2,526 results
Search Results
2. Corrections and complements to my paper 'On a class of operator monotone functions of several variables'
- Author
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A. R. Mirotin
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,Finite-rank operator ,Compact operator ,Strongly monotone ,Shift operator ,01 natural sciences ,Semi-elliptic operator ,Algebra ,Pseudo-monotone operator ,Monotone polygon ,Multiplication operator ,0101 mathematics ,Mathematics - Published
- 2017
3. On D.Y. Gao and X. Lu paper 'On the extrema of a nonconvex functional with double-well potential in 1D'
- Author
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Constantin Zălinescu
- Subjects
021103 operations research ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,General Physics and Astronomy ,Double-well potential ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Maxima and minima ,35J20, 35J60, 74G65, 74S30 ,Optimization and Control (math.OC) ,FOS: Mathematics ,Preprint ,0101 mathematics ,Constant (mathematics) ,Mathematics - Optimization and Control ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3
- Published
- 2017
4. Implementable tensor methods in unconstrained convex optimization
- Author
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Yurii Nesterov, UCL - SSH/LIDAM/CORE - Center for operations research and econometrics, and UCL - SSH/IMMAQ/CORE - Center for operations research and econometrics
- Subjects
tensor mehtods ,90C06 ,General Mathematics ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Upper and lower bounds ,90C25 ,Worst-case complexity bounds ,High-order methods ,Tensor methods ,Tensor (intrinsic definition) ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics ,021103 operations research ,Full Length Paper ,Regular polygon ,Order (ring theory) ,Function (mathematics) ,Lower complexity bounds ,Convex optimization ,Rate of convergence ,Software - Abstract
In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$ O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$ O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.
- Published
- 2021
5. On the Characterizations of Wave Front Sets in Terms of the Short-Time Fourier Transform
- Author
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Stevan Pilipović and Bojan Prangoski
- Subjects
Wavefront ,General Mathematics ,010102 general mathematics ,Short paper ,Mathematical analysis ,Short-time Fourier transform ,02 engineering and technology ,01 natural sciences ,Sobolev space ,symbols.namesake ,020303 mechanical engineering & transports ,Fourier transform ,0203 mechanical engineering ,symbols ,0101 mathematics ,Mathematics - Abstract
© 2019, Pleiades Publishing, Ltd. It is well known that the classical and Sobolev wave fronts were extended to nonequivalent global versions by the use of the short-time Fourier transform. In this very short paper, we give complete characterizations of the former wave front sets in terms of the short-time Fourier transform.
- Published
- 2019
6. A general double-proximal gradient algorithm for d.c. programming
- Author
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Radu Ioan Boț and Sebastian Banert
- Subjects
General Mathematics ,Connection (vector bundle) ,Proximal-gradient algorithm ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,90C26 ,Convergence analysis ,Convergence (routing) ,FOS: Mathematics ,Point (geometry) ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,49M29 ,021103 operations research ,Concave function ,Toland dual ,Full Length Paper ,Regular polygon ,90C26, 90C30, 65K05 ,Numerical Analysis (math.NA) ,Linear map ,Iterated function ,Optimization and Control (math.OC) ,Convex function ,Algorithm ,d.c. programming ,Software ,Kurdyka–Łojasiewicz property - Abstract
The possibilities of exploiting the special structure of d.c. programs, which consist of optimizing the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimization problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka--\L ojasiewicz property. In the last part, we apply the algorithm to an image processing model.
- Published
- 2018
7. U(X) as a ring for metric spaces X
- Author
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Javier Cabello Sánchez
- Subjects
Ring (mathematics) ,021103 operations research ,General Mathematics ,010102 general mathematics ,Short paper ,0211 other engineering and technologies ,02 engineering and technology ,Function (mathematics) ,Space (mathematics) ,01 natural sciences ,Combinatorics ,Uniform continuity ,Metric space ,Bounded function ,0101 mathematics ,Mathematics - Abstract
In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.
- Published
- 2017
8. The Four-Parameter PSS Method for Solving the Sylvester Equation
- Author
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Yan-Ran Li, Xin-Hui Shao, and Hai-Long Shen
- Subjects
Iterative method ,General Mathematics ,lcsh:Mathematics ,Positive and skew-Hermitian iterative method ,Value (computer science) ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,Paper based ,lcsh:QA1-939 ,01 natural sciences ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Sylvester equation ,FPPSS iterative method ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science (miscellaneous) ,Applied mathematics ,Order (group theory) ,Computer Science::Programming Languages ,0101 mathematics ,Coefficient matrix ,Engineering (miscellaneous) ,Mathematics - Abstract
In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.
- Published
- 2019
- Full Text
- View/download PDF
9. Tikhonov regularization of a second order dynamical system with Hessian driven damping
- Author
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
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Hessian matrix ,General Mathematics ,0211 other engineering and technologies ,Dynamical Systems (math.DS) ,02 engineering and technology ,Dynamical system ,01 natural sciences ,Hessian-driven damping ,90C26 ,Tikhonov regularization ,symbols.namesake ,34G25, 47J25, 47H05, 90C26, 90C30, 65K10 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,47J25 ,47H05 ,010102 general mathematics ,Hilbert space ,90C30 ,Function (mathematics) ,Convex optimization ,Optimization and Control (math.OC) ,Second order dynamical system ,34G25 ,symbols ,Fast convergence methods ,Convex function ,Software - Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.
