综合数学
In this paper, we first prove that given a nonnegative integer $m$ and an odd number $t$ not divisible by $3$, there exists a unique Collatz's Sequence \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \]…
The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…
We introduce an explicit logarithmic transformation $T(x) = \{\log_6(x + 1/5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle \(\alpha = \log_6 3\), perturbed by a uniformly bounded error term. We…
We consider the even monic degree-$10$ second cuboid polynomial $Q_{p,q}(t)\in\mathbb{Z}[t]$ depending on coprime integers $p\neq q>0$. We exclude the existence of a splitting of type $5+5$ over $\mathbb{Q}$, i.e., a factorization of…
We obtain an orthogonality space by endowing an implicative-ortholattice with a suitable orthogonality relation; for such spaces, we also investigate the particular case of implicative-orthomodular lattices. Moreover, we define the…
This paper is purely expository. We present short elementary proofs of * the Gauss Theorem on constructibility of regular polygons; * the existence of a cubic equation unsolvable in real radicals; * the existence of a quintic equation…
The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller…
We study the modular resolution method using new tools called polynumbers and polyseries, introduced by Prof. Wildberger N.J. We try to prove an equivalence theorem of the existence and the uniqueness of the solutions of the modular…
Let $q=p^k$ be a prime power, let $\mathbb{F}_q$ be a finite field and let $n\geq2$ be an integer. This note investigates the existence small primitive normal elements in finite field extensions $\mathbb{F}_{q^n}$. It is shown that a small…
Euler totient function $\phi(n)$ plays a central role in number theory and is applied in areas such as cryptography. In this paper, we study iterations of the totient function. We first prove that for any integer $n>2$, iteratively applying…
This paper presents a comprehensive analysis of the driven cubic-quintic Duffing oscillator \[ \ddot{\phi}+\frac{1}{q}\dot{\phi}+\phi^3+\phi^5=A\cos(\omega t), \] advancing both analytical and numerical chaos theory. Using Melnikov analysis…
The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity…
In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(\pi…
In this article, I present a theorem determining a criterion for divisibility of two generalized Mersenne numbers, which are repunits of the same length in base-$a^m$ and base-$a^k$. In addition to the general proof, I present an…
In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic…
In this paper, we introduce and study the quadratic-phase Dunkl transform, a novel integral transform on the real line parameterized by five real numbers $(a, b, c, d, e)$ and a multiplicity parameter $\mu\geq -1/2$. We define the transform…
In this article, we first define the concept of ordered intervals, then introduce ordered fuzzy inner product and describe some of its properties.
In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we…
This study investigates the differential geometric properties of Smarandache curves derived from null curves defined in the ligtlike cone space $% Q_{3}^{2}\subset E_{2}^{4}$. The indefinite metric structure causes the null vectors, and…
This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0…