For $ninmathbb{N}$ we define a double integral begin{equation*} I_n=frac{1}{24}int_0^1int_0^1 frac{-log^3(xy)}{1+xy} (xy(1-xy))^n dxdyend{equation*} We denote $d_n=text{lcm}(1,2,...,n)$ and prove that for all $ninmathbb{N}$, begin{equation*} d_n I_n= 15 (-1)^n 2^{n-4} d_n zeta(5)+(-1)^{n+1} d_n sum_{r=0}^{n} binom{n}{r}left(sum_{k=1}^{n+r} frac{(-1)^{k-1}}{k^5}right) end{equation*} Now if $zeta(5)$ is rational, then $zeta(5)=a/b$, $(a,b)=1$ and $a,binmathbb{N}$. Then we take $ngeq b$ so that $b|d_n$. We show that for all $ngeq 1$, $0
{"title":"A note on the Irrationality of $ζ(5)$ and Higher Odd Zeta Values","authors":"Shekhar Suman","doi":"arxiv-2407.07121","DOIUrl":"https://doi.org/arxiv-2407.07121","url":null,"abstract":"For $ninmathbb{N}$ we define a double integral begin{equation*}\u0000I_n=frac{1}{24}int_0^1int_0^1 frac{-log^3(xy)}{1+xy} (xy(1-xy))^n \u0000dxdyend{equation*} We denote $d_n=text{lcm}(1,2,...,n)$ and prove that for\u0000all $ninmathbb{N}$, begin{equation*} d_n I_n= 15 (-1)^n 2^{n-4} d_n\u0000zeta(5)+(-1)^{n+1} d_n sum_{r=0}^{n} binom{n}{r}left(sum_{k=1}^{n+r}\u0000frac{(-1)^{k-1}}{k^5}right) end{equation*} Now if $zeta(5)$ is rational,\u0000then $zeta(5)=a/b$, $(a,b)=1$ and $a,binmathbb{N}$. Then we take $ngeq b$\u0000so that $b|d_n$. We show that for all $ngeq 1$, $0<d_n I_n<1$. We denote\u0000begin{equation*} S_n=d_n sum_{r=0}^{n} binom{n}{r}left(sum_{k=1}^{n+r}\u0000frac{(-1)^{k-1}}{k^5}right) end{equation*} We prove for all $ngeq b$,\u0000begin{equation*} d_n I_n+(-1)^{n}\u0000left{S_nright}inmathbb{Z}end{equation*} where ${x}=x-[x]$ denotes the\u0000fractional part of $x$ and $[x]$ denotes the greatest integer less than or\u0000equal to $x$. Later we show that the only integer value admissible is\u0000begin{equation*} d_n I_n+(-1)^{n} left{S_nright}=0 text{or} d_n\u0000I_n+(-1)^{n} left{S_nright}=1end{equation*} Finally we prove that these\u0000above values are not possible. This gives a contradiction to our assumption\u0000that $zeta(5)$ is rational. Similarly for all $mgeq 3$ and $ngeq 1$, by\u0000defining begin{equation*} I_{n,m}=frac{1}{Gamma(2m+1)}int_0^1int_0^1\u0000frac{-log^{2m-1}(xy)}{1+xy} (xy(1-xy))^n dxdyend{equation*} where\u0000$Gamma(s)$ denotes the Gamma function, we give an extension of the above proof\u0000to prove the irrationality of higher odd zeta values $zeta(2m+1)$ for all\u0000$mgeq 3$","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584702","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 discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it to analyze chaotic behavior of the logistic map, comparing the $r$-varying Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the idea of the Lyapunov exponent to an $n$-dimensional system and explore the mathematical background behind the analytic calculation of the Lyapunov spectrum. We also outline a method to numerically calculate the maximal Lyapunov exponent using the periodic renormalization of a perturbation vector and a method to numerically calculate the entire Lyapunov spectrum using QR factorization. Finally, we apply both these methods to calculate the Lyapunov exponents of the H'enon map, a multi-dimensional chaotic system.
