Recent zbMATH articles in MSC 41https://zbmath.org/atom/cc/412023-09-22T14:21:46.120933ZWerkzeugHow Leibniz tried to tell the world he had squared the circlehttps://zbmath.org/1517.010072023-09-22T14:21:46.120933Z"Strickland, Lloyd"https://zbmath.org/authors/?q=ai:strickland.lloydIn this paper, the author analyses four strategies through which Leibniz tried to spread his discovery on the approximation of \(\pi\) through an infinite rational series, specifically his identity \(\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\frac{1}{13}\), etc. The author identifies four strategies he calls ``conventional'' (pp. 21--23), ``the grand design'' (pp. 23--28), and ``impostrous'' (pp. 28--32), ``extravagant'' (pp. 33--35), respectively.
The ``conventional strategy'' consisted in divulgating the formula through letters or through the publication in a scientific journal. From an autograph note by Leibniz, we know that his discovery dates to the first half of 1674. In July 1674, he sent a letter to Oldenburg announcing his discovery, and in October a letter on the same subject was addressed to Mariotte (p. 21). In the same month, Leibniz sent Huygens the treatise \textit{Arithmetical quadrature} where he offered not only his formula, but its demonstration too. Huygens greatly appreciated this work of Leibniz. Besides these communications, there is no doubt that Leibniz let know his formula to his friends in France (Ozaman, Malebranche, Gallois, Mariotte himself, p. 22). In the last months of 1675, he expressed to Gallois his intention to publish his formula in a scientific journal, the \textit{Journal des Sçavants}. Leibniz conceived a letter for de La Roche, who was the editor of the \textit{Journal des Sçavants} at that time, but never sent it: it was lengthy and full of explanations, whereas the \textit{Journal} usually published short reports of 1--3 pages. Probably, Leibniz regarded his letter not suitable to be published in the \textit{Journal} (pp. 22--23).
The ``grand design'' consisted in inserting the formula to find \(\pi\) within the treatise \textit{De quadratura arithmetica circuli ellipseos et hyperbolae} [\dots], a text which Leibniz began to write in June 1676 (p. 23). In particular, he found the development in series of arctan (Prop. 31), from which the series which offers the development of \(\frac{\pi}{4}\) can be immediately obtained (p. 24). Starting from 1677, Leibniz tried to publish his treatise. As the author explains, Mariotti and Soudry were involved in this affair, but at the end of 1678 the book had not yet been published and in December Soudry died (p. 25). At that time, the story of the treatise on arithmetic quadrature was connected with Leibniz's attempt to become a member of the Royal Academy. Within the general strategy he developed, Leibniz sent his treatise to Huygens, who urged the former to publish such a work (pp. 26--27). In 1680, Leibniz hoped to publish his \textit{De quadratura} with the Dutch publisher Daniel Elzevier. The initial approach was unsuccessful and, furthermore, on 13th October 1680 Elzevier died, so that the text was not published (pp. 27--28).
Next, the ``impostrous attempt'': Leibniz lost interest in publishing his whole treatise, but he intended to disseminate at least the formula to calculate \(\pi\), which happened in 1682 (\textit{De vera proportione circuli} [\dots], p. 28): in 1681, Leibniz wrote a pseudonymous essay (there are two versions, one in Latin and one in French), referring to him himself in third person. He claimed that Mr. Leibniz gave the arithmetical quadrature of the circle (p. 28). This unpublished paper has several similarities with \textit{De vera proportione circuli} (p. 29). There are other examples in which Leibniz used pseudonyms. On one occasion, he wrote a letter to the \textit{Journal des Sçavants} concerning his own results on infinite series connected with the arithmetical quadratures, pretending the letter's author to be Cassini (pp. 29--30). As the author writes: ``Leibniz may have thought that there was benefit to be had in having his work sponsored [\dots] by luminaries in the Republic of Letters, especially by members of the Academy, the institution he still wished to join'' (p. 30). We ignore whether Cassini was aware of Leibniz's letter and if, in the affirmative case, he had accepted this stratagem. Afterwards, the author analyses the hypothesis that the pseudonymous essay was written for the Royal Academy of Sciences and discusses its plausibility. He concludes that, although Leibniz never sent the essay to the Academy, a priori this hypothesis is not absurd. In this regard, no certain conclusion can be drawn (pp. 31--32).
The fourth attempt by Leibniz is defined the ``extravagant method'' (p. 33): he intended to divulge his result by means of a medal. Leibniz had attempted to immortalise another of his discoveries in a medal: that of binary arithmetic. There are several designs on this (p. 33). The author refers to sketches by Leibniz where drawings appear that show medals depicting results related to binary arithmetic and the squaring of the circle. For, at least until 1682, Leibniz thought of a connection between these two parts of his mathematics, an idea which he abandoned later (p. 34).
