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On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations 偶数二进制 palindromic 词与科拉茨-海尔斯通迭代之间的密切联系
arXiv - MATH - General Mathematics Pub Date : 2024-07-29 DOI: arxiv-2408.00805
T. Raptis
{"title":"On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations","authors":"T. Raptis","doi":"arxiv-2408.00805","DOIUrl":"https://doi.org/arxiv-2408.00805","url":null,"abstract":"The celebrated $3x+1$ problem is reformulated via the use of an analytic\u0000expression of the trailing zeros sequence resulting in a single branch formula\u0000$f(x)+1$ with a unique fixed point. The resultant formula $f(x)$ is also found\u0000to coincide with that of the discrete derivative of the sorted sequence of\u0000fixed points of the reflection operator on even binary palindromes of fixed\u0000even length textit{2k} in any interval $[0cdots2^{2k}-1]$. A set of\u0000equivalent reformulations of the problem are also presented.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948147","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}
引用次数: 0
Inequalities involving Higher Degree Polynomial Functions in $π(x)$ 涉及 $π(x)$ 中高阶多项式函数的不等式
arXiv - MATH - General Mathematics Pub Date : 2024-07-24 DOI: arxiv-2407.18983
Subham De
{"title":"Inequalities involving Higher Degree Polynomial Functions in $π(x)$","authors":"Subham De","doi":"arxiv-2407.18983","DOIUrl":"https://doi.org/arxiv-2407.18983","url":null,"abstract":"The primary purpose of this article is to study the asymptotic and numerical\u0000estimates in detail for higher degree polynomials in $pi(x)$ having a general\u0000expression of the form, begin{align*} P(pi(x)) - frac{e x}{log x} Q(pi(x/e)) + R(x) end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $pi(x)$\u0000denotes the textit{Prime Counting Function}. The proofs require specific order\u0000estimates involving $pi(x)$ and the textit{Second Chebyshev Function}\u0000$psi(x)$, as well as the famous textit{Prime Number Theorem} in addition to\u0000certain meromorphic properties of the textit{Riemann Zeta Function} $zeta(s)$\u0000and results regarding its non-trivial zeros. A few generalizations of these\u0000concepts have also been discussed in detail towards the later stages of the\u0000paper, along with citing some important applications.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867291","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}
引用次数: 0
Semi-primitive roots and irreducible quadratic forms 半原始根与不可还原二次函数形式
arXiv - MATH - General Mathematics Pub Date : 2024-07-24 DOI: arxiv-2407.20269
Marc Wolf, François Wolf
{"title":"Semi-primitive roots and irreducible quadratic forms","authors":"Marc Wolf, François Wolf","doi":"arxiv-2407.20269","DOIUrl":"https://doi.org/arxiv-2407.20269","url":null,"abstract":"Modulo a prime number, we define semi-primitive roots as the square of\u0000primitive roots. We present a method for calculating primitive roots from\u0000quadratic residues, including semi-primitive roots. We then present\u0000progressions that generate primitive and semi-primitive roots, and deduce an\u0000algorithm to obtain the full set of primitive roots without any GCD\u0000calculation. Next, we present a method for determining irreducible quadratic\u0000forms with arbitrarily large conjectured asymptotic density of primes (after\u0000Shanks, [1][2]). To this end, we propose an algorithm for calculating the\u0000square root modulo p, based on the Tonelli-Shanks algorithm [4].","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867290","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}
引用次数: 0
Fixed Point Property in G-Complete Fuzzy Metric Space G 完全模糊度量空间中的定点特性
arXiv - MATH - General Mathematics Pub Date : 2024-07-21 DOI: arxiv-2407.15271
Ismail Tahiri, Ahmed Nuino
