In this article partial modules over rings and tensor product of partial modules and its properties are studied. Left and right partial modules, partial bimodules and their homomorphisms are defined. Next, partial quotient modules are defined and the fundamental homomorphism theorem for partial modules is proven. Also, the tensor product of partial modules and the tensor product of homomorphisms of partial modules is defined. Some properties of the tensor product, the existence of hom-functors and tensor functors are proven. Finally it is shown that the hom-functor and the tensor functor are adjoint functors.

During the third decade of the last century, Arne Beurling introduced the generalise primes as any increasing positive real sequence starting with a real number greater than 1 called ”Beurling primes”. Where the fundamental theorem of arithmetics gives Beurling integers. This work study Beurling’s prime systems and concentrates on the upper bound of Beurling zeta functionin the region (0, 1). This reflects of course on the size of the error term of Beurling counting function of integers Np(x).

Integration in non-integer-dimensional spaces (NIDS) is actively used in quantum field theory, statistical physics, and fractal media physics. The integration over the entire momentum space with non-integer dimensions was first proposed by Wilson in 1973 for dimensional regularization in quantum field theory. However, self-consistent calculus of integrals and derivatives in NIDS and the vector calculus in NIDS, including the fundamental theorems of these calculi, have not yet been explicitly formulated. The construction of precisely such self-consistent calculus is the purpose of this article. The integral and differential operators in NIDS are defined by using the generalization of the Wilson approach, product measure, and metric approaches. To derive the self-consistent formulation of the NIDS calculus, we proposed some principles of correspondence and self-consistency of NIDS integration and differentiation. In this paper, the basic properties of these operators are described and proved. It is proved that the proposed operators satisfy the NIDS generalizations of the first and second fundamental theorems of standard calculus; therefore, these NIDS operators form a calculus. The NIDS derivative satisfies the standard Leibniz rule; therefore, these derivatives are integer-order operators. The calculation of the NIDS integral over the ball region in NIDS gives the well-known equation of the volume of a non-integer dimension ball with arbitrary positive dimension. The volume, surface, and line integrals in D-dimensional spaces are defined, and basic properties are described. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) are proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are proved. Some basic elements of the calculus of differential forms in NIDS are also proposed. The proposed NIDS calculus can be used, for example, to describe fractal media and the fractal distribution of matter in the framework of continuum models by using the concept of the density of states.

While extra-pair mating prevails among socially monogamous birds, it does not occur in all individuals within a population. Then, what underlies this variability? A poorly explored mechanism is the genetic contribution to the behavioral trait, especially for cooperatively breeding species where promiscuity may potentially conflict with the acquisition of indirect benefits to altruistic helpers. We addressed the gap through a quantitative genetic approach with 8 years of data from an individually marked population of Tibetan ground tits (Pseudopodoces humilis). Extra-pair mating was observed in 33.2% of nests, and cooperative breeding occurred in 39.5% of nests. Animal models demonstrated no significant genetic component contributing to the variance in extra-pair mating both during a specific year and over an individual's lifetime. Consequently, the heritabilities were not significantly different from zero. The lack of heritable variation in extra-pair mating can be accounted for by Fisher's fundamental theorem of natural selection, which suggests that genotypes associated with this behavior facilitating reproductive success should have become widespread within the population. Furthermore, the fitness benefits derived from promiscuity were greater for breeders than those from receiving help; for helpers, the fitness benefits from extra-pair mating outweighed the indirect genetic benefits obtained from providing help. This may explain why extra-pair mating and cooperative breeding can coexist in the same population. Our findings imply that individual variation in performing extra-pair mating behavior is more likely to be influenced by environmental factors.

UDC 512.5 We introduce new classes of functions generalizing the well-known classes of functions of complex variable, such as the entire functions, meromorphic functions, rational functions, and polynomial functions, which take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend some recently obtained characterization theorems presented in [B. Q. Li, Amer. Math. Monthly, 122, № 2, 169–172 (2015)] and [B. Q. Li, Amer. Math. Monthly, 132, № 3, 269–271 (2025)] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems generalize a recently established version of the fundamental theorem of algebra presented in [V. M. Abramov, Amer. Math. Monthly, 132, № 4, 356–360 (2025)].

Model fusion aims to integrate several deep neural network (DNN) models' knowledge into one by fusing parameters, and it has promising applications, such as improving the generalization of foundation models and parameter averaging in federated learning. However, models under different settings (data, hyperparameter, etc.) have diverse neuron permutations; in other words, from the perspective of loss landscape, they reside in different loss basins, thus hindering model fusion performances. To alleviate this issue, previous studies highlighted the role of permutation invariance and have developed methods to find correct network permutations for neuron alignment after training. Orthogonal to previous attempts, this paper studies training-time neuron alignment, improving model fusion without the need for post-matching. Training-time alignment is cheaper than post-alignment and is applicable in various model fusion scenarios. Starting from fundamental hypotheses and theorems, a simple yet lossless algorithm called TNA-PFN is introduced. TNA-PFN utilizes partially fixed neuron weights as anchors to reduce the potential of training-time permutations, and it is empirically validated in reducing the barriers of linear mode connectivity and multi-model fusion. It is also validated that TNA-PFN can improve the fusion of pretrained models under the setting of model soup (vision transformers) and ColD fusion (pretrained language models). Based on TNA-PFN, two federated learning methods, FedPFN and FedPNU, are proposed, showing the prospects of training-time neuron alignment. FedPFN and FedPNU reach state-of-the-art performances in federated learning under heterogeneous settings and can be compatible with the server-side algorithm.

