By employing the A−summation process in the B−statistical sense, where A and B are sequences of infinite matrices, we provide new results on the classical Korovkin theorem for a sequence of monotone and sublinear operators. Reported results essentially extend some theorems existing in the literature.

This study introduces and investigates the idea of $S$-pm-rings, a generalization of pm-rings in the context of commutative rings with a multiplicatively closed subset $S$. We prove that a ring $R$ is an $S$-pm-ring if and only if its $S$-maximal spectrum is a retract (specifically, a deformation retract) of its $S$-prime spectrum. Furthermore, we establish the equivalence of the $S$-pm-ring property to the normality of the $S$-prime spectrum and the Hausdorff property of the $S$-maximal spectrum. We also explore the relationship between $S$-pm-rings and $S$-clean rings, demonstrating that every $S$-local ring is $S$-clean, and every $S$-clean ring is an $S$-pm-ring. These results extend classical theorems in commutative algebra and algebraic geometry to the $S$-version context.

Abstract We establish positive energy theorems for complete spin initial data sets with charge in dimensions n ≥ 4, under a dominant energy condition and assuming the existence of at least one asymptotically flat end. Our results, formulated in the purely electric case, extend the classical theorems of Gibbons-Hull [GH82], Gibbons-Hawking-Horowitz-Perry [GHHP83], and Bartnik-Chruściel [BC05].

Abstract We present a simple algorithm to conformally map between two simple and bounded planar domains based on the concept of harmonic measure, which is a conformal invariant. With suitable preprocessing, the algorithm is fast enough to compute all possible conformal maps (having three real degrees of freedom) between the two domains in real time in an interactive system, which seems to be the first to ‘bring to life’ the classical Riemann mapping theorem in this way. The same method can be used to conformally parameterize genus‐0 manifold 3D surfaces with a boundary over arbitrary simple and bounded planar domains and interactively adjust the parameterization.

In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function ℘ ( z ) in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a special case. Furthermore, we give novel two-term addition, three-term addition, duplication and triplication formulas. New identities for elliptic invariants are also proven.

An antlike observer confined to a two-dimensional surface traversed by stripes would wonder whether such a striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem studied in this paper. In a more technical jargon, we determine all possible uniform nematic fields on a smooth surface. It was already known that for such a field to exist, the surface must have constant negative Gaussian curvature. Here we show that all uniform nematic fields on such a surface are parallel transported (in Levi-Civita's sense) by special systems of geodesics, which are termed uniform. We prove that, for every geodesic on the surface, there are two systems of uniform geodesics that include it; they are conventionally called right and left, to evoke handedness. We found explicitly all uniform fields for Beltrami's pseudosphere. Since both geodesics and uniformity are preserved under isometries, by a classical theorem of Minding, the solution for the pseudosphere carries over all other admissible surfaces, thus providing a general solution to the problem (at least in principle). The proved existence of surface nematic uniform fields suggests the definition of a generalized intrinsic elastic energy for fluid membranes with nematic order, which is but one of the many possible applications of our geometric result.

This paper presents a unified study of integer partition theory, a foundational area of number theory and combinatorics. The subject is situated within its historical development, from Euler’s generating function framework to Ramanujan’s profound congruences, and then advances beyond classical exposition by synthesizing these ideas with modern combinatorial perspectives. The work systematically develops essential tools—including Ferrers and Young diagrams, Durfee squares, and generating functions—within a single coherent framework. Classical theorems are rigorously proved using both Algebraic and Combinatorial Techniques, Highlighting the Complementary Nature of these approaches. A novel aspect of this paper lies in its integrative treatment of partitions across different number systems and its qualitative exploration of applications extending beyond pure mathematics, demonstrating how partition theory interfaces with broader mathematical structures. By combining historical insight, illustrative constructions, and formal proofs, this study not only consolidates foundational knowledge but also clarifies pathways toward contemporary research questions in partition theory. The results underscore the continuing relevance of integer partitions as a unifying language in modern mathematics and provide a pedagogically strong and research-oriented framework for future investigations in combinatorics and number theory.

