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2. Rebuttal of Donnelly's paper on the spectral gap
- Author
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Antoine Henrot, Mark S. Ashbaugh, Richard S. Laugesen, Department of Mathematics, University of Missouri Columbia, University of Missouri [Columbia] (Mizzou), University of Missouri System-University of Missouri System, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Urbana], University of Illinois at Urbana-Champaign [Urbana], University of Illinois System-University of Illinois System, CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
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Discrete mathematics ,Sequence ,Conjecture ,General Mathematics ,010102 general mathematics ,Mathematics::History and Overview ,Mathematics::Spectral Theory ,01 natural sciences ,Domain (mathematical analysis) ,Computer Science::Computers and Society ,010101 applied mathematics ,symbols.namesake ,Physics::Popular Physics ,Dirichlet boundary condition ,Euclidean geometry ,symbols ,Calculus ,Convex body ,Quantitative Biology::Populations and Evolution ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Spectral gap ,0101 mathematics ,Mathematics ,Unit interval - Abstract
The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for
- Published
- 2011
3. (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications
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Ana Navarro-Quiles, M.-D. Roselló, José Vicente Romero, and Juan Carlos Cortés
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education.field_of_study ,Differential equation ,Stochastic process ,General Mathematics ,Computation ,010102 general mathematics ,Population ,General Engineering ,Probability density function ,01 natural sciences ,010101 applied mathematics ,Transformation (function) ,Applied mathematics ,0101 mathematics ,Logistic function ,education ,Random variable ,Mathematics - Abstract
The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.
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4. Improving the performance of deep learning models using statistical features: The case study of COVID‐19 forecasting
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Hossein Abbasimehr, Reza Paki, and Aram Bahrini
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2019-20 coronavirus outbreak ,Coronavirus disease 2019 (COVID-19) ,62‐07 ,General Mathematics ,Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ,Context (language use) ,97r40 ,Machine learning ,computer.software_genre ,01 natural sciences ,Convolutional neural network ,Special Issue Paper ,0101 mathematics ,Combined method ,Mathematics ,Special Issue Papers ,business.industry ,Deep learning ,010102 general mathematics ,General Engineering ,deep learning ,COVID‐19 pandemic ,010101 applied mathematics ,hybrid methods ,Memory model ,Artificial intelligence ,business ,computer ,statistical features - Abstract
COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.
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- 2021
5. The r-Hunter-Saxton equation, smooth and singular solutions and their approximation
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Colin J. Cotter, Tristan Pryer, Jacob Deasy, Cotter, Colin J [0000-0001-7962-8324], Apollo - University of Cambridge Repository, and Engineering & Physical Science Research Council (EPSRC)
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Paper ,singular solutions ,GEODESIC-FLOW ,Work (thermodynamics) ,General Mathematics ,Mathematics, Applied ,HYPERBOLIC VARIATIONAL EQUATION ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,Piecewise linear function ,37K06 ,Mathematics - Analysis of PDEs ,0102 Applied Mathematics ,37K05 ,FOS: Mathematics ,Hunter–Saxton equation ,Applied mathematics ,Initial value problem ,Lie symmetries ,0101 mathematics ,nlin.SI ,math.AP ,Mathematical Physics ,Mathematics ,Science & Technology ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Physics ,Applied Mathematics ,010102 general mathematics ,4901 Applied Mathematics ,4904 Pure Mathematics ,Statistical and Nonlinear Physics ,Action (physics) ,Symmetry (physics) ,Physics, Mathematical ,010101 applied mathematics ,35Q53 ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,nonlinear PDEs ,Physical Sciences ,49 Mathematical Sciences ,37K58 ,Exactly Solvable and Integrable Systems (nlin.SI) ,Analysis of PDEs (math.AP) - Abstract
In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation., Revised after referee comments
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- 2019
6. Some significant remarks on multivalued Perov type contractions on cone metric spaces with a directed graph
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Aleksandra Sretenovic, Nicola Fabiano, Ana Savić, Stojan Radenović, and Nikola Mirkov
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Pure mathematics ,General Mathematics ,cone metric space ,010102 general mathematics ,multivalued mapping ,graphic contraction ,Directed graph ,common fixed point ,Fixed point ,Type (model theory) ,Mathematical proof ,directed graph ,01 natural sciences ,Cone (formal languages) ,c-sequence ,010101 applied mathematics ,Metric space ,QA1-939 ,0101 mathematics ,Contraction principle ,perov's type results ,Mathematics ,Complement (set theory) - Abstract
Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.
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- 2022
7. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
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Zhiyue Zhang, Hüseyin Budak, Yu-Ming Chu, Necmettin Alp, Muhammad Ali, and [Belirlenecek]
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Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,Type (model theory) ,Quantum calculus ,quantum calculus ,01 natural sciences ,Midpoint ,26d15 ,Hermite–Hadamard inequality ,QA1-939 ,26a51 ,Differentiable function ,0101 mathematics ,Quantum ,26d10 ,Mathematics ,media_common ,convex function ,hermite-hadamard inequality ,010102 general mathematics ,010101 applied mathematics ,Computer Science::Graphics ,q-integral ,Convex function - Abstract
The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities. © 2021 Muhammad Aamir Ali et al., published by De Gruyter. National Natural Science Foundation of China, NSFC: 11301127, 11601485, 11626101, 11701176, 11971241, 61673169 Funding information : The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241). 2-s2.0-85105011594
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- 2021
8. Generalization of uniqueness and value sharing of meromorphic functions concerning differential polynomials
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Ramya Maligi and Harina P. Waghamore
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Pure mathematics ,Generalization ,primary 30d35 ,General Mathematics ,010102 general mathematics ,uniqueness ,01 natural sciences ,differential polynomials ,010101 applied mathematics ,QA1-939 ,meromorphic functions ,Uniqueness ,sharing value ,0101 mathematics ,[MATH]Mathematics [math] ,Value (mathematics) ,Differential (mathematics) ,Mathematics ,Meromorphic function - Abstract
The motivation of this paper is to study the uniqueness problems of meromorphic functions concerning differential polynomials that share a small function. The results of the paper improve and generalize the recent results due to Fengrong Zhang and Linlin Wu [13]. We also solve an open problem as posed in the last section of [13].
