scholarly journals Consistent Sets of Soft Contexts Defined by Soft Sets

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 71 ◽  
Author(s):  
Won Min
Keyword(s):  
Fundamental Theorem ◽  
Special Class ◽  
Formal Context ◽  
Soft Sets ◽  
Class 1 ◽  
Fundamental Theorems

We introduce the notion of consistent sets of soft contexts and investigate its characterizations. For the purpose of studying the characterizations of consistent sets, we introduce the notions of 1 0 and 2 0 classes of independent attributes. By studying the characterizations, we determined that every consistent set has to contain the special class 1 0 called the Essential Zone of consistent sets. This is very important information that we should consider when constructing consistent sets of a given soft context. Additionally, we study the fundamental theorems necessary to construct the consistent sets of a given soft context. In particular, we apply the Fundamental Theorem 2 to obtain the consistent sets of a given soft context, and show that this fundamental theorem provides a more effective way of constructing the consistent sets of a formal context.

scholarly journals THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Brauer Algebra ◽  
Tensor Space ◽  
General Linear Supergroup ◽  
Special Cases ◽  
Linear Description ◽  
Tensor Representations ◽  
Capelli Identities ◽  
Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

scholarly journals Attribute Reduction in Soft Contexts Based on Soft Sets and Its Application to Formal Contexts

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 689
Author(s):  
Won Keun Min
Keyword(s):  
Attribute Reduction ◽  
Extraction Process ◽  
Formal Context ◽  
Soft Sets ◽  
Step Method ◽  
The Relationship

We introduce the notion of the reduct of soft contexts, which is a special notion of a consistent set for soft contexts. Then, we study its properties and show that this notion is well explained by the two classes, 1 0 and 2 0 , of independent attributes. In particular, we describe in detail how to extract a reduct from a given consistent set. Then, based on this extraction process, we propose a six-step method for constructing a reduct from a given consistent set. Additionally, to apply this method to formal contexts, we examine the relationship between the reducts of a given formal context and the reducts of the associated soft context. We finally illustrate the process of obtaining reducts in a formal context using this relationship and the six-step method using an example.

scholarly journals The Fundamental Theorems of Hyper-Operations

2019 ◽  
Author(s):  
Renny Barrett
Keyword(s):  
Fundamental Theorem ◽  
Natural Numbers ◽  
Higher Rank ◽  
Fundamental Theorem Of Arithmetic ◽  
Fundamental Theorems

We examine the extensions of the basic arithmetical operations of addition and multiplication on the natural numbers into higher-rank hyper-operations also on the natural numbers. We go on to define the concepts of prime and composite numbers under these hyper-operations and derive some results about factorisation, resulting in fundamental theorems analogous to the Fundamental Theorem of Arithmetic.

scholarly journals Indirect genetic effects clarify how traits can evolve even when fitness does not

2018 ◽  
Author(s):  
David N. Fisher ◽  
Andrew G. McAdam
Keyword(s):  
Natural Selection ◽  
Fundamental Theorem ◽  
Genetic Effects ◽  
Social Effects ◽  
Indirect Genetic Effects ◽  
The Social ◽  
Mean Fitness ◽  
Related Individuals ◽  
Fundamental Theorems ◽  
Fisher’S Fundamental Theorem

AbstractThere are many situations in nature where we expect traits to evolve but not necessarily for mean fitness to increase. However, these scenarios are hard to reconcile simultaneously with Fisher’s Fundamental Theorem of Natural Selection and the Price identity. The consideration of indirect genetic effects on fitness reconciles these fundamental theorems with the observation that traits sometimes evolve without any adaptation, by explicitly considering the correlated evolution of the social environment, which is a form of transmission bias. While transmission bias in the Price identity is often assumed to be absent, here we show that explicitly considering indirect genetic effects as a form of transmission bias for fitness has several benefits: 1) it makes clear how traits can evolve while mean fitness remains stationary, 2) it reconciles the fundamental theorem of natural selection with the evolution of maladaptation, 3) it explicitly includes density-dependent fitness through negative social effects that depend on the number of interacting conspecifics, and 4) its allows mean fitness to evolve even when direct genetic variance in fitness is zero, if related individuals interact and/or if there is multilevel selection. In summary, considering fitness in the context of indirect genetic effects aligns important theorems of natural selection with many situations observed in nature and provides a useful lens through which we might better understand evolution and adaptation.

scholarly journals Fundamental Theorems of Pure Mathematics & Absolute Geometry

2021 ◽  
Vol 4 (2) ◽  
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Fundamental Theorem ◽  
Absolute Geometry ◽  
Pure Mathematics ◽  
And Algebra ◽  
Fundamental Theorems

The superunified field theory consists of a row of discoveries in the realm of pure mathematics. It is two centuries ago that Karl Gauss unified higher arithmetic (number theory), algebra and geometry into what is called pure mathematics. The latter, however, still remains without its fundamental theorem despite that arithmetic and algebra, or even analysis, have their own.

scholarly journals The fundamental theorem via derived Morita invariance, localization, and 1-homotopy invariance

2011 ◽  
Vol 9 (3) ◽  
pp. 407-420 ◽  
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Fundamental Theorem ◽  
Cyclic Homology ◽  
Homotopy Invariance ◽  
Homotopy Invariant ◽  
Morita Invariant ◽  
Fundamental Theorems

AbstractWe prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and 1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.

Shape Control of Tensegrity Plates: An Experimental Study

Aerospace ◽  
2004 ◽  
Author(s):  
Wai Leung Chan ◽  
Soren Solari ◽  
Robert E. Skelton
Keyword(s):  
Experimental Study ◽  
Flat Plate ◽  
Special Class ◽  
Shape Control ◽  
Smart Structure ◽  
Sensors And Actuators ◽  
Space Telescope ◽  
Class 1 ◽  
Parabolic Dish ◽  
Simple Class

Tensegrity is a special class of tension-trusses. Strings play an important role in the deployment and shape changing of these structures. A simple class-1 tensegrity unit consists of 3 bars and 9 strings. This unit can be arranged to form a tower or a flat plate. With the integration of sensors and actuators, the tower and plate can be a smart structure. In this paper, we demonstrate the shape changing of a 7-unit tensegrity structure from a flat plate to a parabolic dish by controlling the length of strings in each unit. This provides a feasible structure for a next generation shape changing antenna and space telescope.

scholarly journals Sario’s Potentials and Analytic Mappings

1967 ◽  
Vol 29 ◽  
pp. 93-101
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Maximum Principle ◽  
Potential Theory ◽  
Fundamental Theorem ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Value Distribution Theory ◽  
Complete Form ◽  
Fundamental Theorems ◽  
Analytic Mappings

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

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