{"title":"Describing chaotic systems","authors":"Brandon Le","doi":"arxiv-2407.07919","DOIUrl":"https://doi.org/arxiv-2407.07919","url":null,"abstract":"In this paper, we discuss the Lyapunov exponent definition of chaos and how\u0000it can be used to quantify the chaotic behavior of a system. We derive a way to\u0000practically calculate the Lyapunov exponent of a one-dimensional system and use\u0000it to analyze chaotic behavior of the logistic map, comparing the $r$-varying\u0000Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the\u0000idea of the Lyapunov exponent to an $n$-dimensional system and explore the\u0000mathematical background behind the analytic calculation of the Lyapunov\u0000spectrum. We also outline a method to numerically calculate the maximal\u0000Lyapunov exponent using the periodic renormalization of a perturbation vector\u0000and a method to numerically calculate the entire Lyapunov spectrum using QR\u0000factorization. Finally, we apply both these methods to calculate the Lyapunov\u0000exponents of the H'enon map, a multi-dimensional chaotic system.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"226 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611884","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}
Borwein integrals are one of the most popularly known phenomena in contemporary mathematics. They were found in 2001 by David Borwein and Jonathan Borwein and consist of a simple family of integrals involving the cardinal sine function ``sinc'', so that the first integrals are equal to $pi$ until, suddenly, that pattern breaks. The classical explanation for this fact involves Fourier Analysis techniques. In this paper, we show that it is possible to derive an explanation for this result by means of undergraduate Complex Analysis tools; namely, residue theory. Besides, we show that this Complex Analysis scope allows to go a beyond the classical result when studying these kind of integrals. Concretely, we show a new generalization for the classical Borwein result.
{"title":"An approach to Borwein integrals from the point of view of residue theory","authors":"Daniel Cao Labora, Gonzalo Cao Labora","doi":"arxiv-2407.15856","DOIUrl":"https://doi.org/arxiv-2407.15856","url":null,"abstract":"Borwein integrals are one of the most popularly known phenomena in\u0000contemporary mathematics. They were found in 2001 by David Borwein and Jonathan\u0000Borwein and consist of a simple family of integrals involving the cardinal sine\u0000function ``sinc'', so that the first integrals are equal to $pi$ until,\u0000suddenly, that pattern breaks. The classical explanation for this fact involves\u0000Fourier Analysis techniques. In this paper, we show that it is possible to\u0000derive an explanation for this result by means of undergraduate Complex\u0000Analysis tools; namely, residue theory. Besides, we show that this Complex\u0000Analysis scope allows to go a beyond the classical result when studying these\u0000kind of integrals. Concretely, we show a new generalization for the classical\u0000Borwein result.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771304","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 Cantor pairing polynomials are extended to larger 2D sub-domains and more complex mapping, of which the most important property is the bijectivity. If corners are involved inside (but not the borders of) domain, more than one connected polynomials are necessary. More complex patterns need more complex subsequent application of math series to obtain the mapping polynomials which are more and more inconvenient, although elementary. A tricky polynomial fit is introduced (six coefficients are involved like in the original Cantor polynomials with rigorous but simple restrictions on points chosen) to buy out the regular treatment of math series to find the pairing polynomials instantly. The original bijective Cantor polynomial C1(x,y)= (x2+2xy+y2+3x+y)/2: N02 to N0 (=positive integers) which is 2-fold and runs in zig - zag way along lines x+y=N is extended e.g. to the bijective P(x,y)= 2x2+4sgn(x)sgn(y)xy+2y2-2H(x)sgn(y)x-y+1: Z2 to N0 (with sign and Heaviside functions, Z is integers) running in spiral way along concentric rhombuses, or to the bijective P3D(x,y,z)= [x3+y3+z3 +3(xz2+yz2 +zx2+2xyz +zy2+yx2+xy2) +3(2x2+2y2 +z2+2xz +2yz+4xy) +5x+11y+2z]/6: N03 to N0 which is 6-fold and runs along plains x+y+z=N. Storage device for triangle matrices is also commented as cutting the original Cantor domain to half along with related Diophantine equations.