This paper is a valuable source to understand how mathematics was spread in the 17th century and, particularly, how Leibniz attempted to divulge his discovery concerning the squaring of the circle. As a matter of fact, the picture described in the article goes far beyond the specific problem, because the author enters many details on the relations between Leibniz and the French scientific community. For, if you read this work, you will get a good idea of what the practices were at the time for disseminating scientific knowledge and what the relations were between the scholars who gravitated around the Royal Academy of Sciences. Therefore, this paper is surely recommended to achieve a good idea as to the history concerning the spread of science and mathematics in the 17th century.
Reviewer: Paolo Bussotti (Udine)Polynomial inequalities, o-minimality and Denjoy-Carleman classeshttps://zbmath.org/1517.260122023-09-22T14:21:46.120933Z"Pierzchała, Rafał"https://zbmath.org/authors/?q=ai:pierzchala.rafalSummary: We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map \(h\) in certain quasianalytic Denjoy-Carleman classes, we show how to decide whether the image under \(h\) of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.Some remarks on multivariate fractal approximationhttps://zbmath.org/1517.280082023-09-22T14:21:46.120933Z"Pandey, Megha"https://zbmath.org/authors/?q=ai:pandey.megha"Agrawal, Vishal"https://zbmath.org/authors/?q=ai:agrawal.vishal"Som, Tanmoy"https://zbmath.org/authors/?q=ai:som.tanmoySummary: Approximation theory encompasses a vast area of mathematics. The current context is primarily concerned with the concept of dimension preserving approximation for real-valued multivariate continuous functions defined on a domain. This chapter establishes quite a few results similar to well-known results of multivariate constrained approximation in terms of dimension preserving approximants. In particular, this chapter gives indication for construction of multivariate dimension preserving approximants using the concept of fractal interpolation functions. In the last part, some multi-valued fractal operators associated with multivariate -fractal functions are defined and studied.
For the entire collection see [Zbl 1495.26007].Perspective of fractal calculus on types of fractal interpolation functionshttps://zbmath.org/1517.280092023-09-22T14:21:46.120933Z"Priyanka, T. M. C."https://zbmath.org/authors/?q=ai:priyanka.t-m-c"Agathiyan, A."https://zbmath.org/authors/?q=ai:agathiyan.a"Gowrisankar, A."https://zbmath.org/authors/?q=ai:gowrisankar.aSummary: Fractal calculus is the calculus involving -integral and -derivative, where in (0,1] is the dimension of the fractals. In defining -integral and -derivative, the mass function and staircase function plays an important role. This chapter discusses the fractal calculus of non-affine fractal interpolation functions namely hidden variable fractal interpolation function and -fractal function. The fractal integral of the hidden variable fractal interpolation function is examined by predefining the initial conditions. Similarly, by predefining the initial conditions and imposing some necessary conditions on the -fractal function, its fractal integral is explored.
For the entire collection see [Zbl 1495.26007].On stable sampling and interpolation in Bernstein spaceshttps://zbmath.org/1517.320242023-09-22T14:21:46.120933Z"López Nicolás, José Alfonso"https://zbmath.org/authors/?q=ai:lopez-nicolas.jose-alfonsoSummary: We define the concepts of stable sampling set, interpolation set, uniqueness set and complete interpolation set for a quasinormed space of functions and apply these concepts to Paley-Wiener spaces and Bernstein spaces. We obtain a sufficient condition on a uniformly discrete set to be an interpolation set based on a lemma of convergence of series in Paley-Wiener spaces. We also obtain a result of transference, Kadec type, of the property of being a stable sampling set, from a set with this property to other uniformly discrete set, which we apply to Bernstein spaces.Optimal sampling and Christoffel functions on general domainshttps://zbmath.org/1517.410012023-09-22T14:21:46.120933Z"Dolbeault, Matthieu"https://zbmath.org/authors/?q=ai:dolbeault.matthieu"Cohen, Albert"https://zbmath.org/authors/?q=ai:cohen.albertFor functions \(f\) defined with arguments on a general domain \(D\) and that are square-integrable, it is a classical problem in approximation theory to reconstruct it by sampling at finitely many points from the domain. The square-integrability \(\int_D|f(x)|^2\,\mathrm{d}\mu(x)<\infty\) may be subject to a probability measure \(\mu\).
Of course it is desirable to have as few points as possible that are needed for the reconstruction (to have a low ``budget'' of points), and the points must be a allowed to be scattered. In fact, one should be allowed to choose them at random, subject to a probability measure \(\sigma\).
In the past literature, this problem was mostly addressed for tensor-product constructions of domains \(D\) and the measures, and in this paper, a much more general approach is taken by using general domains \(D\). Not only bounds on the number of points are given which are near optimal, but also strategies to compute the reconstructions are provided. On top of that, numerical examples are delivered to illustrate those algorithms.
