Some Applications of New Complex Function Space Constructed by Different Weights and Exponents

In this article, we develop and study a new complex function space formed by varying the weights and exponents under a definite function. We investigate the geometric and topological characteristics of mapping ideals created using s -numbers and this complex function space. Also, the action of shift mappings on this complex function space has been discussed. Finally, we introduced an extension of Caristi’s fixed point theorem on it.

Understanding Transitional Spaces: A Case Study of three different phases of Delhi – Old Delhi, Colonial Delhi and Contemporary Delhi

Any space needs to be conceptualized by thorough study of environment, its surroundings and community needs. These spaces are planned to provide a distinct function but many spaces are created with no definite function and are used as a changeover between two spaces. These spaces are referred as ‘Transition Spaces ’and they generate a ‘Spatial prospect ’for many activities, rather than serving a specific function. In this changing time of urbanization, the skyline of the city is changing from traditional buildings to glittering glass and steel structures, overshadowing the existing fabric of the city. This change is sudden not gradual. One perceives the landmarks and left behind are the unrecognizable edges and nodes. These nodes and edges are spaces where people interact and intermingle and thus transition spaces are formed. These transition spaces play a vital role in environmental behavior. The idea of this study is to understand the essence of a space in which one experiences a shift. This shift is important because that is the area where most of the activities happen. Space, like man, needs an identity else it would be lost in time. It is necessary for us to be able to distinguish between the ideas of such places, else understanding the transitions would be difficult. ‘People and space depend on one another; they share each other their true colours. ’(Hertzberger, 2000)

Kannan nonexpansive maps on generalized Cesàro backward difference sequence space of non-absolute type with applications to summable equations

In this article, we investigate the notion of the pre-quasi norm on a generalized Cesàro backward difference sequence space of non-absolute type $(\Xi (\Delta,r) )_{\psi }$ ( Ξ ( Δ , r ) ) ψ under definite function ψ. We introduce the sufficient set-up on it to form a pre-quasi Banach and a closed special space of sequences (sss), the actuality of a fixed point of a Kannan pre-quasi norm contraction mapping on $(\Xi (\Delta,r) )_{\psi }$ ( Ξ ( Δ , r ) ) ψ , it supports the property (R) and has the pre-quasi normal structure property. The existence of a fixed point of the Kannan pre-quasi norm nonexpansive mapping on $(\Xi (\Delta,r) )_{\psi }$ ( Ξ ( Δ , r ) ) ψ and the Kannan pre-quasi norm contraction mapping in the pre-quasi Banach operator ideal constructed by $(\Xi (\Delta,r) )_{\psi }$ ( Ξ ( Δ , r ) ) ψ and s-numbers has been determined. Finally, we support our results by some applications to the existence of solutions of summable equations and illustrative examples.

ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

Evenly positive definite function of Hilbert space. Self-adjoint operators which are connected by algebraic relationship

A generalization of P. A. Minlos, V. V. Sazonov’s theorem is proved in a case of bounded evenly positive definite function given in Hilbert space. The integral representation is obtained for a family of bounded commutative self-adjoint operators which are connected by algebraic relationship.

Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

<p style='text-indent:20px;'>In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the <i>kinetic</i> and <i>potential</i> equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the <inline-formula><tex-math id="M1">\begin{document}$ C^{\infty} $\end{document}</tex-math></inline-formula> category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.</p>

ROUSSEAU’S “SOCIAL CONTRACT” AND THE CONCEPT OF ALIENATION

The article actualizes the problem of the possibilities of achieving freedom in civil society. The authors consider this problem through the analysis of the conceptual essence of the phenomenon of “social contract” in its relationship with the concept of alienation. Purpose of the article: to analyze the concept of alienation in versatile historical interpretations of historical and socio-philosophical thought. The article examines the views of Rousseau on the relationship between citizens and the state. The texts of Hobbes, Locke, Hegel and Marx, which considered the concept of alienation, are analyzed. The article substantiates the escalation significance of the socio-philosophical understanding of the state of “civil” freedom, analyzes the essence of the general logic of legal consciousness, identifies the main positions of Rousseau’s concept, which defines a social contract as a dialectical unity of alienated potentials that form a dynamic whole “political machine”. The article also details the positions of Hobbes, Locke, Hegel and Marx, given in comparison with the views of Rousseau. The category of “alienation” is analyzed in the context of specific relations between the subject and his definite function, arising as a result of the loss of the initial integrity/unity and being a predictor of the impoverishment of the nature of the subject itself, leading to the transformation of the function itself. The article concludes about the relevance of Rousseau’s theory of alienation for socio-philosophical knowledge. The authors come to the conclusion that the concept of alienation in versatile historical interpretations makes a full turn before returning to its most “balanced” interpretation - Rousseau’s “social contract”. These provisions remain relevant today. The social contract destroys the “natural” generic quality of a person - to be free, alienating her arbitrary, and often just random gifts in favor of a voluntary association that rationally uses all possible and ultimate values of this ideal and real state of a person burdened with social duty.

Positive definite functions and cut-off for discrete groups

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

INTEGRAL REPRESENTATION OF EVEN POSITIVE DEFINITE BOUNDED FUNCTIONS OF AN INFINITE NUMBER OF VARIABLES

In this article the integral representation for bounded even positive functions $k(x)$\linebreak $\left(x\in \mathbb{R}^\infty=\mathbb{R}\times\mathbb{R}\times\dots \right)$ is proved. We understand the positive the positive definite in the integral sense with integration respects to measure $d\theta(x)= p(x_1)dx_1\otimes p(x_2)dx_2\otimes \dots$\linebreak $\left(p(x)=\sqrt{\frac{1}{\pi}}e^{-x^2} \right)$. This integral representation has the form \begin{equation}\label{ovl1.0} k(x)=\int\limits_{l_2^+} {\rm Cos}\,\lambda_ix_id\rho(\lambda) \end{equation} Equality stands to reason for almost all $x\in \mathbb{R}^\infty$. $l_2^+$ space consists of those vectors $\lambda\in\mathbb{R}^\infty_+=\mathbb{R}^1_+\times \mathbb{R}^1_+\times\dots\left| \sum\limits_{i=1}^\infty \lambda_i^2 <\infty\right.$. Conversely, every integral of form~\eqref{ovl1.0} is bounded by even positively definite function $k(x)$ $x\in\mathbb{R}^\infty$. As a result, from this theorem we shall get generalization of theorem of R.~A.~Minlos--V.~V.~Sazonov \cite{lov2,lov3} in case of bounded even positively definite functions $k(x)$ $(x\in H)$, which are continuous in $O$ in $j$"=topology.