In this paper, some of the properties of continuous linear operators between fuzzifying topological linear space, are studied. Particularly, the equivalence between continuity and boundedness is investigated and the c...
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In this paper, some of the properties of continuous linear operators between fuzzifying topological linear space, are studied. Particularly, the equivalence between continuity and boundedness is investigated and the closed graph theorem is generalized to the fuzzifying setting. Finally some properties of the initial fuzzifying topologies determined by a family of linear operators are investigated. (C) 2006 Elsevier B.V. All rights reserved.
The performance estimation problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving lin...
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The performance estimation problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This extension requires an explicit formulation of interpolation conditions for those linear operators. We consider the class of linear operators M : x H- Mx, where matrix M has bounded singular values, and the class of linear operators, where M is symmetric and has bounded eigenvalues. We describe interpolation conditions for these classes, i.e., necessary and sufficient conditions that, given a list of pairs {(xi, yi)}, characterize the existence of a linear operator mapping xito yifor all i. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective h \circ M, and observe that it always corresponds to M being a scaling operator. We then investigate the Chambolle-Pock method applied to f + g \circ M, and improve the existing analysis to obtain a proof of the exact convergence rate of the primal-dual gap. In addition, we study how this method behaves on Lipschitz convex functions, and obtain a numerical convergence rate for the primal accuracy of the last iterate. We also show numerically that averaging iterates is beneficial in this setting.
Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such a per...
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Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such a perturbing operator to the original one reduces the eigenspace dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind Fredholm operators, the perturbing operators are analyzed such that the spectrum points for an original and the perturbed operators coincide except a spectrum point considered, which can be removed for the perturbed operator. A relation between resolvents of original and perturbed operators is obtained. Effective procedures are described for calculation of the undetermined constants in the right-hand side of an operator equation for the case when these constants must be chosen to satisfy the solvability conditions not written explicitly. implementation of the methods is illustrated on a boundary integral equation of elasticity. (C) 2000 Elsevier Science Ltd. All rights reserved.
This paper presents new results on approximate Birkhoff–James orthogonality in normed spaces. Mainly, by extending the idea of norm derivatives, we establish a characterization of this concept in general complex norm...
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We prove that for any p epsilon [1, + infinity] a finite irreducible family of linear operators possesses an extrernal norm corresponding to the p-radius of these operators. As a corollary, we derive a criterion for t...
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We prove that for any p epsilon [1, + infinity] a finite irreducible family of linear operators possesses an extrernal norm corresponding to the p-radius of these operators. As a corollary, we derive a criterion for the L-p-contractibility property of linear operators and estimate the asymptotic growth of orbits for any point. These results are applied to the study of functional difference equations with linear contractions of the argument (self-similarity equations). We obtain a sharp criterion for the existence and uniqueness of solutions ill various functional spaces, compute the exponents of regularity, and estimate moduli of continuity. This, in particular, gives a geometric interpretation of the p-radius in terms of spectral radii of certain operators in the space L-p[0, 1]. (c) 2007 Elsevier Inc. All rights reserved.
We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of...
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We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.(c) 2022 Elsevier Inc. All rights reserved.
Let C-m,C-omega(R-n) be the space of functions on R-n whose m(th) derivatives have modulus of continuity w. For E subset of R-n, let C-m,C-omega(E) be the space of all restrictions to E of functions in C-m,C-omega(R-n...
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Let C-m,C-omega(R-n) be the space of functions on R-n whose m(th) derivatives have modulus of continuity w. For E subset of R-n, let C-m,C-omega(E) be the space of all restrictions to E of functions in C-m,C-omega(R-n). We show that there exists a bounded linear operator T : C-m,C-omega(E) -> C-m,C-omega(R-n) such that, for any f is an element of C-m,C-omega (E), we have Tf = f on E.
We study the extent to which several classical results relating linear or multilinear forms and their zero-sets can be generalised to linear or bilinear operators with values in Rn. We find some analogues of the class...
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We study the extent to which several classical results relating linear or multilinear forms and their zero-sets can be generalised to linear or bilinear operators with values in Rn. We find some analogues of the classical theorems, and also some restrictions.
Let T, U be two linear operators mapped onto the function f such that U(T(f)) = f, but T(U(f)) f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce fou...
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Let T, U be two linear operators mapped onto the function f such that U(T(f)) = f, but T(U(f)) f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators T and U. In the third example, we apply the basic expansion U(T(f)) = f (x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f)) and U(T(f)). (C) 2009 Elsevier B.V. All rights reserved.
Making use of a certain linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses P-a,P-c(A,B;p,lambda) and P-a,c(+)(A,B;p,lambda) of the class A(p) of...
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Making use of a certain linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses P-a,P-c(A,B;p,lambda) and P-a,c(+)(A,B;p,lambda) of the class A(p) of normalized p-valent analytic functions in the open unit disk. The main objective of the present paper is to investigate the various important properties and characteristics of each of these subclasses. Furthermore, several properties involving neighborhoods of functions in these subclasses are investigated. We also derive many results for the modified Hadamard products of functions belonging to the class P-a,c(+)(A, B;p, lambda). Finally, some applications of fractional calculus operators are considered. (c) 2007 Elsevier Ltd. All rights reserved.
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