We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set of periodic solutions is bounded.
{"title":"Existence of periodic solutions for a scalar differential equation modelling optical conveyor belts","authors":"Luis Carretero, José Valero","doi":"arxiv-2407.10843","DOIUrl":"https://doi.org/arxiv-2407.10843","url":null,"abstract":"We study a one-dimensional ordinary differential equation modelling optical\u0000conveyor belts, showing in particular cases of physical interest that periodic\u0000solutions exist. Moreover, under rather general assumptions it is proved that\u0000the set of periodic solutions is bounded.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In his classical paper [5], Koornwinder studied a family of orthogonal polynomials of two variables, derived from symmetric polynomials. This family possesses a rare property that orthogonal polynomials of degree $n$ have $n(n+1)/2$ real common zeros, which leads to important examples in the theory of minimal cubature rules. This paper aims to give an account of the minimal cubature rules of two variables and examples originating from Koornwinder polynomials, and we will also provide further examples.
{"title":"Minimal cubature rules and Koornwinder polynomials","authors":"Yuan Xu","doi":"arxiv-2407.09903","DOIUrl":"https://doi.org/arxiv-2407.09903","url":null,"abstract":"In his classical paper [5], Koornwinder studied a family of orthogonal\u0000polynomials of two variables, derived from symmetric polynomials. This family\u0000possesses a rare property that orthogonal polynomials of degree $n$ have\u0000$n(n+1)/2$ real common zeros, which leads to important examples in the theory\u0000of minimal cubature rules. This paper aims to give an account of the minimal\u0000cubature rules of two variables and examples originating from Koornwinder\u0000polynomials, and we will also provide further examples.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are concerned with solvability of a non-potential system involving two relativistic operators, subject to boundary conditions expressed in terms of maximal monotone operators. The approach makes use of a fixed point formulation and relies on a priori estimates and convergent to zero matrices.
{"title":"Non-potential systems with relativistic operators and maximal monotone boundary conditions","authors":"Petru Jebelean, Calin Serban","doi":"arxiv-2407.09425","DOIUrl":"https://doi.org/arxiv-2407.09425","url":null,"abstract":"We are concerned with solvability of a non-potential system involving two\u0000relativistic operators, subject to boundary conditions expressed in terms of\u0000maximal monotone operators. The approach makes use of a fixed point formulation\u0000and relies on a priori estimates and convergent to zero matrices.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural extension of Lie's reduction of order algorithm for ordinary differential equations by looking at this algorithm from a different point of view. Many examples exhibit various situations that can arise.
{"title":"The natural extension to PDEs of Lie's reduction of order algorithm for ODEs","authors":"George W. Bluman, Rafael de la Rosa","doi":"arxiv-2407.09063","DOIUrl":"https://doi.org/arxiv-2407.09063","url":null,"abstract":"In this paper, we further consider the symmetry-based method for seeking\u0000nonlocally related systems for partial differential equations. In particular,\u0000we show that the symmetry-based method for partial differential equations is\u0000the natural extension of Lie's reduction of order algorithm for ordinary\u0000differential equations by looking at this algorithm from a different point of\u0000view. Many examples exhibit various situations that can arise.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove a restriction estimate for a hyperbolic paraboloid in $mathbb{R}^5$ by the polynomial partitioning method.
本文通过多项式分割法证明了$mathbb{R}^5$中双曲抛物面的限制估计值。
{"title":"A restriction estimate for a hyperbolic paraboloid in $mathbb{R}^5$","authors":"Zhuoran Li","doi":"arxiv-2407.08549","DOIUrl":"https://doi.org/arxiv-2407.08549","url":null,"abstract":"In this paper, we prove a restriction estimate for a hyperbolic paraboloid in\u0000$mathbb{R}^5$ by the polynomial partitioning method.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141615069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present multiplicity results for the periodic and Neumann-type boundary value problems associated with coupled Hamiltonian systems. For the periodic problem, we couple a system having twist condition with another one whose nonlinearity lies between the gradients of two positive and positively 2-homogeneous Hamiltonain functions. Concerning the Neumann-type problem, we treat the same system without any twist assumption. We examine the cases of nonresonance, simple resonance, and double resonance by imposing some kind of Landesman--Lazer conditions.
{"title":"Landesman-Lazer conditions for systems involving twist and positively homogeneous Hamiltonian systems","authors":"Natnael Gezahegn Mamo, Wahid Ullah","doi":"arxiv-2407.08389","DOIUrl":"https://doi.org/arxiv-2407.08389","url":null,"abstract":"We present multiplicity results for the periodic and Neumann-type boundary\u0000value problems associated with coupled Hamiltonian systems. For the periodic\u0000problem, we couple a system having twist condition with another one whose\u0000nonlinearity lies between the gradients of two positive and positively\u00002-homogeneous Hamiltonain functions. Concerning the Neumann-type problem, we\u0000treat the same system without any twist assumption. We examine the cases of\u0000nonresonance, simple resonance, and double resonance by imposing some kind of\u0000Landesman--Lazer conditions.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the theory of time scales, given $mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration theory on this scale from the point of view of the Lebesgue integral, which generalizes the previous perspective. A great tool obtained when studying the integral of a scale $mathbb{T}$ as a Lebesgue integral is the possibility of converting the ``$Delta$-integral of $mathbb{T}$'' to a classical integral of $mathbb{R}$. In this way, we are able to migrate from a calculation that is sometimes not so intuitive to a more friendly calculation. A question that arises, then, is whether the same result can be obtained just using the ideas of integration via Riemann sums, without the need to develop the Lebesgue integral for $mathbb{T}$. And, in this article, we answer this question affirmatively: In fact, for integrable functions an analogous result is valid by converting a $Delta$-integral over $mathbb{T}$ to a riemannian integral of $mathbb{R}$.
