{"title":"Fourier Transform of Anisotropic Hardy Spaces Associated with Ball Quasi-Banach Function Spaces and Its Applications to Hardy-Littlewood Inequalities","authors":"Chao-an Li, Xian-jie Yan, Da-chun Yang","doi":"10.1007/s10255-024-1124-5","DOIUrl":"https://doi.org/10.1007/s10255-024-1124-5","url":null,"abstract":"<p>Let <i>A</i> be a general expansive matrix and <i>X</i> be a ball quasi-Banach function space on ℝ<sup><i>n</i></sup>, whose certain power (namely its convexification) supports a Fefferman-Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy-Littlewood maximal operator. Let <i>H</i><span>\u0000<sup><i>A</i></sup><sub><i>X</i></sub>\u0000</span>(ℝ<sup><i>n</i></sup>) be the anisotropic Hardy space associated with <i>A</i> and <i>X</i>. The authors first prove that the Fourier transform of <i>f</i> ∈ <i>H</i><span>\u0000<sup><i>A</i></sup><sub><i>X</i></sub>\u0000</span>(ℝ<sup><i>n</i></sup>) coincides with a continuous function <i>F</i> on ℝ<sup><i>n</i></sup> in the sense of tempered distributions. Moreover, the authors obtain a pointwise inequality that the function <i>F</i> is less than the product of the anisotropic Hardy space norm of <i>f</i> and a step function with respect to the transpose matrix of the expansive matrix <i>A</i>. Applying this, the authors further induce a higher order convergence for the function <i>F</i> at the origin and give a variant of the Hardy-Littlewood inequality in <i>H</i><span>\u0000<sup><i>A</i></sup><sub><i>X</i></sub>\u0000</span>(ℝ<sup><i>n</i></sup>). All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to classical (variable and mixed-norm) Lebesgue spaces, Lorentz spaces, Orlicz spaces, Orlicz-slice spaces, and local generalized Herz spaces and, even on the last four function spaces, the obtained results are completely new.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"63 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems","authors":"Yan-ping Chen, Jian-wei Zhou, Tian-liang Hou","doi":"10.1007/s10255-024-1099-2","DOIUrl":"https://doi.org/10.1007/s10255-024-1099-2","url":null,"abstract":"<p>This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"54 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform Large Deviations for Stochastic Generalized Burgers-Huxley Equation Driven by Multiplicative Noise","authors":"Hui Guo, Xi-liang Li, Jin Ma","doi":"10.1007/s10255-024-1129-0","DOIUrl":"https://doi.org/10.1007/s10255-024-1129-0","url":null,"abstract":"<p>In this paper, we establish a Freidlin-Wentzell type large deviation principle uniformly with respect to initial condition in bounded subsets, that do not necessarily belongs to compact sets, of an infinite dimensional Banach space for stochastic 1D generalized Burgers-Huxley equation driven by multiplicative small noise. The proof is based on the weak convergence approach obtained by [Budhiraja, Dupuis and Salins; Trans. Amer. Math. Soc., 2019].</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The P5-saturation Game","authors":"Zhen He, Mei Lu","doi":"10.1007/s10255-024-1125-4","DOIUrl":"https://doi.org/10.1007/s10255-024-1125-4","url":null,"abstract":"<p>Let <i>F, G</i> and <i>H</i> be three graphs with <i>G</i> ⊆ <i>H</i>. We call <i>G</i> an <i>F</i>-saturated graph relative to <i>H</i>, if there is no copy of <i>F</i> in <i>G</i> but there is a copy of <i>F</i> in <i>G</i> + <i>e</i> for any <i>e</i> ∈ <i>E</i>(<i>H</i>) <i>E</i>(<i>G</i>). The <i>F</i>-saturation game on host graph <i>H</i> consists of two players, named Max and Min, who alternately add edges of <i>H</i> to <i>G</i> such that each chosen edge avoids creating a copy of <i>F</i> in <i>G</i>, and the players continue to choose edges until <i>G</i> becomes <i>F</i>-saturated relative to <i>H</i>. Max wishes to maximize the length of the game, while Min wishes to minimize the process. Let sat<sub><i>g</i></sub>(<i>F, H</i>) (resp. sat′<sub><i>g</i></sub>(<i>F, H</i>)) denote the number of edges chosen when Max (resp. when Min) starts the game and both players play optimally. In this article, we show that sat<sub><i>g</i></sub>(<i>P</i><sub>5</sub>, <i>K</i><sub><i>n</i></sub>) = sat′<sub><i>g</i></sub>(<i>P</i><sub>5</sub>, <i>K</i><sub><i>n</i></sub>) = <i>n</i> + 2 for <i>n</i> ≥ 15, and sat<sub><i>g</i></sub>(<i>P</i><sub>5</sub>, <i>K</i><sub><i>m,n</i></sub>), sat′<sub><i>g</i></sub>(<i>P</i><sub>5</sub>, <i>K</i><sub><i>m,n</i></sub>) lie in <span>(left{ {m + n - leftlfloor {{{m - 2} over 4}} rightrfloor ,,m + n - leftlceil {{{m - 3} over 4}} rightrceil } right})</span> if <span>(n ge {5 over 2}m)</span> and <i>m</i> ≥ 4, respectively.