{"title":"Generalization of Cantor Pairing Polynomials (Bijective Mapping Among Natural Numbers) from N02 to N0 to Z2 to N0 and N03 to N","authors":"Sandor Kristyan","doi":"arxiv-2407.05073","DOIUrl":"https://doi.org/arxiv-2407.05073","url":null,"abstract":"The Cantor pairing polynomials are extended to larger 2D sub-domains and more\u0000complex mapping, of which the most important property is the bijectivity. If\u0000corners are involved inside (but not the borders of) domain, more than one\u0000connected polynomials are necessary. More complex patterns need more complex\u0000subsequent application of math series to obtain the mapping polynomials which\u0000are more and more inconvenient, although elementary. A tricky polynomial fit is\u0000introduced (six coefficients are involved like in the original Cantor\u0000polynomials with rigorous but simple restrictions on points chosen) to buy out\u0000the regular treatment of math series to find the pairing polynomials instantly.\u0000The original bijective Cantor polynomial C1(x,y)= (x2+2xy+y2+3x+y)/2: N02 to N0\u0000(=positive integers) which is 2-fold and runs in zig - zag way along lines\u0000x+y=N is extended e.g. to the bijective P(x,y)=\u00002x2+4sgn(x)sgn(y)xy+2y2-2H(x)sgn(y)x-y+1: Z2 to N0 (with sign and Heaviside\u0000functions, Z is integers) running in spiral way along concentric rhombuses, or\u0000to the bijective P3D(x,y,z)= [x3+y3+z3 +3(xz2+yz2 +zx2+2xyz +zy2+yx2+xy2)\u0000+3(2x2+2y2 +z2+2xz +2yz+4xy) +5x+11y+2z]/6: N03 to N0 which is 6-fold and runs\u0000along plains x+y+z=N. Storage device for triangle matrices is also commented as\u0000cutting the original Cantor domain to half along with related Diophantine\u0000equations.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570922","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 divisibility restrictions in the famous equation a n+bn=cn in Fermat Last Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat triple (i.e. solutions) if n greater than 2, decreasing the cardinality of Fermat triples. In our analysis, the restriction on positive integer (PI) solutions ((a,b,c,n) up to the point when there is no more) is not along with restriction on power n in PI as decreasing sets {PI } containing {odd} containing {primes} containing {regular primes}, etc. as in the literature, but with respect to exclusion of more and more c in PI as increasing sets {primes p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation is analyzed in relation to exclusion of solutions, and the effect of simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on the decrease of cardinality of solutions is exhibited. Again, our derivation focuses mainly on the variable c rather than on variable n, oppositely to the literature in which the FLT is historically separated via the values of power n. Among the most famous are the known, about 2500 years old, existing Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases (of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat) less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion (a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over quaternions breaks.
本文分析了费马最后定理（FLT，1637 年）中著名方程 a n+bn=cn 的可分性限制，即如果 n 大于 2，它如何选择出许多三元组作为费马三元组（即解），从而降低了费马三元组的万有引力。在我们的分析中，对正整数（PI）解（（a,b,c,n）直到没有解为止）的限制并不是像文献中那样对 PI 中的幂 n 作为包含{odd}包含{primes}包含{regular primes}等的递减集 {PI } 的限制，而是对 PI 中越来越多的 c 作为 {PI} 中 {p k} 的递增集 {primesp} 的排除。我们分析了费马方程中的可分性和同素数性质与解的排除关系，并展示了 gcd（a,b,c）、gcd（a+b,cn）、gcd（c-a,bn）和 gcd（c-b,an）的同时取值对解的万有引力下降的影响。同样，我们的推导主要集中在变量 c 而不是变量 n 上，这与历史上通过幂的值来区分 FLT 的文献正好相反。其中最有名的是已知的、已有约 2500 年历史的毕达哥拉斯三段论（a,b,c,n=2），以及不到 400 年前的第一个里程碑式的证明案例（高斯证明了 n=3，后来欧拉（1753 年）证明了 n=4，费马证明了 n=4）。众所周知，怀尔斯在 1995 年以一种抽象的迂回方式证明了 FLT。本文对 n<0，n:=1/m，以及复数和四元数（a,b,c）的情况进行了评论，重点放在毕达哥拉斯上。奇数幂 FLT 超四元数断裂。
