Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

On the Best Constant Problem for a Class of Mixed-Norm Hardy Inequalities

In this paper, we obtain the best constant and the equality condition for a class of mixed-norm Hardy inequalities when the weight is a power function. By building and solving the corresponding Euler equation, we look for the best constant and the optimal function. One of the main ingredients is to introduce two key auxiliary functions so that the corresponding equalities are derived.

Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes

We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship dH/dt≤0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems.

An answer to a conjecture on the sum of element orders

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

The Case against Different-Sex Marriage in Kant

Recently, a number of Kantians have argued that despite Kant’s own disparaging comments about same-sex intercourse and marriage, his ethical and legal philosophy lacks the resources to show that they are impermissible. I go further by arguing that his framework is in fact more open to same-sex than to different-sex marriage. Central is Kant’s claim that marriage requires equality between spouses. Kant himself thought that men and women are not equal, and some of his more insightful remarks on the issue reveal that he was also aware that, as a matter of fact, women were disenfranchised by society, and suffer legal and other forms of discrimination. Kant, according to his own account, cannot approve of heterosexual marriage. Same-sex couples, by contrast, can satisfy the crucial equality condition. I conclude with a suggestion for refocus with respect to the issues at hand, calling for attention to more complex and insidious forms of inequality than deprivation of rights and full civil participation.

Equality case for an elliptic area condenser inequality and a related Schwarz type lemma

We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

A Direct Link between Rényi–Tsallis Entropy and Hölder’s Inequality—Yet Another Proof of Rényi–Tsallis Entropy Maximization

The well-known Hölder’s inequality has been recently utilized as an essential tool for solving several optimization problems. However, such an essential role of Hölder’s inequality does not seem to have been reported in the context of generalized entropy, including Rényi–Tsallis entropy. Here, we identify a direct link between Rényi–Tsallis entropy and Hölder’s inequality. Specifically, we demonstrate yet another elegant proof of the Rényi–Tsallis entropy maximization problem. Especially for the Tsallis entropy maximization problem, only with the equality condition of Hölder’s inequality is the q-Gaussian distribution uniquely specified and also proved to be optimal.

CALCULATION OF POLYNOMIAL RATIO APPROXIMATED SURFACES OF BIFOCAL LENS

General imperfection of existent synthesis methods of bifocal lens surface is use complexity emerging over lack of analytical solution. Method of successive approximations is relatively simple, its main problem is determination of power polynomial coefficient approximating illuminated and shadow surface of being synthesized lens. To solve this problem it is suggest analytical determination method of polynomial coefficient approximating illuminated and shadow surface of bifocal lens. This method based on equality condition of electric path length of edge beam and beam passing through lens center. It is obtained analytical equation which permit to determine interdependence polynomials coefficients approximating illuminated and shadow surface of bifocal lens. Moreover, been finded equation allow to calculate bifocal lens thickness, that impossible using known methods of synthesis methods of bifocal lens surface. Cross-section of bifocal lens was be calculated with being suggested analytical method.

A refinement and an exact equality condition for the basic inequality of f-divergences

f-divergences play important role in probability theory, especially in information theory and in mathematical statistics. Remarkable divergences can be found among them. Inequalities for f-divergences are very useful and applicable in information theory. In this paper we give a precise equality condition and a refinement for one of the basic inequalities of f-divergences. The results are illustrated by some applications.

Calculations of the complex network’s steady-state modes by brining it to equivalent open

The paper considers the development of the idea of diakoptics as applied to the calculation of the steady-state modes of energy system’s complex electrical networks. The well-known goal of diakoptics is to obtain the equations of state for the dedicated part of the system, the study of which is much simpler than the study of the initial system and can be achieved by improving its steady state equations. Technique for dividing a complex-closed system into a set of uncomplicated subsystems was developed based on the inverse form of nodal equations. Analytic expressions for the intersection circuits obtained based on identical equality of voltage at the nodes of the system division into subsystems are proposed. Using the example of 110kV network calculation, the technique for determining the matrixes of generalized parameters of the dedicated subsystems, the sizes of which depend on the number of their broken link is shown. Analytical determination of the equality condition of the voltage of subsystems intersections nodes, allowed analyzing a complex closed network by bringing it to an equivalent open.