobolev space $W^{2,1}$($Ω$; $\mathcal{T}_h$) with respect to a geometrically conforming, simplicial triagulation $\mathcal{T}_h$ of a bounded Lipschitz domain $Ω$ in $\mathbb{R}^d$ , $d$ $∈$ $\mathbb{N}$. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as ${\rm C}^0$ Discontinuous Galerkin (${\rm C}^0$${\rm DG}$) approximations of minimization problems in the Sobolev space $W^{2,1}$($Ω$), or more generally, in the Banach space $BV^2$($Ω$) of functions of bounded second order total variation. As an application, we consider a ${\rm C}^0$${\rm DG}$ approximation of a minimization problem in$BV^2$($Ω$) which is useful for texture analysis and management in image restoration.     

-summand vectors in Banach spaces

The aim of this paper is to study the set of all -summand vectors of a real Banach space . We provide a characterization of -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an -sum of a Hilbert space and a Banach space without nontrivial -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.

     

Estimates of the derivatives for parabolic operators with unbounded coefficients

We consider a class of second-order uniformly elliptic operators with unbounded coefficients in . Using a Bernstein approach we provide several uniform estimates for the semigroup generated by the realization of the operator in the space of all bounded and continuous or Hölder continuous functions in . As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation (0$">) and the nonhomogeneous Dirichlet Cauchy problem . Then, we prove two different kinds of pointwise estimates of that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup in weighted -spaces related to the invariant measure associated with the semigroup.

     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

Regularity of the free boundary for the porous medium equation

We study the regularity of the free boundary for solutions of the porous medium equation , , on , with initial data nonnegative and compactly supported. We show that, under certain assumptions on the initial data , the pressure will be smooth up to the interface , when , for some . As a consequence, the free-boundary is smooth.

     

Fixed point theorems for multivalued non-linear $F$-contractions on quasi metric spaces with an application

In this paper, we introduce $(\alpha,\beta)$-type $F-\tau$ contraction and utilize the same to prove some fixed point results for multivalued mappings in quasi metric spaces. Furthermore, we furnish with some examples to exhibit the utility of our results. As an application, we establish the existence of a solution for a non-linear integral equation.     

右型A半群的F-覆盖

A right adequate semigroup of type F is defined as a right adequate semigroup which is an F-rpp semigroup. A right adequate semigroup T of type F is called an F-cover for a right type-A semigroup S if S is the image of T under an L＊-homomorphism. In this paper, we will prove that any right type-A monoid has F-covers and then establish the structure of F-covers for a given right type-A monoid. Our results extend and enrich the related results for inverse semigroups.     

Fixed point property and the Fourier algebra of a locally compact group

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) spaces and use this for the Fourier algebra of a locally compact group In particular we show that if is an IN-group, then has the weak fpp if and only if is compact. We also show that if is any locally compact group, then has the fixed point property (fpp) if and only if is finite. Furthermore if a nonzero closed ideal of has the fpp, then must be discrete.

     

Arithmetic inflection formulae for linear series on hyperelliptic curves

Over the complex numbers, Plücker's formula computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in ${\mathbb{A}}^1$-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.     

The asymptotic efficiency of randomized nets for quadrature

An -type discrepancy arises in the average- and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by , which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized -nets in base . For moderately smooth the discrepancy is , and for with greater smoothness the discrepancy is , where is the number of points in the net. Numerical experiments indicate that the -nets of Faure, Niederreiter and Sobol' do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.

     

Singularities of the hypergeometric system associated with a monomial curve

We compute, using -module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.

     

Fixed Point Theorems for Weakly Contractive Mappings in Ordered Metric Spaces with an Application

In this paper, we prove fixed point theorem for weakly contractive mappings using locally $T$-transitivity of binary relation and presenting an analogous version of Harjani and Sadarangani theorem involving more general relation theoretic metrical notions. Our fixed point results under universal relation reduces to Harjani and Sadarangani [Nonlinear Anal., 71 (2009), 3403–3410] fixed point theorems. In this way we also generalize some of the recent fixed point theorems for weak contraction in the existing literature.     

A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function at such a point where is smooth. We also consider the extension of the Zygmund classes and to the two-dimensional torus .