2,237 results
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2. Corrigendum to the paper 'Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings'
- Author
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Moosa Gabeleh
- Subjects
47h09 ,Pure mathematics ,uniformly convex banach space ,46b20 ,General Mathematics ,QA1-939 ,best proximity (point) pair ,Equivalence (measure theory) ,Mathematics ,noncyclic (cyclic) contraction - Abstract
The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings,” Demonstr. Math. 53 (2020), 38–43.
- Published
- 2021
3. A report of serious and multiple ethical misconducts.
- Author
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Mortini, Raymond, Sal Moslehian, Mohammad, Révész, Szilárd Gy., and Tomilov, Yuri
- Subjects
MATHEMATICAL periodicals ,PLAGIARISM ,MATHEMATICS ,MATHEMATICAL research ,SCIENCE periodicals ,ELECTRONIC systems - Abstract
The article informs about case study of a serious and multiple ethical misconduct after an author submitted a paper to five journals for publication and committed some form of plagiarism. Topics include mathematics and in sciences in general, multiple submissions of the same results or parallel papers are prohibited as serious ethical misconduct; and mathematics simultaneous or concurrent submission of a manuscript describing the same research publication in the electronic online systems.
- Published
- 2021
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4. The Quasi-Empirical Epistemology of Mathematics.
- Author
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Shi, Ellen Yunjie
- Subjects
PHILOSOPHY of mathematics ,THEORY of knowledge ,MATHEMATICS - Abstract
This paper clarifies and discusses Imre Lakatos' claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that (1) Lakatos' motivation for classifying mathematics as a quasi-empirical theory is epistemological; (2) what can be called the quasi-empirical epistemology of mathematics is not correct; (3) analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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5. Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
- Author
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Cingolani Silvia, Gallo Marco, and Tanaka Kazunaga
- Subjects
fractional laplacian ,nonlinear choquard pekar equation ,double nonlocality ,normalized solutions ,infinitely many solutions ,pohozaev identity ,35a01 ,35a15 ,35b06 ,35b38 ,35d30 ,35j15 ,35j20 ,35j61 ,35q40 ,35q55 ,35q60 ,35q70 ,35q75 ,35q85 ,35q92 ,35r09 ,35r11 ,45k05 ,47g10 ,47j30 ,49j35 ,58e05 ,58j05 ,Mathematics ,QA1-939 - Abstract
In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.
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- 2024
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6. Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture
- Author
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Guo Yahong, Ma Lingwei, and Zhang Zhenqiu
- Subjects
gibbons conjecture ,sliding method ,dual fractional nonlinear parabolic equations ,one-dimensional symmetry ,monotonicity ,35r11 ,35b50 ,35b06 ,35a01 ,35k58 ,26a33 ,Mathematics ,QA1-939 - Abstract
In this paper, we consider the general dual fractional parabolic problem ∂tαu(x,t)+Lu(x,t)=f(t,u(x,t))inRn×R. ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂tα ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.
- Published
- 2024
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7. On the functional ∫Ωf + ∫Ω*g
- Author
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Guang Qiang, Li Qi-Rui, and Wang Xu-Jia
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minkowski type problem ,geometric flow ,variational method ,primary 35j20, 35k96 ,secondary 53a07 ,Mathematics ,QA1-939 - Abstract
In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form J(Ω,Ω*)=∫Ωf+∫Ω*g $\mathcal{J}\left({\Omega},{{\Omega}}^{{\ast}}\right)={\int }_{{\Omega}}f+{\int }_{{{\Omega}}^{{\ast}}}g$ , where f, g are given nonnegative functions in a manifold. The duality is a relation α(x, y) ≤ 0 ∀ x ∈ Ω, y ∈ Ω*, for a suitable function α. This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.
