Let S be a semigroup and (z_{0}) a fixed element in S. We determine the complex-valued solutions of the following Kannappan-sine addition law (f(xyz_{0})=f(x)g(y)+f(y)g(x),x,yin S.)
Let S be a semigroup and (z_{0}) a fixed element in S. We determine the complex-valued solutions of the following Kannappan-sine addition law (f(xyz_{0})=f(x)g(y)+f(y)g(x),x,yin S.)
In this article, we study the set-valued dynamics related to some Euler-Lagrange type functional equations of convex-valued m-mappings. We deal with perturbations of these equations. In order to do this, we use the Banach contraction principle and the Hausdorff distance. Several outcomes on approximate solutions of a few important classic equations are discussed and some applications are given.
In this paper we present a complete asymptotic expansion of the Archimedean compound of two symmetric homogeneous means and derive recursive algorithms for coefficients in this expansion. We also show some examples and obtain explicit expansions for the Archimedean compounds of the arithmetic, geometric and harmonic means.
Apollonius defined the circle as the set of points that have a given ratio (mu ) of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and (K'), which define two norms. Their unit balls are K and (K'). The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and (K') with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere.
In this paper, we establish mathematical models for an arbitrarily fixed functional extreme learning machine (FELM). From a FELM ({mathfrak {M}}), we construct a direct graph G induced by ({mathfrak {M}}), and then define the graph groupoid ({mathbb {G}}) of G. Then the graph-groupoid (C^{*})-algebra (M_{G}) of G generated by ({mathbb {G}}) is well-determined. This (C^{*})-algebra (M_{G}) is realized on a certain Hilbert space (H_{G}) up to a canonical representation. It means that the FELM ({mathfrak {M}}) is analyzed in a representation-depending structure in terms of operator algebra theory. By defining a natural free probability on (M_{G}), one can have an assessment tool of the operator algebra on (M_{G}), too.
In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n-metric which assigns a value to a tuple of (n ge 2) points. Some elementary properties of pseudo n-metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n-metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n-metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n-metric.
Using Schauder’s fixed point theorem and the Banach contraction principle, the existence, uniqueness, and stability of monotone solutions and uniformly-like convex solutions of the polynomial-like iterative functional equation are studied in Banach spaces. Furthermore, the approximate solutions of the corresponding solutions are considered. Some examples are considered for our results.
Let S be a semigroup, (z_0) a fixed element in S and (sigma :S longrightarrow S) an involutive automorphism. We determine the complex-valued solutions of the Kannappan-sine subtraction law
$$begin{aligned} f(xsigma (y)z_0)=f(x)g(y)-f(y)g(x),; x,y in S. end{aligned}$$As an application we solve the following variant of the Kannappan-sine subtraction law viz.
$$begin{aligned} f(xsigma (y)z_0)=f(x)g(y)-f(y)g(x)+lambda g(xsigma (y)z_0),;x,y in S, end{aligned}$$where (lambda in mathbb {C}^{*}). The continuous solutions on topological semigroups are given and an example to illustrate the main results is also given.
In this note, we study a family of polynomials that appear naturally when analysing the characteristic functions of the one-dimensional elephant random walk. These polynomials depend on a memory parameter p attached to the model. For certain values of p, these polynomials specialise to classical polynomials, such as the Chebychev polynomials in the simplest case, or generating polynomials of various combinatorial triangular arrays (e.g. Eulerian numbers). Although these polynomials are generically non-orthogonal (except for (p=frac{1}{2}) and (p=1)), they have interlacing roots. Finally, we relate some algebraic properties of these polynomials to the probabilistic behaviour of the elephant random walk. Our methods are reminiscent of classical orthogonal polynomial theory and are elementary.
In this paper we provide a review of the concept of center of an n-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for n-gons for (nge 3) and establish its basic properties.
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