- Published
- 2020
10. Global optimization in Hilbert space
- Author
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Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
- Subjects
Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
11. New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness
- Author
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Fleszar, Krzysztof, Mnich, Matthias, Spoerhase, Joachim, QE Operations research, and RS: GSBE Theme Data-Driven Decision-Making
- Subjects
90C27 ,FOS: Computer and information sciences ,Vertex deletion ,FLOW ,0211 other engineering and technologies ,02 engineering and technology ,Disjoint sets ,68Q17 ,01 natural sciences ,Upper and lower bounds ,05C05 ,05C85 ,Data Structures and Algorithms (cs.DS) ,05C40 ,Feedback vertex set ,Mathematics ,90B18 ,90C39 ,021103 operations research ,Full Length Paper ,68Q87 ,Approximation algorithm ,68W40 ,90B10 ,Binary logarithm ,90C35 ,Graph ,68W05 ,010201 computation theory & mathematics ,Randomized rounding ,90C05 ,90C49 ,68-02 ,General Mathematics ,68R10 ,68-06 ,0102 computer and information sciences ,90C46 ,Combinatorics ,Computer Science - Data Structures and Algorithms ,THEOREM ,49L20 ,Disjoint paths ,0101 mathematics ,05C21 ,000 Computer science, knowledge, general works ,010102 general mathematics ,INTEGER ,68Q25 ,90C10 ,68W20 ,68W25 ,90C59 ,05C38 ,Fixed-parameter algorithm ,Computer Science ,Software - Abstract
We study the classical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}}$$\end{document}NP-hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^{\varOmega (\sqrt{\log n})}}$$\end{document}2Ω(logn), assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}\not \subseteq \mathsf {DTIME}(n^{\mathcal {O}(\log n)})}$$\end{document}NP⊈DTIME(nO(logn)). This constitutes a significant gap to the best known approximation upper bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{n})}$$\end{document}O(n) due to Chekuri et al. (Theory Comput 2:137–146, 2006), and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica 7(4):365–374, 1987) introduce the technique of randomized rounding for LPs; their technique gives an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(1)}$$\end{document}O(1)-approximation when edges (or nodes) may be used by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}\left( \log n/\log \log n\right) }$$\end{document}Ologn/loglogn paths. In this paper, we strengthen the fundamental results above. We provide new bounds formulated in terms of the feedback vertex set number r of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following results:For MaxEDP, we give an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{r} \log ({k}r))}$$\end{document}O(rlog(kr))-approximation algorithm. Up to a logarithmic factor, our result strengthens the best known ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\sqrt{n})}$$\end{document}O(n) due to Chekuri et al., as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r\le n}$$\end{document}r≤n.Further, we show how to route \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varOmega ({\text {OPT}}^{*})}$$\end{document}Ω(OPT∗) pairs with congestion bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}(\log (kr)/\log \log (kr))}$$\end{document}O(log(kr)/loglog(kr)), strengthening the bound obtained by the classic approach of Raghavan and Thompson.For MaxNDP, we give an algorithm that gives the optimal answer in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(k+r)^{\mathcal {O}(r)}\cdot n}$$\end{document}(k+r)O(r)·n. This is a substantial improvement on the run time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^kr^{\mathcal {O}(r)}\cdot n}$$\end{document}2krO(r)·n, which can be obtained via an algorithm by Scheffler. We complement these positive results by proving that MaxEDP is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {NP}}$$\end{document}NP-hard even for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r=1}$$\end{document}r=1, and MaxNDP is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {W}[1]}$$\end{document}W[1]-hard when r is the parameter. This shows that neither problem is fixed-parameter tractable in r unless \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {FPT}= \mathsf {W}[1]}$$\end{document}FPT=W[1] and that our approximability results are relevant even for very small constant values of r.
- Published
- 2016
- Full Text
- View/download PDF
12. Theoretical and empirical analysis of trading activity
- Author
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Ludovic Tangpi, Walter Schachermayer, Mathias Pohl, and Alexander Ristig
- Subjects
021103 operations research ,Quantitative Finance - Trading and Market Microstructure ,Full Length Paper ,General Mathematics ,Financial market ,0211 other engineering and technologies ,Sigma ,Time scaling ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Trading and Market Microstructure (q-fin.TR) ,Universality (dynamical systems) ,FOS: Economics and business ,Stock exchange ,91G80 ,0101 mathematics ,Volatility (finance) ,Empirical evidence ,Mathematical economics ,Scaling ,Software ,Mathematics - Abstract
Understanding the structure of financial markets deals with suitably determining the functional relation between financial variables. In this respect, important variables are the trading activity, defined here as the number of trades N, the traded volume V, the asset price P, the squared volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document}σ2, the bid-ask spread S and the cost of trading C. Different reasonings result in simple proportionality relations (“scaling laws”) between these variables. A basic proportionality is established between the trading activity and the squared volatility, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim \sigma ^2$$\end{document}N∼σ2. More sophisticated relations are the so called 3/2-law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{3/2} \sim \sigma P V /C$$\end{document}N3/2∼σPV/C and the intriguing scaling \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \sim (\sigma P/S)^2$$\end{document}N∼(σP/S)2. We prove that these “scaling laws” are the only possible relations for considered sets of variables by means of a well-known argument from physics: dimensional analysis. Moreover, we provide empirical evidence based on data from the NASDAQ stock exchange showing that the sophisticated relations hold with a certain degree of universality. Finally, we discuss the time scaling of the volatility \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}σ, which turns out to be more subtle than one might naively expect.