{"title":"On the Simple Divisibility Restrictions by Polynomial Equation a n+bn=cn Itself in Fermat Last Theorem for Integer/Complex/Quaternion Triples","authors":"Sandor Kristyan","doi":"arxiv-2407.05068","DOIUrl":"https://doi.org/arxiv-2407.05068","url":null,"abstract":"The divisibility restrictions in the famous equation a n+bn=cn in Fermat Last\u0000Theorem (FLT, 1637) is analyzed how it selects out many triples to be Fermat\u0000triple (i.e. solutions) if n greater than 2, decreasing the cardinality of\u0000Fermat triples. In our analysis, the restriction on positive integer (PI)\u0000solutions ((a,b,c,n) up to the point when there is no more) is not along with\u0000restriction on power n in PI as decreasing sets {PI } containing {odd}\u0000containing {primes} containing {regular primes}, etc. as in the literature, but\u0000with respect to exclusion of more and more c in PI as increasing sets {primes\u0000p} in {p k} in {PI}. The divisibility and co-prime property in Fermat equation\u0000is analyzed in relation to exclusion of solutions, and the effect of\u0000simultaneous values of gcd(a,b,c), gcd(a+b,cn), gcd(c-a,bn) and gcd(c-b,an) on\u0000the decrease of cardinality of solutions is exhibited. Again, our derivation\u0000focuses mainly on the variable c rather than on variable n, oppositely to the\u0000literature in which the FLT is historically separated via the values of power\u0000n. Among the most famous are the known, about 2500 years old, existing\u0000Pythagorean triples (a,b,c,n=2) and the first milestones as the proved cases\u0000(of non-existence as n=3 by Gauss and later by Euler (1753) and n=4 by Fermat)\u0000less than 400 years ago. As it is known, Wiles has proved the FLT in 1995 in an\u0000abstract roundabout way. The n<0, n:=1/m, as well as complex and quaternion\u0000(a,b,c) cases focusing on Pythagoreans are commented. Odd powers FLT over\u0000quaternions breaks.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570921","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 proposes a graph linear canonical transform (GLCT) by decomposing the linear canonical parameter matrix into fractional Fourier transform, scale transform, and chirp modulation for graph signal processing. The GLCT enables adjustable smoothing modes, enhancing alignment with graph signals. Leveraging traditional fractional domain time-frequency analysis, we investigate vertex-frequency analysis in the graph linear canonical domain, aiming to overcome limitations in capturing local information. Filter design methods, including optimal design and learning with stochastic gradient descent, are analyzed and applied to image classification tasks. The proposed GLCT and vertex-frequency analysis present innovative approaches to signal processing challenges, with potential applications in various fields.
{"title":"Graph Linear Canonical Transform: Definition, Vertex-Frequency Analysis and Filter Design","authors":"Jian Yi Chen, Bing Zhao Li","doi":"arxiv-2407.12046","DOIUrl":"https://doi.org/arxiv-2407.12046","url":null,"abstract":"This paper proposes a graph linear canonical transform (GLCT) by decomposing\u0000the linear canonical parameter matrix into fractional Fourier transform, scale\u0000transform, and chirp modulation for graph signal processing. The GLCT enables\u0000adjustable smoothing modes, enhancing alignment with graph signals. Leveraging\u0000traditional fractional domain time-frequency analysis, we investigate\u0000vertex-frequency analysis in the graph linear canonical domain, aiming to\u0000overcome limitations in capturing local information. Filter design methods,\u0000including optimal design and learning with stochastic gradient descent, are\u0000analyzed and applied to image classification tasks. The proposed GLCT and\u0000vertex-frequency analysis present innovative approaches to signal processing\u0000challenges, with potential applications in various fields.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745228","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}
An approximate formula for complex Riemann Xi function, previously developed, is used to refine Backlund's estimate of the number of zeros till a chosen imaginary coordinate
利用之前开发的复黎曼奚函数近似公式，可以完善巴克伦德对所选虚数坐标零点个数的估计值
{"title":"About zero counting of Riemann Z function","authors":"Giovanni Lodone","doi":"arxiv-2407.07910","DOIUrl":"https://doi.org/arxiv-2407.07910","url":null,"abstract":"An approximate formula for complex Riemann Xi function, previously developed,\u0000is used to refine Backlund's estimate of the number of zeros till a chosen\u0000imaginary coordinate","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611886","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}
Two sums over the inverse of the product of an integer n and its greatest prime factor G(n), are computed to first 13 decimal digits. These sums converge, but converge very slowly. They are transformed into sums involving Mertens' prime product with the remainder term which are estimated by means of Chebyshev's {theta}-function.