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- 2020
9. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
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Patrizia Pucci and Letizia Temperini
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General Mathematics ,variational methods ,35b08 ,nonlinear system ,01 natural sciences ,(p ,Heisenberg group ,0101 mathematics ,Q system ,Geometry and topology ,Mathematics ,Mathematical physics ,Q) Laplacian ,35b33 ,lcsh:Mathematics ,010102 general mathematics ,heisenberg group ,(p,q) laplacian ,35j50 ,lcsh:QA1-939 ,Exponential function ,010101 applied mathematics ,Nonlinear system ,35j47 ,(p,Q) Laplacian, Nonlinear system, Critical exponential nonlinearities, Variational methods, Heisenberg group ,35b09 ,critical exponential nonlinearities ,35r03 - Abstract
The paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}.
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- 2020
10. Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes
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Salim Bouzebda, Sultana Didi, Laboratoire de Mathématiques Appliquées de Compiègne (LMAC), Université de Technologie de Compiègne (UTC), and Qassim University [Kingdom of Saudi Arabia]
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60A10 ,Measurable function ,General Mathematics ,Context (language use) ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,Conditional mode ,01 natural sciences ,Article ,Combinatorics ,Mixing (mathematics) ,Martingale difference arrays ,62G08 ,60F05 ,62G07 ,Ergodic theory ,Conditional density ,62G05 ,60E05 ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Strong consistency ,Mathematics ,62E20 ,Smoothness (probability theory) ,Kernel (set theory) ,010102 general mathematics ,Ergodicity ,Confidence regions ,[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] ,Rate of convergence ,010101 applied mathematics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Nadaraya–Watson estimators ,Continuous time processes ,Ergodic processes ,Kernel regression ,Kernel estimate ,Prediction - Abstract
In the present paper, we consider the nonparametric regression model with random design based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}\ge 0}$$\end{document}(Xt,Yt)t≥0 a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}\times \mathbb {R}^{q}$$\end{document}Rd×Rq-valued strictly stationary and ergodic continuous time process, where the regression function is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))$$\end{document}m(x,ψ)=E(ψ(Y)∣X=x)), for a measurable function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}$$\end{document}ψ:Rq→R. We focus on the estimation of the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Theta }}$$\end{document}Θ (mode) of a unique maximum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) by the location \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T of a maximum of the Nadaraya–Watson kernel estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{m}_\mathrm{T}(\cdot , \psi )$$\end{document}m^T(·,ψ) for the curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ). Within this context, we obtain the consistency with rate and the asymptotic normality results for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$\end{document}Θ^T under mild local smoothness assumptions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(\cdot , \psi )$$\end{document}m(·,ψ) and the design density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\cdot )$$\end{document}f(·) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{X}$$\end{document}X. Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.
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- 2020
11. Fixed point theorem for new type of auxiliary functions
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Vishal Gupta, Arslan Hojat Ansari, and Naveen Mani
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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.
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- 2020
12. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
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Wasfi Shatanawi, Hacer Dağ, Ishak Altun, and KKÜ
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Pure mathematics ,General Mathematics ,010102 general mathematics ,primary 47h10 ,Fixed point ,Type (model theory) ,multivalued mappings ,01 natural sciences ,010101 applied mathematics ,Metric space ,fixed point ,QA1-939 ,secondary 54h25 ,0101 mathematics ,quasi metric space ,Mathematics - Abstract
Altun, Ishak/0000-0002-7967-0554 WOS:000537813000001 In this paper, we present some fixed point results for multivalued mappings with both closed values and proximinal values on left K-complete quasi metric spaces. We also provide a nontrivial example to illustrate our results. Prince Sultan University [RG-DES-2017-01-17] The authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper. This work was supported by the Prince Sultan University through the Research Group NAMAM under Grant RG-DES-2017-01-17.
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- 2020
13. Linear representation of a graph
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Eduardo Peña Cabrera, José A. González Campos, Eduardo Montenegro, and Ronald A. Manríquez Peñafiel
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Abstract algebra ,Linear representation ,Group (mathematics) ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Order (ring theory) ,lcsh:QA1-939 ,01 natural sciences ,Graph ,law.invention ,010101 applied mathematics ,Combinatorics ,Invertible matrix ,Simple (abstract algebra) ,law ,0101 mathematics ,Graphs ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area
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- 2019
14. McKay Quivers and Lusztig Algebras of Some Finite Groups
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Ragnar-Olaf Buchweitz, Matthew Lewis, Colin Ingalls, and Eleonore Faber
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General Mathematics ,Field (mathematics) ,Group Theory (math.GR) ,01 natural sciences ,Combinatorics ,Elementary algebra ,Symmetric group ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics ,Finite group ,05E10 16G20 16S35 16S37 20F55 20C30 ,010102 general mathematics ,Quiver ,Mathematics - Rings and Algebras ,010101 applied mathematics ,Clifford theory ,Rings and Algebras (math.RA) ,Combinatorics (math.CO) ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Vector space ,Group ring - Abstract
We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$., Comment: v2: minor revision, final version to appear in Algebr. Represent. Theory
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- 2021
15. Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws
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Giovanni Russo, Irene Gómez-Bueno, Carlos Parés, and Manuel Jesús Castro Díaz
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finite volume methods ,Computer science ,General Mathematics ,systems of balance laws ,reconstruction operators ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,QA1-939 ,Computer Science (miscellaneous) ,0101 mathematics ,Engineering (miscellaneous) ,Shallow water equations ,Collocation ,shallow water equations ,Basis (linear algebra) ,Numerical analysis ,Euler equations ,high order methods ,Quadrature (mathematics) ,Burgers' equation ,010101 applied mathematics ,Law ,collocation methods ,symbols ,well-balanced methods ,Mathematics - Abstract
In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.