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- 2024
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8. A Rheological Model for Cupuassu (Theobroma grandiflorum) Pulp at Different Concentrations and Temperatures
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Arianne Dantas Viana, Luciano Brito Rodrigues, Jadir Noqueira da Silva, Fátima Baptistia, Modesto Antonio Chaves, and Chen, Xiao Dong
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Polynomial regression ,biology ,Pulp (paper) ,Rheometer ,Thermodynamics ,Regression analysis ,engineering.material ,biology.organism_classification ,shear stress ,Shear rate ,Rheology ,viscosity ,Shear stress ,engineering ,Theobroma grandiflorum ,Engineering (miscellaneous) ,Food Science ,Biotechnology ,Mathematics - Abstract
This work was made aiming at studying the best model for the rheological properties of Cupuassu (Theobroma grandiflorum, Schum) pulps with 14 (in nature), 17, 19, 23 and 25°Brix of total soluble solids (TSS) which were measured at 20, 30, 40, 50 and 60°C temperature using a concentric cylinder rheometer. The results were adjusted to the following nine models: Ostwald-de-Waele (power law), Bingham, Casson, Generalized Casson, Heinz–Casson, Herschel–Bulkley, Mizrahi–Berk, Schulmann–Haroske–Reher and Windhab. The parameters of the best model were correlated with pulp temperature and TSS by polynomial regression analysis and were kept in the regression equation only those parameters that contributed more than 1% to the variation of the independent variable. The results indicate that the rheological behavior of Cupuassu pulp in different concentrations and temperatures can be modeled by the Windhab model, although other models can be used in a narrower band of shear stress.
- Published
- 2013
9. MULTI-SORTED LOGIC AND LOGICAL GEOMETRY: SOME PROBLEMS.
- Author
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Plotkin, B. and Plotkin, E.
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ALGEBRAIC geometry ,MATHEMATICS ,CATEGORIES (Mathematics) - Abstract
The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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10. On I. Meghea and C. S. Stamin review article 'Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,' Demonstratio Mathematica 2022; 55: 354–379
- Author
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Göpfert Alfred, Tammer Christiane, and Zălinescu Constantin
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minimal point ,ekeland variational principle ,49j27 ,49j40 ,Mathematics ,QA1-939 - Abstract
Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”
- Published
- 2023
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11. Lakatos' Quasi-Empiricism Revisited.
- Author
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Zeng, Wei
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PHILOSOPHY of mathematics ,HISTORY of mathematics ,EMPIRICISM ,MATHEMATICS ,LOGICAL prediction - Abstract
The central idea of Lakatos' quasi-empiricism view of the philosophy of mathematics is that truth values are transmitted bottom-up, but only falsity can be transmitted from basic statements. As it is falsity but not truth that flows bottomup, Lakatos emphasizes that observation and induction play no role in both conjecturing and proving phases in mathematics. In this paper, I argue that Lakatos' view that one cannot obtain primitive conjectures by induction contradicts the history of mathematics, and therefore undermines his quasi-empiricism theory. I argue that his misconception of induction causes this view of Lakatos. Finally, I propose that Wittgenstein's view that "mathematics does have a grammatical nature, but it is also rooted in empirical regularities" suggests the possibility to improve Lakatos' view by maintaining his quasi-empiricism while accepting the role induction plays in the conjecturing phase. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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12. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
- Author
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López-Gómez Julián, Muñoz-Hernández Eduardo, and Zanolin Fabio
- Subjects
periodic predator-prey volterra model ,subharmonic coexistence states ,global structure ,minimal complexity ,chaotic dynamics ,34c25 ,37b55 ,37e40 ,37j12 ,Mathematics ,QA1-939 - Abstract
This article deals with the existence, multiplicity, minimal complexity, and global structure of the subharmonic solutions to a class of planar Hamiltonian systems with periodic coefficients, being the classical predator-prey model of V. Volterra its most paradigmatic example. By means of a topological approach based on techniques from global bifurcation theory, the first part of the paper ascertains their nature, multiplicity and minimal complexity, as well as their global minimal structure, in terms of the configuration of the function coefficients in the setting of the model. The second part of the paper introduces a dynamical system approach based on the theory of topological horseshoes that permits to detect, besides subharmonic solutions, “chaotic-type” solutions. As a byproduct of our analysis, the simplest predator-prey prototype models in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle.
- Published
- 2023
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13. Fixed point theorem for a sequence of multivalued nonself mappings in metrically convex metric spaces.
- Author
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Aron, David and Kumar, Santosh
- Subjects
FIXED point theory ,SET-valued maps ,METRIC spaces ,MATHEMATICS ,MATHEMATICAL programming - Abstract
In this paper, a common fixed point theorem is demonstrated for a sequence of multivalued mappings which satisfy certain requirements in complete metric spaces. The results proved here will generalize and extend the results due to Ćirić [1]. Suitable examples are given at the end to support the results proved herein. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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14. Convergence analysis for split hierachical monotone variational inclusion problem in Hilbert spaces
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Abass H.A., Jolaoso L. O., and Mewomo O. T.