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic Property of Parabolic Equations Involving Pseudo-relativistic Schrödinger Operators","authors":"Chen Qiao, Su-fang Tang","doi":"10.1007/s10255-024-1097-4","DOIUrl":"https://doi.org/10.1007/s10255-024-1097-4","url":null,"abstract":"<p>In this paper, we investigate parabolic equations involving nonlocal pseudo-relativistic Schrödinger operators (−Δ + <i>m</i><sup>2</sup>)<sup><i>s</i></sup> with <i>s</i> ∈ (0, 1) and mass <i>m</i> > 0 in bounded regions. We establish the asymptotic narrow region principle and asymptotic strong maximum principle for anti symmetric function. As applications, employing the method of moving planes, we show the asymptotical radial symmetry and monotonicity of positive solutions in an unit ball.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Itô-Taylor Expansion Method of European Spread Option Pricing for Multivariate Diffusions with Jumps","authors":"Ge Wang, Yu-xuan Lu, Qing Zhou, Wei-lin Xiao","doi":"10.1007/s10255-024-1094-7","DOIUrl":"https://doi.org/10.1007/s10255-024-1094-7","url":null,"abstract":"<p>In this paper, we propose a new method for spread option pricing under the multivariate irreducible diffusions without jumps and with different types of jumps by the expansion of the transition density function. By the quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, and applying the small-time Itô-Taylor expansion method, we derive explicit recursive formulas for the expansion coefficients of transition densities and spread option prices for multivariate diffusions with jumps in return. It is worth mentioning that we also give the closed-form formula of spread option price whose underlying asset price processes contain a Merton jump and a double exponential jump, which is innovative compared with current literature. The theoretical proof of convergence is presented in detail.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"86 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Privacy-Preserving Frank-Wolfe on Shuffle Model","authors":"Ling-jie Zhang, Shi-song Wu, Hai Zhang","doi":"10.1007/s10255-024-1095-6","DOIUrl":"10.1007/s10255-024-1095-6","url":null,"abstract":"<div><p>In this paper, we design the differentially private variants of the classical Frank-Wolfe algorithm with shuffle model in the optimization of machine learning. Under weak assumptions and the generalized linear loss (GLL) structure, we propose a noisy Frank-Wolfe with shuffle model algorithm (NoisyFWS) and a noisy variance-reduced Frank-Wolfe with the shuffle model algorithm (NoisyVRFWS) by adding calibrated laplace noise under shuffling scheme in the <i>ℓ</i><sub><i>p</i></sub>(<i>p</i> ∈ [1, 2])-case, and study their privacy as well as utility guarantees for the Hölder smoothness GLL. In particular, the privacy guarantees are mainly achieved by using advanced composition and privacy amplification by shuffling. The utility bounds of the NoisyFWS and NoisyVRFWS are analyzed and obtained the optimal excess population risks <span>({cal O}({n^{ - {{1 + alpha } over {4alpha }}}} + {{log (d)sqrt {log ({1 mathord{left/ {vphantom {1 delta }} right.} delta })} } over {nepsilon,}}))</span> and <span>({cal O}({n^{ - {{1 + alpha } over {4alpha }}}} + {{log (d)sqrt {log ({1 mathord{left/ {vphantom {1 delta }} right.} delta })} } over {{n^2epsilon},}}))</span> with gradient complexity <span>({cal O}({n^{ - {{{{(1 + alpha )}^2}} over {4{alpha ^2}}}}}))</span> for <span>(alpha in left[ {{1 mathord{left/ {vphantom {1 {sqrt 3 ,,1}}} right.