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- 2021
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16. Approximation of Endpoints for α—Reich–Suzuki Nonexpansive Mappings in Hyperbolic Metric Spaces
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Afrah An Abdou, Izhar Uddin, and Sajan Aggarwal
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Pure mathematics ,General Mathematics ,010102 general mathematics ,endpoint ,MathematicsofComputing_GENERAL ,Fixed-point theorem ,Fixed point ,01 natural sciences ,fixed point theorems ,010101 applied mathematics ,Metric space ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,α—Riech–Suzuki nonexpansive mapping ,Convergence (routing) ,Computer Science (miscellaneous) ,QA1-939 ,Computer Science::Programming Languages ,hyperbolic metric space ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving α—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong and Δ—convergence theorems in a hyperbolic metric space. Thus, our results generalize and improve many existing results.
- Published
- 2021
17. Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings
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Mohammed Said Souid, Mohammed K. A. Kaabar, Zailan Siri, Shahram Rezapour, Francisco Martínez, Sina Etemad, and Ahmed Refice
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Ulam–Hyers–Rassias stability ,Mathematics::Functional Analysis ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,variable-order operators ,implicit problem ,01 natural sciences ,Stability (probability) ,fixed point theorems ,010101 applied mathematics ,Nonlinear fractional differential equations ,piecewise constant functions ,Nonlinear system ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Order (group theory) ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Variable (mathematics) - Abstract
In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.
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- 2021
18. On the Convergence of a New Family of Multi-Point Ehrlich-Type Iterative Methods for Polynomial Zeros
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Petko D. Proinov and Milena Petkova
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Polynomial ,iteration functions ,Iterative method ,General Mathematics ,010103 numerical & computational mathematics ,Construct (python library) ,multi-point iterative methods ,Type (model theory) ,01 natural sciences ,Local convergence ,010101 applied mathematics ,error estimates ,Convergence (routing) ,semilocal convergence ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,local convergence ,0101 mathematics ,polynomial zeros ,Engineering (miscellaneous) ,Multi point ,Mathematics - Abstract
In this paper, we construct and study a new family of multi-point Ehrlich-type iterative methods for approximating all the zeros of a uni-variate polynomial simultaneously. The first member of this family is the two-point Ehrlich-type iterative method introduced and studied by Trićković and Petković in 1999. The main purpose of the paper is to provide local and semilocal convergence analysis of the multi-point Ehrlich-type methods. Our local convergence theorem is obtained by an approach that was introduced by the authors in 2020. Two numerical examples are presented to show the applicability of our semilocal convergence theorem.
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- 2021
19. A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams
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Snježana Maksimović and Aleksandar Borković
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Basis (linear algebra) ,Plane curve ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Static analysis ,Space (mathematics) ,01 natural sciences ,Computer Science::Digital Libraries ,010101 applied mathematics ,analytical solution ,Bernoulli–Euler beam ,special functions ,Special functions ,Computer Science (miscellaneous) ,QA1-939 ,arc-length parametrization ,Development (differential geometry) ,0101 mathematics ,Sturm–Liouville differential equation ,Engineering (miscellaneous) ,Arc length ,Parametrization ,Mathematics - Abstract
The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.
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- 2021
20. Hermite–Hadamard Inclusions for Co-Ordinated Interval-Valued Functions via Post-Quantum Calculus
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Jessada Tariboon, Sotiris K. Ntouyas, Hüseyin Budak, Muhammad Ali, and [Belirlenecek]
- Subjects
Pure mathematics ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Quantum calculus ,co-ordinated convexity ,quantum calculus ,01 natural sciences ,Interval valued ,Hadamard transform ,(p ,Hermite–Hadamard inequality ,Hermite–Hadamard inclusion ,Computer Science (miscellaneous) ,QA1-939 ,0101 mathematics ,interval-valued functions ,Mathematics ,Hermite polynomials ,010102 general mathematics ,Regular polygon ,(p, q)-integral ,Convex ,010101 applied mathematics ,Hermite-Hadamard inequality ,Chemistry (miscellaneous) ,Hermite-Hadamard inclusion ,q)-integral ,Midpoint Type Inequalities ,Symmetry (geometry) - Abstract
In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite-Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the findings in this paper are important extensions of previous findings in the literature. Finally, we present a few examples of our new findings. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. WOS:000677046700001 2-s2.0-85110868353
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- 2021
21. Entropy in the cusp and phase transitions for geodesic flows
- Author
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Godofredo Iommi, Felipe Riquelme, Anibal Velozo, Facultad de Matemáticas [Santiago de Chile], Pontificia Universidad Católica de Chile (UC), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics - Princeton University, Princeton University, Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Guillemer, Marie-Annick, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
- Subjects
Phase transition ,Markov chain ,Geodesic ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,[ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS] ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,15A15, 26E60, 37A30, 60B20, 60F15 ,01 natural sciences ,010101 applied mathematics ,Geodesic flow ,FOS: Mathematics ,Sectional curvature ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics - Abstract
In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups., 34 pages, 4 figures. In this new version we have improved the organization of the paper and the clarity of some statements
- Published
- 2015
22. On Coefficient Problems for Functions Connected with the Sine Function
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Katarzyna Tra̧bka-Wiȩcław
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Pure mathematics ,Class (set theory) ,functions starlike with respect to symmetric points ,Physics and Astronomy (miscellaneous) ,Logarithm ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,generalized Zalcman coefficient functional ,Hankel determinant ,Chemistry (miscellaneous) ,coefficients of analytic functions ,Computer Science (miscellaneous) ,QA1-939 ,Sine ,0101 mathematics ,Mathematics ,Analytic function - Abstract
In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.
