Setting Of Vector Spaces Research Articles

The Study on the (<i>L,M</i>)-Fuzzy Independent Set Systems

Independent sets play an important role in matroid theory. In this paper, the definitions of pre-independent fuzzy set system and independent fuzzy set system in L-fuzzy setting are presented. Independent M-fuzzifying set system is introduced and some of its properties are discussed. Further independent (L,M)-fuzzy set system is given and some of its properties are obtained. The relations of these independent set systems in the setting of fuzzy vector spaces and fuzzy graphs are showed.

Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses

Simplified neutrosophic set Single valued neutrosophic set Interval neutrosophic set Cosine similarity measure Medical diagnosis a b s t r a c t Objective: In pattern recognition and medical diagnosis, similarity measure is an important mathematical tool. To overcome some disadvantages of existing cosine similarity measures of simplified neutrosophic sets (SNSs) in vector space, this paper proposed improved cosine similarity measures of SNSs based on cosine function, including single valued neutrosophic cosine similarity measures and interval neutrosophic cosine similarity measures. Then, weighted cosine similarity measures of SNSs were introduced by taking into account the importance of each element. Further, a medical diagnosis method using the improved cosine similarity measures was proposed to solve medical diagnosis problems with simplified neutrosophic information. Materials and methods: The improved cosine similarity measures between SNSs were introduced based on cosine function. Then, we compared the improved cosine similarity measures of SNSs with existing cosine similarity measures of SNSs by numerical examples to demonstrate their effectiveness and rationality for overcoming some shortcomings of existing cosine similarity measures of SNSs in some cases. In the medical diagnosis method, we can find a proper diagnosis by the cosine similarity measures between the symptoms and considered diseases which are represented by SNSs. Then, the medical diagnosis method based on the improved cosine similarity measures was applied to two medical diagnosis problems to show the applications and effectiveness of the proposed method. Results: Two numerical examples all demonstrated that the improved cosine similarity measures of SNSs based on the cosine function can overcome the shortcomings of the existing cosine similarity measures between two vectors in some cases. By two medical diagnoses problems, the medical diagnoses using various similarity measures of SNSs indicated the identical diagnosis results and demonstrated the effectiveness and rationality of the diagnosis method proposed in this paper. Conclusions: The improved cosine measures of SNSs based on cosine function can overcome some drawbacks of existing cosine similarity measures of SNSs in vector space, and then their diagnosis method is very suitable for handling the medical diagnosis problems with simplified neutrosophic information and demonstrates the effectiveness and rationality of medical diagnoses.

Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses.

In pattern recognition and medical diagnosis, similarity measure is an important mathematical tool. To overcome some disadvantages of existing cosine similarity measures of simplified neutrosophic sets (SNSs) in vector space, this paper proposed improved cosine similarity measures of SNSs based on cosine function, including single valued neutrosophic cosine similarity measures and interval neutrosophic cosine similarity measures. Then, weighted cosine similarity measures of SNSs were introduced by taking into account the importance of each element. Further, a medical diagnosis method using the improved cosine similarity measures was proposed to solve medical diagnosis problems with simplified neutrosophic information. The improved cosine similarity measures between SNSs were introduced based on cosine function. Then, we compared the improved cosine similarity measures of SNSs with existing cosine similarity measures of SNSs by numerical examples to demonstrate their effectiveness and rationality for overcoming some shortcomings of existing cosine similarity measures of SNSs in some cases. In the medical diagnosis method, we can find a proper diagnosis by the cosine similarity measures between the symptoms and considered diseases which are represented by SNSs. Then, the medical diagnosis method based on the improved cosine similarity measures was applied to two medical diagnosis problems to show the applications and effectiveness of the proposed method. Two numerical examples all demonstrated that the improved cosine similarity measures of SNSs based on the cosine function can overcome the shortcomings of the existing cosine similarity measures between two vectors in some cases. By two medical diagnoses problems, the medical diagnoses using various similarity measures of SNSs indicated the identical diagnosis results and demonstrated the effectiveness and rationality of the diagnosis method proposed in this paper. The improved cosine measures of SNSs based on cosine function can overcome some drawbacks of existing cosine similarity measures of SNSs in vector space, and then their diagnosis method is very suitable for handling the medical diagnosis problems with simplified neutrosophic information and demonstrates the effectiveness and rationality of medical diagnoses.

