Singular Sylvester equation in Banach spaces and its applications: Fredholm theory approach

In this paper, we will use some new properties of non-compactness measure, in order to establish a description of the M-essential spectrum for some matrix operators on Banach spaces.

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Common solutions of linear equations in a ring, with applications

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2910 ◽

2015 ◽

Vol 30 ◽

Cited By ~ 2

Author(s):

Alegra Dajic

Keyword(s):

General Solution ◽

Banach Spaces ◽

Associative Ring ◽

Linear Equations ◽

Sufficient Conditions ◽

Sylvester Equation ◽

Solvability Conditions ◽

Common Solution ◽

Generalized Sylvester Equation ◽

Necessary And Sufficient

This paper gives necessary and sufficient conditions for the existence of a common solution, and two expressions for the general common solution of the equation pair a_1xb_1 = c1, a_2xb_2 = c2, via a simpler equation p_1xp_2 + q_1yq_2=c, when each element belongs to an associative ring with unit. The paper also considers the same problem in the setting of a strongly â-reducing ring. Solutions of the generalized Sylvester equation are also presented. Both the solvability conditions and the expression for the general solution are given in terms of inner inverses. The paper uses the results obtained in the ring setting to give equivalent results for operators between Banach spaces, thus also recovering some of the well known matrix results.

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Ab initio band theory approach to electron energy loss near edge structures

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100104595 ◽

1988 ◽

Vol 46 ◽

pp. 506-507

Author(s):

Xudong Weng ◽

O.F. Sankey ◽

Peter Rez

Keyword(s):

Ab Initio ◽

Local Density ◽

Interpolation Method ◽

Atomic Orbital ◽

Plane Waves ◽

Large Set ◽

Theory Approach ◽

Band Theory ◽

Atomic Orbitals ◽

Pseudopotential Calculations

Single electron band structure techniques have been applied successfully to the interpretation of the near edge structures of metals and other materials. Among various band theories, the linear combination of atomic orbital (LCAO) method is especially simple and interpretable. The commonly used empirical LCAO method is mainly an interpolation method, where the energies and wave functions of atomic orbitals are adjusted in order to fit experimental or more accurately determined electron states. To achieve better accuracy, the size of calculation has to be expanded, for example, to include excited states and more-distant-neighboring atoms. This tends to sacrifice the simplicity and interpretability of the method.In this paper. we adopt an ab initio scheme which incorporates the conceptual advantage of the LCAO method with the accuracy of ab initio pseudopotential calculations. The so called pscudo-atomic-orbitals (PAO's), computed from a free atom within the local-density approximation and the pseudopotential approximation, are used as the basis of expansion, replacing the usually very large set of plane waves in the conventional pseudopotential method. These PAO's however, do not consist of a rigorously complete set of orthonormal states.

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Perceptions of a Collaborative Professional Learning Program: Seeing the “Bigger Picture”

Perspectives of the ASHA Special Interest Groups ◽

10.1044/2019_persp-19-00041 ◽

2020 ◽

Vol 5 (1) ◽

pp. 290-303

Author(s):

P. Charlie Buckley ◽

Kimberly A. Murza ◽

Tami Cassel

Keyword(s):

Data Analysis ◽

Professional Learning ◽

Social Communication ◽

Special Educators ◽

Service Providers ◽

Integrative Approach ◽

Theory Approach ◽

Learning Program ◽

Speech Language Pathologists ◽

Related Service

Purpose The purpose of this study was to explore the perceptions of special education practitioners (i.e., speech-language pathologists, special educators, para-educators, and other related service providers) on their role as communication partners after participation in the Social Communication and Engagement Triad (Buckley et al., 2015 ) yearlong professional learning program. Method A qualitative approach using interviews and purposeful sampling was used. A total of 22 participants who completed participation in either Year 1 or Year 2 of the program were interviewed. Participants were speech-language pathologists, special educators, para-educators, and other related service providers. Using a grounded theory approach (Glaser & Strauss, 1967 ) to data analysis, open, axial, and selective coding procedures were followed. Results Three themes emerged from the data analysis and included engagement as the goal, role as a communication partner, and importance of collaboration. Conclusions Findings supported the notion that educators see the value of an integrative approach to service delivery, supporting students' social communication and engagement across the school day but also recognizing the challenges they face in making this a reality.

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