- Published
- 2018
13. Halfspace type Theorems for Self-Shrinkers
- Author
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Marcos P. Cavalcante and José M. Espinar
- Subjects
Mathematics - Differential Geometry ,0209 industrial biotechnology ,Minimal surface ,General Mathematics ,010102 general mathematics ,Short paper ,02 engineering and technology ,Radius ,Type (model theory) ,Lambda ,01 natural sciences ,Combinatorics ,020901 industrial engineering & automation ,Hypersurface ,Differential Geometry (math.DG) ,Hyperplane ,Catenoid ,FOS: Mathematics ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
In this short paper, we extend the classical Hoffman-Meeks Halfspace Theorem [Hoffman and Meeks, 'The strong halfspace theorem for minimal surfaces', Invent. Math. 101 (1990) 373-377] to self-shrinkers, that is: Let $P$ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma $ contained in one of the closed half-space determined by $P$ is $\Sigma = P$. Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller [Kleene and Moller, 'Self-shrinkers with a rotational symmetry', Trans. Amer. Math. Soc. 366 (2014) 3943-3963]. Also, using a similar geometric idea, we obtain that the only self-shrinker properly immersed in an closed cylinder $ \overline {B^{k+1} (R)} \times {\mathbb R}^{n-k}\subset {\mathbb R}^{n+1}$, for some $k\in \{1, \ldots, n\}$ and radius $R$, $R \leqslant \sqrt {2k}$, is the cylinder ${\mathbb S}^k (\sqrt {2k}) \times {\mathbb R}^{n-k}$. We also extend the above results for $\lambda $-hypersurfaces.
- Published
- 2014
14. Metric Rectifiability of ℍ-regular Surfaces with Hölder Continuous Horizontal Normal
- Author
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Katrin Fässler, Daniela Di Donato, and Tuomas Orponen
- Subjects
0209 industrial biotechnology ,020901 industrial engineering & automation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Metric (mathematics) ,Mathematics::Metric Geometry ,Hölder condition ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Two definitions for the rectifiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on ${\mathbb{H}}$-regular surfaces and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole–Pauls, Bigolin–Vittone, and Antonelli–Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of ${\mathbb{H}}$-regular surfaces. We prove that ${\mathbb{H}}$-regular surfaces in $\mathbb{H}^{n}$ with $\alpha $-Hölder continuous horizontal normal, $\alpha> 0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for $C^{\infty }$-surfaces. In $\mathbb{H}^{1}$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $\epsilon $ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between “big pieces” of metric spaces.
- Published
- 2021
15. A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process
- Author
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Birgit Sollie, Michel Mandjes, and Mathematics
- Subjects
Statistics and Probability ,Mathematical optimization ,General Mathematics ,0211 other engineering and technologies ,Markov process ,Context (language use) ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,SDG 3 - Good Health and Well-being ,0101 mathematics ,Mathematics ,021103 operations research ,Series (mathematics) ,Markov chain ,Model selection ,Quasi birth-death processes ,Maximum likelihood estimation ,Uniformization (probability theory) ,Quasi-birth–death process ,symbols ,Matrix exponential ,Time-dependent probabilities ,Erlang distribution - Abstract
This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.
- Published
- 2022
16. On curves with circles as their isoptics
- Author
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Waldemar Cieślak and Witold Mozgawa
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Pure mathematics ,Class (set theory) ,Plane curve ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,Characterization (mathematics) ,Ellipse ,01 natural sciences ,Discrete Mathematics and Combinatorics ,0101 mathematics ,021101 geological & geomatics engineering ,Mathematics - Abstract
In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.
- Published
- 2021
17. Estimation of Heat Flux in Two-Dimensional Nonhomogeneous Parabolic Equation Based on a Sufficient Descent Levenberg–Marquard Algorithm
- Author
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Haibo Li, Jinhui Zhao, Xinfu Pang, Yang Yu, and Yuan Wang
- Subjects
Work (thermodynamics) ,Article Subject ,020209 energy ,General Mathematics ,Weak solution ,Field (mathematics) ,010103 numerical & computational mathematics ,02 engineering and technology ,Function (mathematics) ,Inverse problem ,01 natural sciences ,Heat flux ,Rate of convergence ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Constant (mathematics) ,Algorithm ,Mathematics - Abstract
The main work of this paper focuses on identifying the heat flux in inverse problem of two-dimensional nonhomogeneous parabolic equation, which has wide application in the industrial field such as steel-making and continuous casting. Firstly, the existence of the weak solution of the inverse problem is discussed. With the help of forward solution and dual equation, this paper proves the Lipchitz continuity of the cost function and derives the Lipchitz constant. Furthermore, in order to accelerate the convergence rate and reduce the running time, this paper presents a sufficient descent Levenberg–Marquard algorithm with adaptive parameter (SD-LMAP) to solve this inverse problem. At last, compared with other methods, the results of simulation experiment show that this algorithm can obviously reduce the running time and iterative number.
- Published
- 2021
18. Stationary Wavelet with Double Generalised Rayleigh Distribution
- Author
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Hassan M. Aljohani
- Subjects
021103 operations research ,Article Subject ,Computer science ,Rayleigh distribution ,General Mathematics ,0211 other engineering and technologies ,General Engineering ,Wavelet transform ,Markov chain Monte Carlo ,02 engineering and technology ,Inverse problem ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Noise ,Wavelet ,Multicollinearity ,Gaussian noise ,QA1-939 ,symbols ,TA1-2040 ,0101 mathematics ,Algorithm ,Mathematics - Abstract
Statistics are mathematical tools applying scientific investigations, such as engineering and medical and biological analyses. However, statistical methods are often improved. Nowadays, statisticians try to find an accurate way to solve a problem. One of these problems is estimation parameters, which can be expressed as an inverse problem when independent variables are highly correlated. This paper’s significant goal is to interpret the parameter estimates of double generalized Rayleigh distribution in a regression model using a wavelet basis. It is difficult to use the standard version of the regression methods in practical terms, which is obtained using the likelihood. Since a noise level usually makes the result of estimation unstable, multicollinearity leads to various estimates. This kind of problem estimates that features of the truth are complicated. So it is reasonable to use a mixed method that combines a fully Bayesian approach and a wavelet basis. The usual rule for wavelet approaches is to choose a wavelet basis, where it helps to compute the wavelet coefficients, and then, these coefficients are used to remove Gaussian noise. Recovering data is typically calculated by inverting the wavelet coefficients. Some wavelet bases have been considered, which provide a shift-invariant wavelet transform, simultaneously providing improvements in smoothness, in recovering, and in squared-error performance. The proposed method uses combining a penalized maximum likelihood approach, a penalty term, and wavelet tools. In this paper, real data are involved and modeled using double generalized Rayleigh distributions, as they are used to estimate the wavelet coefficients of the sample using numerical tools. In practical applications, wavelet approaches are recommended. They reduce noise levels. This process may be useful since the noise level is often corrupted in real data, as a significant cause of most numerical estimation problems. A simulation investigation is studied using the MCMC tool to estimate the underlying features as an essential task statistics.