Extended Vector Equilibrium Problem

In this paper, we consider an extended vector equilibrium problem and prove some existence results in the setting of Hausdorff topological vector spaces and reflexive Banach spaces. Our results extend and improve some known results in the literature. Some examples are given.

Algebraic Separation and Shadowing of Arbitrary Sets

In this paper we consider a generalization of the separation technique proposed in [10,4,7] for the separation of finitely many compact convex sets \(A_i,~i \in I \) by another compact convex set \(S\) in a locally convex vector space to arbitrary sets in real vector spaces. Then we investigate the notation of shadowing set which is a generalization of the notion of separating set and construct separating sets by means of a generalized Demyanov-difference in locally convex vector spaces.

On distance sets and product sets in vector spaces over finite rings

On distance sets and product sets in vector spaces over finite rings

Spread decoding in extension fields

A spread code is a set of vector spaces of a fixed dimension over a finite field Fq with certain properties used for random network coding. It can be constructed in different ways which lead to different decoding algorithms. In this work we consider one such representation of spread codes and present a minimum distance decoding algorithm which is efficient when the code words, the received space and the error space have small dimension.

Representing finite convex geometries by relatively convex sets

A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in n-dimensional vector space and their finite sub-geometries satisfy the n-Carousel Rule, which is the strengthening of the n-Carathéodory property. We also find another property, that is similar to the simplex partition property and independent of 2-Carousel Rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.

On the volume set of point sets in vector spaces over finite fields

We show that if E \mathcal {E} is a subset of the d d -dimensional vector space over a finite field F q \mathbb {F}_q ( d ≥ 3 d \geq 3 ) of cardinality | E | ≥ ( d − 1 ) q d − 1 |\mathcal {E}| \geq (d-1)q^{d - 1} , then the set of volumes of d d -dimensional parallelepipeds determined by E \mathcal {E} covers F q \mathbb {F}_q . This bound is sharp up to a factor of ( d − 1 ) (d-1) , as taking E \mathcal {E} to be a ( d − 1 ) (d - 1) -hyperplane through the origin shows.

(F, K, b)-vex sets and some related properties

In this paper, we introduce the concept of (F, K, b)-vex sets in vector spaces which is a generalization of (E-b)-vex sets given by Mishra, Mohapatra and Youness. Some properties of (F, K, b)-vex sets are investigated.

Integral Cayley Graphs Generated by Distance Sets in Vector Spaces over Finite Fields

Si Li and the fourth listed author (2008) considered unitary graphs attached to the vector spaces over finite rings using an analogue of the Euclidean distance. These graphs are shown to be integral when the cardinality of the ring is odd or the dimension is even. In this paper, we show that the statement also holds for the remaining case: the cardinality of the ring is even and the dimension is odd, by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields to be integral.

Product sets and distance sets of random point sets in vector spaces over finite rings

Product sets and distance sets of random point sets in vector spaces over finite rings

Refinements of existence results for relaxed quasimonotone equilibrium problems

We consider a general equilibrium problem in a normed vector space setting and we establish sufficient conditions for the existence of solutions in compact and non compact cases. Our approach is based on the concept of upper sign property for bifunctions, which turns out to be a very weak assumption for equilibrium problems. In the framework of variational inequalities, this notion coincides with the upper sign continuity for a set-valued operator introduced by Hadjisavvas. More in general, it allows to strengthen a number of existence results for the class of relaxed $$\mu $$ -quasimonotone equilibrium problems.

A soft introduction to algebraic entropy

The goal of this mainly expository paper is to develop the theory of the algebraic entropy in the basic setting of vector spaces V over a field K. Many complications encountered in more general settings do not appear at this first level. We will prove the basic properties of the algebraic entropy of linear transformations \({\phi:V \to V}\) of vector spaces and its characterization as the rank of V viewed as module over the polynomial ring K[X] through the action of \({\phi}\) . The two main theorems on the algebraic entropy, namely, the Addition Theorem and the Uniqueness Theorem, whose proofs require many efforts in more general settings, are easily deduced from the above characterization. The adjoint algebraic entropy of a linear transformation, its connection with the algebraic entropy of the adjoint map of the dual space and the dichotomy of its behavior are also illustrated.

