scholarly journals Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis

CALCOLO ◽  
2020 ◽  
Vol 57 (3) ◽  
Author(s):  
Lidia Aceto ◽  
Mariarosa Mazza ◽  
Stefano Serra-Capizzano
Keyword(s):  
Spectral Analysis ◽  
Laplace Operator ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Two Dimensions ◽  
Spatial Variables ◽  
The Matrix ◽  
Constant Coefficients ◽  
Fractional Laplace Operator ◽  
Concise Description

Abstract In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

CTCS Schemes for Second Order Wave Equation: Numerical Results and Spectral Analysis

2021 ◽  
Vol 55 ◽  
pp. 47-65
Author(s):  
Appanah Rao Appadu ◽  
Gysbert Nicolaas de Waal
Keyword(s):  
Spectral Analysis ◽  
Wave Equation ◽  
Numerical Experiments ◽  
Finite Difference Methods ◽  
Second Order ◽  
Numerical Results ◽  
Order Accuracy ◽  
Constant Coefficients ◽  
Second Order Wave Equation ◽  
The One

IIn this paper, two finite difference methods are used to solve the one-dimensional second order wave equation with constant coefficients subject to specified initial and boundary conditions. Two numerical experiments are considered. The two methods are Central in Time and Central in Space scheme with second order accuracy in both time and space, abbreviated as CTCS (2,2) and Central in Time and Central in Space scheme with second order accuracy in time and fourth order accuracy in space, abbreviated as CTCS (2,4). Properties such as consistency and stability are studied. We also perform spectral analysis of dispersive and dissipative properties of the two methods. Two numerical experiments are considered, and the numerical results are displayed.

A Weak Formulation for Solving the Elliptic Interface Problems with Imperfect Contact

2017 ◽  
Vol 9 (5) ◽  
pp. 1189-1205 ◽  
Author(s):  
Liqun Wang ◽  
Songming Hou ◽  
Liwei Shi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Two Dimensions ◽  
Interface Problems ◽  
Weak Formulation ◽  
Imperfect Contact ◽  
Elliptic Interface Problems ◽  
The Matrix

AbstractWe propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with imperfect contact in two dimensions, which has not been well-studied in the literature. Numerical experiments demonstrated the effectiveness of our method.

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma&gt;0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

scholarly journals Magnetar X-ray emission mechanisms

2012 ◽  
Vol 8 (S291) ◽  
pp. 160-160
Author(s):  
Silvia Zane
Keyword(s):  
Spectral Analysis ◽  
Physical Interpretation ◽  
Spectral Properties ◽  
Gamma Ray ◽  
X Ray ◽  
Spectral Components ◽  
Self Consistent ◽  
Photon Index ◽  
Index 2 ◽  
Surface Fields

AbstractSoft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) are peculiar X-ray sources which are believed to be magnetars: ultra-magnetized neutron stars which emission is dominated by surface fields (often in excess of 1E14 G, i.e. well above the QED threshold).Spectral analysis is an important tool in magnetar astrophysics since it can provide key information on the emission mechanisms. The first attempts at modelling the persistent (i.e. outside bursts) soft X-ray (¡10 keV) spectra of AXPs proved that a model consisting of a blackbody (kT 0.3-0.6 keV) plus a power-law (photon index 2-4) could successfully reproduce the observed emission. Moreover, INTEGRAL observations have shown that, while in quiescence, magnetars emit substantial persistent radiation also at higher energies, up to a few hundreds of keV. However, a convincing physical interpretation of the various spectral components is still missing.In this talk I will focus on the interpretation of magnetar spectral properties during quiescence. I will summarise the present status of the art and the currents attempts to model the broadband persistent emission of magnetars (from IR to hard Xrays) within a self consistent, physical scenario.

scholarly journals p-Laplace Operators for Oriented Hypergraphs

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Dong Zhang
Keyword(s):  
Laplace Operator ◽  
Spectral Properties ◽  
General Setting ◽  
Laplace Operators

AbstractThe p-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex p-Laplacian and a hyperedge p-Laplacian are defined for oriented hypergraphs, for all p ≥ 1. Several spectral properties of these operators are investigated.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

Identifying the Basis for Segmenting Higher Education

2012 ◽  
Vol 2 (2) ◽  
pp. 42-54
Author(s):  
Maha Mourad ◽  
Hamed M. Shamma
Keyword(s):  
Higher Education ◽  
Focus Groups ◽  
Perceived Quality ◽  
Two Dimensions ◽  
Current Position ◽  
The Matrix ◽  
Depth Interviews ◽  
Degree Of Internationalization

This paper reviews the developments taking place in the Higher Education (HE) industry. The focus of the research is to identify the main variables used to segment universities in Egypt. The research is qualitative in nature as the dimensions for segmentation were derived based on a series of in-depth interviews followed by two focus groups. Perceived quality and level of internationalization were found to be the two most important dimensions for classifying HE institutions in Egypt. These two dimensions formed the basis of a two-by-two matrix, which was used to segment HE universities into four main segments. The four HE segments that were identified are: legacy, prestigious, imitators, and the uncertain. Each quadrant was identified based on the level of perceived quality and the degree of internationalization. The matrix is useful for universities’ administrators to identify their current position and assess their future positioning strategies.

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