In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.

We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of $\mathcal{L}^p$ spaces for $1< p<\infty$. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.

This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyze the finite-time synchronization problem of neural networks with stochasticity. Finally, numerical simulations are provided to demonstrate the effectiveness and superiority of the obtained results.

In this paper, we have established some new bounds of Fejér type Hermite-Hadamard inequality for k−fractional integrals involving r−times differentiable preinvex functions. It is noteworthy that in the past there was no weighted version of left and right sides of the Hermite-Hadamard inequality for k-fractional integrals for generalized convex functions available in literature.

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that excludes the existence of non-Shannon inequalities. It is demonstrated that planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.

(i) Investigations of stellar systems known as wide binaries suggest that Newton’s law of gravitation breaks down at low accelerations. This can be explained by proposing a fifth force with $U(1)$ gauge symmetry, whose corresponding gauge boson is the sought for dark matter candidate. (ii) Relativistic causality (i.e. no faster than light signalling) permits nonlocal correlations stronger than those permitted by quantum mechanics. This can be explained if our universe possesses a second 4D spacetime with a signature opposite to ours. (iii) Such a fifth force, and a second spacetime, arise naturally in a recently proposed theory of unification based on an $E_8\times E_8$ symmetry. Our spacetime arises from breaking of an $SU(2)_R\times U(1)_{YDEM}$ symmetry, and the second spacetime arises from broken electroweak symmetry.

We present a new penalized method for estimation in sparse logistic regression models with group structure. Group sparsity suggests us to consider the Group Lasso penalty. Being different from penalized log-likelihood estimation, our method can be viewed as a penalized weighted score function method. Under some mild conditions, we provide non-asymptotic oracle inequalities promoting group sparsity of predictors. A modified block coordinate descent algorithm based on a weighted score function is also employed. The net advantage of our algorithm over the existing Group Lasso-type procedures is that the tuning parameter can be pre-specified. The simulations show that this algorithm is considerably faster and more stable than competing methods. Finally, we illustrate our methodology with two real data sets.

We establish a new Gagliardo-Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behaviour. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of local or non-local nature.

In this paper, we suggest an expansion of the Planck distribution function derived from the Boltzmann principle. Thereby, we consider expanding Planck's law with a new distribution function, and discuss fine structure constant. Furthermore, using ideas applied to the expansion of the Planck distribution function, we show that Von Koch's inequality can be derived without using the Riemann Hypothesis, that is, the Riemann Hypothesis is true, and that the abc conjecture is not true. Furthermore, we define a generalization of Entropy and discuss that Entropy is relevant to dynamical systems described by logistic function models, such as the growth of bacteria and populations.

In this study, we define new classes of convexity called of h-Godunova-Levin and h-Godunova-Levin preinvexity, through which some new inequalities of Hermite-Hadamard type were established. These new classes are the generalisation of several known convexity including the s-convex, P-function, Godunova-Levin. Further, the properties of h-Godunova-Levin function were also discussed. Meanwhile, the applications of h-Godunova-Levin Preinvex function are given

In this article, we analyse the ESG determinants of the “Elderly People Treated in Integrated Home Care”-EPIHC in the Italian regions between 2004 and 2022. We used data from the ISTAT-BES database. We used different econometric techniques i.e.: Panel Data with Random Effects, Panel Data with Fixed Effects, Pooled Ordinary Least Squares-OLS and Weighted Least Squares-WLS. The results show that the EPIHC is positively associated with “Nurses, midwives, and Soil sealing by artificial cover" and negatively associated with "Museum heritage density and relevance" and "Trust in law enforcement agencies and firefighters fire". Furthermore, we have applied a k-Means algorithm with the Silhouette Coefficient and we find the presence of two clusters. Finally, we propose a confrontation among eight different machine-learning algorithms and we find that Linear Regression is the best predictive algorithm.

We introduce the notion of p-adic magic contraction on p-adic Hilbert space. We derive p-adic Halmos dilation, p-adic Egervary N-dilation, p-adic von Neumann inequality and p-adic Sz.-Nagy dilation for p-adic magic contraction.

