Related Theorems Research Articles

On The Anti Fuzzy k-Ideals of Ternary Semi-rings

In this paper, we introduce the notion of anti fuzzy k-ideal of ternary semi ring and we study some of its elementary properties by many related theorems and proofs, with many illustrated examples that explain our work.

On fundamental algebraic characterizations of complex intuitionistic <i>Q</i>-fuzzy subfield

<abstract> <p>The main objective of this study is to propose a new notion of a complex intuitionistic $ Q $-fuzzy subfield of a field $ F $ that is developed from the concept of a complex fuzzy subfield of a field $ F $ by adding the notion of intuitionistic $ Q $-fuzzy into a complex fuzzy subfield. We establish a new structure of complex fuzzy subfields which is called complex intuitionistic $ Q $-fuzzy subfield. The most significant advantage of this addition appears to be that it broadens the scope of the investigation from membership function values to membership and non-membership function values. The range of complex fuzzy subfields is expanded to the unit disc in the complex plane for both membership and non-membership functions. Some fundamental operations, especially the intersection, union, and complement of complex intuitionistic $ Q $-fuzzy subfields are studied. We define the necessity and possibility operators on a complex intuitionistic $ Q $-fuzzy subfield. Moreover, we show that each complex intuitionistic $ Q $-fuzzy subfield generates two intuitionistic $ Q $-fuzzy subfields. Subsequently, several related theorems are proven.</p> </abstract>

Solving Geometry Problems via Feature Learning and Contrastive Learning of Multimodal Data

This paper presents an end-to-end deep learning method to solve geometry problems via feature learning and contrastive learning of multimodal data. A key challenge in solving geometry problems using deep learning is to automatically adapt to the task of understanding single-modal and multimodal problems. Existing methods either focus on single-modal or multimodal problems, and they cannot fit each other. A general geometry problem solver should obviously be able to process various modal problems at the same time. In this paper, a shared feature-learning model of multimodal data is adopted to learn the unified feature representation of text and image, which can solve the heterogeneity issue between multimodal geometry problems. A contrastive learning model of multimodal data enhances the semantic relevance between multimodal features and maps them into a unified semantic space, which can effectively adapt to both single-modal and multimodal downstream tasks. Based on the feature extraction and fusion of multimodal data, a proposed geometry problem solver uses relation extraction, theorem reasoning, and problem solving to present solutions in a readable way. Experimental results show the effectiveness of the method.

Two-dimensional quantum Bernoulli process and the related central limit theorem

In this paper, we introduce a quantum decomposition of a two-dimensional Bernoulli random variable [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. Based on this quantum decomposition, we defined the two-dimensional quantum Bernoulli process and set a corresponding central limit theorem, both in the sense of quantum moments and in the sense of characteristic function.

Rate of homogenization for fully-coupled McKean–Vlasov SDEs

In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest.

Application of intuitionistic fuzzy sets in determining the major in senior high school

Intuitionistic Fuzzy Set (IFS) is useful to construct a model with elaborate uncertainty and ambiguity involved in decision-making. In this paper, the concept relation and operation of intuitionistic fuzzy set and the application in major of senior high school determination using the normalized Euclidean distance method will be reviewed. Some theorem of relation and operation of intuitionistic fuzzy set are proved. In general, to prove the theorem the definition and some basic relation and operation laws of IFS are needed. The distance measure between IFS indicates the difference in grade between the information carried by IFS. There are science, social, and language majors in senior high school. The normalized Euclidean distance method is used to measure the distance between each student and each major. The major, which each student can choose, has been determined depending on test evaluations. The solution provided is the smallest distance between each student and each major using the normalized Euclidean distance method.