- Subjects
split hierachical monotone variational inclusion problem ,fixed point problem ,strictly pseudocontractive mapping ,nonexpansive mapping ,hilbert spaces ,47h06 ,47h09 ,47j05 ,47j25 ,Mathematics ,QA1-939 - Abstract
In this paper, we introduce a new iterative algorithm for approximating a common solution of Split Hierarchical Monotone Variational Inclusion Problem (SHMVIP) and Fixed Point Problem (FPP) of k-strictly pseudocontractive mappings in real Hilbert spaces. Our proposed method converges strongly, does not require the estimation of operator norm and it is without imposing the strict condition of compactness; these make our method to be potentially more applicable than most existing methods in the literature. Under standard and mild assumption of monotonicity of the SHMVIP associated mappings, we establish the strong convergence of the iterative algorithm.We present some applications of our main result to approximate the solution of Split Hierarchical Convex Minimization Problem (SHCMP) and Split Hierarchical Variational Inequality Problem (SHVIP). Some numerical experiments are presented to illustrate the performance and behavior of our method. The result presented in this paper extends and complements some related results in literature.
- Published
- 2022
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15. Wrong Sigma metric causes chaos.
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Coskun, Abdurrahman
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PATHOLOGICAL laboratories ,CLINICAL pathology ,AUTOANALYZERS ,LEUCOCYTES ,MATHEMATICS ,BLOOD cell count ,DIAGNOSTIC errors ,MEDICAL practice - Published
- 2022
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16. Erinnerungen an Eberhard Kirchberg.
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Cuntz, Joachim and Rørdam, Mikael
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C*-algebras ,MATHEMATICIANS ,MATHEMATICS ,UNEMPLOYED people ,CLASSIFICATION - Abstract
Copyright of Mitteilungen der DMV is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2023
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17. Asymptotic stability of solutions for a diffusive epidemic model
- Author
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Bouaziz Khelifa, Douaifia Redouane, and Abdelmalek Salem
- Subjects
equilibrium points ,reaction-diffusion ,reproductive number ,local stability ,global stability ,global existence ,35k45 ,35k57 ,Mathematics ,QA1-939 - Abstract
The aim of this paper is to study the existence and the asymptotic stability of solutions for an epidemiologically emerging reaction-diffusion model. We show that the model has two types of equilibrium points to resolve the proposed system for a fairly broad class of nonlinearity that describes the transmission of an infectious disease between individuals. The model is analyzed by using the basic reproductive number R0{R}_{0}. Finally, we present the numerical examples simulations that clarifies and confirms the results of the study throughout the paper.
- Published
- 2022
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18. Optimal Polynomial Approximants in Lp
- Author
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Centner Raymond
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optimal polynomial approximant ,lpspace ,hardy space ,digital filter ,shanks’ conjecture ,primary 30e10 ,secondary 46e30 ,Mathematics ,QA1-939 - Abstract
Over the past several years, optimal polynomial approximants (OPAs) have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this paper, we introduce the notion of optimal polynomial approximant in the space Lp, 1 ≤ p ≤ ∞. We begin our treatment by showing existence and uniqueness for 1 < p < ∞. For the extreme cases of p = 1 and p = ∞, we show that uniqueness does not necessarily hold. We continue our development by elaborating on the special case of L2. Here, we create a test to determine whether or not a given 1st degree OPA is zero-free in ̄𝔻. Afterward, we shed light on an orthogonality condition in Lp. This allows us to study OPAs in Lp with the additional tools from the L2 setting. Throughout this paper, we focus many of our discussions on the zeros of OPAs. In particular, we show that if 1 < p < ∞, f ∈ Hp, and f(0) ≠ 0, then there exists a disk, centered at the origin, in which all the associated OPAs are zero-free. Toward the end of this paper, we use the orthogonality condition to compute the coefficients of some OPAs in Lp. To inspire further research in the general theory, we pose several open questions throughout our discussions.
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- 2022
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19. Equity returns and sentiment
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Huang Zibin and Ibragimov Rustam
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sentiment ,asset prices ,asset returns ,dependence ,granger causality ,predictive regressions ,autoregressive distributed lag models ,garch models ,volatility ,62p20 ,91b84 ,Science (General) ,Q1-390 ,Mathematics ,QA1-939 - Abstract
This paper analyzes approximately 100 Gigabytes of raw text data from Twitter with keywords “AAPL,” “S&P 500,” “FTSE100” and “NASDAQ” to explore the relationship between sentiment and the returns and prices on the Apple stock and the S&P 500, FTSE 100 and NASDAQ indices. The findings point to significant relationship and dependence between sentiment measures and the S&P 500 and FTSE 100 indices’ returns and prices. The econometric analysis of dependence between the aforementioned variables in the paper is presented in some detail for illustration of the methodology employed.