- Published
- 2021
19. An outer-approximation guided optimization approach for constrained neural network inverse problems
- Author
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Myun-Seok Cheon
- Subjects
FOS: Computer and information sciences ,Computer Science - Machine Learning ,Mathematical optimization ,021103 operations research ,Optimization problem ,Artificial neural network ,General Mathematics ,Computation ,Numerical analysis ,0211 other engineering and technologies ,Process (computing) ,010103 numerical & computational mathematics ,02 engineering and technology ,Inverse problem ,01 natural sciences ,Machine Learning (cs.LG) ,Optimization and Control (math.OC) ,Convergence (routing) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Gradient method ,Software ,Mathematics - Abstract
This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method.
- Published
- 2021
20. Inequalities for Riemann–Liouville Fractional Integrals of Strongly <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <mfenced open='(' close=')' separators='|'> <mrow> <mi>s</mi> <mo>,</mo> <mi>m</mi> </mrow> </mfenced> </math>-Convex Functions
- Author
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Saira Bano Akbar, Fuzhen Zhang, and Ghulam Farid
- Subjects
Pure mathematics ,Hadamard transform ,General Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,010103 numerical & computational mathematics ,02 engineering and technology ,0101 mathematics ,Riemann liouville ,Convex function ,Computer Science::Digital Libraries ,01 natural sciences ,Mathematics - Abstract
The results of this paper provide two Hadamard-type inequalities for strongly s , m -convex functions via Riemann–Liouville fractional integrals and error estimations of well-known fractional Hadamard inequalities. Their special cases are given and connected with the results of some published papers.
- Published
- 2021
21. A Multiple-Fault Localization Method for Embedded Software with Applications in Engineering
- Author
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Lu Kong, Meng-Ru Wang, Jinbo Wang, and Shan Zhou
- Subjects
Article Subject ,Computer science ,General Mathematics ,media_common.quotation_subject ,Reliability (computer networking) ,Emphasis (telecommunications) ,General Engineering ,020207 software engineering ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,Fault (power engineering) ,01 natural sciences ,010104 statistics & probability ,Embedded software ,Transmission (telecommunications) ,Computer engineering ,Debugging ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,Cache ,TA1-2040 ,0101 mathematics ,Cluster analysis ,Mathematics ,media_common - Abstract
Embedded software is increasingly being used with high reliability. However, the fault localization of embedded software is still largely dependent on the experience of engineers. Besides, faults in embedded software programs are not independent individuals; they are related to each other and affect each other, which may lead to more complex interaction behavior. These uncertainties render the traditional methods for single-fault localization with limited practical value. This paper has proposed a multiple-fault localization method to be applied to the embedded software, with emphasis on the cache-based program spectra-acquiring method and the hybrid clustering-based fault partition method. Through case studies on 108 groups of the subject program, it has been proved that the hybrid clustering-based fault partition method has significantly improved the effectiveness of multiple-fault localization in comparison with the traditional fault localization methods. Experiments on three embedded software programs in engineering have revealed that the cache-based program spectra-acquiring method saves nearly half of the running-time cost compared with the traditional spectrum-acquiring method based on real-time transmission. Therefore, the multiple-fault localization method proposed in this paper can be applied in embedded software debugging and testing in engineering.
- Published
- 2021
22. Numerical Investigation on Improving the Computational Efficiency of the Material Point Method
- Author
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Keyi Chen, Weidong Chen, Shengzhuo Lu, Jingxin Ma, and Yaqin Shi
- Subjects
Article Subject ,Computer simulation ,Computer science ,General Mathematics ,General Engineering ,Rotational symmetry ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Shock (mechanics) ,Domain (software engineering) ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,QA1-939 ,Boundary value problem ,TA1-2040 ,0101 mathematics ,Algorithm ,Mathematics ,Material point method - Abstract
Based on the basic theory of the material point method (MPM), the factors affecting the computational efficiency are analyzed and discussed, and the problem of improving calculation efficiency is studied. This paper introduces a mirror reflection boundary condition to the MPM to solve axisymmetric problems; to improve the computational efficiency of solving large deformation problems, the concept of "dynamic background domain (DBD)" is also proposed in this paper. Taking the explosion and/or shock problems as an example, the numerical simulation are calculated, and the typical characteristic parameters and the CPU time are compared. The results show that the processing method introducing mirror reflection boundary condition and MPM with DBD can improve the calculation efficiency of the corresponding problems, which, under the premise of ensuring its calculation accuracy, provide useful reference for further promoting the engineering application of this method.
- Published
- 2021
23. Univalence criteria for a general integral operator
- Author
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Daniel Breaz and Camelia Barbatu
- Subjects
Algebra ,0209 industrial biotechnology ,020901 industrial engineering & automation ,Operator (computer programming) ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
"The main object of this paper is to give sufficient conditions for the general integral operator Tn, to be univalent in the open disk U, when gi, hi, ki ∈ Gbi for all i = 1, n. This general integral operator was considered in a recent work [Barbatu, C. and Breaz, D., ˘ Classes of an univalent integral operator, Studia Univ. Babes¸-Bolyai Math., accepted]. The results derived in this paper are shown to follow upon specializing the parameters involved in our results. Several corollaries of the main results are also considered."