Reviewer: Martin D. Buhmann (Gießen)Failure of approximation of odd functions by odd polynomialshttps://zbmath.org/1517.410022023-09-22T14:21:46.120933Z"Mashreghi, Javad"https://zbmath.org/authors/?q=ai:mashreghi.javad"Parisé, Pierre-Olivier"https://zbmath.org/authors/?q=ai:parise.pierre-olivier"Ransford, Thomas"https://zbmath.org/authors/?q=ai:ransford.thomas-jSummary: We construct a Hilbert holomorphic function space \(H\) on the unit disk such that the polynomials are dense in \(H\), but the odd polynomials are not dense in the odd functions in \(H\). As a consequence, there exists a function \(f\) in \(H\) that lies outside the closed linear span of its Taylor partial sums \(s_n(f)\), so it cannot be approximated by any triangular summability method applied to the \(s_n(f)\). We also show that there exists a function \(f\) in \(H\) that lies outside the closed linear span of its radial dilates \(f_r\), \(r<1\).Cosine polynomials with restrictions on their algebraic representationhttps://zbmath.org/1517.410032023-09-22T14:21:46.120933Z"Oganesyan, Kristina"https://zbmath.org/authors/?q=ai:oganesyan.kristinaSummary: We prove that for any even algebraic polynomial \(p\) one can find a cosine polynomial with an arbitrary small \(l_1\)-norm of coefficients such that the first coefficients of its representation as an algebraic polynomial in \(\cos x\) coincide with those of \(p\).Approximation by fuzzy \((p,q)\)-Bernstein-Chlodowsky operatorshttps://zbmath.org/1517.410042023-09-22T14:21:46.120933Z"Yildiz Ozkan, Esma"https://zbmath.org/authors/?q=ai:yildiz-ozkan.esmaSummary: In this study, we purpose to extend approximation properties of the \((p,q)\)-Bernstein-Chlodowsky operators from real function spaces to fuzzy function spaces. Firstly, we define fuzzy \((p,q)\)-Bernstein-Chlodowsky operators, and we give some auxiliary results. Later, we give a fuzzy Korovkin-type approximation theorem for these operators. Additionally, we investigate rate of convergence by using first order fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Eventually, we give an estimate for fuzzy asymptotic expansions of the fuzzy \((p,q)\)-Bernstein-Chlodowsky operators.On discretizing uniform norms of exponential sumshttps://zbmath.org/1517.410052023-09-22T14:21:46.120933Z"Kroó, András"https://zbmath.org/authors/?q=ai:kroo.andrasA classical problem in approximation theory is the discretisation of \(L^p\)-norms of functions \(f\), say, for various \(p\)s, by replacing the infinite sets \(K\) of the functions' arguments over which the norm is taken by finitely many points. Of course it is desirable to have bounds (e.g., upper and lower ones) as close to \(\|f\|_p\) as possible on one hand, and to need to take as few points as possible on the other hand. It is particularly worthwhile to have a bound on the number of points needed depending for instance on the dimensionality.
In this paper, remarkable results are provided with highly precise bounds on the number of points (best possible subject to potential removal of some small logarithmic terms) in any dimension for the uniform norm on compact domains. The mentioned functions \(f\) have a fairly general form, namely they are so-called exponential sums in any dimension. Of course the said bounds will depend in a natural way on the dimension of the ambient space and on the number of entries in the exponential sums (or rather the not necessarily integral ``degree'' of the exponential sums, i.e., the factors of the argument in the inner product in the exponent).
Reviewer: Martin D. Buhmann (Gießen)A note on the applications of one primary function in deep neural networkshttps://zbmath.org/1517.410062023-09-22T14:21:46.120933Z"Chen, Hengjie"https://zbmath.org/authors/?q=ai:chen.hengjie"Li, Zhong"https://zbmath.org/authors/?q=ai:li.zhong.4Summary: By applying fundamental mathematical knowledge, this paper proves that the function \(y= x^m\) (\(m\) is an integer no less than 2), \(x\in[0,1]\) has the property that the difference between the function value of middle point of arbitrarily two adjacent equidistant distribution nodes on \([0,1]\) and the mean of function values of these two nodes is a constant depending only on the number of nodes if and only if \(m=2\). By them, we establish an important result about deep neural networks that the function \(y= x^2\), \(x\in[0,1]\) can be interpolated by a deep Rectified Linear Unit (ReLU) network with depth \(n+1\) on the equidistant distribution nodes in interval \([0,1]\), and the error of approximation is \(2^{-(2n+2)}\). Then based on the main result that has just been proven and the Chebyshev orthogonal polynomials, we construct a deep network and give the error estimate of approximation to polynomials and continuous functions, respectively. In addition, this paper constructs one deep network with local sparse connections, shared weights and activation function \(y= x^2\), and discusses its density and complexity.Multivariate approximation in \(\varphi\)-variation for nonlinear integral operators via summability methodshttps://zbmath.org/1517.410072023-09-22T14:21:46.120933Z"Aslan, İsmail"https://zbmath.org/authors/?q=ai:aslan.ismail|aslan.ismail.1Summary: We consider convolution-type nonlinear integral operators endowed with Musielak-Orlicz \(\varphi\)-variation. Our aim is to get more powerful approximation results with the help of summability methods. In this study, we use \(\varphi\)-absolutely continuous functions for our convergence results. Moreover, we study the order of approximation using suitable Lipschitz class of continuous functions. A general characterization theorem for \(\varphi\)-absolutely continuous functions is also obtained. We also give some examples of kernels in order to verify our approximations. At the end, we indicate our approximations in figures together with some numerical computations.Construction of the Kantorovich variant of the Bernstein-Chlodovsky operators based on parameter \(\alpha\)https://zbmath.org/1517.410082023-09-22T14:21:46.120933Z"Lian, Bo-Yong"https://zbmath.org/authors/?q=ai:lian.boyong"Cai, Qing-Bo"https://zbmath.org/authors/?q=ai:cai.qingboThe authors introduce a new family of Kantorovich variants of Chlodovsky operators. They establish approximation theorems, such as a direct approximation result by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. Furthermore, a Voronovskaja type asymptotic estimate formula is presented. Finally, the rate of convergence for some absolutely continuous functions is obtained.