- Published
- 2022
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20. Predictability of cryptocurrency returns: evidence from robust tests
- Author
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He Siyun and Ibragimov Rustam
- Subjects
bitcoin ,cryptocurrencies ,predictive regressions ,robust inference ,hac ,t-statistic inference ,62p20 ,91b84 ,Science (General) ,Q1-390 ,Mathematics ,QA1-939 - Abstract
The paper provides a comparative empirical study of predictability of cryptocurrency returns and prices using econometrically justified robust inference methods. We present robust econometric analysis of predictive regressions incorporating factors, which were suggested by Liu, Y., & Tsyvinski, A. (2018). Risks and returns of cryptocurrency. NBER working paper no. 24877; Liu, Y., & Tsyvinski, A. (2021). Risks and returns of cryptocurrency. The Review of Financial Studies, 34(6), 2689–2727, as useful predictors for cryptocurrency returns, including cryptocurrency momentum, stock market factors, acceptance of Bitcoin, and Google trends measure of investors’ attention. Due to inherent heterogeneity and dependence properties of returns and other time series in financial and crypto markets, we provide the analysis of the predictive regressions using both heteroskedasticity and autocorrelation consistent (HAC) standard-errors and also the recently developed tt-statistic robust inference approaches, Ibragimov, R., & Müller, U. K. (2010). t-statistic based correlation and heterogeneity robust inference. Journal of Business and Economic Statistics, 28, 453–468; Ibragimov, R., & Müller, U. K. (2016). Inference with few heterogeneous clusters. Review of Economics and Statistics, 98, 83–96. We provide comparisons of robust predictive regression estimates between different cryptocurrencies and their corresponding risk and factor exposures. In general, the number of significant factors decreases as we use more robust t-tests, and the t-statistic robust inference approaches appear to perform better than the t-tests based on HAC standard errors in terms of pointing out interpretable economic conclusions. The results in this paper emphasize the importance of the use of robust inference approaches in the analysis of economic and financial data affected by the problems of heterogeneity and dependence.
- Published
- 2022
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21. Dynamical behaviors of a k-order fuzzy difference equation
- Author
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Han Caihong, Li Lue, Su Guangwang, and Sun Taixiang
- Subjects
fuzzy difference equation ,solution ,boundedness ,convergence ,39a10 ,65k10 ,Mathematics ,QA1-939 - Abstract
Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: xn+1=xnA+Bxn−k(n=0,1,2,…),{x}_{n+1}=\frac{{x}_{n}}{A+B{x}_{n-k}}\hspace{0.33em}\left(n=0,1,2,\ldots ), where parameters A,BA,B and initial value x−k,x−k+1,…,x−1,x0{x}_{-k},{x}_{-k+1},\ldots ,{x}_{-1},{x}_{0}, k∈{0,1,…}k\in \{0,1,\ldots \} are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.
- Published
- 2022
- Full Text
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22. Laplacian spectrum of comaximal graph of the ring ℤn
- Author
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Banerjee Subarsha
- Subjects
comaximal graph ,laplacian eigenvalues ,vertex connectivity ,algebraic connectivity ,laplacian spectral radius ,finite ring ,05c25 ,05c50 ,Mathematics ,QA1-939 - Abstract
In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) of the ring Zn{{\mathbb{Z}}}_{n} for n>2n\gt 2. We first determine the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{\alpha }{q}^{\beta }, where p,qp,q are primes and α,β\alpha ,\beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for n=pαqβn={p}^{\alpha }{q}^{\beta }. The algebraic and vertex connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.
- Published
- 2022
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23. Transcendental operators acting on slice regular functions
- Author
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de Fabritiis Chiara
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slice-regular functions ,-product of slice-regular functions ,-exponential ,sine and cosine ,-logarithm ,primary 47j05 ,secondary: 30g35, 39b52 ,Mathematics ,QA1-939 - Abstract
The aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely *-exponential, *-sine and *-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of *-exponential were investigated in a previous paper by Altavilla and the author. We show how exp*(f ), sin*(f ), cos*(f ), sinh*(f ) and cosh*(f ) can be written in terms of the real and the vector part of the function f and we examine the relation between cos* and cosh* when the domain Ω is product and when it is slice. In particular we prove that when Ω is slice, then cos*(f ) = cosh*(f * I) holds if and only if f is ℂI preserving, while in the case Ω is product there is a much larger family of slice regular functions for which the above relation holds.