- Published
- 2021
24. On Isotypic Decompositions for Non-Semisimple Hopf Algebras
- Author
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Vincent Koppen, Christoph Schweigert, and Ehud Meir
- Subjects
Large class ,Pure mathematics ,Property (philosophy) ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,0211 other engineering and technologies ,Zero (complex analysis) ,Haar ,021107 urban & regional planning ,02 engineering and technology ,Hopf algebra ,01 natural sciences ,Set (abstract data type) ,Character (mathematics) ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial., Comment: 29 pages
- Published
- 2021
25. A New Study on Halpern and Nonconvex Combination Algorithm for Nonlinear Mappings in Banach Spaces with Applications
- Author
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Gaobo Li
- Subjects
021103 operations research ,Article Subject ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,Combination algorithm ,02 engineering and technology ,01 natural sciences ,Bounded operator ,Algebra ,Nonlinear system ,QA1-939 ,Equilibrium problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi- ϕ -nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.
- Published
- 2021
26. Prediction of Power Outage Quantity of Distribution Network Users under Typhoon Disaster Based on Random Forest and Important Variables
- Author
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Jufang Yu, Hao Geng, Ling Zhu, Yong Huang, Xianqiang Li, Min Li, and Hui Hou
- Subjects
021110 strategic, defence & security studies ,Data collection ,Article Subject ,Computer science ,General Mathematics ,0211 other engineering and technologies ,General Engineering ,Decision tree ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Random forest ,Support vector machine ,010104 statistics & probability ,Variable (computer science) ,Linear regression ,Statistics ,QA1-939 ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
Typhoons can have disastrous effects on power systems. They may lead to a large number of power outages for distribution network users. Therefore, this paper establishes a model to predict the power outage quantity of distribution network users under a typhoon disaster. Firstly, twenty-six explanatory variables (called global variables) covering meteorological factors, geographical factors, and power grid factors are considered as the input variables. On this basis, the correlation between each explanatory variable and response variable is analyzed. Secondly, we established a global variable model to predict the power outage quantity of distribution network users based on Random Forest (RF) algorithm. Then the importance of each explanatory variable is mined to extract the most important variables. To reduce the complexity of the model and ease the burden of data collection, eight variables are eventually selected as important variables. Afterward, we predict the power outage quantity of distribution network users again using the eight important variables. Thirdly, we compare the prediction accuracy of a model called the No-model that has been used before, Linear Regression (LR), Support Vector Regression (SVR), Decision Tree Regression (DTR), RF-global variable model, and RF-important variable model. Simulation results show that the RF-important variable model proposed in this paper has a better effect. Since fewer variables can save prediction time and make the model simplified, it is recommended to use the RF-important variable model.
- Published
- 2021
27. Estimating the moments of a random forcing field of 2D fluid from image sequences using energy minimisation method
- Author
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Vishal Kumar Pandey, Harish Parthasarathy, and Jyotsna Singh
- Subjects
Random field ,Field (physics) ,Force field (physics) ,General Mathematics ,Mathematical analysis ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Moment (mathematics) ,Flow (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Vector field ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.
- Published
- 2021
28. On Bourgain’s Counterexample for the Schrödinger Maximal Function
- Author
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Lillian B. Pierce
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,021001 nanoscience & nanotechnology ,01 natural sciences ,symbols.namesake ,symbols ,Maximal function ,0101 mathematics ,0210 nano-technology ,Schrödinger's cat ,Counterexample ,Mathematics - Abstract
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.
- Published
- 2020
29. Further study on the Brück conjecture and some non-linear complex differential equations
- Author
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Dilip Chandra Pramanik and Kapil Roy
- Subjects
Monomial ,021103 operations research ,Current (mathematics) ,Conjecture ,Complex differential equation ,General Mathematics ,Entire function ,0211 other engineering and technologies ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,0101 mathematics ,Complex number ,Differential (mathematics) ,Mathematics - Abstract
PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.
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- 2020
30. Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon
- Author
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Gaurav Dhar, Loïc Bourdin, Mathématiques & Sécurité de l'information (XLIM-MATHIS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), and Dhar, Gaurav
- Subjects
function of bounded variations ,General Mathematics ,Stieltjes integral ,0211 other engineering and technologies ,[MATH] Mathematics [math] ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,digital control ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,optimal control ,Variational principle ,Pontryagin maximum principle ,Applied mathematics ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics ,021103 operations research ,state constraints ,Numerical analysis ,[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC] ,Riemann–Stieltjes integral ,Ekeland variational principle ,shooting method ,Optimal control ,indirect method ,Nonlinear system ,[SPI.AUTO] Engineering Sciences [physics]/Automatic ,Sampled-data control ,Control system ,Bounded function ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Riemann-Stieltjes integral ,Software ,Hamiltonian (control theory) - Abstract
International audience; Sampled-data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled-data control systems and certain generalizations. In this paper we derive a Pontryagin maximum principle for general nonlinear finite-dimensional optimal sampled-data control problems with running inequality state constraints. In particular, we obtain a nonpositive averaged Hamiltonian gradient condition with the adjoint vector being a function of bounded variations. Our proof is based on the Ekeland variational principle. In general, optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities (the jumps and the singular part) of the adjoint vector. However in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times. We call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper. Finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions.