} {sqrt 3 ,,1}}} right])</span>. It turns out that the risk rates under shuffling scheme are a nearly-dimension independent rate, which is consistent with the previous work in some cases. In addition, there is a vital tradeoff between (<i>α, L</i>)-Hölder smoothness GLL and the gradient complexity. The linear gradient complexity <span>({cal O}(n))</span> is showed by the parameter <i>α</i> = 1.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 4","pages":"887 - 907"},"PeriodicalIF":0.9,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Meromorphic Solutions to Higher Order Nonlinear Delay Differential Equations","authors":"Ye-zhou Li, Ming-yue Wu, He-qing Sun","doi":"10.1007/s10255-024-1098-3","DOIUrl":"https://doi.org/10.1007/s10255-024-1098-3","url":null,"abstract":"<p>Let <i>w</i>(<i>z</i>) be non-rational meromorphic solutions with hyper-order less than 1 to a family of higher order nonlinear delay differential equations </p><span>$$matrix{{wleft( {z + 1} right)wleft( {z - 1} right), + ,aleft( z right){{{w^{left( k right)}}left( z right)} over {wleft( z right)}} = Rleft( {z,,wleft( z right)} right),} & {k in mathbb{N}{^ + },} cr}$$</span><p> where <i>a</i>(<i>z</i>) is rational, <span>(Rleft( {z,,wleft( z right)} right) = {{Pleft( {z,,w,,left( z right)} right)} over {Qleft( {z,,w,,left( z right)} right)}})</span> is an irreducible rational function in <i>w</i> with rational coefficients in <i>z</i>. This paper mainly show the relationships of the degree of <i>P</i>(<i>z,w</i>(<i>z</i>)) and <i>Q</i>(<i>z,w</i>(<i>z</i>)) when the above equations exist such solutions <i>w</i>(<i>z</i>). There are also some examples to show that our results are sharp.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"49 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Information Heterogeneity: Implications for Liquidity and Price Efficiency","authors":"Zhi-hua Li, Hong Liu, Qing-shan Yang","doi":"10.1007/s10255-024-1092-9","DOIUrl":"https://doi.org/10.1007/s10255-024-1092-9","url":null,"abstract":"<p>This paper discusses a model about information heterogeneity. The market size, information attention and information asymmetry affect market liquidity and price information. Larger market size can lead to an increase in market liquidity and the release of price information; however, traders who possess higher information attention can quickly affect price efficiency through their trading, leading to increased price fluctuations. In addition, the increase in asymmetry scale will lead to a decrease in liquidity and a reduction in the release of price information. Under information asymmetry, large trading groups attract followers, thereby increasing market liquidity and trading intensity.</p>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Novel Analysis Approach of Uniform Persistence for an Epidemic Model with Quarantine and Standard Incidence Rate","authors":"Song-bai Guo, Yu-ling Xue, Xi-liang Li, Zuo-huan Zheng","doi":"10.1007/s10255-023-1078-y","DOIUrl":"10.1007/s10255-023-1078-y","url":null,"abstract":"<div><p>Inspired by the transmission characteristics of the Coronavirus disease 2019 (COVID-19), an epidemic model with quarantine and standard incidence rate is first developed, then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals, which means that the epidemic is uniformly persistent if the control reproduction number <span>({{cal R}_c} > 1)</span>. This approach can be applied to the related biomathematical models, and some existing works can be improved by using that. In addition, the infection-free equilibrium <i>V</i><sup>0</sup> of the model is locally asymptotically stable (LAS) if <span>({{cal R}_c} < 1)</span> and linearly stable if <span>({{cal R}_c} = 1)</span>; while <i>V</i><sup>0</sup> is unstable if <span>({{cal R}_c} > 1)</span>.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 3","pages":"695 - 707"},"PeriodicalIF":0.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}