- Published
- 2022
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24. Poly-falling factorial sequences and poly-rising factorial sequences
- Author
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Kim Hye Kyung
- Subjects
falling factorials ,rising factorials ,modified polyexponential functions ,stirling numbers of the first and second kind ,poly-bernoulli polynomials ,poly-genocchi polynomials ,11b73 ,11b83 ,05a19 ,Mathematics ,QA1-939 - Abstract
In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k=1k=1, necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k=1k=1. We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of xx; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.
- Published
- 2021
- Full Text
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25. Some new results on the weaving of K-g-frames in Hilbert spaces
- Author
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Xiang Zhong-Qi
- Subjects
k-g-frame ,k-woven ,perturbation ,pseudo-inverse ,42c15 ,47b40 ,Mathematics ,QA1-939 - Abstract
In this paper, we provide some conditions for a K-woven pair of K-g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K-woven pair of K-g-frames can leave them K-woven. Several new methods on the construction of K-woven pair of K-g-frames are also given. We end the paper with a new perturbation result on the weaving of K-g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K-g-frames can be K-woven in the whole space, not merely in the subspace Range(K).
- Published
- 2021
- Full Text
- View/download PDF
26. Generalized split null point of sum of monotone operators in Hilbert spaces
- Author
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Mebawondu Akindele A., Abass Hammed A., Oyewole Olalwale K., Aremu Kazeem O., and Narain Ojen K.
- Subjects
generalized split monotone variational inclusion ,inertial iterative scheme ,firmly nonexpansive ,fixed point problem ,47h06 ,47h09 ,47j05 ,47j25 ,Mathematics ,QA1-939 - Abstract
In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.
- Published
- 2021
- Full Text
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27. so-metrizable spaces and images of metric spaces
- Author
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Yang Songlin and Ge Xun
- Subjects
so-network ,so-metrizable space ,so-open mapping ,compact-covering mapping ,σ-mapping ,54e35 ,54e40 ,54e45 ,54e50 ,Mathematics ,QA1-939 - Abstract
so-metrizable spaces are a class of important generalized metric spaces between metric spaces and snsn-metrizable spaces where a space is called an so-metrizable space if it has a σ\sigma -locally finite so-network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so-metrizable spaces by images of metric spaces. This paper gives such characterizations for so-metrizable spaces. More precisely, this paper introduces so-open mappings and uses the “Pomomarev’s method” to prove that a space XX is an so-metrizable space if and only if it is an so-open, compact-covering, σ\sigma -image of a metric space, if and only if it is an so-open, σ\sigma -image of a metric space. In addition, it is shown that so-open mapping is a simplified form of snsn-open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so-metrizable spaces and establish some equivalent relations among so-open mapping, snsn-open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.
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- 2021
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28. Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument
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Li Jungang, Lu Guozhen, and Zhu Maochun
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trudinger–moser inequality ,heisenberg group ,concentration-compactness principles ,riemannian manifolds ,exponential growth ,46e35 ,53c21 ,42b35 ,Mathematics ,QA1-939 - Abstract
The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.
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- 2021
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29. Gesamtheft 32(2).
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ARTIFICIAL intelligence ,GEOMETRIC connections ,SOCIAL learning ,BLENDED learning ,CONCEPT learning ,MATHEMATICS - Abstract
Copyright of Mitteilungen der DMV is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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- 2024
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30. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
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Rasham Tahair, Mustafa Arjumand, Mukheimer Aiman, Nazam Muhammad, and Shatanawi Wasfi
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fixed point ,orbitally b-metric space ,new extensions of nashine, wardowski, feng-liu and ćirić-type contraction ,two families of set-valued dominated mappings ,integral equations ,fractional differential equations ,47h10 ,47h04 ,45p05 ,Mathematics ,QA1-939 - Abstract
In this manuscript, we prove new extensions of Nashine, Wardowski, Feng-Liu, and Ćirić-type contractive inequalities using orbitally lower semi-continuous functions in an orbitally complete bb-metric space. We accomplish new multivalued common fixed point results for two families of dominated set-valued mappings in an ordered complete orbitally bb-metric space. Some new definitions and illustrative examples are given to validate our new results. To show the novelty of our results, applications are given to obtain the solution of nonlinear integral and fractional differential equations. Our results expand the hypothetical consequences of Nashine et al. (Feng–Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation, Nonlinear Anal. Model. Control. 26 (2021), no. 3, 522–533) and Rasham et al. (Common fixed point results for new Ciric-type rational multivalued-contraction with an application, J. Fixed Point Theory Appl. 20 (2018), no. 1, Paper No. 45).
