The celebrated $3x+1$ problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula $f(x)+1$ with a unique fixed point. The resultant formula $f(x)$ is also found to coincide with that of the discrete derivative of the sorted sequence of fixed points of the reflection operator on even binary palindromes of fixed even length textit{2k} in any interval $[0cdots2^{2k}-1]$. A set of equivalent reformulations of the problem are also presented.
{"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}
The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $pi(x)$ having a general expression 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 $pi(x)$ denotes the textit{Prime Counting Function}. The proofs require specific order estimates involving $pi(x)$ and the textit{Second Chebyshev Function} $psi(x)$, as well as the famous textit{Prime Number Theorem} in addition to certain meromorphic properties of the textit{Riemann Zeta Function} $zeta(s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.
{"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}
Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any GCD calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [4].
我们将半原始根定义为原始根的平方。我们提出了一种从二次残差（包括半原始根）计算原始根的方法。然后，我们提出了生成初等根和半初等根的级数，并推导出无需任何 GCD 计算即可获得全套初等根的类似算法。接下来，我们提出了一种确定具有任意大的素数猜想渐近密度的不可还原二次型的方法（after Shanks, [1][2]）。为此，我们基于托内利-香克斯算法[4]，提出了一种计算 p 的平方根模的算法。
{"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}
Our purpose of this paper is to focus on fixed point property in fuzzy metric space. To achieve our objective, we will introduce a new contraction condition to examine the fixed point for multi-valued mapping, then we will be investigating the obtained result to ensure the existence and uniqueness of this property for single-valued mapping. To show the use of our main result, we will give the relative result in the ordinary metric space.
{"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}
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker than ZF, that is finitely axiomatized, and that does not have a countable model (if it has a model at all, that is). Here we prove that T is relatively consistent with ZF. We conclude that this is an important step towards showing that T is an advancement in the foundations of mathematics.
最近，在《公理 10(2)：119 (2021)一文中，介绍了一个关于集合与函数的非经典一阶理论T，如果我们想要一个不弱于ZF、有限公理化、没有可解释模型（如果它有模型的话）的（所有）数学基础理论，那么T就是我们必须接受的公理集合。在此，我们证明 T 与 ZF 相对一致。我们的结论是，这是朝着证明 T 是数学基础的进步迈出的重要一步。
{"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}
We investigate the convexity property on $(0,1)$ of the function $$f_a(x)=frac{{cal K}{(sqrt x)}}{a-(1/2)log(1-x)}.$$ We show that $f_a$ is strictly convex on $(0,1)$ if and only if $ageq a_c$ and $1/f_a$ is strictly convex on $(0,1)$ if and only if $aleqlog 4$, where $a_c$ is some critical value. The second main result of the paper is to study the log-convexity and log-concavity of the function $$h_p(x)=(1-x)^p{cal K}(sqrt x).$$ We prove that $h_p$ is strictly log-concave on $(0,1)$ if and only if $pgeq 7/32$ and strictly log-convex if and only if $pleq 0$. This solves some problems posed by Yang and Tian and complete their result and a result of Alzer and Richards that $f_a$ is strictly concave on $(0,1)$ if and only if $a=4/3$ and $1/f_a$ is strictly concave on $(0,1)$ if and only if $ageq 8/5$. As applications of the convexity and concavity, we establish among other inequalities, that for $ageq a_c$ and all $rin(0,1)$ $$frac{2pisqrtpi}{(2a+log 2)Gamma(3/4)^2}leq frac{{cal K}(sqrt r)}{a-frac12log (r)}+frac{{cal K}(sqrt{1-r})}{a-frac12log (1-r)}<1+fracpi{2a},$$ and for $pgeq 3(2+sqrt 2)/8$ and all $rin(0,1)$ $$sqrt{(r-r^2)^p{cal K}(sqrt{1-r}){cal K}(sqrt r)}< frac{pisqrtpi}{2^{p+1}Gamma(3/4)^2}
{"title":"Convexity and concavity of a class of functions related to the elliptic functions","authors":"Mohamed Bouali","doi":"arxiv-2407.14547","DOIUrl":"https://doi.org/arxiv-2407.14547","url":null,"abstract":"We investigate the convexity property on $(0,1)$ of the function\u0000$$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}
Differential equations have void applications in several practical situations, sciences, and non sciences as Euler Lagrange equation in classical mechanics, Radioactive decay in nuclear physics, Navier Stokes equations in fluid dynamics, Verhulst equation in biological population growth, Hodgkin Huxley model in neural action potentials, etc. The cantilever bridge problem is very important in Bridge Engineering and this can be modeled as a homogeneous obstacle problem in Mathematics. Due to this and various other applications, obstacle problems become an important part of our literature. A lot of work is dedicated to the solution of the obstacle problems. However, obstacle problems are not solved by the considered method in the literature we have visited. In this work, we have investigated the finding of the exact solution to several obstacle problems using the optimal homotopy asymptotic method (OHAM). The graphical representation of results represents the symmetry among them.
{"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}
In this paper we give an elementary proof for Bertrand's postulate also known as Bertrand-Chebyshev theorem.
本文给出了贝特朗公设（又称贝特朗-切比雪夫定理）的基本证明。
{"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}
The purpose of this study is to introduce a new class of regular spaces called e*$theta$-regular spaces which is a generalization of the class of $betatheta$-regular spaces. Also, we investigate some basic properties and several characterizations of e*$theta$-regular and e*$theta$-normal spaces. Moreover, some functions such as e*$theta$-closed function, generalized e*$theta$-closed function, generalized e*$theta$-closed function have been defined and studied. Furthermore, we obtain some preservation theorems.
{"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}
This research article provides an unconditional proof of an inequality proposed by textit{Srinivasa Ramanujan} involving the Prime Counting Function $pi(x)$, begin{align*} (pi(x))^{2}
{"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}
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