综合数学
Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of…
The classical Vieta formulas relate the coefficients of a complex scalar polynomial to the elementary symmetric polynomials of its roots. In this paper, we establish analogous spectral identities for complex matrix polynomials. For a monic…
For an odd perfect number $N$, write $q=\min\{p:p\mid N\}$ for its smallest prime divisor. This paper proves a certified branch-closure theorem for the five minimal-prime branches $q=5,7,11,13,17$. The proof combines the exact $q$-adic…
In this paper, picture fuzzy multisets were studied together with their associated properties. We also introduced the concept of picture fuzzy multigroups and established some of their algebraic properties.
Given two sequences $\phi=(\phi_i)_{i\ge 1}$ and $\psi=(\psi_i)_{i\ge 1}$ and numbers $a,b,c$, we introduce the GKP sequence of polynomials $(p_n)_n$ using the following recurrence formula: $p_0 = 1$ and for $n\ge 1$ \[ p_{n}(x) =…
We give an exact, checkable rank-certificate method for realized planar unit-distance frameworks. The method is motivated by Vogel's computations for matchstick graphs and by the insertion-edge tests used in the Matchstick Graphs…
This paper develops a graph fractional uncertainty principle in the graph fractional Fourier transform (GFRFT) domain. We introduce localization operators in the vertex domain and the graph fractional spectral domain, and build an operator…
This article deals with combinatorial identities with two complex parameters. Starting with a fundamental lemma, we derive various polynomial identities, combinatorial sums and related results. For example, we generalize a polynomial…
In this paper, we study the properties of linear second-order recurrences whose coefficients are non-integer rational numbers. We examine the set of indices at which the terms in the sequence are integers. Specifically, we prove two…
Based on the growth patterns of 166 CHO monoclones observed over a 15 day period, we show that the standard population growth in a confined space equation, i.e. the sigmoid/logistic function, is alone does not capture the complex behaviour…
In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in…
The sequence of Mersenne numbers $\{M_n\}_{n\geq 0}$ is defined as $M_n = 2^n-1.$ In this study we introduce the Mersenne-Bernoulli and Mersenne-Euler polynomials. Using the generating functions and $M$-calculus we find some identities…
In this work, we study a normalized remainder $T_{n,\lambda}[\e_\lambda]$ for the degenerate exponential $\e_\lambda(u)=(1+\lambda u)^{1/\lambda}$ ($\lambda>0$). We establish an integral representation, an exact monotonicity threshold at…
We classify a reciprocal degree-five quadrinomial family over the quadratic extension F_{q^2}, where q is an odd prime power. The family has four terms, coefficients in F_q, and a coefficient constraint that makes the induced rational…
We introduce a new mathematical constant $\lambda_\infty = 0.674036183193696139936660007576508455780\ldots$ (OEIS A396695), defined as the unique solution in $(1/4,+\infty)$ of $h(x) := \sum_{p \text{ prime}} 1/(xp^2-p+1) = 1$. This…
This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the…
We calculate the first derivative of the Jacobi polynomials with respect to their order in explicit form. This derivative is not an elementary function, but contains elementary special cases. As an application, we use our result with a…
The ranking of fuzzy numbers has become a challenging task in fuzzy set theory due to their complex, multi-dimensional nature. While the Klir-Yuan partial order provides a natural term-wise comparison of $\alpha$-cuts, it often leaves many…
We describe an artistic project consisting of fabricating the 3532 different soccer balls that can be obtained by randomly assembling the 32 pieces of a classic Telstar soccer ball.
In this paper, we introduce the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$. By employing the $M$-calculus, based on the Mersenne numbers, we establish explicit series representations, addition theorems,…