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- 2024
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31. Some new fixed point theorems of α-partially nonexpansive mappings
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Shukla Rahul
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nonexpansive mapping ,condition (e) ,uniformly convex space ,47h10 ,54h25 ,47h09 ,Mathematics ,QA1-939 - Abstract
In this paper, we introduce a new class of nonlinear mappings and compare it to other classes of nonlinear mappings that have appeared in the literature. We establish various existence and convergence theorems for this class of mappings under different assumptions in Banach spaces, particularly Banach spaces with a normal structure. In addition, we provide examples to substantiate the findings presented in this study. We prove the existence of a common fixed point for a family of commuting α\alpha -partially nonexpansive self-mappings. Furthermore, we extend the results reported by Suzuki (Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), no. 2, 1088–1095), Llorens-Fuster (Partially nonexpansive mappings, Adv. Theory Nonlinear Anal. Appl. 6 (2022), no. 4, 565–573), and Dhompongsa et al. (Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350 (2009), no. 1, 12–17). Finally, we present an open problem concerning the existence of fixed points for α\alpha -partially nonexpansive mappings in the context of uniformly nonsquare Banach spaces.
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- 2024
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32. Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials
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Zhang Caifeng and Zhu Maochun
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trudinger–moser inequalities ,degenerate potential ,nehari manifold ,ground states ,35j10 ,35j91 ,46e35 ,26d10 ,Mathematics ,QA1-939 - Abstract
The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint:(0.1)supu∈H1(R2),∫R2(|∇u|2+V(x)u2)dx≤1∫R2e4π1−β2u2−1|x|βdx
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- 2024
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33. Two solutions for Dirichlet double phase problems with variable exponents
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Amoroso Eleonora, Bonanno Gabriele, D’Aguì Giuseppina, and Winkert Patrick
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bounded solutions ,critical point theory ,double phase operator with variable exponents ,multiplicity results ,superlinear reaction ,35a01 ,35b33 ,35d30 ,35j62 ,35j66 ,Mathematics ,QA1-939 - Abstract
This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.
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- 2024
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34. Generating unfavourable VaR scenarios under Solvency II with patchwork copulas
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Pfeifer Dietmar and Ragulina Olena
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solvency ii ,copulas ,patchwork copulas ,bernstein copulas ,monte carlo methods ,62h05 ,62h12 ,62h17 ,11k45 ,Science (General) ,Q1-390 ,Mathematics ,QA1-939 - Abstract
The central idea of the paper is to present a general simple patchwork construction principle for multivariate copulas that create unfavourable VaR (i.e. Value at Risk) scenarios while maintaining given marginal distributions. This is of particular interest for the construction of Internal Models in the insurance industry under Solvency II in the European Union. Besides this, the Delegated Regulation by the European Commission requires all insurance companies under supervision to consider different risk scenarios in their risk management system for the company’s own risk assessment. Since it is unreasonable to assume that the potential worst case scenario will materialize in the company, we think that a modelling of various unfavourable scenarios as described in this paper is likewise appropriate. Our explicit copula approach can be considered as a special case of ordinal sums, which in two dimensions even leads to the technically worst VaR scenario.
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- 2021
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35. A fixed point theorem involving rational expressions without using Picard iteration
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Fulga Andreea
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picard iteration ,fixed point ,47h10 ,54h25 ,46j10 ,46j15 ,Mathematics ,QA1-939 - Abstract
In this paper, we consider a certain fixed point theorem that contains some rational expressions. The main aim of this paper is to prove a fixed point theorem without using the Picard iteration.
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- 2021
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36. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
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Ali Muhammad Aamir, Alp Necmettin, Budak Hüseyin, Chu Yu-Ming, and Zhang Zhiyue
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hermite-hadamard inequality ,q-integral ,quantum calculus ,convex function ,26d10 ,26d15 ,26a51 ,Mathematics ,QA1-939 - 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.
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- 2021
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37. Explaining predictive models using Shapley values and non-parametric vine copulas
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Aas Kjersti, Nagler Thomas, Jullum Martin, and Løland Anders
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prediction explanation ,shapley values ,conditional distribution ,vine copulas ,non-parametric ,62g05 ,62h05 ,68t01 ,91a12 ,Science (General) ,Q1-390 ,Mathematics ,QA1-939 - Abstract
In this paper the goal is to explain predictions from complex machine learning models. One method that has become very popular during the last few years is Shapley values. The original development of Shapley values for prediction explanation relied on the assumption that the features being described were independent. If the features in reality are dependent this may lead to incorrect explanations. Hence, there have recently been attempts of appropriately modelling/estimating the dependence between the features. Although the previously proposed methods clearly outperform the traditional approach assuming independence, they have their weaknesses. In this paper we propose two new approaches for modelling the dependence between the features. Both approaches are based on vine copulas, which are flexible tools for modelling multivariate non-Gaussian distributions able to characterise a wide range of complex dependencies. The performance of the proposed methods is evaluated on simulated data sets and a real data set. The experiments demonstrate that the vine copula approaches give more accurate approximations to the true Shapley values than their competitors.