- Published
- 2021
- Full Text
- View/download PDF
23. General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications
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Hari M. Srivastava, Kalpana Fatawat, Yashoverdhan Vyas, and Shivani Pathak
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Pure mathematics ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Quantum calculus ,symmetric quantum calculus ,Mathematical proof ,01 natural sciences ,q-Kummer second and third summation theorems ,symbols.namesake ,Heine’s transformation ,QA1-939 ,Computer Science (miscellaneous) ,0101 mathematics ,Invariant (mathematics) ,Hypergeometric function ,Mathematics ,Series (mathematics) ,q-Kummer summation theorem ,010102 general mathematics ,Gauss ,Thomae’s q-integral representation ,010101 applied mathematics ,Number theory ,Chemistry (miscellaneous) ,symbols ,quantum or basic (or q-) hypergeometric series ,Jacobi polynomials ,q-Binomial theorem - Abstract
This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.
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- 2021
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24. k-Version of Finite Element Method for BVPs and IVPs
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Sri Sai Charan Mathi, Karan S. Surana, and Celso H. Carranza
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General Mathematics ,finite element method ,MathematicsofComputing_GENERAL ,higher order global differentiability ,02 engineering and technology ,Isogeometric analysis ,01 natural sciences ,k-version ,0203 mechanical engineering ,Convergence (routing) ,QA1-939 ,Computer Science (miscellaneous) ,Applied mathematics ,Initial value problem ,isogeometric ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,tensor product ,higher order spaces ,Differential operator ,IVPs ,Finite element method ,variational consistency ,010101 applied mathematics ,BVPs ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,020303 mechanical engineering & transports ,Tensor product ,Self-adjoint operator - Abstract
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.
- Published
- 2021
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25. Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses
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Mohamed I. Abbas and Snezhana Hristova
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Physics and Astronomy (miscellaneous) ,Differential equation ,General Mathematics ,010102 general mathematics ,Scalar (physics) ,Function (mathematics) ,Type (model theory) ,01 natural sciences ,Symmetry (physics) ,010101 applied mathematics ,symbols.namesake ,Transformation (function) ,generalized proportional fractional derivatives ,Chemistry (miscellaneous) ,Mittag-Leffler function ,QA1-939 ,Computer Science (miscellaneous) ,symbols ,Initial value problem ,Applied mathematics ,Mittag–Leffler function ,0101 mathematics ,Mathematics - Abstract
The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.
- Published
- 2021
- Full Text
- View/download PDF
26. Industrial Steel Heat Treating: Numerical Simulation of Induction Heating and Aquaquenching Cooling with Mechanical Effects
- Author
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Francisco Ortegón Gallego, José Manuel Díaz Moreno, Giuseppe Viglialoro, María Teresa González Montesinos, Concepción García Vázquez, Matemáticas, Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII), Ministerio de Educación y Ciencia (MEC). España, and Junta de Andalucía
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Austenite ,Induction heating ,Materials science ,Computer simulation ,thermomechanical problem ,General Mathematics ,finite element method ,Joule ,nonlinear coupled system of PDEs/ODEs ,010103 numerical & computational mathematics ,Mechanics ,01 natural sciences ,steel hardening ,phase transitions ,010101 applied mathematics ,Martensite ,Displacement field ,Computer Science (miscellaneous) ,Hardening (metallurgy) ,QA1-939 ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Heat treating - Abstract
This paper summarizes a mathematical model for the industrial heating and cooling processes of a steel workpiece corresponding to the steering rack of an automobile. The general purpose of the heat treatment process is to create the necessary hardness on critical parts of the workpiece. Hardening consists of heating the workpiece up to a threshold temperature followed by a rapid cooling such as aquaquenching. The high hardness is due to the steel phase transformation accompanying the rapid cooling resulting in non-equilibrium phases, one of which is the hard microconstituent of steel, namely martensite. The mathematical model describes both processes, heating and cooling. During the first one, heat is produced by Joule's effect from a very high alternating current passing through the rack. This situation is governed by a set of coupled PDEs/ODEs involving the electric potential, the magnetic vector potential, the temperature, the austenite transformation, the stresses and the displacement field. Once the workpiece has reached the desired temperature, the current is switched off an the cooling stage starts by aquaquenching. In this case, the governing equations involve the temperature, the austenite and martensite phase fractions, the stresses and the displacement field. This mathematical model has been solved by the FEM and 2D numerical simulations are discussed along the paper., This research was partially supported by Ministerio de Educacion y Ciencia under grants MTM2010-16401 and TEC2017-86347-C2-1-R with the participation of FEDER, and Consejeria de Educacion y Ciencia de la Junta de Andalucia, research group FQM-315. Giuseppe Viglialoro is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and he is partially supported by the research projects Evolutive and stationary Partial Differential Equations with a focus on biomathematics, funded by Fondazione di Sardegna (2019), and by MIUR (Italian Ministry of Education, University and Research) Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Grant Number: 2017AYM8XW).
- Published
- 2021
27. Error Estimations for Total Variation Type Regularization
- Author
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Chun Huang, Ziyang Yuan, and Kuan Li
- Subjects
Series (mathematics) ,General Mathematics ,Stability (learning theory) ,010103 numerical & computational mathematics ,Inverse problem ,Type (model theory) ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,regularization ,total variation ,Rate of convergence ,Consistency (statistics) ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,A priori and a posteriori ,inverse problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper provides several error estimations for total variation (TV) type regularization, which arises in a series of areas, for instance, signal and imaging processing, machine learning, etc. In this paper, some basic properties of the minimizer for the TV regularization problem such as stability, consistency and convergence rate are fully investigated. Both a priori and a posteriori rules are considered in this paper. Furthermore, an improved convergence rate is given based on the sparsity assumption. The problem under the condition of non-sparsity, which is common in practice, is also discussed, the results of the corresponding convergence rate are also presented under certain mild conditions.