{"title":"Fixed Point Property in G-Complete Fuzzy Metric Space","authors":"Ismail Tahiri, Ahmed Nuino","doi":"arxiv-2407.15271","DOIUrl":"https://doi.org/arxiv-2407.15271","url":null,"abstract":"Our purpose of this paper is to focus on fixed point property in fuzzy metric\u0000space. To achieve our objective, we will introduce a new contraction condition\u0000to examine the fixed point for multi-valued mapping, then we will be\u0000investigating the obtained result to ensure the existence and uniqueness of\u0000this property for single-valued mapping. To show the use of our main result, we\u0000will give the relative result in the ordinary metric space.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771302","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}
引用次数: 0
Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF 包含 ZF 的有限非经典理论与包含 ZF 的范畴理论的相对一致性
arXiv - MATH - General Mathematics Pub Date : 2024-07-21 DOI: arxiv-2407.18969
Marcoen J. T. F. Cabbolet, Adrian R. D. Mathias
{"title":"Relative consistency of a finite nonclassical theory incorporating ZF and category theory with ZF","authors":"Marcoen J. T. F. Cabbolet, Adrian R. D. Mathias","doi":"arxiv-2407.18969","DOIUrl":"https://doi.org/arxiv-2407.18969","url":null,"abstract":"Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of\u0000sets and functions has been introduced as the collection of axioms we have to\u0000accept if we want a foundational theory for (all of) mathematics that is not\u0000weaker than ZF, that is finitely axiomatized, and that does not have a\u0000countable model (if it has a model at all, that is). Here we prove that T is\u0000relatively consistent with ZF. We conclude that this is an important step\u0000towards showing that T is an advancement in the foundations of mathematics.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867295","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}
引用次数: 0
Convexity and concavity of a class of functions related to the elliptic functions 与椭圆函数有关的一类函数的凸性和凹性
arXiv - MATH - General Mathematics Pub Date : 2024-07-16 DOI: arxiv-2407.14547
Mohamed Bouali
{"title":"Convexity and concavity of a class of functions related to the elliptic functions","authors":"Mohamed Bouali","doi":"arxiv-2407.14547","DOIUrl":"https://doi.org/arxiv-2407.14547","url":null,"abstract":"We investigate the convexity property on $(0,1)$ of the function\u0000$$f_a(x)=frac{{cal K}{(sqrt x)}}{a-(1/2)log(1-x)}.$$ We show that $f_a$ is\u0000strictly convex on $(0,1)$ if and only if $ageq a_c$ and $1/f_a$ is strictly\u0000convex on $(0,1)$ if and only if $aleqlog 4$, where $a_c$ is some critical\u0000value. The second main result of the paper is to study the log-convexity and\u0000log-concavity of the function $$h_p(x)=(1-x)^p{cal K}(sqrt x).$$ We prove\u0000that $h_p$ is strictly log-concave on $(0,1)$ if and only if $pgeq 7/32$ and\u0000strictly log-convex if and only if $pleq 0$. This solves some problems posed\u0000by Yang and Tian and complete their result and a result of Alzer and Richards\u0000that $f_a$ is strictly concave on $(0,1)$ if and only if $a=4/3$ and $1/f_a$ is\u0000strictly concave on $(0,1)$ if and only if $ageq 8/5$. As applications of the\u0000convexity and concavity, we establish among other inequalities, that for $ageq\u0000a_c$ and all $rin(0,1)$ $$frac{2pisqrtpi}{(2a+log 2)Gamma(3/4)^2}leq\u0000frac{{cal K}(sqrt r)}{a-frac12log (r)}+frac{{cal\u0000K}(sqrt{1-r})}{a-frac12log (1-r)}<1+fracpi{2a},$$ and for $pgeq 3(2+sqrt\u00002)/8$ and all $rin(0,1)$ $$sqrt{(r-r^2)^p{cal K}(sqrt{1-r}){cal K}(sqrt\u0000r)}< frac{pisqrtpi}{2^{p+1}Gamma(3/4)^2}<frac{r^p{cal\u0000K}(sqrt{1-r})+(1-r)^p{cal K}(sqrt r)}{2}.$$","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141771303","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}
引用次数: 0
Solving Obstacle Problems using Optimal Homotopy Asymptotic Method 用最优同调渐近法解决障碍问题
arXiv - MATH - General Mathematics Pub Date : 2024-07-13 DOI: arxiv-2407.09863
Muhammad Amjad, Haider Ali