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- 2021
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38. Sharp Hardy Identities and Inequalities on Carnot Groups
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Flynn Joshua, Lam Nguyen, and Lu Guozhen
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bessel pair ,hardy inequality ,hardy–sobolev inequality ,weights ,carnot groups ,carnot–carathéodory metric ,42b35 ,46e35 ,35j15 ,22e30 ,43a15 ,Mathematics ,QA1-939 - Abstract
In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.
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- 2021
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39. Schrödinger’s tridiagonal matrix
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Kovačec Alexander
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tridiagonal matrix ,eigenvalues ,partial fraction decomposition ,rational function identities ,orthogonal polynomials ,quantum theory ,history ,15a15 ,15b99 ,47b36 ,Mathematics ,QA1-939 - Abstract
In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.
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- 2021
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40. Isogenies on twisted Hessian curves
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Perez Broon Fouazou Lontouo, Dang Thinh, Fouotsa Emmanuel, and Moody Dustin
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elliptic curves ,isogeny ,hessian curves ,vélu's formulas ,14h52 ,14k02 ,Mathematics ,QA1-939 - Abstract
Elliptic curves are typically defined by Weierstrass equations. Given a kernel, the well-known Vélu's formula shows how to explicitly write down an isogeny between Weierstrass curves. However, it is not clear how to do the same on other forms of elliptic curves without isomorphisms mapping to and from the Weierstrass form. Previous papers have shown some isogeny formulas for (twisted) Edwards, Huff, and Montgomery forms of elliptic curves. Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form. In addition, we examine the numbers of operations in the base field to compute the formulas. In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing an input point in affine coordinates.
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- 2021
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41. Michael-Simon type inequalities in hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows
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Cui Jingshi and Zhao Peibiao
- Subjects
locally constrained curvature flow ,michael-simon type inequality ,kth mean curvatures ,primary 53e99 ,secondary 52a20 ,35k96 ,Mathematics ,QA1-939 - Abstract
In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows(0.1)∫Mλ′f2E12+|∇Mf|2−∫M∇̄fλ′,ν+∫∂Mf≥ωn1n∫Mfnn−1n−1n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in Hn+1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below(0.2)∫Mλ′f2Ek2+|∇Mf|2Ek−12−∫M∇̄fλ′,ν⋅Ek−1+∫∂Mf⋅Ek−1≥pk◦q1−1(W1(Ω))1n−k+1∫Mfn−k+1n−k⋅Ek−1n−kn−k+1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k(r) = ω n(λ′)k−1, W1(Ω)=1n|M| ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q1(r)=W1Srn+1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q1−1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).
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- 2024
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42. Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces
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Taheri Ali and Vahidifar Vahideh
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smooth metric measure spaces ,nonlinear parabolic equations ,witten laplacian ,li-yau estimates ,differential harnack inequalities ,liouville type results ,53c44 ,58j60 ,58j35 ,60j60 ,Mathematics ,QA1-939 - Abstract
In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.
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- 2024
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43. Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system
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Ortegón Gallego Francisco, Rhoudaf Mohamed, and Talbi Hajar
- Subjects
coupled system ,bilateral solution ,nonlinear elliptic equation ,thermistor problem ,penalization techniques ,Mathematics ,QA1-939 - Abstract
In this paper, we analyze the following nonlinear elliptic problem A(u)=ρ(u)|∇φ|2 in Ω,div(ρ(u)∇φ)=0 in Ω,u=0 on ∂Ω,φ=φ0 on ∂Ω. $\begin{cases}A\left(u\right)=\rho \left(u\right)\vert \nabla \varphi {\vert }^{2}\,\text{in}\,{\Omega},\quad \hfill \\ \text{div}\left(\rho \left(u\right)\nabla \varphi \right)=0\,\text{in}\,{\Omega},\quad \hfill \\ u=0\,\text{on}\,\partial {\Omega},\quad \hfill \\ \varphi ={\varphi }_{0}\,\text{on}\,\partial {\Omega}.\quad \hfill \end{cases}$ where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L 1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.
