Akbar H. Borzabadi, Mohammad Heidari, Delfim F. M. Torres
We consider fuzzy valued functions from two parametric representations of $alpha$-level sets. New concepts are introduced and compared with available notions. Following the two proposed approaches, we study fuzzy differential equations. Their relation with Zadeh's extension principle and the generalized Hukuhara derivative is discussed. Moreover, we prove existence and uniqueness theorems for fuzzy differential equations. Illustrative examples are given.
{"title":"Alternative views on fuzzy numbers and their application to fuzzy differential equations","authors":"Akbar H. Borzabadi, Mohammad Heidari, Delfim F. M. Torres","doi":"arxiv-2407.07906","DOIUrl":"https://doi.org/arxiv-2407.07906","url":null,"abstract":"We consider fuzzy valued functions from two parametric representations of\u0000$alpha$-level sets. New concepts are introduced and compared with available\u0000notions. Following the two proposed approaches, we study fuzzy differential\u0000equations. Their relation with Zadeh's extension principle and the generalized\u0000Hukuhara derivative is discussed. Moreover, we prove existence and uniqueness\u0000theorems for fuzzy differential equations. Illustrative examples are given.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611888","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 linear 1D transport equation is likely the most solved transport equation in radiative transfer and neutron transport investigations. Nearly every method imaginable has been applied to establish solutions, including Laplace and Fourier transforms, singular eigenfunctions, solutions of singular integral equation, PN expansions, double PN expansions, Chebychev expansions, Lagrange polynomial expansions, numerical discrete ordinates with finite difference, analytical discrete ordinates, finite elements, solutions to integral equations, adding and doubling, invariant imbedding, solution of Ricatti equations and response matrix methods -- and probably more methods of which the authors are unaware. Of those listed, the response matrix solution to the discrete ordinates form of the 1D transport equation is arguably the simplest and most straightforward. Here, we propose another response of exponential solutions but to the first order equation enabled by matrix scaling.
{"title":"Response Matrix Benchmark for the 1D Transport Equation with Matrix Scaling","authors":"B. D. Ganapol, J. K. Patel","doi":"arxiv-2407.07905","DOIUrl":"https://doi.org/arxiv-2407.07905","url":null,"abstract":"The linear 1D transport equation is likely the most solved transport equation\u0000in radiative transfer and neutron transport investigations. Nearly every method\u0000imaginable has been applied to establish solutions, including Laplace and\u0000Fourier transforms, singular eigenfunctions, solutions of singular integral\u0000equation, PN expansions, double PN expansions, Chebychev expansions, Lagrange\u0000polynomial expansions, numerical discrete ordinates with finite difference,\u0000analytical discrete ordinates, finite elements, solutions to integral\u0000equations, adding and doubling, invariant imbedding, solution of Ricatti\u0000equations and response matrix methods -- and probably more methods of which the\u0000authors are unaware. Of those listed, the response matrix solution to the\u0000discrete ordinates form of the 1D transport equation is arguably the simplest\u0000and most straightforward. Here, we propose another response of exponential\u0000solutions but to the first order equation enabled by matrix scaling.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611890","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 present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form ${0,1}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 times 2 times cdots times 2$ ($k$ times) chessboards concerning the wazir, the threeleaper, and the zebra, for all $k geq 15$. Our result considers the three above-mentioned leapers and replicates for each of them the recent discovery of Euclidean knight's tours for the same set of $2 times 2 times cdots times 2$ grids, opening a new research path on the topic by studying different fairy chess leapers that perform jumps of fixed Euclidean length on given regular grids, visiting all their vertices exactly once before coming back to the starting one.