- Published
- 2021
28. A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions
- Author
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Sama Arjika and Hari M. Srivastava
- Subjects
Pure mathematics ,rogers type formulas ,basic (or q-) hypergeometric series ,General Mathematics ,Bilinear interpolation ,q-binomial theorem ,Type (model theory) ,Computer Science::Digital Libraries ,01 natural sciences ,Combinatorics ,Identity (mathematics) ,symbols.namesake ,Mathematics - Analysis of PDEs ,QA1-939 ,FOS: Mathematics ,Computer Science (miscellaneous) ,Mathematics - Combinatorics ,Point (geometry) ,homogeneous q-difference operator ,0101 mathematics ,Srivastava-Agarwal type generating functions ,Engineering (miscellaneous) ,Mathematics ,010102 general mathematics ,Generating function ,Al-Salam-Carlitz q-polynomials ,General family ,010101 applied mathematics ,symbols ,05A30, 33D15, 33D45, 05A40, 11B65 ,Jacobi polynomials ,Combinatorics (math.CO) ,cauchy polynomials ,Analysis of PDEs (math.AP) - Abstract
In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant., 14 pages
- Published
- 2021
29. Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings
- Author
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Vasile Berinde and Cristina Ţicală
- Subjects
Physics and Astronomy (miscellaneous) ,Computer science ,General Mathematics ,Perturbation (astronomy) ,enriched demicontractive operator ,01 natural sciences ,Edge detection ,Brain ct ,QA1-939 ,Computer Science (miscellaneous) ,Numerical tests ,admissible perturbation ,ant-based algorithm ,0101 mathematics ,Complement (set theory) ,edge detection ,symmetric medical image ,010102 general mathematics ,Process (computing) ,test function ,010101 applied mathematics ,Image edge ,Chemistry (miscellaneous) ,Test functions for optimization ,Algorithm ,Mathematics ,asymmetric medical image - Abstract
The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second author and her collaborators, where they used admissible perturbations of demicontractive mappings as test functions. To illustrate this fact, we first consider some typical properties of demicontractive mappings and of their admissible perturbations and then present some appropriate numerical tests to illustrate the improvement brought by the admissible perturbations of demicontractive mappings when they are taken as test functions in ant-based algorithms for medical image edge detection. The edge detection process reported in our study considers both symmetric (Head CT and Brain CT) and asymmetric (Hand X-ray) medical images. The performance of the algorithm was tested visually with various images and empirically with evaluation of parameters.
- Published
- 2021
- Full Text
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30. Multiple Diamond-Alpha Integral in General Form and Their Properties, Applications
- Author
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Zhong-Xuan Mao, Shi-Pu Liu, Jun-Ping Hou, Chun-Ping Ma, and Ya-Ru Zhu
- Subjects
Pure mathematics ,Delta integral ,General Mathematics ,010102 general mathematics ,Ostrowski type inequalities ,multiple Diamond-Alpha integral ,Diamond ,engineering.material ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Alpha (programming language) ,Nabla integral ,Computer Science (miscellaneous) ,engineering ,QA1-939 ,Nabla symbol ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.
- Published
- 2021
- Full Text
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31. On Certain Differential Subordination of Harmonic Mean Related to a Linear Function
- Author
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Anna Dobosz, Piotr Jastrzębski, and Adam Lecko
- Subjects
Subordination (linguistics) ,Pure mathematics ,Physics and Astronomy (miscellaneous) ,Generalization ,General Mathematics ,Harmonic mean ,harmonic mean ,01 natural sciences ,arithmetic mean ,Mathematics::Probability ,Computer Science (miscellaneous) ,QA1-939 ,0101 mathematics ,Mathematics ,convex function ,Linear function (calculus) ,010102 general mathematics ,010101 applied mathematics ,geometric mean ,Chemistry (miscellaneous) ,Geometric mean ,Convex function ,differential subordination ,Differential (mathematics) ,Arithmetic mean - Abstract
In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.
- Published
- 2021
32. Smooth kNN Local Linear Estimation of the Conditional Distribution Function
- Author
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Ali Laksaci, Zouaoui Chikr Elmezouar, Ibrahim M. Almanjahie, and Mustapha Rachdi
- Subjects
General Mathematics ,01 natural sciences ,conditional predictive region ,010104 statistics & probability ,Mixing (mathematics) ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,0101 mathematics ,distribution function ,Engineering (miscellaneous) ,Mathematics ,Variable (mathematics) ,Sequence ,Series (mathematics) ,kernel weighting ,k nearest neighbors smoothing (kNN) ,functional mixing data ,Estimator ,Linearity ,Function (mathematics) ,Conditional probability distribution ,complete convergence (a.co.) ,010101 applied mathematics ,Local Linear Fitting (LLM) - Abstract
Previous works were dedicated to the functional k-Nearest Neighbors (kNN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of (Xi,Yi)i=1,…,n of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of Yi given functional input variable Xi. Precisely, we combine the kNN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.
- Published
- 2021
33. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?
- Author
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B. Cano and Nuria Reguera
- Subjects
Order reduction ,General Mathematics ,Krylov methods ,010103 numerical & computational mathematics ,Exponential integrator ,01 natural sciences ,Exponential function ,010101 applied mathematics ,efficiency ,QA1-939 ,Computer Science (miscellaneous) ,Spite ,Applied mathematics ,avoiding order reduction ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.
- Published
- 2021
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34. A new way to represent functions as series
- Author
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Manuel Norman
- Subjects
values of infinite series ,Series (mathematics) ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Function series ,function series ,primary 26a06 ,secondary 41a58 ,lcsh:QA1-939 ,01 natural sciences ,010101 applied mathematics ,Algebra ,lagrange’s mean value theorem ,0101 mathematics ,40a30 ,Geometry and topology ,Mathematics - Abstract
In this paper we will show a new way to represent functions as infinite series, finding some conditions under which a function is expandable with this method, and showing how it allows us to find the values of many interesting series. At the end, we will prove one of the main results of the paper, a Representation Theorem.