Reviewer: Ioan Raşa (Cluj-Napoca)On King type modification of \((p,q)\)-Lupaş Bernstein operators with improved estimateshttps://zbmath.org/1517.410092023-09-22T14:21:46.120933Z"Nisar, K. S."https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Sharma, V."https://zbmath.org/authors/?q=ai:sharma.vinita"Khan, A."https://zbmath.org/authors/?q=ai:khan.asif|khan.aftab|khan.aziz|khan.abdul-hakimSummary: This paper aims to modify the \((p,q)\)-Lupaş Bernstein operators using King's technique and to establish convergence results of these operators by using of modulus of continuity and Lipschitz class functions. Some approximation results for this new sequence of operators are obtained. It has been shown that the convergence rate of King type modification is better than the \((p,q)\)-Lupaş Bernstein operators. King type modification of operators also provide better error estimation within some subinterval of \([0,1]\) in comparison to \((p,q)\)-Lupaş Bernstein operators. In the last section, some graphs and tables provided for simulation purposes using MATLAB (R2015a).Parametric generalization of the Meyer-König-Zeller operatorshttps://zbmath.org/1517.410102023-09-22T14:21:46.120933Z"Sofyalıoğlu, Melek"https://zbmath.org/authors/?q=ai:sofyalioglu.melek"Kanat, Kadir"https://zbmath.org/authors/?q=ai:kanat.kadir"Çekim, Bayram"https://zbmath.org/authors/?q=ai:cekim.bayramSummary: The current paper deals with the parametric modification of Meyer-König-Zeller operators which preserve constant and Korovkin's other test functions in the form of \((\frac{x}{1-x})^u\), \(u=1,2\) in limit case. The uniform convergence of the newly defined operators is investigated. The rate of convergence is studied by means of the modulus of continuity and by the help of Peetre-\(\mathcal{K}\) functionals. Also, a Voronovskaya type asymptotic formula is given. Finally, some numerical examples are illustrated to show the effectiveness of the newly constructed operators for computing the approximation of function.On the approximation of functions in \(C^1 ( I )\) by a class of positive linear operatorshttps://zbmath.org/1517.410112023-09-22T14:21:46.120933Z"Xiang, Jim X."https://zbmath.org/authors/?q=ai:xiang.jim-xThe author generalizes a former theorem of \textit{F. Schurer} and \textit{F. W. Steutel} [J. Approx. Theory 19, 69--82 (1977; Zbl 0339.41007)] to a class of positive linear operators. The obtained result is applied to some well-known operators as Bernstein operator, Szász-Mirakyan operator, Gamma operator, Baskakov operator and B-spline operator, respectively.
Reviewer: Zoltán Finta (Cluj-Napoca)The Lebesgue constants on projective spaceshttps://zbmath.org/1517.410122023-09-22T14:21:46.120933Z"Kushpel, Alexander"https://zbmath.org/authors/?q=ai:kushpel.alexander-kSummary: We give the solution of a classical problem of Approximation Theory on sharp asymptotic of the Lebesgue constants or norms of the Fourier-Laplace projections on the real projective spaces \(\mathrm{P}^d(\mathbb{R})\). In particular, these results extend sharp asymptotic found by \textit{L. Fejér} [J. Reine Angew. Math. 138, 22--53 (1910; JFM 41.0288.01)] in the case of \(\mathbb{S}^1\) in 1910 and by \textit{T. H. Gronwall} [Trans. Am. Math. Soc. 15, 1--30 (1914; JFM 45.0413.02)]
in 1914 in the case of \(\mathbb{S}^2\). The case of spheres, \(\mathbb{S}^d\), complex and quaternionic projective spaces, \(\mathrm{P}^d(\mathbb{C})\), \(\mathrm{P}^d(\mathbb{H})\) and the Cayley elliptic plane \(\mathrm{P}^{16}(\mathrm{Cay})\) was considered by \textit{A. K. Kushpel} [Ukr. Math. J. 71, No. 8, 1224--1233 (2020; Zbl 07622912); translation from Ukr. Mat. Zh. 71, No. 8, 1073--1081 (2019)].Strict protosuns in asymmetric spaces of continuous functionshttps://zbmath.org/1517.410132023-09-22T14:21:46.120933Z"Alimov, Alexey R."https://zbmath.org/authors/?q=ai:alimov.alexey-rThis paper considers the approximation by the strict protosun (Kolmogorov sets) in asymmetric spaces of continuous functions, and presents results on characterization of strict protosuns (Kolmogorov sets) in spaces \(C_\psi(Q)\) and \(C_{0;\psi}(Q)\) in terms of Brosowski-Wegmann connectedness, ORL-continuity of the metric projection operator, unimodality, and lunarity.