本文旨在将$k$维网格的${0,1}^k$形式的马巡游问题扩展到其他仙女棋跳跃者。相应地，我们构造性地证明了在所有$k geq 15$的2 times2 times cdots times 2$（$k$次）棋盘中存在关于瓦齐尔、三跃马和斑马的封闭巡游。我们的结果考虑了上述三个跳跃者，并为它们中的每一个复制了最近在同一组 $2 times 2 times cdots times2$ 网格中发现的欧几里得骑士巡游，通过研究在给定正则网格上执行固定欧几里得长度跳跃的不同仙棋跳跃者，开辟了一条新的研究路径，这些跳跃者在回到起点之前会准确地访问它们的所有顶点一次。
{"title":"Euclidean Tours in Fairy Chess","authors":"Gabriele Di Pietro, Marco Ripà","doi":"arxiv-2407.07903","DOIUrl":"https://doi.org/arxiv-2407.07903","url":null,"abstract":"The present paper aims to extend the knight's tour problem for\u0000$k$-dimensional grids of the form ${0,1}^k$ to other fairy chess leapers.\u0000Accordingly, we constructively show the existence of closed tours in $2 times\u00002 times cdots times 2$ ($k$ times) chessboards concerning the wazir, the\u0000threeleaper, and the zebra, for all $k geq 15$. Our result considers the three\u0000above-mentioned leapers and replicates for each of them the recent discovery of\u0000Euclidean knight's tours for the same set of $2 times 2 times cdots times\u00002$ grids, opening a new research path on the topic by studying different fairy\u0000chess leapers that perform jumps of fixed Euclidean length on given regular\u0000grids, visiting all their vertices exactly once before coming back to the\u0000starting one.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611891","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 1999, Molodtsov cite{1} developed the idea of soft set theory, proving it to be a flexible mathematical tool for dealing with uncertainty. Several researchers have extended the framework by combining it with other theories of uncertainty, such as fuzzy set theory, intuitionistic fuzzy soft set theory, rough soft set theory, and so on. These enhancements aim to increase the applicability and expressiveness of soft set theory, making it a more robust tool for dealing with complex, real-world problems characterized by uncertainty and vagueness. The notion of fuzzy soft sets and their associated operations were introduced by Maji et al. cite{7}. However, Molodtsov cite{3} identified numerous incorrect results and notions of soft set theory that were introduced in the paper cite{7}. Therefore, the derived concept of fuzzy soft sets is equally incorrect since the basic idea of soft sets in cite{7} is flawed. Consequently, it is essential to address these incorrect notions and provide an exact and formal definition of the idea of fuzzy soft sets. This reevaluation is important to guarantee fuzzy soft set theory's theoretical stability and practical application across a range of domains. In this paper, we propose fuzzy soft set theory based on Molodtsov's correct notion of soft set theory and demonstrate a fuzzy soft set in matrix form. Additionally, we derive several significant findings on fuzzy soft sets.
1999 年，莫洛佐夫（Molodtsov）提出了软集合理论的思想，证明它是处理不确定性的一种灵活的数学工具。一些研究者将该框架与其他不确定性理论相结合，如模糊集合理论、直觉模糊软集合理论、粗糙软集合理论等，从而扩展了该框架。这些改进旨在提高软集合理论的适用性和表达能力，使其成为处理以不确定性和模糊性为特征的复杂现实问题的更强大工具。模糊软集的概念及其相关运算是由 Maji 等人提出的。然而，莫洛佐夫（Molodtsov）指出了论文（cite{7}）中引入的软集合理论的许多错误结果和概念。因此，由于 cite{7}中关于软集合的基本思想存在缺陷，因此衍生出的模糊软集合概念也同样是不正确的。这种重新评价对于保证模糊软集理论的理论稳定性和在一系列领域的实际应用是非常重要的。本文基于莫洛佐夫正确的软集合理论概念，提出了模糊软集合理论，并展示了矩阵形式的模糊软集合。此外，我们还得出了关于模糊软集的几个重要发现。
{"title":"The correct structures in fuzzy soft set theory","authors":"Santanu Acharjee, Sidhartha Medhi","doi":"arxiv-2407.06203","DOIUrl":"https://doi.org/arxiv-2407.06203","url":null,"abstract":"In 1999, Molodtsov cite{1} developed the idea of soft set theory, proving it\u0000to be a flexible mathematical tool for dealing with uncertainty. Several\u0000researchers have extended the framework by combining it with other theories of\u0000uncertainty, such as fuzzy set theory, intuitionistic fuzzy soft set theory,\u0000rough soft set theory, and so on. These enhancements aim to increase the\u0000applicability and expressiveness of soft set theory, making it a more robust\u0000tool for dealing with complex, real-world problems characterized by uncertainty\u0000and vagueness. The notion of fuzzy soft sets and their associated operations\u0000were introduced by Maji et al. cite{7}. However, Molodtsov cite{3} identified\u0000numerous incorrect results and notions of soft set theory that were introduced\u0000in the paper cite{7}. Therefore, the derived concept of fuzzy soft sets is\u0000equally incorrect since the basic idea of soft sets in cite{7} is flawed.\u0000Consequently, it is essential to address these incorrect notions and provide an\u0000exact and formal definition of the idea of fuzzy soft sets. This reevaluation\u0000is important to guarantee fuzzy soft set theory's theoretical stability and\u0000practical application across a range of domains. In this paper, we propose\u0000fuzzy soft set theory based on Molodtsov's correct notion of soft set theory\u0000and demonstrate a fuzzy soft set in matrix form. Additionally, we derive\u0000several significant findings on fuzzy soft sets.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570926","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}
Prime factorization has been a buzzing topic in the field of number theory since time unknown. However, in recent years, alternative avenues to tackle this problem are being explored by researchers because of its direct application in the arena of cryptography. One of such applications is the cryptanalysis of RSA numbers, which requires prime factorization of large semiprimes. Based on numerical experiments, this paper proposes a conjecture on the distribution of digits on prime of infinite length. This paper infuses the theoretical understanding of primes to optimize the search space of prime factors by shrinking it upto 98.15%, which, in terms of application, has shown 26.50% increase in the success rate and 41.91% decrease of the maximum number of generations required by the genetic algorithm used traditionally in the literature. This paper also introduces a variation of the genetic algorithm named Sieve Method that is fine-tuned for factorization of big semi-primes, which was able to factor numbers up to 23 decimal digits with 84% success rate. Our findings shows that sieve methods on average has achieved 321.89% increase in success rate and 64.06% decrement in the maximum number of generations required for the algorithm to converge compared to the existing literatures.
