https://math-sci.ui.ac.ir/ Mathematics and Society en daily 1 Sun, 11 Jan 2026 00:00:00 +0330 Sun, 11 Jan 2026 00:00:00 +0330 https://math-sci.ui.ac.ir/article_30163.html If \( n \) is an integer, the set of all prime divisors of \( n \) is denoted by \( \pi(n) \). Let \( G \) be a finite group. Then \( \pi(|G|) \) is denoted by \( \pi(G) \). The prime graph of \( G \), denoted by \( \Gamma(G) \), is constructed as follows: the vertex set is \( \pi(G) \), and two distinct primes \( p \) and \( q \) are connected by an edge if and only if \( G \) has an element of order \( pq \). A finite nonabelian simple group $P$ is called quasirecognizable by prime graph, if each finite group $G$ with $\Gamma(G) = \Gamma(P)$ has a unique composition factor isomorphic to $P$. We denote by $k(\Gamma(G))$ the number of isomorphism classes of finite groups $H$ satisfying $\Gamma(G)=\Gamma(H)$. Given a natural number $r$, a finite group $G$ is called $r$-recognizable by prime graph if $k(\Gamma(G))=r$ and if $k(\Gamma(G))=1$, is called recognizable . In this paper, as the main result, we show that if \( G \) is a finite group such that \( \Gamma(G) = \Gamma(E_6(5)) \), then \( G \cong E_6(5) \). https://math-sci.ui.ac.ir/article_30273.html In this paper, we study comparison theorems for a Riemannian manifold $‎M^n‎$ under a lower bound condition on the Ricci curvature involving vector fields and gradient vector fields. We first establish Laplacian and volume comparison theorems for Riemannian manifolds endowed with a modified Ricci curvature as follow‎\begin{align}\nonumber‎‎\tilde{Ric}:= Ric‎ + ‎\frac{1}{2}\mathcal{L}_{V}g \geq (n-1)k,‎\end{align}‎and then for manifolds satisfying the Bakry–Émery Ricci curvature condition as follow‎\begin{equation}‎\nonumber‎‎Ric+Hess h\geq (n-1)k‎,‎‏‎\end{equation}for some smooth function $‎h‎$‎‏‎‏‏‏. Furthermore, we show that these comparison theorems remain valid, in particular, for the shrinking, steady, and expanding gradient Ricci solitons under the non-collapsing volume condition, and even without this condition when the soliton potential function is bounded. Consequently, we extend all of these results to almost Ricci solitons and gradient almost Ricci solitons. By using the obtained comparison results, we also derive a segment inequality for Riemannian manifolds $‎M^n‎$ with bounded Ricci curvature. https://math-sci.ui.ac.ir/article_30366.html In this paper, a new class of maps on the two-dimensional torus $\mathbb{T}^2$ is introduced that are not uniformly expanding and have no type of dominated splitting; nevertheless, we show that these maps stably possess a finite number of absolutely continuous invariant probability measures (ACIP) with respect to Lebesgue measure. The proof is based on the concept of "virtually expanding", recently introduced by Tsujii. Our method provides a framework for explicitly constructing these examples. To find new examples, we use a geometric interpretation of the virtually expanding property. This geometric perspective allows us to provide a sufficient condition for an endomorphism to be virtually expanding. The construction of this category of examples is based on a precise perturbation and surgery. These perturbations are supported on a small ball. In fact, by using the bump function technique, we change a linear endomorphism near a fixed point and replace its derivative with a suitably chosen matrix. This perturbation is designed in such a way that it preserves the virtual expanding property but eliminates any dominant splitting. https://math-sci.ui.ac.ir/article_30490.html In this paper, ergodic convergence for a sequence of nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces is studied. By introducing another definition of an almost orbit, an attempt is made to replace the uniform convergence condition with a weaker one. Our results provide a generalization of some classical ergodic convergence theorems for nonexpansive mappings. https://math-sci.ui.ac.ir/article_30519.html Neural Style Transfer (NST) stands as a remarkable example of the synthesis of mathematics, statistics, deep learning, and art. In this article, we present a historical and analytical exploration of the mathematical evolution of image-processing tools---from classical and derivative-based filters to frequency-domain transforms and multiscale representations---and trace how these concepts ultimately paved the way for convolutional neural networks, which form the foundation for understanding style transfer. We show how these mathematical developments culminated in the formulation of the NST algorithm. We then examine the mathematical structure of style transfer with a focus on the role of Gram matrices, learned feature spaces, and the optimization-based objective function. A central insight of NST is that its loss function becomes meaningful only through the representational power of deep neural networks: the notions of ``style'' and ``content'' are defined in learned feature spaces rather than in pixel space. Grounded in statistical concepts and solved through classical optimization methods, this formulation illustrates the deep and elegant interplay between applied mathematics, statistics, and deep learning in creating an artistic computational technique. https://math-sci.ui.ac.ir/article_30528.html This paper presents a comprehensive analysis of the interaction between homomorphisms and decomposition theorems in finite generalized groups. By synthesizing the theory of homomorphisms and external structures with decomposition theorems, we obtain new results about the structure of finite strongly normal generalized groups. We introduce the concept of homomorphic decomposition and prove that every finite strongly normal generalized group can be decomposed into the direct product of its image under a homomorphism and a kernel structure. We also establish conditions under which the decomposition preserves homomorphic properties. Our results provide a unified framework for understanding the structure of finite generalized groups and their homomorphisms. \end{abstract}