scholarly journals On Convergence of Infinite Matrix Products with Alternating Factors from Two Sets of Matrices

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Victor Kozyakin
Keyword(s):  
Suitable Choice ◽  
Infinite Matrix ◽  
Open Questions ◽  
Matrix Products ◽  
The Matrix

We consider the problem of convergence to zero of matrix products AnBn⋯A1B1 with factors from two sets of matrices, Ai∈A and Bi∈B, due to a suitable choice of matrices {Bi}. It is assumed that for any sequence of matrices {Ai} there is a sequence of matrices {Bi} such that the corresponding matrix products AnBn⋯A1B1 converge to zero. We show that, in this case, the convergence of the matrix products under consideration is uniformly exponential; that is, AnBn⋯A1B1≤Cλn, where the constants C>0 and λ∈(0,1) do not depend on the sequence {Ai} and the corresponding sequence {Bi}. Other problems of this kind are discussed and open questions are formulated.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

Cosserat Modeling of Size Effects in Crystalline Solids

2000 ◽  
Vol 653 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Stress State ◽  
Size Effects ◽  
Elastic Inclusion ◽  
Characteristic Size ◽  
Infinite Matrix ◽  
Crystalline Solids ◽  
Generalized Continua ◽  
Absolute Size ◽  
Kink Bands ◽  
The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

Matrix transformations between some classes of sequences

1970 ◽  
Vol 68 (1) ◽  
pp. 99-104 ◽  
Author(s):  
C. G. Lascarides ◽  
I. J. Maddox
Keyword(s):  
Matrix Transformation ◽  
Matrix Transformations ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
The Matrix ◽  
Complex Sequences ◽  
Class Of Matrices

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

scholarly journals Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Linear Operator ◽  
Sufficient Conditions ◽  
Sequence Spaces ◽  
General Linear ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
Complex Field ◽  
Infinite Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the fieldC*and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets overC*to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

Quantitative stress determination in SiC reinforced Al2O3

1990 ◽  
Vol 48 (4) ◽  
pp. 190-191
Author(s):  
Bernhard Caspers ◽  
Werner Mader ◽  
Peter Angelini
Keyword(s):  
Strain Analysis ◽  
Elastic Field ◽  
Elastic Interaction ◽  
Ceramic Composites ◽  
Infinite Matrix ◽  
Free Surfaces ◽  
Elastic Boundary ◽  
The Matrix ◽  
Quantitative Stress ◽  
Elastic Boundary Conditions

Whisker and fiber reinforced ceramic composites offer the potential for increased fracture toughness and fracture strength. However, differences in thermal expansion between whisker and matrix generate residual stresses which may affect the mechanical properties of such composites. The microscopic stress can be determined by analyzing the strain contrast around SiC whiskers quantitatively, similar to the strain analysis performed on ZrO2 inclusions in AI2O3. Previously obtained experimental TEM results on strain contrast around SiC whiskers embedded in Al2O3 were only qualitatively. In the present study the experimental observations are quantitatively evaluated.Contrast simulation requires the knowledge of the matrix displacement field in the vicinity of the whisker. Due to two force-free surfaces adjacent to the inclusion, Eshelby’s method for an inclusion in an infinite matrix can not be used for the evaluation of the elastic field in a TEM foil. Therefore, the Green’s function method was applied to fulfill the elastic boundary conditions for both surfaces neglecting only the elastic interaction between the surfaces.

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