CyberSec Research

Let $S$ be a semigroup, $\mu$ a discrete measure on $S$ and $\sigma:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(x\sigma(y)t)d\mu(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(x\sigma(y)t)d\mu(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)\chi(y)+f(y)\chi(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(x\sigma(y)t)d\mu(t)=[f(x)\chi(y)+f(y)\chi(x)]\int_{S}\chi(t)d\mu(t), $ $x,y\in S$, and where $\chi$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.

Our work began as an effort to understand calculations by Morris & Szekeres (1961) and Walker (1991) regarding fractional iteration.

We consider the perturbed Mann's iterative process \begin{equation} x_{n+1}=(1-\theta_n)x_n+\theta_n f(x_n)+r_n, \end{equation} where $f:[0,1]\rightarrow[0,1]$ is a continuous function, $\{\theta_n\}\in [0,1]$ is a given sequence, and $\{r_n\}$ is the error term. We establish that if the sequence $\{\theta_n\}$ converges relatively slowly to $0$ and the error term $r_n$ becomes enough small at infinity, any sequences $\{x_n\}\in [0,1]$ satisfying the process converges to a fixed point of the function $f$. We also study the asymptotic behavior of the trajectories $x(t)$ as $t\rightarrow\infty$ of a continuous version of the the considered. We investigate the similarities between the asymptotic behaviours of the sequences generated by the considered discrete process and the trajectories $x(t)$ of its corresponding continuous version.

The Kolakoski sequence K(1,3) over {1, 3} is known to be structured, unlike K(1,2), with symbol frequency d approx. 0.397 linked to the Pisot number alpha (real root of x^3 - 2x^2 - 1 = 0). We reveal an explicit nested recursion defining block sequences B(n) and pillar sequences P(n) via B(n+1) = B(n) P(n) B(n) and P(n+1) = G(R(P(n)), 3), where G generates runs from vector R(P(n)). We prove B(n) are prefixes of K(1,3) converging to it, and B(n+1) = G(R(B(n)), 1), directly reflecting the Kolakoski self-encoding property. We derive recurrences for lengths |B(n)|, |P(n)| and symbol counts, confirming growth governed by alpha (limit |B(n+1)|/|B(n)| = alpha as n -> infinity). If block/pillar densities converge, they must equal d. This constructive framework provides an alternative perspective on K(1,3)'s regularity, consistent with known results from substitution dynamics.

One-dimensional and two-dimensional integrals containing $E_b(-u)$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ are considered. $E_b(-u)$ is the Mittag-Leffler function and the integral is taken over the rectangle $0 \leq x < \infty, 0 \leq u < \infty$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ is the generalized Mittag-Leffler function and the integral is over $0\leq x \leq b$ with infinite intervals explored. A representation in terms of the Hurwitz-Lerch zeta function and other special functions are derived for the double and single integrals, from which special cases can be evaluated in terms of special function and fundamental constants.

In this paper, we introduce algebraic theories such as set theory and group theory into the analysis of event execution order. We propose concepts like "optional intervals event" and "sequential operation", summarize their algebraic properties and draw Cayley tables. Based on these efforts, we offer new interpretations for certain physical phenomena and computer application scenarios. Finally, we present other issues derived from this paradigm. These concepts can deepen our understanding of motion and find applications in areas such as event arrangement, physical simulation, and computer modeling

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 nonstructured subset $\mathcal{A}\subset \mathbb{F}_{q^n}$ of cardinality $\#\mathcal{A}\gg q^{\varepsilon}$, where $\varepsilon>0$ is a small number, contains a primitive normal element.

A solitary number is a positive integer that shares its abundancy index only with itself. $10$ is the smallest positive integer suspected to be solitary, but no proof has been established so far. In this paper, we prove that not all half of the exponents of the prime divisors of a friend of 10 are congruent to $1$ modulo $3$. Furthermore, we prove that if $F=5^{2a}\cdot Q^2$ ($Q$ is an odd positive integer coprime to $15$) is a friend of $10$, then $\sigma(5^{2a})+\sigma(Q^2)$ is congruent to $6$ modulo $8$ if and only if $a$ is even, and $\sigma(5^{2a}) + \sigma(Q^2)$ is congruent to $2$ modulo $8$ if and only if $a$ is odd. In addition, if we set $Q={\displaystyle \prod_{i=2}^{\omega(F)}}p_{i}^{a_i}$ and $a=a_1$, where $p_i$ are prime numbers, then we establish that $$F>\frac{25}{81}\cdot\prod_{i=1}^{\omega(F)}(2a_i + 1)^2,$$ in particular $F> 625\cdot 9^{\omega(F)-3}.$

Vector similarity measures play a fundamental role in various fields, including machine learning, natural language processing, information retrieval, and data mining. These measures quantify the closeness between two vectors in a high-dimensional space and are vital for tasks such as document similarity, recommendation systems, and clustering. While several vector similarity measures exist, each similarity measure is suited to specific purposes. In addition, some of these measures lack robustness to certain data types and/or probability distributions. This article presents a measure of vector similarity, known as the Joint Distance Measure (JDM). The JDM combines the Minkowski distance measure and the cosine similarity measure to establish a distance measure that captures the advantages of both measures.