Attribute reduction in inconsistent grey decision systems based on variable precision grey multigranulation rough set model

This paper mainly deals with attribute reduction of inconsistent grey decision systems (IGDSs) based on the variable precision grey multigranulation rough set (VP-GMGRS). Firstly, we present two transformation models to transform IGDS into consistent decision confidence system. One is the consistent decision system transformation model, based on which, an IGDS can be transformed into a VP-GMGRS approximate distribution consistent decision system. The other is the decision confidence system transformation model, which can be degenerated to a classical group decision system. Meanwhile, we educe related judgement theorems of approximation distribution consistent set in IGDS. Following that, a theoretical attribute reduction approach is presented by employing discernibility attribute sets and function based on VP-GMGRS approximate distributions. In addition, algorithms and illustrative examples with IGDS are employed and assisted to understand and explain the mechanism of attribute reduction theoretical approaches. Finally, comparison experiments are organized to verify the validity and feasibility of the proposed reduction method.

A nonlinear projection theorem for Assouad dimension and applications

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.

NAPOLEON'S THEOREM FROM THE VIEW POINT OF GRÖBNER BASES

In this article, we present a new proof of the Napoleon's theorem using algorithmic commutative algebra and algebraic geometry. We also show that, by using the same technique, several related theorems, with the same basic set of objects can be proved. Thus, from the new proof of Napoleon's theorem, we prove the Relative of Napoleon's theorem (result given by B. Gr\"unbaum). Then, we present a new theorem related to Napoleon's theorem. In this theorem the existence of two more quadruplets of equilateral triangles associated with a given triangle was established.

ROUGH RINGS, ROUGH SUBRINGS, AND ROUGH IDEALS

The basic concept in algebra that is set theory can be expanded into rough sets. Basic operations on the set such as intersections, unions, differences, and complements can still apply to rough sets. In addition, one of the applications of rough sets is the use of rough matrices in decision making. Furthermore, mathematical or informatics researchers who work on rough sets connect the concept of rough sets with algebraic structures (groups, rings, and modules) so that a concept called rough algebraic structures is obtained. Since the research related to rough sets is mostly carried out at the same time, different concepts have emerged related to rough sets and rough algebraic structures. In this paper, other definitions of rough ring and rough subring will be given along with related examples and theorems. Furthermore, it will also be defined the left ideal and the right ideal of the rough ring along with examples. Finally, we will discuss the theorem regarding rough ideals.

Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

AbstractIn this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter λ and with Dirichlet boundary conditions. For the Hamiltonian problems, we do not assume any controllability and strict normality assumptions and also omit the classical Legendre condition for their Hamiltonians. The main result of the paper, the relative oscillation theorem, relates the difference of the numbers of finite eigenvalues of the two problems in the intervals and , respectively, with the so‐called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for and . As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in . The consideration is based on the comparative index theory applied to the continuous case.

The Relative Index Theorem for General First-Order Elliptic Operators

The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov–Lawson for generalised Dirac operators as well as the result of Bär–Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär–Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.

JORDAN RIGHT DERIVATIONS ON SEMIPRIME Γ-RING

In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then . In this paper, we have analyzed the basic properties and related theorems of Jordan’s right derivations on semiprime -rings with their mathematical simulation. We mainly focused on the characterizations of and -torsion-free semiprime -ring by using Jordan Right Derivations. Important lemmas and theorems related to Jordan derivation on semiprime -ring have been derived here with sufficient calculations. Our main objective is to prove that if is a -torsion free semiprime -ring and , be the Jordan right derivations on provided that then .

Filtrations on combinatorial intersection cohomology and invariants of subdivisions

Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed h-polynomial, the refined limit mixed h⁎-polynomial, and the mixed cd-index, defined by Katz–Stapledon and Dornian–Katz–Tsang. Previously, only the refined limit mixed h⁎-polynomial had a geometric interpretation, which came from filtrations on the cohomology of a schön hypersurface. Consequently, we generalize a positivity result on the mixed h-polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu.

Study on a Strong and Weak n-Connected Total Perfect k-Dominating set in Fuzzy Graphs

In this paper, the concept of a strong n-Connected Total Perfect k-connected total perfect k-dominating set and a weak n-connected total perfect k-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an n-connected total perfect k-dominating set nctpkD(G) and number γnctpD(G). New definitions are compared with old ones. Strong and weak n-connected total perfect k-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak nctpkD(G) set of fuzzy graphs. The order and size of the strong and weak nctpkD(G) fuzzy set are studied. Additionally, a few related theorems and statements are analyzed.