EXISTENCE OF SOLUTIONS OF STRONG VECTOR EQUILIBRIUM PROBLEMS

In this paper, we consider strong form of a vector equilibrium problem and establish some existence results for solutions of such a problem in the setting of topological vector spaces. We provide several coercivity conditions under which strong vector equilibrium problem has a solution. Our results generalize and extend the results of Bianchi and Pini [10] to the topological vector space setting.

Kakeya-type sets in finite vector spaces

For a finite vector space V and a nonnegative integer r≤dim V, we estimate the smallest possible size of a subset of V, containing a translate of every r-dimensional subspace. In particular, we show that if K⊆V is the smallest subset with this property, n denotes the dimension of V, and q is the size of the underlying field, then for r bounded and r<n≤rq r−1, we have |V∖K|=Θ(nq n−r+1); this improves the previously known bounds |V∖K|=Ω(q n−r+1) and |V∖K|=O(n 2 q n−r+1).

Minkowski-Voronoi diagrams as a method to generate random packings of spheropolygons for the simulation of soils

Minkowski operators (dilation and erosion of sets in vector spaces) have been extensively used in computer graphics, image processing to analyze the structure of materials, and more recently in molecular dynamics. Here, we apply those mathematical concepts to extend the discrete element method to simulate granular materials with complex-shaped particles. The Voronoi-Minkowski diagrams are introduced to generate random packings of complex-shaped particles with tunable particle roundness. Contact forces and potentials are calculated in terms of distances instead of overlaps. By using the Verlet method to detect neighborhood, we achieve CPU times that grow linearly with the body's number of sides. Simulations of dissipative granular materials under shear demonstrate that the method maintains conservation of energy in accord with the first law of thermodynamics. A series of simulations for biaxial test, shear band formation, hysteretic behavior, and ratcheting show that the model can reproduce the main features of real granular-soil behavior.

An accurate numerical integration scheme for finite rotations using rotation vector parametrization

A numerical integration procedure for rotational motion using a rotation vector parametrization is explored from an engineering perspective by using rudimentary vector analysis. The incremental rotation vector, angular velocity and acceleration correspond to different tangent spaces of the rotation manifold at different times and have a non-vectorial character. We rewrite the equation of motion in terms of vectors lying in the same tangent space, facilitating vector space operations consistent with the underlying geometric structure. While any integration algorithm (that works within a vector space setting) may be used, we presently employ a family of explicit Runge–Kutta algorithms to solve this equation. While this work is primarily motivated out of a need for highly accurate numerical solutions of dissipative rotational systems of engineering interest, we also compare the numerical performance of the present scheme with some of the invariant preserving schemes, namely ALGO-C1, STW, LIEMID[EA] and SUBCYC-M. Numerical results show better local accuracy via the present approach vis-á-vis the preserving algorithms. It is also noted that the preserving algorithms do not simultaneously preserve all constants of motion. We incorporate adaptive time-stepping within the present scheme and this in turn enables still higher accuracy and a ‘near preservation’ of constants of motion over significantly longer intervals.

Existence of Solutions for η-Generalized Vector Variational-Like Inequalities

We introduce and study a class of -generalized vector variational-like inequalities and a class of -generalized strong vector variational-like inequalities in the setting of Hausdorff topological vector spaces. An equivalence result concerned with two classes of -generalized vector variational-like inequalities is proved under suitable conditions. By using FKKM theorem, some new existence results of solutions for the -generalized vector variational-like inequalities and -generalized strong vector variational-like inequalities are obtained under some suitable conditions.

GENERALIZED F-IMPLICIT MULTIVALUED VARIATIONAL INEQUALITY PROBLEMS AND COMPLEMENTARITY PROBLEMS

In this paper, we study generalized F-implicit multivalued variational inequality problems on a real normed vector space setting. As an application, we study generalized F-implicit multivalued complementarity problems.