We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We pro...
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We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is R. We also show compatibility with the concept of co-local weak differential introduced by Convent and Van Schaftingen. (C) 2019 Published by Elsevier Inc.
Let C(I) be the set of all continuous self-maps on I=[0, 1] with the topology of uniformly convergence. A map f is an element of C(I) is called transitive if for every pair of non-empty open sets U, V in I, there exis...
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Let C(I) be the set of all continuous self-maps on I=[0, 1] with the topology of uniformly convergence. A map f is an element of C(I) is called transitive if for every pair of non-empty open sets U, V in I, there exists a positive integer nsuch that U boolean AND f(-n)(V) not equal empty set. The set of all transitive maps and its closure in the space C(I) are denoted by <(T(I))over bar> and <(T(I))over bar>. It is shown that both subspaces T(I) and <(T(I))over bar> of C(I) are homeomorphic to the separable Hilbert space l(2). (C) 2020 Elsevier B.V. All rights reserved.
In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties o...
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In this article, we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF's which only admit constant multipliers. To do that, using a method from Mashreghi and Ransford (Anal Math Phys 9(2):899-905, 2019), we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space of infinite cardinality. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.
We develop a constructive theory of continuous domains from the perspective of program extraction. Our goal that programs represent (provably correct) computation without witnesses of correctness is achieved by formul...
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We develop a constructive theory of continuous domains from the perspective of program extraction. Our goal that programs represent (provably correct) computation without witnesses of correctness is achieved by formulating correctness assertions classically. Technically, we start from a predomain base and construct a completion. We then investigate continuity with respect to the Scott topology, and present a construction of the function space. We then discuss our main motivating example in detail, and instantiate our theory to real numbers that we conceptualise as the total elements of the completion of the predomain of rational intervals, and prove a representation theorem that precisely delineates the class of representable continuous functions.
We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are ...
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We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic - and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere. (C) 2021 Elsevier Inc. All rights reserved.
In this paper l(p)*-invariant properties of metric spaces are presented. These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces C-p*(X), where X is any...
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In this paper l(p)*-invariant properties of metric spaces are presented. These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces C-p*(X), where X is any countable metric space of scattered height less than or equal to omega. Examples are presented to show that this isomorphic classification differs from the isomorphic classification of function spaces C-p (X) . (C) 2021 Elsevier B.V. All rights reserved.
Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2(c)-sized collection of sub-posets in 2(omega) which forms an antichain in the sense of Tukey ordering....
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Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2(c)-sized collection of sub-posets in 2(omega) which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of omega(omega) is a Tukey quotient of omega(omega), we answer two open questions published in [14]. The relation between P-base and strong Pytkeev* property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P-base has the strong Pytkeev* property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P-base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given. (C) 2021 Elsevier B.V. All rights reserved.
In this paper, we consider an inverse scattering problem of a time-harmonic wave from a penetrable obstacle. A Dirichlet-to-Neumann map is defined to convert this problem into an exterior boundary value problem with a...
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In this paper, we consider an inverse scattering problem of a time-harmonic wave from a penetrable obstacle. A Dirichlet-to-Neumann map is defined to convert this problem into an exterior boundary value problem with an impedance-like boundary condition. The problem is then solved without the need of solving an interior problem which is inevitable in the usual way. This process reduces the size of the problem and thus enables an efficient treatment of this inverse scattering problem.
In this note the (j*)-property, being a generalization of the (*)-property, (lambda*)-property and (d*)-property, is defined. There is proved that if J is an ideal of subsets of the real line containing all singletons...
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In this note the (j*)-property, being a generalization of the (*)-property, (lambda*)-property and (d*)-property, is defined. There is proved that if J is an ideal of subsets of the real line containing all singletons and a topology T has the (J*)-property then the family of Darboux functions continuous with respect to the topology T in the domain is strongly porous set in the space of all functions having T-Darboux property with the supremum metric. (C) 2019 Elsevier B.V. All rights reserved.
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