For an index set Gamma and a cardinal number kappa the Sigma(kappa)-product of real lines Sigma(kappa)(R-Gamma) consist of all elements of R Gamma with <kappa nonzero coordinates. A compact space is kappa-Corson if...
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For an index set Gamma and a cardinal number kappa the Sigma(kappa)-product of real lines Sigma(kappa)(R-Gamma) consist of all elements of R Gamma with spaces wider than the class of omega-Corson compact spaces, investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and Zakrzewski called NY compact spaces. For a Tychonoff space X, let C-p(X) be the space of real continuous functions on the space X, endowed with the pointwise convergence topology. We present here a characterisation of kappa-Corson compact spaces K for regular, uncountable cardinal numbers kappa in terms of function spaces C-p(K), extending a theorem of Bell and Marciszewski and a theorem of Pol. We also prove that classes of NY compact spaces and omega-Corson compact spaces K are preserved by linear homeomorphisms of function spaces C-p(K).
In this article, we study the novel concept of non-stationary iterated function systems (IFSs) introduced by Massopust in 2019. At first, using a sequence of different contractive operators, we construct non-stationar...
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In this article, we study the novel concept of non-stationary iterated function systems (IFSs) introduced by Massopust in 2019. At first, using a sequence of different contractive operators, we construct non-stationary alpha-fractal functions on the space of all continuous functions. Next, we provide some elementary properties of the fractal operator associated with the non-stationary alpha-fractal functions. Further, we show that the proposed interpolant generalizes the existing stationary interpolant in the sense of IFS. For a class of functions defined on an interval, we derive conditions on the IFS parameters so that the corresponding non-stationary alpha-fractal functions are elements of some standard spaces like bounded variation space, convex Lipschitz space, and other function spaces. Finally, we discuss the dimensional analysis of the corresponding non-stationary alpha-fractal functions on these spaces. (c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong soluti...
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We present a new proof of the caloric smoothing related to the fractional Gauss-Weierstrass semi-group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.
Due to the successful applications in abundant practical application problems involving fuzzy sets and systems, overlap functions on unit closed interval have attracted continuous attentions of many scholars since the...
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Due to the successful applications in abundant practical application problems involving fuzzy sets and systems, overlap functions on unit closed interval have attracted continuous attentions of many scholars since they were proposed. In particular, recently, the author extended the concept of overlap functions on unit closed interval to the so-called quasi-overlap functions on bounded partially ordered sets. In this paper, we pay attention to the extension constructions of quasi-overlap functions and their derivative concepts on function spaces with bounded partially ordered sets as underlying sets. Specifically, first, based on quasi-overlap functions and their derivative concepts on any bounded partially ordered set, we give the way to construct quasi-overlap functions and their derivative concepts on function space composed of all fuzzy sets with that bounded partially ordered set as the truth values set along with the function space composed of all order-preserving functions from arbitrary sup semilattice to that bounded partially ordered set, respectively. Second, we introduce the concepts of representable quasi-overlap functions and representable derivative concepts of quasi-overlap functions on function spaces and obtain their equivalent characterizations. Third, we discuss some vital properties of representable quasi-overlap functions and representable derivative concepts of quasi-overlap functions on function spaces. Fourth, it should be mentioned that the obtained results cover the cases of quasi-overlap functions and their derivative concepts on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when the underlying bounded partially ordered set is taken as the corresponding truth values set, respectively.(c) 2022 Elsevier Inc. All rights reserved.
Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperpla...
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Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective torus. While the generalization error bounds attained via Vapnik-Chervonenkis (VC) dimensions in a distribution-free manner still depend on the dimension, we also show numerically and theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).
The paper deals with some recent assertions about truncations f bar right arrow vertical bar f vertical bar and compositions f bar right arrow g omicron f in the spaces A(p,q)(s)(R-n), A is an element of {B, F}.
The paper deals with some recent assertions about truncations f bar right arrow vertical bar f vertical bar and compositions f bar right arrow g omicron f in the spaces A(p,q)(s)(R-n), A is an element of {B, F}.
In this paper, some elementary and useful terminologies associated to Lebedev-Skalskaya transform (LS-transform) are notified and then estimates for R kappa(1/2+i tau) (x), F kappa(1/2+i tau) (x), translation and conv...
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In this paper, some elementary and useful terminologies associated to Lebedev-Skalskaya transform (LS-transform) are notified and then estimates for R kappa(1/2+i tau) (x), F kappa(1/2+i tau) (x), translation and convolution operators associated to LS-transform are obtained. The continuity of LS-transforms, translation and convolution operators on functions spaces S-alpha,S-kappa, H-a,H-k, and G(alpha,kappa) are discussed. The continuity of pseudo-differential operators in terms of LS-transforms is discussed on functions spaces S-alpha,S-kappa, H-a,H-k, and G(alpha,kappa) and Lebesgue space. At the end, an application of LS-transform is shown by solving a differential equation of convolution type.
We study traces of weighted Triebel-Lizorkin spaces Fp,qs(Rn,w)$Fs_{p,q}(\mathbb {R}n,w)$ on hyperplanes Rn-k$\mathbb {R}{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight w alpha(x)...
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We study traces of weighted Triebel-Lizorkin spaces Fp,qs(Rn,w)$F<^>s_{p,q}(\mathbb {R}<^>n,w)$ on hyperplanes Rn-k$\mathbb {R}<^>{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight w alpha(x)=|xn|alpha$w_\alpha (x) = {\big\vert x_n\big\vert }<^>\alpha$ when |xn|<= 1$\big\vert x_n\big\vert \le 1$, x is an element of Rn$x\in \mathbb {R}<^>n$, and w alpha(x)=1$w_\alpha (x)=1$ otherwise, where alpha>-1$\alpha >-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space (Bp1,p1s1(Rn-k),Bp2,p2s2(Rn-k))theta,r$\big (B<^>{s_1}_{p_1,p_1}\big (\mathbb {R}<^>{n-k}\big ), B<^>{s_2}_{p_2,p_2}{\big (\mathbb {R}<^>{n-k}\big )\big )}_{\theta ,r}$, 00$s>0$ sufficiently large, 0
We present a general constructive approach to blossoming for arbitrary finite dimensional univariate function spaces by altering the diagonal property of the classical blossom. We apply this approach to blossoming to ...
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We present a general constructive approach to blossoming for arbitrary finite dimensional univariate function spaces by altering the diagonal property of the classical blossom. We apply this approach to blossoming to develop the corresponding theory of Bernstein bases and B & eacute;zier curves including de Casteljau type algorithms for evaluation and subdivision as well as a Marsden identity for arbitrary finite dimensional univariate function spaces.
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
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