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- 2020
31. Risk and complexity in scenario optimization
- Author
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Marco C. Campi and Simone Garatti
- Subjects
Structure (mathematical logic) ,Mathematical optimization ,021103 operations research ,Optimization problem ,Data-driven optimization ,Probabilistic constraints ,Scenario approach ,Stochastic optimization ,General Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Constraint satisfaction ,01 natural sciences ,Joint probability distribution ,Convex optimization ,Scenario optimization ,0101 mathematics ,Empirical evidence ,Software ,Mathematics - Abstract
Scenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called “scenarios”, for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimization, a solid theory has been developed that provides guarantees of performance, and constraint satisfaction, of the scenario solution. In this paper, we open a new direction of investigation: the risk that a performance is not achieved, or that constraints are violated, is studied jointly with the complexity (as precisely defined in the paper) of the solution. It is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk. This result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios; in particular, no extra knowledge on the distribution by which scenarios are generated is assumed, so that the result is broadly applicable. This deep-seated result unveils a fundamental and general structure of data-driven optimization and suggests practical approaches for risk assessment.
- Published
- 2022
32. On QZ steps with perfect shifts and computing the index of a differential-algebraic equation
- Author
-
Paul Van Dooren and Nicola Mastronardi
- Subjects
Index (economics) ,Applied Mathematics ,General Mathematics ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Mathematics::Numerical Analysis ,Computational Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Differential algebraic equation ,QZ algorithm ,eigenvalues ,perfect shifts ,index ,Mathematics - Abstract
In this paper we revisit the problem of performing a $QZ$ step with a so-called ‘perfect shift’, which is an ‘exact’ eigenvalue of a given regular pencil $\lambda B-A$ in unreduced Hessenberg triangular form. In exact arithmetic, the $QZ$ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the $QZ$ step gets ‘blurred’ and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the $QZ$ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.
- Published
- 2020
33. Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime
- Author
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Anton Braverman
- Subjects
Mathematical optimization ,021103 operations research ,Steady state ,General Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Computer Science Applications ,010104 statistics & probability ,Idle ,Exponential ergodicity ,Server ,Join (sigma algebra) ,0101 mathematics ,Queue ,Mathematics - Abstract
This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.
- Published
- 2020
34. Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems
- Author
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Ying Xu, Fengxin Sun, and Jufeng Wang
- Subjects
Correctness ,Article Subject ,General Mathematics ,Mathematical analysis ,General Engineering ,Boundary (topology) ,02 engineering and technology ,Radius ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,QA1-939 ,Boundary value problem ,TA1-2040 ,0101 mathematics ,Coefficient matrix ,Boundary element method ,Condition number ,Mathematics ,Interpolation - Abstract
The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.
- Published
- 2020
35. Fixed point theorem for new type of auxiliary functions
- Author
-
Vishal Gupta, Arslan Hojat Ansari, and Naveen Mani
- Subjects
Pure mathematics ,021103 operations research ,metric spaces ,General Mathematics ,0211 other engineering and technologies ,Fixed-point theorem ,02 engineering and technology ,Auxiliary function ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,54h25 ,fixed point ,auxiliary function ,QA1-939 ,0101 mathematics ,47h10 ,Mathematics - Abstract
In this paper, we present some fixed point results satisfying generalized contractive condition with new auxiliary function in complete metric spaces. More precisely, the structure of the paper is the following. In the first section, we present some useful notions and results. The main aim of second section is to establish some new fixed point results in complete metric spaces. Finally, in the third section, we show the validity and superiority of our main results by suitable example. Also, as an application of our main result, some interesting corollaries have been included, which make our concepts and results effective. Our main result generalizes some well known existing results in the literature.
- Published
- 2020
36. Lattice of Definability in the Order of Rational Numbers
- Author
-
Alexei Semenov and An. A. Muchnik
- Subjects
Rational number ,Pure mathematics ,Automorphism group ,020303 mechanical engineering & transports ,0203 mechanical engineering ,General Mathematics ,Lattice (order) ,010102 general mathematics ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Linear subspace ,Mathematics - Abstract
A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: “equality,” “less,” “between,” “cycle,” and “linkage.” For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description of the automorphism group is given. Although the structure of this lattice was known previously, the proof in the paper is of syntactic nature and avoids the use of a group-theoretical method.
- Published
- 2020
37. 2-Colorings of Hypergraphs with Large Girth
- Author
-
Yu. A. Demidovich
- Subjects
Hypergraph ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,Girth (graph theory) ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Has property ,Homogeneous ,Simple (abstract algebra) ,0101 mathematics ,Mathematics - Abstract
A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.
- Published
- 2020
38. A Net with Applications for Continuity in a Fuzzy Soft Topological Space
- Author
-
Jianrong Wu and Rui Gao
- Subjects
Mathematical optimization ,Article Subject ,Basis (linear algebra) ,Mathematics::General Mathematics ,Computer science ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,010102 general mathematics ,General Engineering ,02 engineering and technology ,Topological space ,Engineering (General). Civil engineering (General) ,Net (mathematics) ,01 natural sciences ,Fuzzy logic ,Field (computer science) ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Point (geometry) ,ComputingMethodologies_GENERAL ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
In this paper, the concept of a fuzzy soft point is redefined, and the definition of a fuzzy soft net in a fuzzy soft topological space is given. On this basis, the convergence of a fuzzy soft net is defined by using the Q-neighborhood theory, and the continuity of fuzzy soft mappings is characterized by the net. The obtained results demonstrate that the concepts proposed in this paper are very suitable and will provide powerful research tools for further research in this field.
