Pub Date : 2024-12-01DOI: 10.1016/S0034-4877(24)00084-3
Begzod M. Isakov
We shall construct a class of boundary conditions which will produce any given translationinvariant splitting Gibbs measure (TISGM) of the Ising–Potts model on the binary tree.
{"title":"Boundary condition problems for the Ising-Potts model on the binary tree","authors":"Begzod M. Isakov","doi":"10.1016/S0034-4877(24)00084-3","DOIUrl":"10.1016/S0034-4877(24)00084-3","url":null,"abstract":"<div><div>We shall construct a class of boundary conditions which will produce any given translationinvariant splitting Gibbs measure (TISGM) of the Ising–Potts model on the binary tree.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 3","pages":"Pages 353-363"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143317064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00073-9
Lajos Diósi
The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker–Planck equation using the standard toolbox of Riemann geometry. The covariant form of the adapted Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free.
{"title":"The Covariant Langevin Equation of Diffusion on Riemannian Manifolds","authors":"Lajos Diósi","doi":"10.1016/S0034-4877(24)00073-9","DOIUrl":"10.1016/S0034-4877(24)00073-9","url":null,"abstract":"<div><div>The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker–Planck equation using the standard toolbox of Riemann geometry. The covariant form of the adapted Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 143-148"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00076-4
Alcides Garat
In previous works it has been demonstrated that all the standard model local gauge groups are isomorphic to local groups of special tetrad transformations. The skeleton-gauge-vector tetrad vector structure enables to prove all of these isomorphism theorems. These new tetrads have been specially constructed for Yang–Mills theories, Abelian and non-Abelian in four-dimensional Lorentzian spacetimes. In the present paper a new tetrad is employed for the Yang–Mills SU(2) × U(1) formulation. These new tetrads establish a connection between local groups of gauge transformations and local groups of spacetime tetrad transformations. We will prove that these Yang–Mills tetrads under the local Yang-Mills gauge transformations not only transform a local group into another local group but also satisfy the group law.
{"title":"The Group Law for The New Internal-Spacetime Mapping Between The Group of Internal Yang-Mills Gauge Transformations and The Groups (õLB1)3 and (õLB2)3 of Spacetime Tetrad Transformations","authors":"Alcides Garat","doi":"10.1016/S0034-4877(24)00076-4","DOIUrl":"10.1016/S0034-4877(24)00076-4","url":null,"abstract":"<div><div>In previous works it has been demonstrated that all the standard model local gauge groups are isomorphic to local groups of special tetrad transformations. The skeleton-gauge-vector tetrad vector structure enables to prove all of these isomorphism theorems. These new tetrads have been specially constructed for Yang–Mills theories, Abelian and non-Abelian in four-dimensional Lorentzian spacetimes. In the present paper a new tetrad is employed for the Yang–Mills SU(2) × U(1) formulation. These new tetrads establish a connection between local groups of gauge transformations and local groups of spacetime tetrad transformations. We will prove that these Yang–Mills tetrads under the local Yang-Mills gauge transformations not only transform a local group into another local group but also satisfy the group law.</div><div><strong>PACS numbers:</strong> 12.10.-g, 04.40.Nr, 04.20.Cv, 11.15.-q, 02.40.Ky, 02.20.Qs, MSC2010, 51H25, 53c50, 20F65, 70s15, 70G65, 70G45.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 189-218"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00077-6
Xiu Han, Dengyin Wang, Chunguang Xia
We classify extensions between finite irreducible conformal modules over Lie conformal algebras Bℌ(a, b) of planar Galilean type, where a and b are complex numbers. We find that although finite irreducible conformal modules over Bℌ(a, b) are simply conformal modules over its Heisenberg–Virasoro conformal subalgebra, there exist more nontrivial extensions between conformal Bℌ(a, b)-modules.
