COMPARATOR IDENTIFICATION IN THE CONDITIONS OF BIFUZZY INITIAL DATA

When solving a large number of problems in the study of complex systems, it becomes necessary to establish a relationship between a variable that sets the level of efficiency of the system's functioning and a set of other variables that determine the state of the system or the conditions of its operation. To solve this problem, the methods of regression analysis are traditionally used, the application of which in many real situations turns out to be impossible due to the lack of the possibility of direct measurement of the explained variable. However, if the totality of the results of the experiments performed can be ranked, for example, in descending order, thus forming a system of inequalities, the problem can be presented in such a way as to determine the coefficients of the regression equation in accordance with the following requirement. It is necessary that the results of calculating the explained variable using the resulting regression equation satisfy the formed system of inequalities. This task is called the comparator identification task. The paper proposes a method for solving the problem of comparator identification in conditions of fuzzy initial data. A mathematical model is introduced to describe the membership functions of fuzzy parameters of the problem based on functions (L–R) – type. The problem is reduced to a system of linear algebraic equations with fuzzy variables. The analytical relationships required for the formation of a quality criterion for solving the problem of comparator identification in conditions of fuzzy initial data are obtained. As a result, a criterion for the effectiveness of the solution is proposed, based on the calculation of membership functions of the results of experiments, and the transformation of the problem to a standard problem of linear programming is shown. The desired result is achieved by solving a quadratic mathematical programming problem with a linear constraint. The proposed method is generalized to the case when the fuzzy initial data are given bifuzzy

Liouville-type theorem for high order degenerate Lane-Emden system

<p style='text-indent:20px;'>In this paper, we are concerned with the following high order degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE111000">\begin{document}$\left\{ \begin{align} &amp; {{(-A)}^{m}}u={{v}^{p}} \\ &amp; {{(-A)}^{m}}v={{u}^{q}}\quad \text{ in }\mathbb{R}_{+}^{n+1}:=\left\{ (x,y)|x\in {{\mathbb{R}}^{n}},y&gt;0 \right\}, \\ &amp; u\ge 0,v\ge 0 \\ \end{align} \right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>where the operator <inline-formula><tex-math id="M1">\begin{document}$ A: = y\partial_{y}^2+a\partial_{y}+\Delta_{x}, \;a\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ n+2a&gt;2m, m\in \mathbb{Z}^+,\;p,\,q\geq 1 $\end{document}</tex-math></inline-formula>. We prove the non-existence of positive smooth solutions for <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p,\, q&lt;\frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>, and classify positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p = q = \frac{n+2a+2m}{n+2a-2m} $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{p+1}+\frac{1}{q+1}&gt;\frac{n+2a-2m}{n+2a} $\end{document}</tex-math></inline-formula>, we show the non-existence of positive, ellipse-radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities <inline-formula><tex-math id="M6">\begin{document}$ (-A)^{m}u\geq v^p, (-A)^{m}v\geq u^q, u\geq 0, v\geq 0, $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}_+^{n+1} $\end{document}</tex-math></inline-formula> for either <inline-formula><tex-math id="M8">\begin{document}$ (n+2a-2m)q&lt;\frac{n+2a}{p}+2m $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M9">\begin{document}$ (n+2a-2m)p&lt;\frac{n+2a}{q}+2m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M10">\begin{document}$ p,q&gt;1 $\end{document}</tex-math></inline-formula>.</p>

PC-solutions and quasi-solutions of the interval system of linear algebraic equations

The problem of solving the interval system of linear algebraic equations (ISLAEs) is one of the well-known problems of interval analysis, which is currently undergoing intensive development. In general, this solution represents a set, which may be given differently, de- pending on which quantifiers are related to the elements of the left and right sides of this system. Each set of solutions of ISLAE to be determined is described by the domain of compatibility of the corresponding system of linear inequalities and, normally, one nonlinear condition of the type of complementarity. It is difficult to work with them when solving specific problems. Therefore, in the case of nonemptiness in the process of solving the problem it is recommended to find a so-called PC-solution, based on the application of the technique known in the theory of multi-criterial choice, that presumes maximization of the solving capacity of the system of inequalities. If this set is empty, it is recommended to find a quasi- solution of ISLAE. The authors compare the approach proposed for finding PC- and/or quasi-solutions to the approach proposed by S. P. Shary, which is based on the application of the recognizing functional.

KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( X ) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$ NT ( X ) .

Milan’s inclusive entrepreneurship policies: the risk of ‘differentiated inclusion’

Using Milan’s entrepreneurship policies as a case-study, this paper explores the diversity that the broad policy category of ‘inclusive entrepreneurship’ may entail in terms of policy instruments and participants’ experiences. To this end, the study considers policymakers, practitioners and young adults receiving local support to establish their own businesses. It does so by drawing on qualitative interviews. It argues that inclusive entrepreneurship policies may entail a ‘differentiated inclusion’ that is produced by the simultaneous interaction between diversified policy instruments and a system of inequalities that affect individual capacities to strategise entrepreneurial risks and opportunities.

Classification of Skills of Students of Senior High School at the Study of Algebra and the Beginning of the Analysis

The classification of skills is developed, which is the basis for the development of the pupils’ mathematical abilities for basic and in-depth study of the algebra course and the beginning of analysis in the format of numerical, algebraic, trigonometric and stochastic lines. Block 1 (skills, students at the basic level, aimed at mastering the methods of learning mathematical activity (mastering the basic techniques of general learning activity, operating (rational expressions, functions, power with a rational exponent, square roots and polynomials), mastering the technology to solve (equation, inequality and a system of equations with two variables), operation (sequences, elements of combinatorics, and probabilistic properties of the surrounding phenomena.) Block 2 (students’ abilities in depth (algebraic and irrational expressions, functions, degrees with rational and irrational exponent, n-th roots and polynomials), mastering the technology to solve (equation, system of equations and system of inequalities with two variables), operating sequences, elements of combinatorics, and probabilistic properties of surrounding phenomena).

Bell’s theorem

Bell’s theorem is concerned with the outcomes of a special type of ‘correlation experiment’ in quantum mechanics. It shows that under certain conditions these outcomes would be restricted by a system of inequalities (the ‘Bell inequalities’) that contradict the predictions of quantum mechanics. Various experimental tests confirm the quantum predictions to a high degree and hence violate the Bell inequalities. Although these tests contain loopholes due to experimental inefficiencies, they do suggest that the assumptions behind the Bell inequalities are incompatible not only with quantum theory but also with nature. A central assumption used to derive the Bell inequalities is a species of no-action-at-a-distance, called ‘locality’: roughly, that the outcomes in one wing of the experiment cannot immediately be affected by measurements performed in another wing (spatially distant from the first). For this reason the Bell theorem is sometimes cited as showing that locality is incompatible with the quantum theory, and the experimental tests as demonstrating that nature is nonlocal. These claims have been contested.

The Yamada polynomial of spatial graphs obtained by edge replacements

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labeled graphs is obtained.

The Uniqueness Theorem of the Solution for a Class of Differential Systems with Coupled Integral Boundary Conditions

We discuss the uniqueness of the solution to a class of differential systems with coupled integral boundary conditions under a Lipschitz condition. Our main method is the linear operator theory and the solvability for a system of inequalities. Finally, an example is given to demonstrate the validity of our main results.