{"title":"A Different Demonstration for Integral Identity Across Distinct Time Scales","authors":"Patrick Oliveira","doi":"arxiv-2407.08144","DOIUrl":"https://doi.org/arxiv-2407.08144","url":null,"abstract":"In the theory of time scales, given $mathbb{T}$ a time scale with at least\u0000two distinct elements, an integration theory is developed using ideas already\u0000well known as Riemann sums. Another, more daring, approach is to treat an\u0000integration theory on this scale from the point of view of the Lebesgue\u0000integral, which generalizes the previous perspective. A great tool obtained\u0000when studying the integral of a scale $mathbb{T}$ as a Lebesgue integral is\u0000the possibility of converting the ``$Delta$-integral of $mathbb{T}$'' to a\u0000classical integral of $mathbb{R}$. In this way, we are able to migrate from a\u0000calculation that is sometimes not so intuitive to a more friendly calculation.\u0000A question that arises, then, is whether the same result can be obtained just\u0000using the ideas of integration via Riemann sums, without the need to develop\u0000the Lebesgue integral for $mathbb{T}$. And, in this article, we answer this\u0000question affirmatively: In fact, for integrable functions an analogous result\u0000is valid by converting a $Delta$-integral over $mathbb{T}$ to a riemannian\u0000integral of $mathbb{R}$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the operator begin{equation*} mathcal{C} f(x,y) := sup_{vin mathbb{R}} Big|mathrm{p.v.} int_{mathbb{R}} f(x-t, y-t^2) e^{i v t^3} frac{mathrm{d} t}{t} Big| end{equation*} is bounded on $L^p(mathbb{R}^2)$ for every $1 < p < infty$. This gives an affirmative answer to a question of Pierce and Yung.
我们证明算子f(x,y) := sup_{vin mathbb{R}}f(x-t, y-t^2) e^{i v t^3}f(x-t, y-t^2) e^{i v t^3}对于每$1 < p < infty$，$L^p(mathbb{R}^2)$都是有界的。
{"title":"On a planar Pierce--Yung operator","authors":"David Beltran, Shaoming Guo, Jonathan Hickman","doi":"arxiv-2407.07563","DOIUrl":"https://doi.org/arxiv-2407.07563","url":null,"abstract":"We show that the operator begin{equation*} mathcal{C} f(x,y) := sup_{vin mathbb{R}} Big|mathrm{p.v.}\u0000int_{mathbb{R}} f(x-t, y-t^2) e^{i v t^3} frac{mathrm{d} t}{t} Big|\u0000end{equation*} is bounded on $L^p(mathbb{R}^2)$ for every $1 < p < infty$.\u0000This gives an affirmative answer to a question of Pierce and Yung.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The application of variational structure for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is subjected to a fixed value and the other end is free, have been less studied, especially in terms of their stability despite their abundance. In this article, we develop the stability conditions for these problems by examining the second variation of the energy functional using the generalized Jacobi condition, which includes computing conjugate points. These conjugate points are determined by solving a set of initial value problems from the resulting linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to a tip load. Kirchhoff rod theory is employed to model the elastic rod deformations. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. This snap-back instability is demonstrated using various examples, highlighting its dependence on various system parameters. The presented examples illustrate the potential applications in the design of flexible soft robotic arms and mechanisms.
{"title":"Stability Analysis of Cantilever-like Structures with Applications to Soft Robotic Arms","authors":"Siva Prasad Chakri Dhanakoti","doi":"arxiv-2407.07601","DOIUrl":"https://doi.org/arxiv-2407.07601","url":null,"abstract":"The application of variational structure for analyzing problems in the\u0000physical sciences is widespread. Cantilever-like problems, where one end is\u0000subjected to a fixed value and the other end is free, have been less studied,\u0000especially in terms of their stability despite their abundance. In this\u0000article, we develop the stability conditions for these problems by examining\u0000the second variation of the energy functional using the generalized Jacobi\u0000condition, which includes computing conjugate points. These conjugate points\u0000are determined by solving a set of initial value problems from the resulting\u0000linearized equilibrium equations. We apply these conditions to investigate the\u0000nonlinear stability of intrinsically curved elastic cantilevers subject to a\u0000tip load. Kirchhoff rod theory is employed to model the elastic rod\u0000deformations. The role of intrinsic curvature in inducing complex nonlinear\u0000phenomena, such as snap-back instability, is particularly emphasized. This\u0000snap-back instability is demonstrated using various examples, highlighting its\u0000dependence on various system parameters. The presented examples illustrate the\u0000potential applications in the design of flexible soft robotic arms and\u0000mechanisms.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was introduced in cite{alves2024mode}, motivated by a question in diffusive relaxation. The main purpose of this note is to compare asymptotic $L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of the results obtained provides a characterization of convergence in measure on finite measure spaces in terms of asymptotic $L_p$-convergence.
{"title":"Relation between asymptotic $L_p$-convergence and some classical modes of convergence","authors":"Nuno J. Alves, Giorgi G. Oniani","doi":"arxiv-2407.06830","DOIUrl":"https://doi.org/arxiv-2407.06830","url":null,"abstract":"Asymptotic $L_p$-convergence, which resembles convergence in $L_p$, was\u0000introduced in cite{alves2024mode}, motivated by a question in diffusive\u0000relaxation. The main purpose of this note is to compare asymptotic\u0000$L_p$-convergence with convergence in measure and in weak $L_p$ spaces. One of\u0000the results obtained provides a characterization of convergence in measure on\u0000finite measure spaces in terms of asymptotic $L_p$-convergence.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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