k, t, d-Proximities in Rough Set

The present paper is devoted to the study of k, t, d −proximity on rough sets from the relation point of view. The operator O0(φ,Α) and O(φ,Α) is defined using the upper approximation and closure operators derived from the relation. The properties of induced operator and their connections are henceforth obtained. Moreover, our approach represents a new generalization of the operator using upper approximation only.

Solution formulas for differential Sylvester and Lyapunov equations

The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches when applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $${\mathcal {S}}(X)=AX+XB$$S(X)=AX+XB and derive a formula for its norm using an induced operator norm based on the spectrum of A and B. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

Induced operators on the space of homogeneous polynomials

Let [Formula: see text] be the complex vector space of homogeneous polynomials of degree [Formula: see text] with the independent variables [Formula: see text]. Let [Formula: see text] be the complex vector space of homogeneous linear polynomials in the variables [Formula: see text]. For any linear operator [Formula: see text] acting on [Formula: see text], there is a (unique) induced operator [Formula: see text] acting on [Formula: see text] satisfying [Formula: see text] In this paper, we study some algebraic and geometric properties of induced operator [Formula: see text]. Also, we obtain the norm of the derivative of the map [Formula: see text] in terms of the norm of [Formula: see text].

Numerical Range of the Derivation of an Induced Operator

Let V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that⊆ W*(DK(T)) ⊆ W(DK(T)) = con where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

A knowledge-Induced Operator Model

Learning systems are in the forefront of analytical investigation in the sciences. In the social sciences they occupy the study of complexity and strongly interactive world-systems. Sometimes they are diversely referred to as symbiotics and semiotics when studied in conjunction with logical expressions. In the mathematical sciences the methodology underlying learning systems with complex behavior is based on formal logic or systems analysis. In this paper relationally learning systems are shown to transcend the space-time domain of scientific investigation into the knowledge dimension. Such a knowledge domain is explained by pervasive interaction leading to integration and followed by continuous evolution as complementary processes existing between entities and systemic domains in world-systems, thus the abbreviation IIE-processes. This paper establishes a mathematical characterization of the properties of knowledge-induced process-based world-systems in the light of the epistemology of unity of knowledge signified in this paper by extensive complementarities caused by the epistemic and ontological foundation of the text of unity of knowledge, the prime example of which is the realm of the divine laws. The result is formalism in mathematical generalization of the learning phenomenon by means of an operator. This operator summarizes the properties of interaction, integration and evolution (IIE) in the continuum domain of knowledge formation signified by universal complementarities across entities, systems and sub-systems in unifying world-systems. The opposite case of ‘de-knowledge’ and its operator is also briefly formalized.