Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V...

We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class. 1. Introduction. Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded operators on H into itself. Given A, B ∈ B(H), we define the generalized derivation δ A,B : B(H) B(H) b...

In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain non-integrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper boun...

We are interested in the investigation of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel. 1. Introduction. Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded operators on H into itself. Given A, B ∈ B(H), we define the generalized derivation δ A,B : B(H) B(H) by δ A,B (X) = AX −XB and...

Abstract We consider a non-Hermitian matrix orthogonality on contour in the complex plane. Given diagonalizable and rational valued weight, we show that Christoffel–Darboux (CD) kernel, which is built terms of orthogonal polynomials, equivalent to scalar reproducing kernel meromorphic functions Riemann surface. If this surface has genus $0$, then CD polynomials Interestingly, not necessarily ke...

We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman [SIGMA 4 (2008), 089, 18 pages] as a limit of the 9− j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a “poker dice” type probability model. The proof in this paper derives and make...

Multimodal imaging has transformed neuroscience research. While it presents unprecedented opportunities, also imposes serious challenges. Particularly, is difficult to combine the merits of interpretability attributed a simple association model with flexibility achieved by highly adaptive nonlinear model. In this article, we propose an orthogonalized kernel debiased machine learning approach, w...

In this paper we investigate how the short-time Fourier transform can be extended in a Clifford setting. We prove some of main properties such as orthogonality relation, reconstruction property and reproducing kernel formula. Moreover, show effects modulating translating signal window function, respectively. Finally, demonstrate Lieb’s uncertainty principle for transform.

We present a trust management kernel that clearly separates authorization and structured distributed naming. Given an access request and supporting credentials, the kernel determines whether the request is authorized. We prove soundness and completeness of the authorization system without names and prove that naming is orthogonal to authorization in a precise sense. The orthogonality theorem gi...