This article presents several fundamental theorems relating to convex integral inequalities. Each theorem has the potential to serve as a valuable intermediary tool in a wide range of analytical applications. The two main categories of results considered are convex simple integral theorems, involving single integrals, and convex double integral theorems, based on double integrals. Full, detailed proofs are provided that are designed to be easily reproducible, requiring only minimal preliminary knowledge. Comprehensive examples illustrate the theory.

On the Fundamental Theorem of Submanifold Theory and Isometric Immersions with Supercritical Low Regularity

Analytical investigation of fundamental theorems and wave propagation in microstretch thermoelastic diffusion with multi-phase-lag effects

The study of finite extension of Galois rings in the recent past have given rise to commutative completely primary finite rings that have attracted much attention as they have yielded important results towards classification of finite rings into well-known structures. In this paper, we give a construction of a class of completely primary finite ringof characteristicwhose subsets of zero divisors satisfy the condition . The ring is constructed over its subring as an idealization of the - modules. A thorough determination and classification of the structure of the group of invertible elements using fundamental theorem of finitely generated abelian groups is given.

Fisher's Fundamental Theorem of Natural Selection continues to be widely cited in the literature but there is still misunderstanding about its interpretation and significance. Even though it is now recognized that the additive genetic variance in its statement captures only a partial rate of change in mean fitness, the original terms and arguments used to present it remain unclear, not to mention its real meaning. Here we revisit the interpretation of this partial rate of change. Applying the properties of the additive genetic values and residual addends of a quantitative trait to the relative growth rate of genotype frequency in a diploid population, and comparing two reproductive systems, clonal reproduction and sexual reproduction with either random union of gametes or random mating with additive fecundities of mating types, we argue that this additive genetic rate of change corresponds to the change that is invariant under reshuffling of genes. We show that this is actually the case for the partial rate of change in the mean of any measurement given by the additive genetic covariance with fitness. We focus on the one-locus multiallele setting in continuous time without age effects for simplicity, but the conclusion can be extended to multilocus settings with age effects in continuous time as well as discrete time.

Polyploidy and whole genome duplication (WGD) are widespread biological phenomena with substantial cellular, meiotic, and genetic effects. Despite their prevalence and significance across the tree of life, population genetics theory for polyploids is not well developed. The lack of theoretical models limits our understanding of polyploid evolution and restricts our ability to harness polyploidy for crop improvement amidst increasing environmental stress. To address this gap, we developed and analyzed deterministic models of mutation-selection balance for tetraploids under polysomic (autotetraploid) and disomic (allotetraploid) inheritance patterns and arbitrary dominance relationships. We also introduced a new mathematical framework based on ordinary differential equations and nonlinear dynamics for analyzing the models. We find that autotetraploids approach Hardy-Weinberg Equilibrium 33% faster than allotetraploids, but the different tetraploid inheritance models show little differences in mutation load and allele frequency at mutation-selection balance. Our model also reveals two bistable points of mutation-selection balance for dominant alleles with biased mutation rates over a wide range of selection coefficients in the tetraploid models compared to bistability in only a narrow range for diploids. Finally, using discrete time simulations, we explore the temporal dynamics of allele frequency and fitness change and compare these dynamics to the predictions of Fisher’s Fundamental Theorem of Natural Selection. While Fisher’s predictions generally hold, we show that the bistable dynamics for dominant mutations fundamentally alter the associated temporal dynamics. Overall, this work develops foundational theoretical models that will facilitate the development of population genetic models and methodologies to study evolution in empirical tetraploid populations.

The q-rung orthopair fuzzy set (q-ROFS) has been developed as an extension of the Pythagorean fuzzy set (PFS) to address ambiguity in various decision-making contexts. Group theory is a significant area of mathematics with numerous applications across various scientific fields. This paper examines q-rung orthopair fuzzy group theory, emphasizing the importance of q-ROFS and group theory. The concept of a q-rung orthopair fuzzy subgroup (q-ROFSG) is introduced, and its various algebraic properties are examined. A comprehensive investigation into q-rung orthopair fuzzy cosets (q-ROFCs) and q-rung orthopair fuzzy normal subgroups (q-ROFNSGs) has been conducted. The definitions of q-rung orthopair fuzzy homomorphism and isomorphism are presented. We extend the concept of the quotient group of a classical group V in relation to its normal subgroup U by introducing a q-ROFSG of V⁄U. The q-rung orthopair fuzzy variant of the three fundamental isomorphism theorems has been demonstrated.