Reviewer: Jin Liang (Shanghai)Unified error estimate for weak biorthogonal greedy algorithmshttps://zbmath.org/1517.410142023-09-22T14:21:46.120933Z"Jiang, Bing"https://zbmath.org/authors/?q=ai:jiang.bing"Ye, Peixin"https://zbmath.org/authors/?q=ai:ye.peixin"Zhang, Wenhui"https://zbmath.org/authors/?q=ai:zhang.wenhuiSummary: In this paper, we obtain the unified error estimate for some weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces by using some kind of \(K\)-functional. From this estimate, we derive the sufficient conditions for the convergence and the convergence rates on sparse classes induced by the \(K\)-functional. The results on convergence and the convergence rates are sharp.Fourier coefficients of functions in power-weighted \(L_2\)-spaces and conditionality constants of bases in Banach spaceshttps://zbmath.org/1517.420022023-09-22T14:21:46.120933Z"Ansorena, J. L."https://zbmath.org/authors/?q=ai:ansorena.jose-luisSummary: We prove that, given \(2< p<\infty \), the Fourier coefficients of functions in \(L_2(\mathbb{T}, |t|^{1-2/p}\,{\mathrm{d}}t)\) belong to \(\ell_p\), and that, given \(1< p<2\), the Fourier series of sequences in \(\ell_p\) belong to \(L_2(\mathbb{T}, \vert{t}\vert^{2/p-1} \mathrm{d}t)\). Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every \(1< p<\infty\) and every \(0\leqslant \alpha <1\), there is a Schauder basis of \(\ell_p\) whose conditionality constants grow as \((m^{\alpha })_{m=1}^{\infty } \), and there is an almost greedy basis of \(\ell_p\) whose conditionality constants grow as \(((\log m)^{\alpha })_{m=2}^{\infty } \).Jackson theorems for the quaternion linear canonical transformhttps://zbmath.org/1517.440022023-09-22T14:21:46.120933Z"Achak, A."https://zbmath.org/authors/?q=ai:achak.azzedine"Ahmad, O."https://zbmath.org/authors/?q=ai:ahmad.owias|ahmad.owais|ahmad.ozair|ahmad.ola-suleiman"Belkhadir, A."https://zbmath.org/authors/?q=ai:belkhadir.a"Daher, R."https://zbmath.org/authors/?q=ai:daher.radouanSummary: In this paper, we establish Bernstein inequality, Jackson's direct and inverse theorems for quaternion linear canonical transform using the functions with bounded spectrum.On a conjecture by Mbekhta about best approximation by polar factorshttps://zbmath.org/1517.470022023-09-22T14:21:46.120933Z"Chiumiento, Eduardo"https://zbmath.org/authors/?q=ai:chiumiento.eduardoLet \(\mathcal{B}(\mathcal{H})\) be the algebra of all bounded linear operators on a complex separable Hilbert space \(\mathcal{H}\), and \(\mathcal{I}\) be the collection of all partial isometries on \(\mathcal{H}\). The polar factor of an operator \(T\in \mathcal{B}(\mathcal{H})\) is the unique \(V\in \mathcal{I}\) such that \(T=V|T|\) and \(\ker(V)=\ker(T)\), where \(|T|:=(T^*T)^{1/2}\) is the absolute value of \(T\). In [J. Math. Anal. Appl. 487, No. 1, Article ID 123954, 12 p. (2020; Zbl 1486.47014)], \textit{M. Mbekhta} obtained explicit formulas the polar factor of any operator \(T\in\mathcal{B}(\mathcal{H})\) and conjectured that, if \(X_0\in\mathcal{I}\) such that \(\ker(X_0)=\ker(T)\), then \(X_0\) is the polar factor of \(T\) if and only if \[\|T-X_0\|=\min\{\|T-X\|:X\in\mathcal{I},\ \ker(X)=\ker(T)\}.\] In the aforecited paper, Mbekhta showed that the ``only if'' part holds true provided that \(T\) is injective.