- Published
- 2019
35. Infinitely many weak solutions for fourth-order equations depending on two parameters
- Author
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Ghasem A. Afrouzi, H. Zahmatkesh, and Saeid Shokooh
- Subjects
Ricceri variational principle ,infinitely many solutions ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,lcsh:QA1-939 ,01 natural sciences ,010101 applied mathematics ,Fourth order ,Variational principle ,Order (business) ,Applied mathematics ,fourth-order equations ,0101 mathematics ,Mathematics - Abstract
In this paper, by employing Ricceri variational principle, we prove the existence of infinitely many weak solutions for fourth-order problems depending on two real parameters. We also provide some particular cases and a concrete example in order to illustrate the main abstract results of this paper.
- Published
- 2018
36. General fractional integrals and derivatives of arbitrary order
- Author
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Yuri Luchko
- Subjects
Pure mathematics ,Sonine kernel ,Physics and Astronomy (miscellaneous) ,Integrable system ,Generalization ,General Mathematics ,second fundamental theorem of fractional calculus ,general fractional derivative of arbitrary order ,general fractional integral of arbitrary order ,01 natural sciences ,26A33, 26B30, 44A10, 45E10 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,QA1-939 ,Computer Science (miscellaneous) ,Order (group theory) ,Point (geometry) ,0101 mathematics ,Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Fractional calculus ,010101 applied mathematics ,Chemistry (miscellaneous) ,Mathematics - Classical Analysis and ODEs ,first fundamental theorem of fractional calculus ,Gravitational singularity - Abstract
In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives with these kernels possess the integrable singularities at the point zero, the kernels of the general fractional integrals can be - depending on their order - both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved., 15 pages
- Published
- 2021
37. Common Fixed Points Technique for Existence of a Solution of Urysohn Type Integral Equations System in Complex Valued b-metric Spaces
- Author
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Naeem Saleem, Liliana Guran, Monica-Felicia Bota, Muhammad Suhail Aslam, and Mohammad S. R. Chowdhury
- Subjects
Pure mathematics ,lcsh:Mathematics ,General Mathematics ,010102 general mathematics ,Complex valued ,common fixed point ,Fixed point ,Type (model theory) ,lcsh:QA1-939 ,01 natural sciences ,Integral equation ,010101 applied mathematics ,Metric space ,single-valued mappings ,Computer Science (miscellaneous) ,Common fixed point ,Order (group theory) ,nonlinear integral equations ,0101 mathematics ,complex valued b-metric spaces ,Engineering (miscellaneous) ,Contraction (operator theory) ,Mathematics - Abstract
In this paper we give some common fixed point theorems for Ćirić type operators in complex valued b-metric spaces. Also, some corollaries under this contraction condition are obtained. Our results extend and generalize the results of Hammad et al. In the second part of the paper, in order to strengthen our main results, an illustrative example and some applications are given.
- Published
- 2021
- Full Text
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38. Hölder regularity for the spectrum of translation flows
- Author
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Boris Solomyak, Alexander I. Bufetov, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Bar-Ilan], Bar-Ilan University [Israël], ANR-18-CE40-0035,REPKA,Noyaux reproduisants en Analyse et au-delà(2018), and European Project: 647133,H2020,ERC-2014-CoG,IChaos(2016)
- Subjects
[PHYS]Physics [physics] ,Pure mathematics ,Mathematics::Dynamical Systems ,Property (philosophy) ,Markov chain ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Dynamical Systems (math.DS) ,Translation (geometry) ,37Axx, 37Dxx ,01 natural sciences ,010101 applied mathematics ,Genus (mathematics) ,FOS: Mathematics ,0101 mathematics ,Abelian group ,Mixing (physics) ,Mathematics - Abstract
The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera., Corrects the formulation of Proposition 5.4 in the published version (reversed order of quantifiers); the results and proofs are unchanged, with a minor rewording
- Published
- 2021
39. New Design of Composite Structures Used in Automotive Engineering
- Author
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Vasile Gheorghe, Mihai Ulea, Eliza Chircan, Virgil Barbu Ungureanu, and Maria Luminita Scutaru
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Physics and Astronomy (miscellaneous) ,Computer science ,General Mathematics ,Composite number ,composite materials ,Automotive industry ,02 engineering and technology ,01 natural sciences ,Automotive engineering ,Computer Science (miscellaneous) ,vehicles ,metal materials ,plastics ,impact ,0101 mathematics ,Car door ,business.industry ,lcsh:Mathematics ,021001 nanoscience & nanotechnology ,lcsh:QA1-939 ,Finite element method ,010101 applied mathematics ,Chemistry (miscellaneous) ,0210 nano-technology ,business - Abstract
The paper proposes composite materials for the manufacturing of parts of the car body structure, namely a door. This work aims to analyze the possibility of replacing the metal door of a vehicle with a door made of composite materials. Specific issues related to this replacement are analyzed in the paper. Test specimens were made of composite materials of different sizes, using several types of constituents to determine which material might be most suitable to replace metal in the manufacturing of the door. The choice of materials for the car door was made starting from the characteristics of the analyzed composite materials, but also taking into account the manufacturing possibilities and other engineering limitations. The behavior of the automotive structure as analyzed, using the finite element method for determining the stresses in the structure. Experimental verifications were performed on an experimental stand which has been specially designed for this purpose, to validate the proposed model.
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- 2021
40. On the Reciprocal Sums of Products of Balancing and Lucas-Balancing Numbers
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Younseok Choo
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Fibonacci number ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,reciprocal ,Fibonacci numbers ,floor function ,lcsh:QA1-939 ,01 natural sciences ,Lucas-balancing numbers ,010101 applied mathematics ,Combinatorics ,Computer Science (miscellaneous) ,balancing numbers ,0101 mathematics ,Engineering (miscellaneous) ,Reciprocal ,Mathematics - Abstract
Recently Panda et al. obtained some identities for the reciprocal sums of balancing and Lucas-balancing numbers. In this paper, we derive general identities related to reciprocal sums of products of two balancing numbers, products of two Lucas-balancing numbers and products of balancing and Lucas-balancing numbers. The method of this paper can also be applied to even-indexed and odd-indexed Fibonacci, Lucas, Pell and Pell–Lucas numbers.