This paper establishes a comprehensive framework for fixed point theory in MR-metric spaces,a generalization of standard metric spaces that incorporates three-point relations. We present fourfundamental theorems:(1) A Banach contraction principle with optimal contraction constant k < 1/3R(2) A solvability theorem for Fredholm-type integral equations(3) A Krasnoselskii-type hybrid fixed point theorem(4) A Leray-Schauder alternative for generalized contractionsThe theoretical results are applied to:• Nonlinear integral equations in neutron transport theory• Optimization problems in neural networks• Boundary value problems for nonlinear ODEsKey innovations include the development of error estimates in the MR-metric framework and the derivation of precise existence conditions for operator equations. The work bridges theoretical mathematics with practical applications in physics and machine learning.

While discrete (metric) connections have become a staple of n -vector field design and analysis on simplicial meshes, the notion of torsion of a discrete connection has remained unstudied. This is all the more surprising as torsion is a crucial component in the fundamental theorem of Riemannian geometry, which introduces the existence and uniqueness of the Levi-Civita connection induced by the metric. In this paper, we extend the existing geometry processing toolbox by providing torsion control over discrete connections. Our approach consists in first introducing a new discrete Levi-Civita connection for a metric with locally-constant curvature to replace the hinge connection of a triangle mesh whose curvature is concentrated at singularities; from this reference connection, we define the discrete torsion of a connection to be the discrete dual 1-form by which a connection deviates from our discrete Levi-Civita connection. We discuss how the curvature and torsion of a discrete connection can then be controlled and assigned in a manner consistent with the continuous case. We also illustrate our approach through theoretical analysis and practical examples arising in vector and frame design.

Abstract We introduce the Frenet theory of curves in dual space $$\mathbb {D}^3$$ D 3 . After defining the curvature and the torsion of a curve, we classify all curves in dual plane with constant curvature. We also establish the fundamental theorem of existence in the theory of dual curves, proving that there is a dual curve with prescribed curvature and torsion. Finally we classify all dual curves with constant curvature and torsion.

The primary aim of this paper is to explore novel concepts of convergence and summability for sequences of numbers within the framework of octonion-valued metric spaces, drawing inspiration from modulus functions. Specifically, the study focuses on [Formula: see text]-lacunary statistical convergence of order [Formula: see text] and strong [Formula: see text]-lacunary summability of order [Formula: see text] for sequences of numbers. These innovative concepts are supported by several fundamental theorems, properties, and interrelations.

Abstract In this paper we introduce the notion of generalized invertible 1-cocycle in a strict braided monoidal category $$\textsf{C}$$ C , and we prove that the category of Hopf trusses is equivalent to the category of generalized invertible 1-cocycles. On the other hand, we also introduce the notions of module for a Hopf truss and for a generalized invertible 1-cocycle. We prove some functorial results involving these categories of modules and we show that the category of modules associated to a generalized invertible 1-cocycle is equivalent to a category of modules associated to a suitable Hopf truss. Finally, assuming that in $$\textsf{C}$$ C we have equalizers, we introduce the notion of Hopf-module in the Hopf truss setting and we obtain the Fundamental Theorem of Hopf modules associated to a Hopf truss.

Triangular shapes have been studied from different perspectives over a wide temporal frame since ancient times. Initially, fundamental theorems were formulated to demonstrate their geometrical properties. Philosophy and art leveraged the peculiar aspects of triangles as building blocks for more complex geometrical shapes. This paper will review triangles by adopting a multidisciplinary approach, recalling ancient science and Plato’s arguments in relation to their connection with philosophy. It will then consider the artistic utilization of triangles, particularly in compositions created during the medieval era, as exemplified by the Cosmati Italian family’s masterpieces. Various scientific environments have explored triangular 2D and 3D shapes for different purposes, which will be briefly reviewed here. After that, Sierpiński geometry and its properties will be introduced, focusing on the equilateral shape and its internal complexity generated by subdividing the entire triangle into smaller sub-triangles. Finally, examples of triangular planar shapes that fulfill the Sierpiński geometry will be presented as an application in signal processing for high-frequency signals in the microwave and millimeter-wave range.

In this paper, we derive novel results concerning third-order differential subordinations for meromorphic functions, utilizing a newly defined linear operator that involves the inverse of the Legendre chi function in conjunction with the Mittag-Leffler identity. To establish these results, we introduce several families of admissible functions tailored to this operator and formulate sufficient conditions under which the subordinations hold. Our study presents three fundamental theorems that extend and generalize known results in the literature. Each theorem is accompanied by rigorous proofs and further supported by corollaries and illustrative examples that validate the applicability and sharpness of the derived results. In particular, we highlight special cases and discuss their implications through both analytical evaluations and graphical interpretations, demonstrating the strength and flexibility of our framework. This work contributes meaningfully to the field of geometric function theory by offering new insights into the behavior of third-order differential operators acting on p-valent meromorphic functions. Furthermore, the involvement of the Mittag-Leffler function positions the results within the broader context of fractional calculus, suggesting potential for applications in the mathematical modeling of complex and nonlinear phenomena. We hope this study stimulates further research in related domains.