\par Let \(P\) and \(Q\) be two orthogonal projections on \(\mathcal{H}\) and set \[j(P,Q):=\dim(\operatorname{ran}(P)\cap\ker(Q))-\dim(\ker(P)\cap\operatorname{ran}(Q))\] with the convention that \(j(P,Q)=0\) if both dimensions are infinite. Here, \(\operatorname{ran}(T)\) denotes the range of any operator \(T\in\mathcal{B}(\mathcal{H})\). In the paper under review, the author shows that, if \(T\in\mathcal{B}(\mathcal{H})\) with polar decomposition \(T=V|T|\), then
\[
\begin{aligned}
\|T-V\|&=\min\left\{\|T-X\|:X\in\mathcal{I},~j(V^*V,X^*X)\leq0 \right\}\\
&=\min\left\{\|T-X\|:X\in\mathcal{I},~j(VV^*,XX^*)\leq0 \right\}.
\end{aligned}
\]
Since \(\ker(T)=\ker(V)\) and \(j(V^*V,X^*X)=0\) for any \(X\in\mathcal{I}\) for which \(\ker(X)=\ker(T)\), it then follows that \[\|T-V\|=\min\{\|T-X\|:X\in\mathcal{I},~\ker(X)=\ker(T)\}.\] This shows that the ``only if'' part in Mbekhta's conjecture always holds without the injectivity condition on \(T\). He also shows that the ``if'' part in such a conjecture is, in general, false. Moreover, he gives necessary and sufficient conditions on the spectrum of an operator to guarantee that its polar factor becomes a best approximate in the set of all partial isometries. Furthermore, he provides several examples and remarks which nicely illustrate the results obtained.
Reviewer: Abdellatif Bourhim (Syracuse)A refinement of non-uniform estimates of the rate of convergence in the central limit theorem under the existence of moments of orders no higher than twohttps://zbmath.org/1517.600312023-09-22T14:21:46.120933Z"Korolev, V. Yu."https://zbmath.org/authors/?q=ai:korolev.victor-yu"Popov, S. V."https://zbmath.org/authors/?q=ai:popov.sergey-vLet \(X_1,\ldots,X_n\) be independent and identically distributed random variables with \(\mathbf{E}X_i=0\) for each \(i\), and \(0<\mathbf{E}X_i^2=\sigma_i^2<\infty\) with \(\sigma_1^2+\cdots+\sigma_n^2=1\). The authors establish new bounds in the non-uniform central limit theorem for \(W_n=X_1+\cdots+X_n\). In particular, they establish the following bounds on
\[
\Delta_x=|\mathbf{P}(W_n<x)-\Phi(x)|\,,
\]
where \(\Phi\) is the standard normal distribution function: there are functions \(C(x)\), \(C_2(x)\) and \(C_3(x)\) such that
\begin{align*}
\Delta_x&\leq C(x)\sum_{i=1}^n\left[\frac{\mathbf{E}X_i^2\mathbb{I}(|X_i|\geq1+|x|)}{(1+|x|)^2}+\frac{\mathbf{E}|X_i|^3\mathbb{I}(|X_i|<1+|x|)}{(1+|x|)^3}\right]\,,\\
\Delta_x&\leq\frac{C_2(x)}{(1+|x|)^2}\sum_{i=1}^n\mathbf{E}X_i^2\mathbb{I}(|X_i|\geq1+|x|)+\frac{C_3(x)}{(1+|x|)^3}\sum_{i=1}^n\mathbf{E}|X_i|^3\mathbb{I}(|X_i|<1+|x|)\,,
\end{align*}
where \(C(x)\leq39.317\), \(C_2(x)\leq14.262\) and \(C_3(x)\leq41.229\).
Reviewer: Fraser Daly (Edinburgh)An averaging principle for neutral stochastic fractional order differential equations with variable delays driven by Lévy noisehttps://zbmath.org/1517.600682023-09-22T14:21:46.120933Z"Shen, Guangjun"https://zbmath.org/authors/?q=ai:shen.guangjun"Wu, Jiang-Lun"https://zbmath.org/authors/?q=ai:wu.jianglun"Xiao, Ruidong"https://zbmath.org/authors/?q=ai:xiao.ruidong"Yin, Xiuwei"https://zbmath.org/authors/?q=ai:yin.xiuweiIn present paper, the authors are concerned with an averaging principle (see [\textit{R. Z. Has'minskij}, Kybernetika 4, 260--279 (1968; Zbl 0231.60045)]) for a neutral stochastic fractional (in Riemann-Liouville sense, see for example [\textit{St. G. Samko} et al., Fractional integrals and derivatives: theory and applications. Transl. from the Russian. New York, NY: Gordon and Breach (1993; Zbl 0818.26003)]) differential equation driven by Lévy process with variable delay. The main results is given in Theorem 3.3 where it is proved that the solution of the considered equations (1.1) can be approximated by solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square under some non-Lipschitz coefficients. This result seems to generalize and improve those of \textit{W. Xu} and \textit{W. Xu} [Appl. Math. Lett. 106, Article ID 106344, 5 p. (2020; Zbl 1462.60081)], \textit{D. Luo} et al. [Appl. Math. Lett. 105, Article ID 106290, 8 p. (2020; Zbl 1436.34072)], \textit{W. Xu} et al. [Fract. Calc. Appl. Anal. 23, No. 3, 908--919 (2020; Zbl 1474.60168)] or \textit{Y. Xu} et al. [Physica D 240, No. 17, 1395--1401 (2011; Zbl 1236.60060)]. The paper ends with an interesting example for one-dimensional neutral stochastic fractional differential equation driven by Lévy process with variable delay.
