Related Theorems Research Articles

Klein–Maskit combination theorem for Anosov subgroups: Amalgams

Abstract The classical Klein–Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones. There are two distinct but closely related combination theorems: the first deals with amalgamated free products, whereas the second deals with HNN extensions. This article gives analogs of both combination theorems for Anosov subgroups.

Orthogonal Laurent Polynomials of Two Real Variables

ABSTRACTIn this paper, we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable and , and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research.

Bipolar complex fuzzy near rings

In this article, we devise the novel concept of bipolar complex fuzzy (BCF) near rings (BCFNR), to fill a momentous research gap in connection of bipolar complex fuzzy sets with the theory of near rings in the existing literature. We expand the theory of near ring into the structure of BCF set (BCFS), offering a more suitable approach for the representation of algebraic systems with inherent ambiguity, bipolarity, and 2nd dimension information. Further, we introduce the concept of the bipolar complex fuzzy sub-near ring (BCFSNR), bipolar complex fuzzy left ideal (BCFLI), bipolar complex fuzzy right ideal (BCFRI), and bipolar complex fuzzy ideal (BCFI) in the near ring. After that, we prove the related theorem and results of these devised concepts. We also introduce theorems based on the homomorphism theory and Noetherian theory of near rings within the frame of bipolar complex fuzzy near rings. At the end of the manuscript, we reveal the application of BCFNRs in decision-making (DM) and illustrate it through an example.

Approximation of $$L^\infty $$ functionals with generalized Orlicz norms

AbstractThe aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. We generalize results proven by Bertazzoni, Harjulehto and Hästö in Journ. of Math. Anal. and Appl. (2024) for integral type energies (in generalized Orlicz spaces), considering milder convexity assumptions. $$\Gamma $$ Γ -convergence results and related representation theorems in terms of $$L^\infty $$ L ∞ functionals are proven. The convexity hypotheses are completely removed in the variable exponent setting, thus extending the results in Eleuteri-Prinari in Nonlinear Anal. Real. World Appl. (2021) and Prinari-Zappale in JOTA (2020).

Relative monadicity

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense ▪-functor j:A→E, a ▪-functor r:D→E is j-monadic if and only if r admits a left j-relative adjoint and creates j-absolute colimits. This provides a refinement of the classical monadicity theorem – characterising those categories whose objects are given by those of E equipped with algebraic structure – in which the arities of the algebraic operations are valued in A. In particular, when j=1, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the (j-relative) monadicity of the composite of a ▪-functor with a (j-relatively) monadic ▪-functor.

A fusion model for multiple partial-order quotient spaces based on three-way decision

With a quotient space, a complex problem can be transformed into an appropriate granularity for processing. However, within a quotient space, there are no partial order relations between information granules. In practical decision-making, it may be necessary to deal with quotient spaces that have partial order relations between information granules. In this paper, first, the concept of a partial-order quotient space is proposed and some of its properties are studied. Second, a fusion model for multiple partial-order quotient spaces based on three-way decision is proposed, and some related theorems are proved. Finally, a case study of teacher evaluation, an experiment on real data, and several simulation experiments are presented to demonstrate the practicality of the proposed model in this paper.

Tightness of Harary Graphs

In the design of real-world networks, researchers evaluate various structural parameters to assess vulnerability, including connectivity, toughness, and tenacity. Recently, the tightness metric has emerged as a potentially superior vulnerability measure, although many related theorems remain unknown due to its novelty. Harary graphs, known for their maximum connectivity, are an important class of graph models for network design. Prior work has evaluated the vulnerability of three types of Harary graphs using different parameters, but the tightness metric has not been thoroughly explored. This article aims to calculate the tightness values for all three types of Harary graphs. First, it will attempt to calculate the lower bound for the value of the tightness parameter in Harary graphs using existing lemmas and theorems. Then, by presenting new lemmas and theorems, we will try to find the exact value or upper bound for this parameter in Harary graphs. For the first type of Harary graph, the tightness is precisely determined, while for the second and third types, upper bounds are provided due to structural complexity. The lemmas, theorems, and proof methods presented in this research may be used to calculate other graph and network parameters. However, the newness of the tightness parameter means that further research is needed to fully characterize its properties.