{"title":"Solving Obstacle Problems using Optimal Homotopy Asymptotic Method","authors":"Muhammad Amjad, Haider Ali","doi":"arxiv-2407.09863","DOIUrl":"https://doi.org/arxiv-2407.09863","url":null,"abstract":"Differential equations have void applications in several practical\u0000situations, sciences, and non sciences as Euler Lagrange equation in classical\u0000mechanics, Radioactive decay in nuclear physics, Navier Stokes equations in\u0000fluid dynamics, Verhulst equation in biological population growth, Hodgkin\u0000Huxley model in neural action potentials, etc. The cantilever bridge problem is\u0000very important in Bridge Engineering and this can be modeled as a homogeneous\u0000obstacle problem in Mathematics. Due to this and various other applications,\u0000obstacle problems become an important part of our literature. A lot of work is\u0000dedicated to the solution of the obstacle problems. However, obstacle problems\u0000are not solved by the considered method in the literature we have visited. In\u0000this work, we have investigated the finding of the exact solution to several\u0000obstacle problems using the optimal homotopy asymptotic method (OHAM). The\u0000graphical representation of results represents the symmetry among them.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722215","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}
引用次数: 0
An Elementary proof for Bertrand's Postulate 贝特朗公设的基本证明
arXiv - MATH - General Mathematics Pub Date : 2024-07-10 DOI: arxiv-2407.07620
Pranav Narayan Sharma
{"title":"An Elementary proof for Bertrand's Postulate","authors":"Pranav Narayan Sharma","doi":"arxiv-2407.07620","DOIUrl":"https://doi.org/arxiv-2407.07620","url":null,"abstract":"In this paper we give an elementary proof for Bertrand's postulate also known\u0000as Bertrand-Chebyshev theorem.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584704","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}
引用次数: 0
On e*$θ$-regular and e*$θ$-normal Spaces 论 e*$θ$ 不规则空间和 e*$θ$ 正则空间
arXiv - MATH - General Mathematics Pub Date : 2024-07-09 DOI: arxiv-2407.07927
Burcu Sünbül Ayhan
{"title":"On e*$θ$-regular and e*$θ$-normal Spaces","authors":"Burcu Sünbül Ayhan","doi":"arxiv-2407.07927","DOIUrl":"https://doi.org/arxiv-2407.07927","url":null,"abstract":"The purpose of this study is to introduce a new class of regular spaces\u0000called e*$theta$-regular spaces which is a generalization of the class of\u0000$betatheta$-regular spaces. Also, we investigate some basic properties and\u0000several characterizations of e*$theta$-regular and e*$theta$-normal spaces.\u0000Moreover, some functions such as e*$theta$-closed function, generalized\u0000e*$theta$-closed function, generalized e*$theta$-closed function have been\u0000defined and studied. Furthermore, we obtain some preservation theorems.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611883","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}
引用次数: 0
On proving an Inequality of Ramanujan using Explicit Order Estimates of the Mertens Function 论利用梅腾斯函数的显式阶次估计值证明拉马努扬不等式
arXiv - MATH - General Mathematics Pub Date : 2024-07-08 DOI: arxiv-2407.12052
Subham De
{"title":"On proving an Inequality of Ramanujan using Explicit Order Estimates of the Mertens Function","authors":"Subham De","doi":"arxiv-2407.12052","DOIUrl":"https://doi.org/arxiv-2407.12052","url":null,"abstract":"This research article provides an unconditional proof of an inequality\u0000proposed by textit{Srinivasa Ramanujan} involving the Prime Counting Function\u0000$pi(x)$, begin{align*} (pi(x))^{2}<frac{ex}{log\u0000x}pileft(frac{x}{e}right) end{align*} for every real $xgeq exp(1486)$,\u0000using specific order estimates of the textit{Mertens Function}, $M(x)$. The\u0000proof primarily hinges upon investigating the underlying relation between\u0000$M(x)$ and the textit{Second Chebyshev Function}, $psi(x)$, in addition to\u0000applying the meromorphic properties of the textit{Riemann Zeta Function},\u0000$zeta(s)$ with an intention of deriving an improved approximation for\u0000$pi(x)$.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"172 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745226","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}
引用次数: 0
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