Reviewer: Romeo Negrea (Timişoara)Simultaneous approximation of multivariate functions by superposition of a sigmoidal functionhttps://zbmath.org/1517.650132023-09-22T14:21:46.120933Z"Hahm, N. W."https://zbmath.org/authors/?q=ai:hahm.n-wSummary: Since \textit{G. Cybenko} [Math. Control Signals Syst. 2, No. 4, 303--314 (1989; Zbl 0679.94019)] showed a density result of multivariate functions by neural networks, many authors investigated a complexity result of multivariate functions by neural networks. Some complexity results were suggested by Chen but target functions were univariate functions. Simultaneous approximation by neural networks with a sigmoidal activation function has been investigated by some authors but target functions were also univariate functions. \textit{N. Hahm} et al. [J. Korean Phys. Soc. 58, No. 2, 174--181 (2011; \url{10.3938/jkps.58.174})] showed constructive approximation of multivariate functions by generalized translation networks. Using algorithms in [Hahm and Hong, loc. cit.], we examine a constructive simultaneous approximation of multivariate function by superposition of a sigmoidal activation function in this paper.Mimetic finite difference operators and higher order quadratureshttps://zbmath.org/1517.650752023-09-22T14:21:46.120933Z"Srinivasan, Anand"https://zbmath.org/authors/?q=ai:srinivasan.anand"Dumett, Miguel"https://zbmath.org/authors/?q=ai:dumett.miguel-a"Paolini, Christopher"https://zbmath.org/authors/?q=ai:paolini.christopher-p"Miranda, Guillermo F."https://zbmath.org/authors/?q=ai:miranda.guillermo-f"Castillo, José E."https://zbmath.org/authors/?q=ai:castillo.jose-eSummary: Mimetic finite difference operators \(\mathbf{D},\mathbf{G}\) are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss' divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth-order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings.Accurate boundary treatment for Riesz space fractional diffusion equationshttps://zbmath.org/1517.650802023-09-22T14:21:46.120933Z"Tang, Shaoqiang"https://zbmath.org/authors/?q=ai:tang.shaoqiang"Pang, Gang"https://zbmath.org/authors/?q=ai:pang.gangThe Cauchy problem for the Riesz space fractional diffusion equation with compact initial data in one and two space dimension(s) is considered and numerically addressed for the purpose of boundary treatment. First, the Riesz space fractional equation is semi-discretized into a lattice system. Using kernel functions, an equivalent decoupled form for its dynamics is derived and for its numerical evaluation series expansions and path integration are proposed to numerically evaluate the kernel functions with high accuracy. This allows an accurate numerical boundary treatment for the Riesz space fractional diffusion equation. The theoretical findings are supported by numerical results and these results confirms the effectiveness of the method.
