Recent zbMATH articles in MSC 26Bhttps://www.zbmath.org/atom/cc/26B2021-04-16T16:22:00+00:00WerkzeugImplicit parametrizations and applications in optimization and control.https://www.zbmath.org/1456.490082021-04-16T16:22:00+00:00"Tiba, Dan"https://www.zbmath.org/authors/?q=ai:tiba.danThe subject is the characterization (with numerical applications in mind) of the manifold \(V\) of solutions of the nonlinear system
\[
F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1}
\]
in the vicinity of \(x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,\) under the Jacobian assumption
\[
\frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0
\quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, .
\]
The first step involves the underdetermined linear system
\[
v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l)
\]
which is used to obtain bases \((v_1(x), v_2(x), \dots, v_{d - l}(x))\) for the tangent spaces of \(V.\) Next, the chain of differential equations
\begin{align*}
\frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0
\cr
\frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1)
\cr
& \hskip 2em \dots \dots \dots \dots
\cr
\frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}}
&= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, ,
\cr
& \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1})
\end{align*}
is set up, thus constructing a parametrization of \(V\) which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function
\(g(x_0, x_2, \dots , x_d)\) subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem
of minimizing \(l(x(0), x(1))\) among the trajectories of the system
\(x'(t) = f(t, x(t), u(t))\) subject to the state-control constraint \(h(x(t), u(t)) = 0.\) There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines.
Reviewer: Hector O. Fattorini (Los Angeles)Monotone Sobolev functions in planar domains: level sets and smooth approximation.https://www.zbmath.org/1456.260152021-04-16T16:22:00+00:00"Ntalampekos, Dimitrios"https://www.zbmath.org/authors/?q=ai:ntalampekos.dimitriosThe aim of the present work is to describe the structure of the level sets of Sobolev functions defined in a planar domain. The author establishes that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. Furthermore, a stronger result is proved in the case that the Sobolev function in the plane is monotone in the Lebesgue sense (i.e., the maximum and minimum of the function in an open set are attained at the boundary), namely almost every level set is an embedded 1-dimensional (in the topological sense) submanifold of the plane. The obtained result is an analog of Sard's theorem for \(C^2\)-smooth functions in a planar domain, which asserts that almost every value is a regular value. Using the theory of \(p\)-harmonic functions, as an application of the obtained results, he proves that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.
Reviewer: Andrey Zahariev (Plovdiv)Stream functions for divergence-free vector fields.https://www.zbmath.org/1456.350812021-04-16T16:22:00+00:00"Kelliher, James P."https://www.zbmath.org/authors/?q=ai:kelliher.james-pSummary: In 1990, von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field \(u\) in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that \(u\) can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain.A strong form of Hardy type inequalities on domains of the Euclidean space.https://www.zbmath.org/1456.260162021-04-16T16:22:00+00:00"Avkhadiev, F. G."https://www.zbmath.org/authors/?q=ai:avkhadiev.farit-gabidinovichThe author offers several new Hardy-type inequalities that contain the scalar product of gradients of the text functions and of the gradient of the distance function from the boundary of an open subset of the Euclidean space. His method is based on interior and exterior approximations of a given domain by sequences of simplest domains. Among others, he gives a new proof in an improved form of a theorem by \textit{A. A. Balinsky} and \textit{W. D. Evans} [Appl. Math. Inf. Sci. 4, No. 2, 191--208 (2010; Zbl 1189.26038)] and also improved versions of his earlier results from [J. Math. Anal. Appl. 442, No. 2, 469--484 (2016; Zbl 1342.26046)]. Basic theorems due to \textit{H. Rademacher} [Math. Ann. 79, 340--359 (1920; JFM 47.0243.01)], \textit{T. Motzkin} [Atti Accad. Naz. Lincei, Rend., VI. Ser. 21, 562--567 (1935; Zbl 0011.41105)] and \textit{H. Hadwiger} [Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Berlin- Göttingen-Heidelberg: Springer-Verlag (1957; Zbl 0078.35703)] are also used in the proofs.
Reviewer: József Sándor (Cluj-Napoca)A simplified proof of CLT for convex bodies.https://www.zbmath.org/1456.520042021-04-16T16:22:00+00:00"Fresen, Daniel J."https://www.zbmath.org/authors/?q=ai:fresen.daniel-jSummary: We present a short proof of Klartag's central limit theorem for convex bodies, using only the most classical facts about log-concave functions. An appendix is included where we give the proof that the thin shell implies CLT. The paper is accessible to anyone.A method for efficient computation of integrals with oscillatory and singular integrand.https://www.zbmath.org/1456.650172021-04-16T16:22:00+00:00"Kurtoǧlu, Dilan Kılıç"https://www.zbmath.org/authors/?q=ai:kurtoglu.dilan-kilic"Hasçelik, A. Ihsan"https://www.zbmath.org/authors/?q=ai:hascelik.ali-ihsan"Milovanović, Gradimir V."https://www.zbmath.org/authors/?q=ai:milovanovic.gradimir-vSummary: A method based on modification of numerical steepest descent method to efficiently compute highly oscillatory integrals having endpoint singularities of algebraic and logarithmic type is proposed in this paper. The three-term recursion coefficients for orthogonal polynomials with respect to Gautschi's weight function \(w^G (t; s) = t^s (t - 1 - \log t) \text{e}^{-t} (s > -1)\) on \((0, \infty)\), as well as the corresponding quadrature formulas of Gaussian type, are used in this method. Finally, in order to illustrate the efficiency of the presented method a few numerical examples are included. The obtained results show that the proposed method is very efficient and economical in terms of computation time.The isoperimetric problem of a complete Riemannian manifold with a finite number of \(C^0\)-asymptotically Schwarzschild ends.https://www.zbmath.org/1456.530302021-04-16T16:22:00+00:00"Muñoz Flores, Abraham Enrique"https://www.zbmath.org/authors/?q=ai:munoz-flores.abraham-enrique"Nardulli, Stefano"https://www.zbmath.org/authors/?q=ai:nardulli.stefanoSummary: We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in \(C^2\)-locally asymptotically Euclidean Riemannian manifolds with a finite number of \(C^0\)-asymptotically Schwarzschild ends. This work extends previous results contained in
[\textit{M. Eichmair} and \textit{J. Metzger}, Invent. Math. 194, No. 3, 591--630 (2013; Zbl 1297.49078); J. Differ. Geom. 94, No. 1, 159--186 (2013; Zbl 1269.53071); \textit{S. Brendle} and \textit{M. Eichmair}, J. Differ. Geom. 94, No. 3, 387--407 (2013; Zbl 1282.53053)]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named \(C^0\)-strongly asymptotic Schwarzschild, extending results of [Zbl 1282.53053]. Such results are of interest in the field of mathematical general relativity.Higher order strongly uniform convex functions.https://www.zbmath.org/1456.260142021-04-16T16:22:00+00:00"Noor, Muhammad Aslam"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://www.zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: Some new concepts of the higher order strongly uniform convex functions with an increasing modulus \(\varphi(\cdot)\) vanishing only at 0 are considered in this paper. Some properties of the higher order strongly uniformly convex functions are investigated under suitable conditions. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine uniform convex functions as novel applications. It is shown that the minimum of the higher order strongly uniform convex functions can be characterized by the variational inequalities. Some important special cases as applications of our results are discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.Loss of double-integral character during relaxation.https://www.zbmath.org/1456.490142021-04-16T16:22:00+00:00"Kreisbeck, Carolin"https://www.zbmath.org/authors/?q=ai:kreisbeck.carolin"Zappale, Elvira"https://www.zbmath.org/authors/?q=ai:zappale.elvira