- Published
- 2020
39. A generic exact solver for vehicle routing and related problems
- Author
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Eduardo Uchoa, Artur Alves Pessoa, Ruslan Sadykov, François Vanderbeck, Universidade Federal Fluminense [Rio de Janeiro] (UFF), Reformulations based algorithms for Combinatorial Optimization (Realopt), Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Atoptima, We would like to thank Teobaldo Bulhoes and Guillaume Marques for a large part of the implementation of the Julia–JuMP interface to the solver, Teobaldo Bulhoes, Guillaume Marques and Eduardo Queiroga for implementing, over that interface, the models corresponding to the examples of this paper, and Laurent Facq for a general support of the computing environment. Experiments presented in this paper were carried out using the PlaFRIM (Federative Platform for Research in Computer Science and Mathematics), created under the Inria PlaFRIM development action with support from Bordeaux INP, LABRI and IMB and other entities: Conseil Régional d’Aquitaine, Université de Bordeaux, CNRS and ANR in accordance to the 'Programme d’Investissements d’Avenir'. This study was financed in part by the Conselho Nacional de Científico e Tecnológico (CNPq), grant 313601/2018-6 (Produtividade 1B), and by the Funda¸c˜ao de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), grant E-26/202.887/2017 (Cientista do Estado)., Plafrim, Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
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Mathematical optimization ,021103 operations research ,Bin packing problem ,General Mathematics ,Column generation ,0211 other engineering and technologies ,Integer programming ,Rule-based system ,[INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO] ,010103 numerical & computational mathematics ,02 engineering and technology ,Solver ,01 natural sciences ,Vehicle routing problem ,Path (graph theory) ,0101 mathematics ,Routing (electronic design automation) ,Software ,Routing ,Mathematics - Abstract
Major advances were recently obtained in the exact solution of vehicle routing problems (VRPs). Sophisticated branch-cut-and-price (BCP) algorithms for some of the most classical VRP variants now solve many instances with up to a few hundreds of customers. However, adapting and reimplementing those successful algorithms for other variants can be a very demanding task. This work proposes a BCP solver for a generic model that encompasses a wide class of VRPs. It incorporates the key elements found in the best existing VRP algorithms: ng-path relaxation, rank-1 cuts with limited memory, path enumeration, and rounded capacity cuts; all generalized through the new concepts of “packing set” and “elementarity set”. The concepts are also used to derive a branching rule based on accumulated resource consumption and to generalize the Ryan and Foster branching rule. Extensive experiments on several variants show that the generic solver has an excellent overall performance, in many problems being better than the best specific algorithms. Even some non-VRPs, like bin packing, vector packing and generalized assignment, can be modeled and effectively solved.
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- 2020
40. Factorization of Noncommutative Polynomials and Nullstellensätze for the Free Algebra
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Igor Klep, J.W. Helton, and Jurij Volčič
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General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,Mathematics - Rings and Algebras ,02 engineering and technology ,01 natural sciences ,Noncommutative geometry ,Algebra ,Factorization ,Rings and Algebras (math.RA) ,Free algebra ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. A polynomial $f$ that depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being “stably associated” to a factor of $h$ or of $h^*$. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above “algebraic certificate” does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.
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- 2020
41. Local Mappings Generated by Multiplication on Rings of Matrices
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N. M. Umrzaqov and F. N. Arzikulov
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Pure mathematics ,Rational number ,Mathematics::Commutative Algebra ,General Mathematics ,Unital ,Mathematics::Rings and Algebras ,010102 general mathematics ,02 engineering and technology ,01 natural sciences ,Matrix ring ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Division ring ,Symmetric matrix ,Multiplier (economics) ,0101 mathematics ,Associative property ,Mathematics - Abstract
In the present paper, the notion of (additive) local multiplier on the ring of matrices over an arbitrary unital associative ring is introduced and investigated. It is proved that every local left multiplier on the matrix ring over a division ring is a left multiplier. Also, in the present paper, the notion of (additive) local Jordan multiplier on a Jordan ring of symmetric matrices is introduced and investigated. It is proved that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier. Also corollaries of these results are proved.
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- 2020
42. Residual Nilpotence of Groups with One Defining Relation
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D. I. Moldavanskii
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Group (mathematics) ,General Mathematics ,010102 general mathematics ,Prime number ,02 engineering and technology ,Residual ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Mathematics::Group Theory ,Nilpotent ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Simple (abstract algebra) ,Order (group theory) ,0101 mathematics ,Mathematics - Abstract
All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.
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- 2020
43. The Monotone Contraction Mapping Theorem
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Joseph Frank Gordon
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Mathematics::Functional Analysis ,Pure mathematics ,Article Subject ,General Mathematics ,010102 general mathematics ,Banach space ,Regular polygon ,02 engineering and technology ,01 natural sciences ,Complete metric space ,Monotone polygon ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Contraction mapping ,0101 mathematics ,Contraction (operator theory) ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper, the fixed-point theorem for monotone contraction mappings in the setting of a uniformly convex smooth Banach space is studied. This paper provides a version of the Banach fixed-point theorem in a complete metric space.
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- 2020
44. Numerical Model for Simulation of the Cathodic Protection System with Dynamic Nonlinear Polarization Characteristics
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Adnan Mujezinović, Slobodan Milojković, Alija Muharemović, Irfan Turkovic, and Sanja Martinez
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010101 applied mathematics ,Physics::Instrumentation and Detectors ,020209 energy ,General Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Nonlinear polarization ,02 engineering and technology ,Mechanics ,0101 mathematics ,Boundary Element Method ,Finite Difference Time Domain Method ,Newton-Raphson Technique ,Nonlinear Systems ,01 natural sciences ,Mathematics ,Cathodic protection - Abstract
Cathodic protection is defined as a method for slowing down or complete elimination of corrosion processes on underground or underwater, insulated or uninsulated metal structures. Protection by cathodic protection system is achieved by polarizing protected object to more negative value, with respect to its equilibrium potential. Design of the cathodic protection system implies determination of the electric potential and current density on the electrode surfaces after installation of the cathodic protection system. Most efficient way for determination of the electric potential and current density in the cathodic protection system is by applying numerical techniques. When modeling cathodic protection systems by numerical techniques, electrochemical reactions that occur on electrode surfaces are taken into account by polarization characteristics. Because of nature of the electrochemical reactions, polarization characteristics are nonlinear and under certain conditions can be time – varying (dynamic nonlinear polarization characteristics). This paper deals with numerical modeling of the cathodic protection system with dynamic nonlinear polarization characteristics. Numerical model presented in this paper is divided in the two parts. First part, which is based on the direct boundary element method, is used for the calculation of the distribution of electric potential and current density on the electrode surfaces in the spatial domain. Second part of the model is based on the finite difference time domain method and is used for the calculation of the electric potential and current density change over time. The use of presented numerical model is demonstrated on two simple geometrically examples.