Reviewer: Petr Sváček (Praha)Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichmenthttps://zbmath.org/1517.651172023-09-22T14:21:46.120933Z"Smetana, Kathrin"https://zbmath.org/authors/?q=ai:smetana.kathrin"Taddei, Tommaso"https://zbmath.org/authors/?q=ai:taddei.tommasoSummary: We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations. CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our methodology and demonstrate its effectiveness.Distributed minimum error entropy algorithmshttps://zbmath.org/1517.683302023-09-22T14:21:46.120933Z"Guo, Xin"https://zbmath.org/authors/?q=ai:guo.xin"Hu, Ting"https://zbmath.org/authors/?q=ai:hu.ting"Wu, Qiang"https://zbmath.org/authors/?q=ai:wu.qiangSummary: Minimum Error Entropy (MEE) principle is an important approach in Information Theoretical Learning (ITL). It is widely applied and studied in various fields for its robustness to noise. In this paper, we study a reproducing kernel-based distributed MEE algorithm, DMEE, which is designed to work with both fully supervised data and semi-supervised data. The divide-and-conquer approach is employed, so there is no inter-node communication overhead. Similar as other distributed algorithms, DMEE significantly reduces the computational complexity and memory requirement on single computing nodes. With fully supervised data, our proved learning rates equal the minimax optimal learning rates of the classical pointwise kernel-based regressions. Under the semi-supervised learning scenarios, we show that DMEE exploits unlabeled data effectively, in the sense that first, under the settings with weak regularity assumptions, additional unlabeled data significantly improves the learning rates of DMEE. Second, with sufficient unlabeled data, labeled data can be distributed to many more computing nodes, that each node takes only \(O(1)\) labels, without spoiling the learning rates in terms of the number of labels. This conclusion overcomes the saturation phenomenon in unlabeled data size. It parallels a recent results for regularized least squares [\textit{S.-B. Lin} and \textit{D.-X. Zhou}, Constr. Approx. 47, No. 2, 249--276 (2018; Zbl 1390.68542)], and suggests that an inflation of unlabeled data is a solution to the MEE learning problems with decentralized data source for the concerns of privacy protection. Our work refers to pairwise learning and non-convex loss. The theoretical analysis is achieved by distributed U-statistics and error decomposition techniques in integral operators.Error bounds for approximations using multichannel deep convolutional neural networks with downsamplinghttps://zbmath.org/1517.683432023-09-22T14:21:46.120933Z"Liu, Xinling"https://zbmath.org/authors/?q=ai:liu.xinling"Hou, Jingyao"https://zbmath.org/authors/?q=ai:hou.jingyaoSummary: Deep learning with specific network topologies has been successfully applied in many fields. However, what is primarily called into question by people is its lack of theoretical foundation investigations, especially for structured neural networks. This paper theoretically studies the multichannel deep convolutional neural networks equipped with the downsampling operator, which is frequently used in applications. The results show that the proposed networks have outstanding approximation and generalization ability of functions from ridge class and Sobolev space. Not only does it answer an open and crucial question of why multichannel deep convolutional neural networks are universal in learning theory, but it also reveals the convergence rates.Reply to: ``Comment on: `On the characteristic polynomial of an effective Hamiltonian{'}''https://zbmath.org/1517.810532023-09-22T14:21:46.120933Z"Zheng, Yong"https://zbmath.org/authors/?q=ai:zheng.yongSummary: In a recent comment by \textit{F. M. Fernández} [Phys. Lett., A 452, Article ID 128456, 3 p. (2022; Zbl 1515.81103)], it has been argued that our solution method of an effective Hamiltonian based on the characteristic polynomial [the author, Phys. Lett., A 443, Article ID 128215, 5 p. (2022; Zbl 1498.81077)] had been developed several years earlier by \textit{L. E. Fried} and \textit{G. S. Ezra} [J. Chem. Phys. 90, 6378--6390 (1989; \url{doi:10.1063/1.456303})]. We show here several important differences between our treatment and the resummation method proposed previously by Fried and Ezra.Single-valued hyperlogarithms, correlation functions and closed string amplitudeshttps://zbmath.org/1517.810852023-09-22T14:21:46.120933Z"Vanhove, Pierre"https://zbmath.org/authors/?q=ai:vanhove.pierre"Zerbini, Federico"https://zbmath.org/authors/?q=ai:zerbini.federicoSummary: We give new proofs of a global and a local property of the integrals which compute closed string theory amplitudes at genus zero. Both kinds of properties are related to the newborn theory of singlevalued periods, and our proofs provide an intuitive understanding of this relation. The global property, known in physics as the KLT formula, is a factorisation of the closed string integrals into products of pairs of open string integrals. We deduce it by identifying closed string integrals with special values of single-valued correlation functions in two dimensional conformal field theory, and by obtaining their conformal block decomposition. The local property is of number theoretical nature. We write the asymptotic expansion coefficients as multiple integrals over the complex plane of special functions known as single-valued hyperlogarithms. We develop a theory of integration of single-valued hyperlogarithms, and we use it to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.Analytic continuation of harmonic sums: dispersion representationhttps://zbmath.org/1517.810912023-09-22T14:21:46.120933Z"Velizhanin, V. N."https://zbmath.org/authors/?q=ai:velizhanin.v-nSummary: We present a simple representation for analytically continued nested harmonic sums for the arbitrary complex arguments. This representation can be obtained for a wide range of nested harmonic sums from a precomputed database for the pole expressions of these sums near negative integers. We describe the procedure for the precise numerical evaluations of the corresponding results from the dispersion representation.Squeezed number state representation of the inflaton and particle production in the FRW universehttps://zbmath.org/1517.830032023-09-22T14:21:46.120933Z"Chand, Karam"https://zbmath.org/authors/?q=ai:chand.karamSummary: We use the single-mode coherent and squeezed number state formalism and analyze the nature of a massive homogeneous scalar field minimally coupled to gravity in the framework of semiclassical gravity in the Friedmann-Robertson-Walker (FRW) universe. We have obtained an estimate leading solution to the semiclassical Einstein equation for the FRW universe which shows the scale factor \(t^{2/3}\) power-law expansion. The mechanism of the particle production and quantum fluctuations are also analyzed in the FRW universe.