Abstract A classical theorem of Weyl states that any polynomial with an irrational coefficient other than the constant term is uniformly distributed mod 1. We prove a new function field analogue of this statement, confirming a conjecture of Lê, Liu, and Wooley.

The optical response of mesoscale metallic nanostructures (MMNSs) with extreme nanoscale feature sizes is largely affected by the nonclassical quantum effects, which can be comprehensively described by the nonclassical electromagnetic boundary conditions (NEBC) formulated via Feibelman d-parameters. In classical electrodynamics, the classical Lorentz reciprocity theorem (CLRT) is of fundamental importance with extensive applications. In this work, we establish the nonclassical Lorentz reciprocity theorem (NLRT) under the NEBC, where the two non-perturbative d-parameters are considered in a rigorous manner. By comparing the NLRT with the CLRT under the classical electromagnetic boundary condition (CEBC), we derive the specific conditions under which the CLRT remains valid within the NEBC framework, and confirm their effectiveness by numerical examples. Our findings pave the way for directly extending the wide applications of the CLRT from classical electromagnetism to nonclassical regimes, and provide a theoretical justification for the rationality of the NEBC in describing the nonclassical effects without violating the CLRT.

In this paper, we investigate contact skew CR-warped product submanifolds of locally conformal almost cosymplectic manifolds, a framework that simultaneously generalizes warped product pseudo-slant, semi-slant, and contact CR-submanifolds. We first establish a necessary and sufficient characterization theorem showing that a proper contact skew CR-submanifold with integrable slant distribution admits a local warped product structure if and only if certain shape operator conditions involving the slant angle and the warping function are satisfied. Subsequently, we derive sharp geometric inequalities for the squared norm of the second fundamental form in terms of the warping function, the slant angle, and the conformal factor of the ambient manifold. The equality cases are completely characterized and lead to strong rigidity results, namely that the base manifold is totally geodesic while the slant fiber is totally umbilical in the ambient space. Several applications are presented, showing that our results recover and extend a number of known inequalities and classification theorems for warped product submanifolds in cosymplectic, Kenmotsu, and Sasakian geometries as special cases.

This paper presents a study of thermoelasticity theory grounded in the recently developed two-temperature Moore–Gibson–Thompson (2TMGT) generalized thermoelasticity theory. We derive the fundamental governing equations for a homogeneous and isotropic medium within the context of 2TMGT. A primary objective is to establish key theoretical results, beginning with the uniqueness theorem for a mixed initial-boundary value problem in linear thermoelasticity under the current framework. Furthermore, we formulate a variational principle based on an appropriate functional corresponding to the governing equations of motion. This variational formulation offers deeper insights into the interactions between the mechanical and thermal aspects of the system. In addition, a reciprocity theorem is derived through the application of the Laplace transform method. The generalized results presented herein not only extend classical thermoelastic theorems but also enhance their applicability to a broader range of physical scenarios.

In this paper, we prove that two normal complex surface germs that are inner bilipschitz—but not necessarily orientation-preserving—homeomorphic, have in fact the same oriented topological type and the same minimal plumbing graph. Along the way, we show that the oriented homeomorphism type of an isolated complex surface singularity germ determines the oriented homeomorphism type of its link, providing a converse to the classical Conical Structure Theorem. These results require to study the topology first, and the inner lipschitz geometry later, of Hirzebruch–Jung and cusp singularities, the normal surface singularities whose links are lens spaces and fiber bundles over the circle.

The nonlinear Bagley–Torvik equation is of fundamental importance, as it captures a realistic and intricate interplay among memory effects, nonlinearity, and functional dependence—making it a powerful model for a wide range of natural and engineered systems. Its analysis contributes significantly to both the theoretical development of fractional differential equations and their practical applications across science and technology. In this paper, we employ the inverse operator method, the multivariate Mittag-Leffler function, and several classical fixed-point theorems to establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions to the nonlinear Bagley–Torvik equation with functional initial conditions. Finally, we present several examples by explicitly computing values of the multivariate Mittag-Leffler functions to illustrate the main results.