We consider a particular generalized Lambert function, $y(x)$, defined by the implicit equation $y^\beta = 1 - e^{-xy}$, with $x>0$ and $ \beta > 1$. Solutions to this equation can be found in terms of a certain continued exponential. Asymptotic and structural properties of a non-trivial solution, $y_\beta(x)$, and its connection to the extinction probability of related branching processes are discussed. We demonstrate that this function constitutes a cumulative distribution function of a previously unknown non-negative absolutely continuous random variable.

The eigenvalues of companion matrices associated with generalized Lucas sequences, denoted as $\mathcal{L}$, exhibit a striking geometric resemblance to the Mandelbrot set $\mathcal{M}$. This work investigates this connection by analyzing the statistical distribution of eigenvalues and constructing a variety of homotopies that map different regions of $\mathcal{L}$ to structurally corresponding subsets of $\mathcal{M}$. In particular, we explore both global and piecewise homotopies, including a sinusoidal interpolation targeting the main cardioid and localized deformations aligned with the periodic bulbs. We also study a variation of the Jungreis map to better capture angular and radial structures. In addition to visual and geometric matching, we classify the eigenvalues according to their dynamical behavior, identifying subsets associated with hyperbolic, parabolic, Misurewicz, and Siegel disk points. Our findings suggest that meaningful correspondences between $\mathcal{L}$ and $\mathcal{M}$ must integrate both geometric deformation and dynamical classification. In light of these observations, we also suggest a conjectural homeomorphism between $\mathcal{L}$ and a dense subset of the Mandelbrot cardioid boundary, based on the behavior of the sinusoidal homotopy and the eigenvalue accumulation.

A brief history and two formulations of the Diophantine problem's requirements are presented. One tier consisting of three two-parameter solutions is studied for its ability to provide examples for the small natural numbers considered. Nested within it is a second tier consisting of five shifted-square solutions of the form $u^2+c$, where $u,c \in Q$. All told, they provide numerical examples for all but two $a \in N[1000]$, the set of natural numbers less than or equal to $1000$. A few open questions remain. Does this scheme of solutions cover every $a \in N[1000]$? If so, might they account for all $a \in N$? Are the three $tier_1$ solutions redundant with respect to the $a's$ they provide? Do other $tier_1$ and shifted-square $tier_2$ solutions exist?

In this short note, an example of a semifield of order 128 containing the Galois field $\mathbb{F}_8$ is given. Up to our knowledge, this is the first example supporting the following problem by Cordero and Chen (2013): ``There exist semifield planes of order $2^t$, for any integers $t$ relatively prime to 3 that admit semifield subplanes of order $2^3$''.

In this work, we develop a method of rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \quad z = x + iy, \] where $K(x,y)$ and $L(x,y)$ are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on values of a sampled function. As a set of the Voigt/complex error function values remains the same, this approach provides rapid computation. Mathematica codes with some examples are presented.

This article contains a new result in Fourier analysis concerning jump type discontinuities.

Building upon specific compatibility conditions, we establish fundamental structural results concerning ordering relations for triangular fuzzy numbers. We demonstrate that orders satisfying compatibility with arithmetic operations, MIN-MAX operators, and the Weak Law of Trichotomy (WLT) are completely determined on the fibers of the natural projection to real numbers. Furthermore, such orders naturally induce - in analogy with real numbers - well-defined notions of fuzzy absolute value and fuzzy distance that preserve the essential properties of their classical counterparts. These results enable us to characterize open and closed balls through interval representations, providing a robust theoretical framework for future studies regarding metric properties of fuzzy numbers.

This article reviews the properties of the self-similar solutions of the Navier-Stokes equation for incompressible fluids. Since any smooth solution can be embedded into a self-similar solution at the identity scale, it follows that under standard flow conditions, the initial solution will remain smooth for all time as long as the self-similar solution is selected to have certain isobaric weight.

This paper introduces a novel generalization of Stirling and Lah numbers, termed ``heterogeneous Stirling numbers," which smoothly interpolate between these classical combinatorial sequences. Specifically, we define heterogeneous Stirling numbers of the second and first kinds, demonstrating their convergence to standard Stirling numbers for lambda=0 and to (signed) Lah numbers for lambda =1. We derive fundamental properties, including generating functions, explicit formulas, and recurrence relations. Furthermore, we extend these concepts to heterogeneous Bell polynomials, obtaining analogous results such as generating function, combinatorial identity and Dobinski-like formula. Finally, we introduce and analyse heterogeneous r-Stirling numbers of the second kind and their associated r-Bell polynomials.