{"title":"Cryptanalysis of RSA Cryptosystem: Prime Factorization using Genetic Algorithm","authors":"Mahadee Al Mobin, Md Kamrujjaman","doi":"arxiv-2407.05944","DOIUrl":"https://doi.org/arxiv-2407.05944","url":null,"abstract":"Prime factorization has been a buzzing topic in the field of number theory\u0000since time unknown. However, in recent years, alternative avenues to tackle\u0000this problem are being explored by researchers because of its direct\u0000application in the arena of cryptography. One of such applications is the\u0000cryptanalysis of RSA numbers, which requires prime factorization of large\u0000semiprimes. Based on numerical experiments, this paper proposes a conjecture on\u0000the distribution of digits on prime of infinite length. This paper infuses the\u0000theoretical understanding of primes to optimize the search space of prime\u0000factors by shrinking it upto 98.15%, which, in terms of application, has shown\u000026.50% increase in the success rate and 41.91% decrease of the maximum number\u0000of generations required by the genetic algorithm used traditionally in the\u0000literature. This paper also introduces a variation of the genetic algorithm\u0000named Sieve Method that is fine-tuned for factorization of big semi-primes,\u0000which was able to factor numbers up to 23 decimal digits with 84% success rate.\u0000Our findings shows that sieve methods on average has achieved 321.89% increase\u0000in success rate and 64.06% decrement in the maximum number of generations\u0000required for the algorithm to converge compared to the existing literatures.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570923","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}
Using Brakke's Evolver, we numerically verify previous conjectures for optimal double bubbles for density $r^p$ in $R^3$ and our own new conjectures for triple bubbles.
{"title":"Numerically Computed Double and Triple Bubbles in $R^3$ for Density $r^p$","authors":"Eve Parrott","doi":"arxiv-2407.07122","DOIUrl":"https://doi.org/arxiv-2407.07122","url":null,"abstract":"Using Brakke's Evolver, we numerically verify previous conjectures for\u0000optimal double bubbles for density $r^p$ in $R^3$ and our own new conjectures\u0000for triple bubbles.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588311","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}
How do people come up with new sets of tiles including new tile shapes that would only tile non-periodically? This paper presents our graphical journey in tilings and provides a new set of three polyominoes named Ax for its relationship with Ammann A4.