对整数 n 及其最大质因数 G(n) 的乘积的倒数的两个和计算到小数点后 13 位。这些和收敛了，但收敛得很慢。它们被转化为涉及梅尔腾斯素数乘积与余项的和，这些和是通过切比雪夫的{theta}函数估算出来的。
{"title":"Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor","authors":"Tengiz O. Gogoberidze","doi":"arxiv-2407.12047","DOIUrl":"https://doi.org/arxiv-2407.12047","url":null,"abstract":"Two sums over the inverse of the product of an integer n and its greatest\u0000prime factor G(n), are computed to first 13 decimal digits. These sums\u0000converge, but converge very slowly. They are transformed into sums involving\u0000Mertens' prime product with the remainder term which are estimated by means of\u0000Chebyshev's {theta}-function.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745230","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 give for the first time a systematic study of the variance of the distance to the boundary for arbitrary bounded convex domains in $mathbb{R}^2$ and $mathbb{R}^3$. In dimension two, we show that this function is strictly convex, which leads to a new notion of the centre of such a domain, called the variocentre. In dimension three, we investigate the relationship between the variance and the distance to the boundary, which mathematically justifies claims made for a recently developed algorithm for classifying interior and exterior points with applications in biology.
{"title":"Variance of the distance to the boundary of convex domains in $mathbb{R}^{2}$ and $mathbb{R}^{3}$","authors":"Alastair N. Fletcher, Alexander G. Fletcher","doi":"arxiv-2407.12041","DOIUrl":"https://doi.org/arxiv-2407.12041","url":null,"abstract":"In this paper, we give for the first time a systematic study of the variance\u0000of the distance to the boundary for arbitrary bounded convex domains in\u0000$mathbb{R}^2$ and $mathbb{R}^3$. In dimension two, we show that this function\u0000is strictly convex, which leads to a new notion of the centre of such a domain,\u0000called the variocentre. In dimension three, we investigate the relationship\u0000between the variance and the distance to the boundary, which mathematically\u0000justifies claims made for a recently developed algorithm for classifying\u0000interior and exterior points with applications in biology.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745229","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}
Mordell equations are celebrated equations within number theory and are named after Louis Mordell, an American-born British mathematician, known for his pioneering research in number theory. In this paper, we discover all Mordell equations of the form $y^2 = x^3 + k$, where $k in mathbb Z$, with exactly $|k|$ integral solutions. We also discover explicit bounds for Mordell equations, parameterized families of elliptic curves and twists on elliptic curves. Using the connection between Mordell curves and binary cubic forms, we improve the lower bound for the number of integral solutions of a Mordell curve by looking at a pair of curves with unusually high rank.
{"title":"On Bounds and Diophantine Properties of Elliptic Curves","authors":"Navvye Anand","doi":"arxiv-2407.09558","DOIUrl":"https://doi.org/arxiv-2407.09558","url":null,"abstract":"Mordell equations are celebrated equations within number theory and are named\u0000after Louis Mordell, an American-born British mathematician, known for his\u0000pioneering research in number theory. In this paper, we discover all Mordell\u0000equations of the form $y^2 = x^3 + k$, where $k in mathbb Z$, with exactly\u0000$|k|$ integral solutions. We also discover explicit bounds for Mordell\u0000equations, parameterized families of elliptic curves and twists on elliptic\u0000curves. Using the connection between Mordell curves and binary cubic forms, we\u0000improve the lower bound for the number of integral solutions of a Mordell curve\u0000by looking at a pair of curves with unusually high rank.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722102","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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