Towards a rigorous analysis of mutual information in contrastive learning

Contrastive learning has emerged as a cornerstone in unsupervised representation learning. Its primary paradigm involves an instance discrimination task utilizing InfoNCE loss where the loss has been proven to be a form of mutual information. Consequently, it has become a common practice to analyze contrastive learning using mutual information as a measure. Yet, this analysis approach presents difficulties due to the necessity of estimating mutual information for real-world applications. This creates a gap between the elegance of its mathematical foundation and the complexity of its estimation, thereby hampering the ability to derive solid and meaningful insights from mutual information analysis. In this study, we introduce three novel methods and a few related theorems, aimed at enhancing the rigor of mutual information analysis. Despite their simplicity, these methods can carry substantial utility. Leveraging these approaches, we reassess three instances of contrastive learning analysis, illustrating the capacity of the proposed methods to facilitate deeper comprehension or to rectify pre-existing misconceptions. The main results can be summarized as follows: (1) While small batch sizes influence the range of training loss, they do not inherently limit learned representation’s information content or affect downstream performance adversely; (2) Mutual information, with careful selection of positive pairings and post-training estimation, proves to be a superior measure for evaluating practical networks; and (3) Distinguishing between task-relevant and irrelevant information presents challenges, yet irrelevant information sources do not necessarily compromise the generalization of downstream tasks.

Integral theorems for the gradient of a vector field, with a fluid dynamical application

The familiar divergence and Kelvin–Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the tensor product of the vector field with the exterior normal. The importance of this long-established yet relatively little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the two-dimensional divergence and Kelvin–Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential extension of the generalized two-dimensional identity to curved surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems.

New solutions of time-fractional cancer tumor models using modified He-Laplace algorithm

Cancer develops through cells when mutations build up in different genes that control cell proliferation. To treat these abnormal cells and minimize their growth, various cancer tumor samples have been modeled and analyzed in literature. The current study is focused on the investigation of more generalized cancer tumor model in fractional environment, where net killing rate is taken into account in different domains. Three types of killing rates are considered in the current study including time and position dependent killing rates, and concentration of cells based killing rate. A hybrid mechanism is proposed in which different homotopies are used with perturbation technique and Laplace transform. This leads to a convenient algorithm to tackle all types of fractional derivatives efficiently. The convergence and error bounds of the proposed scheme are computed theoretically by proving related theorems. In the next phase, convergence and validity is analyzed numerically by calculating residual errors round the fractional domain. It is observed that computed errors are very less in the entire fractional domain. Moreover, comparative analysis of Caputo, Caputo-Fabrizio (CF), and Atangana–Baleanu (AB) fractional derivatives is also performed graphically to discern the effect of different fractional approaches on the solution profile. Analysis asserts the reliability of proposed methodology in the matter of intricate fractional tumor models, and hence can be used to other complex physical phenomena.

Common Fixed Point Results for w-α-Distance

In this study, we examined some fixed point theorems in non-full metric spaces. We define the notions of α-lower semi-continuous , w-α-distance, w_0-α– distance, w-α-rational contraction and generalized w-α-rational contraction mapping. We also give related theorem and example. Then, we prove Banach’s fixed-point theorem thanks to the concept w-α- distance in metric spaces equipped with an abritrary binary relation. Also, w-α- rational contraction mapping and generalized w-α-rational contraction mapping are defined and by using these definitions, the theorem related fixed point is expressed and proved.

Neutrosophic Semi <i>β<sup>F</sup></i>-Contra Continuity in a Neutrosophic Topological Space

The main intention of this article is to propose the theory of a neutrosophic SβF contra continuous function, neutrosophic strongly SβF continuous function, neutrosophic strongly SβF contra continuous function and neutrosophic SβF contra irresolute function in neutrosophic topological spaces. Further several properties based on the above concepts have been studied. Many theorems based on the above ideas have been proved with the suitable illustration. Further the relationship connecting neutrosophic SβF contra continuous function, neutrosophic strongly SβF continuous function, neutrosophic strongly SβF contra continuous function and neutrosophic SβF contra irresolute function are studied and illustrated. Moreover, various theorems based on compositions of functions have been proved. In addition to this, theproduct of two neutrosophic sets is defined and related theorems have been proved.