我们对平面伽利略类型的 Lie 共形代数 Bℌ(a,b)上的有限不可还原共形模块之间的扩展进行了分类，其中 a 和 b 是复数。我们发现，虽然 Bℌ(a,b)上的有限不可还原共形模块只是其 Heisenberg-Virasoro 共形子代数上的共形模块，但 Bℌ(a,b)共形模块之间存在更多的非难扩展。
{"title":"Extensions of Conformal Modules Over Finite Lie Conformal Algebras of Planar Galilean Type","authors":"Xiu Han, Dengyin Wang, Chunguang Xia","doi":"10.1016/S0034-4877(24)00077-6","DOIUrl":"10.1016/S0034-4877(24)00077-6","url":null,"abstract":"<div><div>We classify extensions between finite irreducible conformal modules over Lie conformal algebras <strong>B</strong>ℌ(<em>a, b)</em> of planar Galilean type, where <em>a</em> and <em>b</em> are complex numbers. We find that although finite irreducible conformal modules over <strong>B</strong>ℌ(<em>a</em>, <em>b)</em> are simply conformal modules over its Heisenberg–Virasoro conformal subalgebra, there exist more nontrivial extensions between conformal <strong>B</strong>ℌ(<em>a, b</em>)-modules.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 219-233"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00074-0
Nour Elhouda Djaa, Aydin Gezer
The primary objective of this study is to examine harmonic and generalized magnetic vector fields as mappings from an anti-paraKählerian manifold to its associated tangent bundle, endowed with a ciconia metric. Initially, the conditions under which a vector field is harmonic (or magnetic) concerning a ciconia metric are investigated. Subsequently, the mappings between any given Riemannian manifold and the tangent bundle of an anti-paraKählerian manifold are explored. The paper delves into identifying the circumstances under which vector fields exhibit harmonicity or magnetism within the framework of a ciconia metric. Additionally, it explores the relationships between specific harmonic and magnetic vector fields, particularly emphasizing their behaviour under conformal transformations of metrics.
{"title":"Exploring Harmonic and Magnetic Fields on The Tangent Bundle with A Ciconia Metric Over An Anti-Parakähler Manifold","authors":"Nour Elhouda Djaa, Aydin Gezer","doi":"10.1016/S0034-4877(24)00074-0","DOIUrl":"10.1016/S0034-4877(24)00074-0","url":null,"abstract":"<div><div>The primary objective of this study is to examine harmonic and generalized magnetic vector fields as mappings from an anti-paraKählerian manifold to its associated tangent bundle, endowed with a ciconia metric. Initially, the conditions under which a vector field is harmonic (or magnetic) concerning a ciconia metric are investigated. Subsequently, the mappings between any given Riemannian manifold and the tangent bundle of an anti-paraKählerian manifold are explored. The paper delves into identifying the circumstances under which vector fields exhibit harmonicity or magnetism within the framework of a ciconia metric. Additionally, it explores the relationships between specific harmonic and magnetic vector fields, particularly emphasizing their behaviour under conformal transformations of metrics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 149-173"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00078-8
Jeffrey S. Ovall, Hadrian Quan, Robyn Reid, Stefan Steinerberger
Let Ω ⊂ ℝd and consider the magnetic Laplace operator given by H(A) = (–i∇ – A(x))2, where A : Ω → ℝd, subject to Dirichlet boundary conditions. For certain vector fields A, this operator can have eigenfunctions, H(A)ψ = λψ, that are highly localized in a small region of Ω. The main goal of this paper is to show that if |ψ| assumes its maximum at x0 ∈ Ω, then A behaves 'almost' like a conservative vector field in a -neighborhood of x0 in a precise sense. In particular, we expect localization in regions where |curl A| is small. The result is illustrated with numerical examples.
