• Volume 117, Issue 1

February 2007,   pages  1-145

• Frames and Bases in Tensor Products of Hilbert Spaces and Hilbert $C^\ast$-Modules

In this article, we study tensor product of Hilbert $C^∗$-modules and Hilbert spaces. We show that if 𝐸 is a Hilbert 𝐴-module and 𝐹 is a Hilbert 𝐵-module, then tensor product of frames (orthonormal bases) for 𝐸 and 𝐹 produce frames (orthonormal bases) for Hilbert $A \otimes B$-module $E \otimes F$, and we get more results.

For Hilbert spaces 𝐻 and 𝐾, we study tensor product of frames of subspaces for 𝐻 and 𝐾, tensor product of resolutions of the identities of 𝐻 and 𝐾, and tensor product of frame representations for 𝐻 and 𝐾.

• Khinchin's Inequality, Dunford-Pettis and Compact Operators on the Space $C([0, 1], X)$

We prove that if $X, Y$ are Banach spaces, 𝛺 a compact Hausdorff space and $U:C(\Omega,X)\to Y$ is a bounded linear operator, and if 𝑈 is a Dunford–Pettis operator the range of the representing measure $G(\Sigma)\subseteq D P(X, Y)$ is an uniformly Dunford–Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_0$ or $l_p,(1\leq p &lt; \infty)$ be Dunford–Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.

• On the Stability of Jensen's Functional Equation on Groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and $T(2,K)$, where 𝐾 is an arbitrary commutative field with characteristic different from two. We also prove that any group 𝐴 can be embedded into some group 𝐺 such that the Jensen functional equation is stable on 𝐺.

• On the Problem of Isometry of a Hypersurface Preserving Mean Curvature-Erratum

The problem of determining the Bonnet hypersurfaces in $R^{n+1}$, for $n&gt;1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface.

The orthogonal net which is called 𝐴-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in $R^{n+1}$ with no umbilical points is a Bonnet hypersurface if and only if it has an 𝐴-net.

• Quotient Probabilistic Normed Spaces and Completeness Results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.

• Semigroups on Frechet Spaces and Equations with Infinite Delays

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

• Positive Solutions and Eigenvalue Intervals for Nonlinear Systems

This paper deals with the existence of positive solutions for the nonlinear system

$$(q(t)\phi(p(t){u'}_i(t)))'+f^i(t,u)=0, \quad 0 &lt; t &lt; 1, \quad i=1,2,\ldots,n.$$

This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $u=(u_1,...,u_n)$ and $f^i,i=1,2,\ldots,n$ are continuous and nonnegative functions, $p(t), q(t):[0, 1]\to(0,\infty)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for

$$(q(t)\phi(p(t){u'}_i(t)))'+\lambda h_i(t)g^i(u)=0,\quad 0 &lt; t &lt; 1, \quad i=1,2,\ldots,n.$$

The proof is based on a well-known fixed point theorem in cones.

• Strong Convergence of Modified Ishikawa Iterations for Nonlinear Mappings

In this paper, we prove a strong convergence theorem of modified Ishikawa iterations for relatively asymptotically nonexpansive mappings in Banach space. Our results extend and improve the recent results by Nakajo, Takahashi, Kim, $Xu$, Matsushita and some others.

• Nonconforming $h-p$ Spectral Element Methods for Elliptic Problems

In this paper we show that we can use a modified version of the $h-p$ spectral element method proposed in [6,7,13,14] to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in [2]. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in [6,7,14] and the $h-p$ finite element method and stronger error estimates are obtained.

• # Proceedings – Mathematical Sciences

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