{"title":"Ax, 3 polyominoes for tiling the plane non-periodically","authors":"Vincent Van Dongen, Pierre Gradit","doi":"arxiv-2407.06202","DOIUrl":"https://doi.org/arxiv-2407.06202","url":null,"abstract":"How do people come up with new sets of tiles including new tile shapes that\u0000would only tile non-periodically? This paper presents our graphical journey in\u0000tilings and provides a new set of three polyominoes named Ax for its\u0000relationship with Ammann A4.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570756","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}
Michael R. Bacon, Charles K. Cook, Rigoberto Flórez, Robinson A. Higuita, José L. Ramírez
Inspired by the ancient spiral constructed by the greek philosopher Theodorus which is based on concatenated right triangles, we have created a spiral. In this spiral, called emph{Fibonacci--Theodorus}, the sides of the triangles have lengths corresponding to Fibonacci numbers. Towards the end of the paper, we present a generalized method applicable to second-order recurrence relations. Our exploration of the Fibonacci--Theodorus spiral aims to address a variety of questions, showcasing its unique properties and behaviors. For example, we study topics such as area, perimeter, and angles. Notably, we establish a relationship between the ratio of two consecutive areas and the golden ratio, a pattern that extends to angles sharing a common vertex. Furthermore, we present some asymptotic results. For instance, we demonstrate that the sum of the first $n$ areas comprising the spiral approaches a multiple of the sum of the initial $n$ Fibonacci numbers. Moreover, we provide a sequence of open problems related to all spiral worked in this paper. Finally, in his work Hahn, Hahn observed a potential connection between the golden ratio and the ratio of areas between spines of lengths $sqrt{F_{n+1}}$ and $sqrt{F_{n+2}-1}$ and the areas between spines of lengths $sqrt{F_{n}}$ and $sqrt{F_{n+1}-1}$ in the Theodorus spiral. However, no formal proof has been provided in his work. In this paper, we provide a proof for Hahn's conjecture.
{"title":"Fibonacci--Theodorus Spiral and its properties","authors":"Michael R. Bacon, Charles K. Cook, Rigoberto Flórez, Robinson A. Higuita, José L. Ramírez","doi":"arxiv-2407.07109","DOIUrl":"https://doi.org/arxiv-2407.07109","url":null,"abstract":"Inspired by the ancient spiral constructed by the greek philosopher Theodorus\u0000which is based on concatenated right triangles, we have created a spiral. In\u0000this spiral, called emph{Fibonacci--Theodorus}, the sides of the triangles\u0000have lengths corresponding to Fibonacci numbers. Towards the end of the paper,\u0000we present a generalized method applicable to second-order recurrence\u0000relations. Our exploration of the Fibonacci--Theodorus spiral aims to address a variety\u0000of questions, showcasing its unique properties and behaviors. For example, we\u0000study topics such as area, perimeter, and angles. Notably, we establish a\u0000relationship between the ratio of two consecutive areas and the golden ratio, a\u0000pattern that extends to angles sharing a common vertex. Furthermore, we present\u0000some asymptotic results. For instance, we demonstrate that the sum of the first\u0000$n$ areas comprising the spiral approaches a multiple of the sum of the initial\u0000$n$ Fibonacci numbers. Moreover, we provide a sequence of open problems related\u0000to all spiral worked in this paper. Finally, in his work Hahn, Hahn observed a potential connection between the\u0000golden ratio and the ratio of areas between spines of lengths $sqrt{F_{n+1}}$\u0000and $sqrt{F_{n+2}-1}$ and the areas between spines of lengths $sqrt{F_{n}}$\u0000and $sqrt{F_{n+1}-1}$ in the Theodorus spiral. However, no formal proof has\u0000been provided in his work. In this paper, we provide a proof for Hahn's\u0000conjecture.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584701","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 main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE as a generalized power series, shedding light on the choice of the exponent for the monomials. Furthermore, it accommodates situations where terms in the equation are multiplied by $t^{alpha}$for example. The key contribution is how the exponents for these terms are chosen, which is different from traditional methods.
{"title":"A New Method For Solving Fractional And Classical Differential Equations Based On a New Generalized Fractional Power Series","authors":"Youness Assebbane, Mohamed Echchehira, Mohamed Bouaouid, Mustapha Atraoui","doi":"arxiv-2406.16980","DOIUrl":"https://doi.org/arxiv-2406.16980","url":null,"abstract":"The main objective of this paper is to introduce an algorithm for solving\u0000fractional and classical differential equations based on a new generalized\u0000fractional power series. The algorithm relies on expanding the solution of an\u0000FDE or an ODE as a generalized power series, shedding light on the choice of\u0000the exponent for the monomials. Furthermore, it accommodates situations where\u0000terms in the equation are multiplied by $t^{alpha}$for example. The key\u0000contribution is how the exponents for these terms are chosen, which is\u0000different from traditional methods.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504906","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}
This paper investigates the p-adic valuation trees of degree-2 and degree-3 polynomials in two variables over any prime p, building upon prior research outlined in [14].
{"title":"The p-adic valuation of the general degree-2 and degree-3 polynomial in 2 variables","authors":"Shubham","doi":"arxiv-2407.07103","DOIUrl":"https://doi.org/arxiv-2407.07103","url":null,"abstract":"This paper investigates the p-adic valuation trees of degree-2 and degree-3\u0000polynomials in two variables over any prime p, building upon prior research\u0000outlined in [14].","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588467","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}
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：