In this paper, our goal is to establish the relationship between solutions of local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, also Minty local sharp vector variational type inequality and sharp efficient solutions of vector optimization problems, under generalized approximate $\eta$-convexity conditions for locally Lipschitzian functions

We propose a novel tensor-based formalism for inferring causal structures from time series. An information theoretical analysis of transfer entropy (TE), shows that TE results from transmission of information over a set of communication channels. Tensors are the mathematical equivalents of these multi-channel causal channels. A multi-channel causal channel is a generalization of a discrete memoryless channel (DMC). We consider a DMC as a single-channel causal channel. Investigation of a system comprising three variables shows that in our formalism, bivariate analysis suffices to differentiate between direct and indirect relations. For this to be true, we have to combine the output of multi-channel causal channels with the output of single-channel causal channels. We can understand this result when we consider the role of noise. Subsequent transmission of information over noisy channels can never result in less noisy transmission overall. This implies that a Data Processing Inequality (DPI) exists for transfer entropy.

Inspired by the forward and the reverse channels from the image-size characterization problem in network information theory, we introduce a functional inequality which unifies both the Brascamp-Lieb inequality and Barthe's inequality, which is a reverse form of the Brascamp-Lieb inequality. For Polish spaces, we prove its equivalent entropic formulation using the Legendre-Fenchel duality theory. Capitalizing on the entropic formulation, we elaborate on a "doubling trick" used by Lieb and Geng-Nair to prove the Gaussian optimality in this inequality for the case of Gaussian reference measures.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the loss of reality, the behavior of micro particles shows randomness and hopping. The second is the loss of certainty, the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will change the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell's inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell's inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not parallel relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which show the wave-particle duality simultaneously and provide intuitionistic teaching materials for the new interpretation of quantum theory.

A robust explicit model predictive control (EMPC)-based frequency adaptive grid voltage sensorless control is developed for a grid-connected inverter (GCI) via a linear matrix inequality (LMI) approach under the model parameter uncertainties and non-ideal grid environment. The control scheme is achieved by an enhanced prediction model rather than the conventional prediction model provided by fixed parameters to maintain the quality of grid currents injected into the grid when the model parameters are varied. Furthermore, an LMI-based observer is integrated with the disturbance observer to improve the reference tracking performance and reject disturbances. The proposed observer is employed for the grid frequency adaptive control without need of grid voltage sensors. The proposed current controller and observer employ the LMI scheme to maintain a stable and robust operation of the GCI. The discrete-time frequency response and pole-zero map analyses are used to investigate system stability and robustness against parameter uncertainties. Comprehensive simulation, experimental results and analyses clearly demonstrate the robustness of the proposed control scheme under various adverse test conditions with unexpected grid and system uncertainties.

In a p-uniformly convex and uniformly smooth Banach space, let the pair of variational inequality and fixed point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of symmetric structure mutually. Under mild registrations, we prove weak and strong convergence of the suggested algorithms to a common solution of this pair of VIFPPs, respectively. Lastly, an illustrative example is furnished to verify the applicability and implementability of our proposed approaches.

In this paper, we design two inertial-type subgradient extragradient algorithms with line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP) of finite Bregman relatively nonexpansive mapping and a Bregman relatively demicontractive mapping in p-uniformly convex and uniformly smooth Banach spaces, which are more general than Hilbert spaces. Under mild conditions, we derive weak and strong convergence of the suggested algorithms to a common solution of the VIPs and CFPP, respectively. Additionally, an illustrated example is furnished to back up the feasibility and implementability of our proposed methods.

Meixner's historical remark in 1969 "... it can be shown that the concept of entropy in the absence of equilibrium is in fact not only questionable but that it cannot even be defined...." is investigated from today's insight. Several statements --such as the three laws of phenomenological thermodynamics, the embedding theorem and the adiabatical uniqueness-- are used to get rid of non-equilibrium entropy as a primitive concept. In this framework, Clausius inequality of open systems can be derived by use of the defining inequalities which establish the non-equilibrium quantities contact temperature and non-equilibrium molar entropy which allow to describe the interaction between the Schottky system and its controlling equilibrium environment.

In this paper, two double Jordan-type inequalities are introduced based on the papers [1]-[5]. These inequalities generalize the inequalities obtained in [1]-[5]. As a result, some new upper and lower bounds of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions elaborated in [6]. Based on this approach, some optimal approximations of the sinc function are derived by determining corresponding minimax approximants, also described in the paper [6].

In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.