{"title":"On Localization of Eigenfunctions of The Magnetic Laplacian","authors":"Jeffrey S. Ovall, Hadrian Quan, Robyn Reid, Stefan Steinerberger","doi":"10.1016/S0034-4877(24)00078-8","DOIUrl":"10.1016/S0034-4877(24)00078-8","url":null,"abstract":"<div><div>Let Ω ⊂ ℝ<em><sup>d</sup></em> and consider the magnetic Laplace operator given by <em>H</em>(<em>A</em>) = (–<em>i</em>∇ – <em>A</em>(<em>x</em>))<sup>2</sup>, where <em>A</em> : Ω <em>→</em> ℝ<em><sup>d</sup></em>, subject to Dirichlet boundary conditions. For certain vector fields <em>A</em>, this operator can have eigenfunctions, <em>H</em>(<em>A</em>)ψ = λψ, that are highly localized in a small region of Ω. The main goal of this paper is to show that if |ψ| assumes its maximum at <em>x</em><sub>0</sub> ∈ Ω, then <em>A</em> behaves 'almost' like a conservative vector field in a \u0000\t\t\t\t<span><math><mrow><mn>1</mn><mo>/</mo><msqrt><mi>λ</mi></msqrt></mrow></math></span>-neighborhood of <em>x</em><sub>0</sub> in a precise sense. In particular, we expect localization in regions where |curl <em>A</em>| is small. The result is illustrated with numerical examples.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 235-257"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00075-2
Adam R. Szewczyk
This paper deals with the temperature profile of a simple combustion and presents the alternative exact formulas for the temperature profile of the planar vessel. The differential equation that describes this system is referred as a Bratu equation or Poisson's equation in one-dimensional steady state case. In this present study, new solutions with general boundary conditions are developed. The results are compared with numerical solutions using Maxima, a computer algebra system program capable of numerical and symbolic computation. The new solutions yield formula that may provide a valuable information about relationship between terms, variables and coefficients which can be useful for theoretical physics.
{"title":"Exact Solution to Bratu Second Order Differential Equation","authors":"Adam R. Szewczyk","doi":"10.1016/S0034-4877(24)00075-2","DOIUrl":"10.1016/S0034-4877(24)00075-2","url":null,"abstract":"<div><div>This paper deals with the temperature profile of a simple combustion and presents the alternative exact formulas for the temperature profile of the planar vessel. The differential equation that describes this system is referred as a Bratu equation or Poisson's equation in one-dimensional steady state case. In this present study, new solutions with general boundary conditions are developed. The results are compared with numerical solutions using Maxima, a computer algebra system program capable of numerical and symbolic computation. The new solutions yield formula that may provide a valuable information about relationship between terms, variables and coefficients which can be useful for theoretical physics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 175-188"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/S0034-4877(24)00079-X
Ashik A. Kabeer, Dilip Kumar
The paper is devoted to presenting a novel closed-form representation of the resonant thermonuclear functions and the nonrelativistic Voigt function, which are essential tools in nuclear physics. Understanding thermonuclear fusion reaction rates within solar analogs is crucial for understanding stellar evolution and energy production mechanisms. Initially, this paper focuses on evaluating fusion reaction rates, particularly emphasizing resonant reactions, which play pivotal roles in stellar evolution phases. A key challenge lies in solving reaction rate integrals in closed form. The Buschman–Fox H-function of two variables is employed to address this issue. Conventionally, it is assumed that the plasma particles' velocity follows the Maxwell–Boltzmann distribution. However, it is acknowledged that particles may deviate from this assumed equilibrium state in actual scenarios, leading to nonequilibrium situations. The study also aims to address these nonequilibrium situations by utilizing appropriate velocity models from the existing literature. Utilizing the Mellin transform technique, we achieve the closed-form representation of the resonant reaction rate integral. Furthermore, we address the nonrelativistic Voigt profile and, in particular, Voigt function. The Voigt profile, resulting from the convolution of Gaussian and Lorentzian distributions, effectively captures the intricate shapes of spectral lines encountered in spectroscopy. Apart from its significance in spectroscopy, the Voigt function finds application in various areas such as plasma nuclear studies, acoustics, and radiation transfer. Many approximations of the Voigt function can be found in the literature, yet currently, there is no existing closed-form expression. This paper also sets out to fill this gap by deriving the exact closed-form expressions for the Voigt function and its conjugate in terms of Buschman–Fox H-function, employing the Mellin convolution theorem. This paper marks the first instance in the literature where the applications of Buschman Fox's H-function has been documented.