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- 2020
45. Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model
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Lesław Gajek and Marcin Rudź
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Statistics and Probability ,0209 industrial biotechnology ,Insolvency ,Markov chain ,business.industry ,General Mathematics ,Risk measure ,02 engineering and technology ,State (functional analysis) ,Function (mathematics) ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,020901 industrial engineering & automation ,Operator (computer programming) ,Applied mathematics ,0101 mathematics ,business ,Risk management ,Mathematics - Abstract
Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.
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- 2020
46. Uniqueness of the Continuation of a Certain Function to a Positive Definite Function
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A. D. Manov
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Class (set theory) ,Continuous function (set theory) ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,Function (mathematics) ,Positive-definite matrix ,Extension (predicate logic) ,01 natural sciences ,Combinatorics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Positive-definite function ,Interval (graph theory) ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a ≤ 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| ≤ a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a ∈ [1/2,1] and Re ϕ(x) = 1 - |x|, |x| ≤ a, for some ϕ ∈ T, then ϕ(x) = (1 - |x|) +, x G R. In addition, for any a G (0,1/2), there exists a function ϕ ∈ T such that ϕ(x) = 1 - |x|, |x| ≤ a, but ϕ(x) ≠ (1 - |x|)+. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.
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- 2020
47. An $${\mathcal {O}}(n^2 \log {n})$$ algorithm for the weighted stable set problem in claw-free graphs
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Paolo Nobili and Antonio Sassano
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Vertex (graph theory) ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Stable set problem ,Binary logarithm ,01 natural sciences ,Graph ,law.invention ,Computer Science::Discrete Mathematics ,law ,Independent set ,Line graph ,0101 mathematics ,Algorithm ,Software ,Mathematics - Abstract
A graph G(V, E) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into {claw, net}-free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in $$\mathcal{O}(|V|(|V| \log |V| + |E|))$$ time. In this paper, we describe a direct decomposition of a claw-free graph into {claw, net}-free strips and strips with stability number at most three which can be performed in $$\mathcal{O}(|V|^2)$$ time. In two companion papers we showed that the MWSS problem can be solved in $$\mathcal{O}(|E| \log |V|)$$ time in claw-free graphs with $$\alpha (G) \le 3$$ and in $$\mathcal{O}(|V| \sqrt{|E|})$$ time in {claw, net}-free graphs with $$\alpha (G) \ge 4$$ . These results prove that the MWSS problem in a claw-free graph can be solved in $$\mathcal{O}(|V|^2 \log |V|)$$ time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs.
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- 2020
48. GLOBAL RELAXED MODULUS-BASED SYNCHRONOUS BLOCK MULTISPLITTING MULTI-PARAMETERS METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS
- Author
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Xianyu Zuo and Litao Zhang
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Numerical linear algebra ,General Mathematics ,Modulus ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,computer.software_genre ,01 natural sciences ,Complementarity (physics) ,Linear complementarity problem ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Matrix analysis ,0101 mathematics ,System matrix ,computer ,Mathematics - Abstract
Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425Ƀ439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307Ƀ312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954Ƀ1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block $ H_{+}- $matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97Ƀ110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.
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- 2020
49. MMAP/(PH,PH)/1 Queue with Priority Loss through Feedback
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Agassi Melikov, Achyutha Krishnamoorthy, Sevinj Aliyeva, and Divya Velayudhan Nair
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Operations research ,Marked Markovian arrival process ,Computer science ,mmap ,General Mathematics ,MathematicsofComputing_GENERAL ,0211 other engineering and technologies ,feedback ,02 engineering and technology ,Space (commercial competition) ,01 natural sciences ,preemptive ,010104 statistics & probability ,queueing system ,QA1-939 ,Computer Science (miscellaneous) ,Markovian arrival process ,0101 mathematics ,priority loss ,Engineering (miscellaneous) ,Queue ,Service (business) ,Queueing theory ,021103 operations research ,non-pre-emptive ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Line (text file) ,Priority queue ,Mathematics - Abstract
In this paper, we consider two single server queueing systems to which customers of two distinct priorities (P1 and P2) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a P1 customer to feedback is θ on completion of his service. The feedback (P1) customers, as well as P2 customers, join the low priority queue. Low priority (P2) customers are taken for service from the head of the line whenever the P1 queue is found to be empty at the service completion epoch. We assume a finite waiting space for P1 customers and infinite waiting space for P2 customers. Two models are discussed in this paper. In model I, we assume that the service of P2 customers is according to a non-preemptive service discipline and in model II, the P2 customers service follow a preemptive policy. No feedback is permitted to customers in the P2 line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.
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- 2021
- Full Text
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50. Viscosity approximation method for solving variational inequality problem in real Banach spaces
- Author
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Godwin Chidi Ugwunnadi
- Subjects
Sequence ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,Lipschitzian Mapping ,02 engineering and technology ,Nonexpansive Mapping ,Fixed point ,01 natural sciences ,Viscosity (programming) ,Fixed Point ,Variational inequality ,Strongly Accretive Mapping ,QA1-939 ,Applied mathematics ,Hierarchical Fixed Point Problems ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.
- Published
- 2021
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