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- 2024
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44. On the large solutions to a class of k-Hessian problems
- Author
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Wan Haitao
- Subjects
k-hessian problem ,the first expansions ,the radial solutions ,global estimates ,35j60 ,35j25 ,35b40 ,35j67 ,Mathematics ,QA1-939 - Abstract
In this paper, we consider the k-Hessian problem S k(D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in RN ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f∈C1(R) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first expansions (equalities) of k-convex solutions to the above problem when f is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of f and principal curvatures of ∂Ω on the first expansion of solutions. For the latter, we find the principal curvatures of ∂Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including k = N). Moreover, we know the existence of k-convex solutions to the above problem (including k = N) is still an open problem when b possesses high singularity on ∂Ω and f satisfies Keller–Osserman type condition. For the radially symmetric case in the ball, we give a positive answer to this open problem, and then we further show the global estimates for all radial large solutions.
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- 2024
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45. New multiplicity results in prescribing Q-curvature on standard spheres
- Author
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Ben Ayed Mohamed and El Mehdi Khalil
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partial differential equations ,analysis on manifolds ,q curvature problem ,35a15 ,58j05 ,58e05 ,Mathematics ,QA1-939 - Abstract
In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K.
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- 2024
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46. Liouville type theorems involving fractional order systems
- Author
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Liao Qiuping, Liu Zhao, and Wang Xinyue
- Subjects
method of moving spheres ,fractional laplacian ,liouville type theorem ,primary: 35r11 ,secondary: 35b53 ,45g15 ,Mathematics ,QA1-939 - Abstract
In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (−Δ)α/2u(x)=f(u(x),v(x)),x∈Rn,(−Δ)α/2v(x)=g(u(x),v(x)),x∈Rn. $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).
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- 2024
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47. The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients
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Ikoma Norihisa and Yamanobe Mizuki
- Subjects
l2–normalized solution ,multiple solutions ,(symmetric) mountain pass theorem ,palais–smale–cerami condition ,concentration-compactness principle ,35j20 ,35j61 ,35b09 ,35b38 ,Mathematics ,QA1-939 - Abstract
The existence of L 2–normalized solutions is studied for the equation −Δu+μu=f(x,u) inRN,∫RNu2dx=m. $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.
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- 2024
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48. Segregated solutions for nonlinear Schrödinger systems with a large number of components
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Chen Haixia and Pistoia Angela
- Subjects
schrödinger systems ,segregated solutions ,large number of components ,35b44 ,35j47 (primary) ,35b33 (secondary) ,Mathematics ,QA1-939 - Abstract
In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.
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- 2024
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49. Moving planes and sliding methods for fractional elliptic and parabolic equations
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Chen Wenxiong, Hu Yeyao, and Ma Lingwei
- Subjects
direct method of moving planes ,sliding method ,fractional laplacian ,nonlocal elliptic and parabolic equations ,qualitative properties ,transition from elliptic to parabolic ,35r11 ,35b50 ,35b06 ,35a01 ,Mathematics ,QA1-939 - Abstract
In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.
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- 2024
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50. Liouville theorems of solutions to mixed order Hénon-Hardy type system with exponential nonlinearity
- Author
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Dai Wei and Peng Shaolong
- Subjects
mixed order hénon-hardy type system ,exponential nonlinearity ,liouville theorems ,method of scaling spheres ,primary: 35m30 ,secondary: 35b53, 35j91 ,Mathematics ,QA1-939 - Abstract
In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R+2 ${\mathbb{R}}_{+}^{2}$ : (−Δ)α2u(x)=|x|aup1(x)eq1v(x),x∈R+2,(−Δ)v(x)=|x|bup2(x)eq2v(x),x∈R+2, $\begin{cases}{\left(-{\Delta}\right)}^{\frac{\alpha }{2}}u\left(x\right)=\vert x{\vert }^{a}{u}^{{p}_{1}}\left(x\right){e}^{{q}_{1}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \\ \left(-{\Delta}\right)v\left(x\right)=\vert x{\vert }^{b}{u}^{{p}_{2}}\left(x\right){e}^{{q}_{2}v\left(x\right)}, x\in {\mathbb{R}}_{+}^{2},\quad \hfill \end{cases}$ with Dirichlet boundary conditions, where 0 0. First, we derived the integral representation formula corresponding to the above system under the assumption p1≥−2aα−1 ${p}_{1}\ge -\frac{2a}{\alpha }-1$ . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres.
- Published
- 2024
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