A matrix method is suggested for approximating polynomial expansions of both even and odd degrees using symmetric coefficient matrices and column vectors. The approach utilizes binomial coefficients within a matrix framework and articulates the expansion in a matrix notation as This method ensures precision and balance in computations, consistent with the classical binomial theorem. The symbolic numerical implementation in Python demonstrates the method’s scalability and efficacy. The concept enhances awareness of algebraic patterns and provides a computational tool for matrix representation of polynomial formulas.

In the present paper, a class of non-weight modules over loop Ramond algebras R are defined. Denote h = C L 0 ; 0 ⊕ C G 0 ; 0 . We prove that these modules constitute a complete classification of U ( h ) -free modules of rank 1 over R . Based on this result, it is relatively easy to provide the classification theorem for some other loop Lie superalgebras such as the loop twisted N = 2 superconformal algebra, the loop super-Galilean conformal algebra and the loop Ramond-Block algebra.

Spherical density functional theory (DFT) is a reformulation of the classic theorems of DFT, in which the role of the total density of a many-electron system is replaced by a set of sphericalized densities, constructed by spherically averaging the total electron density about each atomic nucleus. In Hohenberg-Kohn DFT and its constrained-search generalization, the electron density suffices to reconstruct the spatial locations and atomic numbers of the constituent atoms, and thus the external potential. However, the original proofs of spherical DFT require knowledge of the atomic locations at which each sphericalized density originates, in addition to the set of sphericalized densities themselves. In the present work, we utilize formal results from geometric algebra-in particular, the subfield of distance geometry-to show that for Coulombic systems, this spatial information is encoded within the ensemble of sphericalized densities themselves and does not require independent specification. Consequently, the set of sphericalized densities uniquely determines the total external potential of the system, exactly as in Hohenberg-Kohn DFT. This theoretical result is illustrated through numerical examples for LiF and for glycine, the simplest amino acid. In addition to establishing a sound practical foundation for spherical DFT as applied to Coulombic systems, the extended theorem provides a rationale for the use of sphericalized atomic basis densities-rather than orientation-dependent basis functions-when designing classical or machine-learned potentials for atomistic simulation.

In this note, we first demonstrate that, aside from planes, oblate ellipsoids are total umbilical surfaces in the three‐dimensional Euclidean space under a general Euclidean metric, and vice versa. We then generalize the total torsion theorem of spheres to oblate ellipsoids. It is proven that the total torsion of regular closed curves on an oblate ellipsoid in equals zero; conversely, if a surface ensures the total torsion of all regular closed curves in is zero, then it must be a part of an oblate ellipsoid or a plane. Compared with the classical total torsion theorem, our results reveal a deterministic relationship between total umbilical surfaces and metrics.

Abstract This paper surveys the impact of Eremenko and Lyubich's paper “Examples of entire functions with pathological dynamics” , published in 1987 in the Journal of the London Mathematical Society . Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics.

Tian’s conjecture states that for any fixed distinct prime numbers p1,…,pm, the Diophantine equation n+12=p1α1·p2α2···pmαm in positive integers n,α1,…,αm has at most m solutions. In this paper, we develop a computational method to verify some special cases of this conjecture. We also give an alternative proof using the classical Zsigmondy theorem. For m=2 and 3, a sharp absolute upper bound for the number of solutions is given.

An extension of the classical Thistlethwaite theorem for links asserts that the Kauffman bracket of a link can be obtained from an evaluation of the Bollobás–Riordan polynomial of a ribbon graph associated to one of the link’s Kauffman states. In this paper, we further extend this result to knotoids, which are a generalization of knots that naturally arise in applications such as DNA and protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollobás–Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.