Construction of line Layout Model Based on Geometric Deduction and Constrained Heuristic Search

Based on problem B of the National Mathematical Contest in Modeling for College Students in 2023, this paper establishes a model through geometric deductive deduction, and establishes and solves an optimal model of survey line layout through constrained heuristic algorithm. In problem 1, a mathematical model of coverage width and overlap rate is established based on triangle relation and triangle sine theorem by finding the mathematical relationship between target quantity and known quantity. In problem 2, the angle between the known direction of the survey line and the projection of the normal direction of the submarine slope on the horizontal plane is . We can imagine a surface perpendicular to the direction of the measurement line, and then find the slope of the imaginary surface according to the geometric relationship, and replace the slope Angle of problem 1 with the imaginary slope Angle to obtain the mathematical model of the coverage width. Based on this, the constraints of the Angle and overlap rate of the line are added, and the problem 3 is simplified into the optimization problem of multiple routes in problem 2. Finally, by changing the parameter setting of slope Angle in the model, the sensitivity analysis is carried out, and it is found that the model is stable and reliable.To sum up, the problems from one to four are gradually deepened and progressively advanced, and the development of Marine survey field is facilitated by establishing a model of depth survey line layout.

Event-triggered prescribed-time consensus for nonholonomic multi-agent systems with external disturbances

This paper considers the prescribed-time event-triggered consensus for a class of nonholonomic multi-agent systems (MASs) with external disturbances. Firstly, the nonholonomic chained MASs are divided into two subsystems, and a switching control scheme is introduced. Secondly, a distributed prescribed-time continuous protocol is proposed to ensure that all agents reach average consensus within a prescribed finite-time. Moreover, to reduce the communication costs, an event-triggered consensus protocol is further established and some related theorems are given. In addition, the Zeno behaviour can be excluded by choosing parameters appropriately. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed protocols.

Similarity measure between generalized intuitionistic fuzzy sets with three kinds of negation

The similarity measure of intuitionistic fuzzy sets is a primary method for dealing with uncertainty and fuzzy problems and is commonly used in fuzzy decision-making and pattern recognition. The current mainstream similarity measure is based on the classical fuzzy set with only one negation, which often violates the intuitionistic problem in applications because the actual semantics of multiple negations are not considered. To solve the inconsistency and irrationality problems in the classical similarity methods, we introduce three negations (contradiction negation, opposition negation, and mediation negation) in the intuitionistic fuzzy set to obtain the generalized intuitionistic fuzzy set and prove its related property theorem. On this basis, our similarity measure adopts a mediational negation to represent non-membership, which fully utilizes the multiple negation information of non-membership and hesitancy and avoids the loss of fuzzy information. We verify the method’s rationality, validity, and originality through pattern recognition experiments and numerical examples, which improves the performance of intuitionistic fuzzy set similarity in practical applications and provides a new approach for future research on intuitionistic fuzzy inference.

General Mean-Field BDSDEs with Stochastic Linear Growth and Discontinuous Generator

In this paper, we consider the general mean-field backward doubly stochastic differential equations (mean-field BDSDEs) whose generator f can be discontinuous in y. We prove the existence theorem of solutions under stochastic linear growth conditions and also obtain the related comparison theorem. Naturally, we present those results under the linear growth condition, which is a special case of the stochastic condition. Finally, a financial claim sale problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance.

Solving a Nonlinear Fractional Integral Equation by Fixed Point Approaches Using Auxiliary Functions Under Measure of Noncompactness

This manuscript is devoted to ensure the existence of a solution to nonlinear fractional integral equations with three variables under a measure of noncompactness. In order to accomplish our main goal, we develop a new fixed point theorem that generalizes Darbo’s fixed point theorem by utilizing a measure of noncompactness and a new contraction operator. A related tripled FP theorem is also obtained. Finally, we use this generalized Darbo’s fixed point theorem to solve a nonlinear fractional integral equation involving three variables, and an example to demonstrate our results is presented.

A Generalized Von Neumann’s Theorem for Linear Relations in Hilbert Spaces

A Generalized Von Neumann’s Theorem for Linear Relations in Hilbert Spaces

The Proper Landau–Ginzburg Potential, Intrinsic Mirror Symmetry and the Relative Mirror Map

Given a smooth log Calabi–Yau pair (X, D), we use the intrinsic mirror symmetry construction to define the mirror proper Landau–Ginzburg potential and show that it is a generating function of two-point relative Gromov–Witten invariants of (X, D). We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of Fan et al. (Sel Math (NS) 25(4): Art. 54, 25, 2019. https://doi.org/10.1007/s00029-019-0501-z) to compute two-point relative invariants. When D is nef, we compute the proper Landau–Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety X, this implies the conjecture of m Gräfnitz et al. (2022) that the proper Landau–Ginzburg potential is the open mirror map. When X is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.

Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory

This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of L2→L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2 \\rightarrow L^2$$\\end{document} Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.