Optimal Control for Parabolic Uncertain System Based on Wavelet Transformation

In this paper, we study a new type of optimal control problem subject to a parabolic uncertain partial differential equation where the expected value criterion is adopted in the objective function. The basic idea of Haar wavelet transformation is to transform the proposed problem into an approximate uncertain optimal control problem with arbitrary accuracy because the dimension of Haar basis tends to infinity. The relative convergence theorem is proved. An application to an optimal control problem with an uncertain heat equation is dealt with to illustrate the efficiency of the proposed method.

On <img src=image/13427444_01.gif>-Ideal Statistically Convergent of Double Sequences in n-Normed Spaces over Non-Archimedean Fields

The main aim of this work is to investigate some important properties of statistical convergence sequence in non-Archimedean fields. Statistical convergence has been discussed in various fields of mathematics namely approximation theory, measure theory, probability theory, trigonometric series, number theory, etc. The concept of summability over valued fields is a significant area of mathematics that has many applications in analytic continuation, quantum mechanics, probability theory, Fourier analysis, approximation theory, and fixed point theory. The theory of statistical convergence plays a notable space in the summability theory and functional analysis. The purpose of this work is to provide certain characterizations of <img src=image/13427444_01.gif> ideal statistical convergence of sequence and <img src=image/13427444_01.gif> ideal statistical Cauchy sequence in n-normed spaces and the establishment of relevant results in non-Archimedean fields. The <img src=image/13427444_01.gif> ideal statistical convergence of sequence and <img src=image/13427444_01.gif> ideal statistically Cauchy sequence are defined. A few related theorems are proved in field <img src=image/13427444_02.gif>. The results of this work are extended to establish statistical convergence of double sequences in n-normed space and some new results have been proved. In this work, the main concept is <img src=image/13427444_01.gif> ideal statistical convergence of double sequences in n-normed space over a complete, non-trivially valued, non-Archimedean field. Throughout this article, <img src=image/13427444_02.gif> is a complete, non-trivially valued, non-Archimedean field.

A Survey on Gruss-Type Inequalities by Means of Different Generalized Fractional Integrals

In this survey paper, we review a generalization of Gruss - type inequality by means of two different operators, one through K - fractional integral and other through Katugampola fractional integral. We state some related theorems, corollaries and their proofs. We study how results are interrelated?&#x0D;

Characterizations of Γ Rings in Terms of Rough Fuzzy Ideals

Fuzzy sets are a major simplification and wing of classical sets. The extended concept of set theory is rough set (RS) theory. It is a formalistic theory based upon a foundational study of the logical features of the fundamental system. The RS theory provides a new mathematical method for insufficient understanding. It enables the creation of sets of verdict rules from data in a presentable manner. An RS boundary area can be created using the algebraic operators union and intersection, which is known as an approximation. In terms of data uncertainty, fuzzy set theory and RS theory are both applicable. In general, as a uniting theme that unites diverse areas of modern arithmetic, symmetry is immensely important and helpful. The goal of this article is to present the notion of rough fuzzy ideals (RFI) in the gamma ring structure. We introduce the basic concept of RFI, and the theorems are proven for their characteristic function. After that, we explain the operations on RFI, and related theorems are given. Additionally, we prove some theorems on rough fuzzy prime ideals. Furthermore, using the concept of rough gamma endomorphism, we propose some theorems on the morphism properties of RFI in the gamma ring.

Bipolar Complex Fuzzy Subgroups

In this study, firstly, we interpret the level set, support, kernel for bipolar complex fuzzy (BCF) set, bipolar complex characteristic function, and BCF point. Then, we interpret the BCF subgroup, BCF normal subgroup, BCF conjugate, normalizer for BCF subgroup, cosets, BCF abelian subgroup, and BCF factor group. Furthermore, we present the associated examples and theorems, and prove these associated theorems. After that, we interpret the image and pre-image of BCF subgroups under homomorphism and prove the related theorems.