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- 2021
41. Refinements of Hermite–Hadamard Inequalities for Continuous Convex Functions via (p,q)-Calculus
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Julalak Prabseang, Sotiris Ntouyas, Jessada Tariboon, and Kamsing Nonlaopon
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Pure mathematics ,convex functions ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,MathematicsofComputing_GENERAL ,(p,q)-derivative ,01 natural sciences ,Computer Science::Digital Libraries ,Hadamard transform ,Hermite–Hadamard inequality ,Computer Science (miscellaneous) ,medicine ,0101 mathematics ,Engineering (miscellaneous) ,Calculus (medicine) ,Mathematics ,Hermite polynomials ,Multiple integral ,lcsh:Mathematics ,010102 general mathematics ,(p,q)-integral ,medicine.disease ,lcsh:QA1-939 ,010101 applied mathematics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Computer Science::Programming Languages ,Convex function - Abstract
In this paper, we present some new refinements of Hermite–Hadamard inequalities for continuous convex functions by using (p,q)-calculus. Moreover, we study some new (p,q)-Hermite–Hadamard inequalities for multiple integrals. Many results given in this paper provide extensions of others given in previous research.
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- 2021
42. A Moreau-Yosida regularization for Markov decision processes
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R. Israel Ortega-Gutiérrez and Hugo Cruz-Suárez
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Mathematical optimization ,Markov chain ,Computer science ,General Mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Regularization (mathematics) ,Convexity ,Complement (complexity) ,Discounted Markov decision processes ,010101 applied mathematics ,Bellman equation ,Markov decision process ,Differentiable function ,0101 mathematics ,Uniqueness of optimal policies, Moreau-Yosida regularization - Abstract
This paper addresses a class of sequential optimization problems known as Markov decision processes. These kinds of processes are considered on Euclidean state and action spaces with the total expected discounted cost as the objective function. The main goal of the paper is to provide conditions to guarantee an adequate Moreau-Yosida regularization for Markov decision processes (named the original process). In this way, a new Markov decision process that conforms to the Markov control model of the original process except for the cost function induced via the Moreau-Yosida regularization is established. Compared to the original process, this new discounted Markov decision process has richer properties, such as the differentiability of its optimal value function, strictly convexity of the value function, uniqueness of optimal policy, and the optimal value function and the optimal policy of both processes, are the same. To complement the theory presented, an example is provided.
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- 2021
43. Dual exponential polynomials and a problem of Ozawa
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Zhi-Tao Wen, Katsuya Ishizaki, Kazuya Tohge, and Janne Heittokangas
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Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Duality (optimization) ,01 natural sciences ,Exponential polynomial ,Dual (category theory) ,010101 applied mathematics ,Linear differential equation ,FOS: Mathematics ,30D15, 30D35 ,Order (group theory) ,Applied mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics - Abstract
Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems., 17 pages
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- 2021
44. On Some New Contractive Conditions in Complete Metric Spaces
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Mirjana Pavlovic, Eugen Ljajko, Jelena Vujaković, and Stojan Radenović
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Pure mathematics ,Current (mathematics) ,General Mathematics ,Fixed point ,01 natural sciences ,α-admissible mappings ,Computer Science (miscellaneous) ,F contraction ,0101 mathematics ,triangularly α-admissible mappings ,F-contraction ,fixed point ,contractive condition ,Engineering (miscellaneous) ,Complement (set theory) ,Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Function (mathematics) ,lcsh:QA1-939 ,010101 applied mathematics ,Nonlinear system ,Metric space ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Line (geometry) - Abstract
One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.
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- 2021
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45. Some New Observations on Generalized Contractive Mappings and Related Results in b-Metric-Like Spaces
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Ljiljana Paunović, Manuel De la Sen, Dušan Rakić, Tatjana Došenović, and Stojan Radenović
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Sequence ,Pure mathematics ,Article Subject ,General Mathematics ,010102 general mathematics ,Cauchy distribution ,Context (language use) ,Fixed point ,Mathematical proof ,01 natural sciences ,010101 applied mathematics ,Metric (mathematics) ,QA1-939 ,0101 mathematics ,Contraction (operator theory) ,Mathematics ,Complement (set theory) - Abstract
In this paper, we consider, discuss, complement, improve, generalize, and enrich some fixed point results obtained forβ−ψ1−ψ2−contractive conditions in ordered b-metric-like spaces. By using our new approach for the proof that one Picard’s sequence isbbl−Cauchy in the context of b-metric-like spaces, we get much shorter proofs than the ones mentioned in the recent papers. Also, by the use of our method, we complement and enrich some common fixed point results forβq,ϕs,ψ−contraction mappings. Our approach in this paper generalizes and modifies several comparable results in the existing literature.