本文专门介绍了共振热核函数和非相对论沃伊特函数的新颖闭式表示法，它们是核物理中的重要工具。了解太阳类似物中的热核聚变反应速率对于理解恒星演化和能量产生机制至关重要。本文最初侧重于评估核聚变反应速率，特别强调在恒星演化阶段起关键作用的共振反应。以封闭形式求解反应速率积分是一项关键挑战。为了解决这个问题，我们采用了双变量的 Buschman-Fox H 函数。传统的假设是等离子体粒子的速度遵循麦克斯韦-玻尔兹曼分布。然而，在实际情况中，粒子可能会偏离这种假定的平衡状态，从而导致非平衡状态。本研究还旨在利用现有文献中的适当速度模型来解决这些非平衡状况。利用梅林变换技术，我们实现了共振反应速率积分的闭式表示。此外，我们还讨论了非相对论的 Voigt 剖面，特别是 Voigt 函数。Voigt 轮廓由高斯分布和洛伦兹分布卷积而成，能有效捕捉光谱学中光谱线的复杂形状。Voigt 函数除了在光谱学中具有重要意义外，还应用于等离子体核研究、声学和辐射传输等多个领域。文献中可以找到许多 Voigt 函数的近似值，但目前还没有现成的闭式表达式。本文利用梅林卷积定理，以 Buschman-Fox H 函数为基础，推导出 Voigt 函数及其共轭函数的精确闭式表达式，从而填补了这一空白。本文是文献中首次记录布施曼-福克斯 H 函数应用的实例。
{"title":"Applications of Buschman–fox H-Function in Nuclear Physics","authors":"Ashik A. Kabeer, Dilip Kumar","doi":"10.1016/S0034-4877(24)00079-X","DOIUrl":"10.1016/S0034-4877(24)00079-X","url":null,"abstract":"<div><div>The paper is devoted to presenting a novel closed-form representation of the resonant thermonuclear functions and the nonrelativistic Voigt function, which are essential tools in nuclear physics. Understanding thermonuclear fusion reaction rates within solar analogs is crucial for understanding stellar evolution and energy production mechanisms. Initially, this paper focuses on evaluating fusion reaction rates, particularly emphasizing resonant reactions, which play pivotal roles in stellar evolution phases. A key challenge lies in solving reaction rate integrals in closed form. The Buschman–Fox <em>H</em>-function of two variables is employed to address this issue. Conventionally, it is assumed that the plasma particles' velocity follows the Maxwell–Boltzmann distribution. However, it is acknowledged that particles may deviate from this assumed equilibrium state in actual scenarios, leading to nonequilibrium situations. The study also aims to address these nonequilibrium situations by utilizing appropriate velocity models from the existing literature. Utilizing the Mellin transform technique, we achieve the closed-form representation of the resonant reaction rate integral. Furthermore, we address the nonrelativistic Voigt profile and, in particular, Voigt function. The Voigt profile, resulting from the convolution of Gaussian and Lorentzian distributions, effectively captures the intricate shapes of spectral lines encountered in spectroscopy. Apart from its significance in spectroscopy, the Voigt function finds application in various areas such as plasma nuclear studies, acoustics, and radiation transfer. Many approximations of the Voigt function can be found in the literature, yet currently, there is no existing closed-form expression. This paper also sets out to fill this gap by deriving the exact closed-form expressions for the Voigt function and its conjugate in terms of Buschman–Fox <em>H</em>-function, employing the Mellin convolution theorem. This paper marks the first instance in the literature where the applications of Buschman Fox's <em>H</em>-function has been documented.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 2","pages":"Pages 259-278"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/S0034-4877(24)00055-7
Alemu Yilma Tefera, Shangshuai Li, Da-jun Zhang
The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {S(i,j)} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {S(i,j)} can be derived. Some identities of {S(i,j)} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.
{"title":"Nonisospectral equations from the Cauchy matrix approach","authors":"Alemu Yilma Tefera, Shangshuai Li, Da-jun Zhang","doi":"10.1016/S0034-4877(24)00055-7","DOIUrl":"10.1016/S0034-4877(24)00055-7","url":null,"abstract":"<div><div>The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {<em>S</em><sup>(<em>i,j</em>)</sup>} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {<em>S</em><sup>(<em>i,j</em>)</sup>} can be derived. Some identities of {<em>S</em><sup>(<em>i,j</em>)</sup>} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 47-72"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：