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- 2021
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46. Perturbations of the scattering resonances of an open cavity by small particles: Part II—the transverse electric polarization case
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Habib Ammari, Alexander Dabrowski, Brian Fitzpatrick, and Pierre Millien
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Pole-pencil decomposition ,Applied Mathematics ,General Mathematics ,Shift of scattering resonances ,Subwavelength resonant nanoparticles ,General Physics and Astronomy ,Open dielectric resonator ,01 natural sciences ,010101 applied mathematics ,0103 physical sciences ,Splitting of scattering resonances ,Exceptional scattering resonances ,0101 mathematics ,010306 general physics - Abstract
This paper is concerned with the scattering resonances of open cavities. It is a follow-up of Ammari et al. (ZAMP 71:102, 2020), where the transverse magnetic polarization was assumed. In that case, using the method of matched asymptotic expansions, the leading-order term in the shifts of scattering resonances due to the presence of small particles of arbitrary shapes was derived and the effect of radiation on the perturbations of open cavity modes was characterized. The derivations were formal. In this paper, we consider the transverse electric polarization and prove a small-volume formula for the shifts in the scattering resonances of a radiating dielectric cavity perturbed by small particles. We show a strong enhancement in the frequency shift in the case of subwavelength particles with dipole resonances. We also consider exceptional scattering resonances and perform small-volume asymptotic analysis near them. A significant observation is the large-amplitude splitting of exceptional scattering resonances induced by small particles. Our method in this paper relies on pole-pencil decompositions of volume integral operators., Zeitschrift für angewandte Mathematik und Physik, 72 (2), ISSN:1420-9039, ISSN:0044-2275
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- 2021
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47. Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions
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Yan Debao
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Class (set theory) ,Article Subject ,General Mathematics ,010102 general mathematics ,General Engineering ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Boundary values ,010101 applied mathematics ,Nonlinear fractional differential equations ,QA1-939 ,Applied mathematics ,Uniqueness ,0101 mathematics ,TA1-2040 ,Mathematics - Abstract
This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D 0 + β υ τ + f τ , υ τ = 0 , τ ∈ 0,1 , υ 0 = υ ′ 0 = υ ″ 0 = υ ″ 1 = 0 , where 3 < β ≤ 4 , D 0 + β is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function f τ , υ τ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.
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- 2021
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48. A New Iterative Construction for Approximating Solutions of a Split Common Fixed Point Problem
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Rudong Chen, Huimin He, and Qinwei Fan
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Mathematical optimization ,Sequence ,021103 operations research ,Article Subject ,General Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Construct (python library) ,01 natural sciences ,010101 applied mathematics ,Norm (mathematics) ,Convergence (routing) ,QA1-939 ,Common fixed point ,0101 mathematics ,Mathematics - Abstract
In this paper, we aim to construct a new strong convergence algorithm for a split common fixed point problem involving the demicontractive operators. It is proved that the vector sequence generated via the Halpern-like algorithm converges to a solution of the split common fixed point problem in norm. The main convergence results presented in this paper extend and improve some corresponding results announced recently. The highlights of this paper shed on the novel algorithm and the new analysis techniques.
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- 2021
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49. An Integrated Genetic Algorithm and Homotopy Analysis Method to Solve Nonlinear Equation Systems
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Hala A. Omar
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Article Subject ,Differential equation ,General Mathematics ,Homotopy ,General Engineering ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Square (algebra) ,010101 applied mathematics ,Linear map ,Nonlinear system ,Genetic algorithm ,Convergence (routing) ,QA1-939 ,Applied mathematics ,0101 mathematics ,TA1-2040 ,Homotopy analysis method ,Mathematics - Abstract
Solving nonlinear equation systems for engineering applications is one of the broadest and most essential numerical studies. Several methods and combinations were developed to solve such problems by either finding their roots mathematically or formalizing such problems as an optimization task to obtain the optimal solution using a predetermined objective function. This paper proposes a new algorithm for solving square and nonsquare nonlinear systems combining the genetic algorithm (GA) and the homotopy analysis method (HAM). First, the GA is applied to find out the solution. If it is realized, the algorithm is terminated at this stage as the target solution is determined. Otherwise, the HAM is initiated based on the GA stage’s computed initial guess and linear operator. Moreover, the GA is utilized to calculate the optimum value of the convergence control parameter (h) algebraically without plotting the h-curves or identifying the valid region. Four test functions are examined in this paper to verify the proposed algorithm’s accuracy and efficiency. The results are compared to the Newton HAM (NHAM) and Newton homotopy differential equation (NHDE). The results corroborated the superiority of the proposed algorithm in solving nonlinear equation systems efficiently.
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- 2021
50. Imaging in Random Media by Two-Point Coherent Interferometry
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Liliana Borcea, Josselin Garnier, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Analyse d’interactions stochastiques intelligentes et coopératives (ASCII), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and ANR-19-CE46-0007,ICCI,Caracterisation des ondes incoherentes pour l'imagerie par correlations croisees(2019)
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Synthetic aperture radar ,General Mathematics ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,02 engineering and technology ,01 natural sciences ,Mathematics - Analysis of PDEs ,Optics ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,0202 electrical engineering, electronic engineering, information engineering ,FOS: Electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Point (geometry) ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Physics ,business.industry ,Scattering ,Applied Mathematics ,35Q93, 45Q05, 49R99, 60H30 ,Image and Video Processing (eess.IV) ,Random media ,Function (mathematics) ,Electrical Engineering and Systems Science - Image and Video Processing ,Reflectivity ,010101 applied mathematics ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Interferometry ,Computer Science::Computer Vision and Pattern Recognition ,020201 artificial intelligence & image processing ,Phase retrieval ,business ,Analysis of PDEs (math.AP) - Abstract
This paper considers wave-based imaging through a heterogeneous (random) scattering medium. The goal is to estimate the support of the reflectivity function of a remote scene from measurements of the backscattered wave field. The proposed imaging methodology is based on the coherent interferometric (CINT) approach that exploits the local empirical cross correlations of the measurements of the wave field. The standard CINT images are known to be robust (statistically stable) with respect to the random medium, but the stability comes at the expense of a loss of resolution. This paper shows that a two-point CINT function contains the information needed to obtain statistically stable and high-resolution images. Different methods to build such images are presented, theoretically analyzed and compared with the standard imaging approaches using numerical simulations. The first method involves a phase-retrieval step to extract the reflectivity function from the modulus of its Fourier transform. The second method involves the evaluation of the leading eigenvector of the two-point CINT imaging function seen as the kernel of a linear operator. The third method uses an optimization step to extract the reflectivity function from some cross products of its Fourier transform. The presentation is for the synthetic aperture radar data acquisition setup, where a moving sensor probes the scene with signals emitted periodically and records the resulting backscattered wave. The generalization to other imaging setups, with passive or active arrays of sensors, is discussed briefly., Comment: 31 pages, 5 figures
- Published
- 2021
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