As effective tools for time series analyzing, a variety of information entropies have been widely applied in engineering, economic, biomedicine and other fields. In this paper, we define a new distance between finite sequence based on inversion and derive a new entropy, Fuzzy Permutation Entropy(FPE). A comparison of recognition performance for WGN, 1/f noise, periodical, or chaotic sequence and sine

With $ p\left(n,k\right) $ denote the numerical value of the number of partitions of the natural number $ n $ on exactly $ k $ parts. Form an arithmetic progression of $ k $ natural numbers with an arbitrary first value $ x_1=p\left(j,k\right)$, and the difference $ d=m \cdot LCM\left(1,2,\dots,k\right) $, where $ j$ and $ m $ an arbitrary natural numbers. Calculate all the values of $ \left\{p\left(x_i

If $G$ is a connected graph, the $distance$ $d(u, v)$ between two vertices $u, v \in V(G)$ is the length of a shortest path between them. Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The $representation$ $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ is called a $resolving set$ or a $locating

Here, the numerical solutions for Laplace equation with nonlinear boundary conditions is studied. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equation. The modified quadrature method is presented for solving the nonlinear equation, which possesses high accuracy order $O(h^3)$ and low computing complexities. A nonlinear system is obtained by discretizing

In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution

Let $G=(V,E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The weight of a Roman dominating function is the value $f(G)=\sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on

The present research studies the boundedness issue of Lévy driven non-autonomous stochastic differential systems with mixed discrete and distributed delays. A set of sufficient conditions of the $p$th moment globally asymptotical boundedness is obtained by combining the Lyapunov function method with the inequality technique. The proposed results reveal that the convergence rate $\lambda$ and the coefficients

Type-reduction (TR) is a key block for interval type-2 fuzzy logic systems (IT2 FLSs). In general, Karnik-Mendel (KM) (or enhanced Karnik-Mendel (EKM)) algorithms are used to perform the TR. These two types of algorithms have the advantage of preserving the uncertainties of membership functions (MFs) flow in IT2 FLSs. This paper gives the initialization explanations of KM and EKM algorithms, and proposes

This paper deals with a Gause-type predator-prey model with Allee effect and Holling type III functional response. We also consider the influence of predator competition and the artificial harvesting on predator-prey system. The existence of multiple positive periodic solutions of the predator-prey model is established by using the Mawhin coincidence degree theory.

In this paper, we shall extend some results regarding the growth estimate of entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.

In the article, we introduce the concept of the exponentially $tgs$-convex function and discover two new conformable fractional integral identities concerning the first-order differentiable convex mappings. By using these identities, we establish several new right-sided Hermite-Hadamard type inequalities for the exponentially $tgs$-convex functions via conformable fractional integrals. Our outcomes

The Hermite-Hadamard inequality by means of the Riemann-Liouville fractional integral operators is already known in the literature. In this paper, it is our purpose to reconstruct this inequality via a relatively new method called the green function technique. In the process, some identities are established. Using these identities, we obtain loads of new results for functions whose second derivative

The key purpose of this study is to suggest a delta Riemann-Liouville (RL) fractional integral operators for deriving certain novel refinements of Pólya-Szegö and Čebyšev type inequalities on time scales. Some new Pólya-Szegö, Čebyšev and extended Čebyšev inequalities via delta-RL fractional integral operator on a time scale that captures some continuous and discrete analogues in the relative literature

In this paper, we prove the existence of at least three periodic solutions for the quasilinear periodic boundary value problem \begin{eqnarray} \left\{ \begin{array}{ll} -p(x')x''+\alpha(t)x=\lambda f(t,x) ~{\rm a.e.} ~t\in[0,1], \\ x(1) -x(0)= x'(1)-x'(0)=0 \end{array} \right. \end{eqnarray} under appropriate hypotheses via a three critical points theorem of B. Ricceri. In addition, we give an example

In this paper, we use the elementary methods, the properties of the Gauss sums and the estimate for character sums to study the calculating problems of a certain cubic residues modulo p, and give some interesting identities and asymptotic formulas for their counting functions.

A family of subgraphs of a finite, simple and connected graph $G$ is called an edge covering of $G$ if every edge of graph $G$ belongs to at least one of the subgraphs. In this manuscript, we define the edge covering of a stacked book graph and its uniform subdivision by cycles of different lengths. If every subgraph of $G$ is isomorphic to one graph $H$ (say) and there is a bijection $\phi:V(G)\cup

In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h − m)-convex functions, exponentially (h − m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results

The aim of this paper is to develop an accurate symmetric collocation scheme for a class of linear stationary singular perturbation problems with two boundary layers. To adapt to the character of solutions, piecewise reproducing kernels is constructed. In the boundary layers intervals, inverse multiquadrics kernel function is employed. In the regular interval, exponential kernel function is used. On

Ulam stability problems have received considerable attention in the field of differential equations. However, how to effectively build the fuzzy model for Ulam stability problems is less attractive due to varies of differentiabilities requirements. The paper discusses the Ulam stability of fuzzy differential equations in Banach spaces. After introducing the new definitions of differentiabilities for

In this paper, we introduce a mixed type finite variable functional equation deriving from quadratic and additive functions and obtain the general solution of the functional equation and investigate the Hyers-Ulam stability for the functional equation in quasi-Banach spaces.

In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert

In this paper, a class of inertial neural networks with bounded time-varying delays and unbounded continuously distributed delays are explored by applying non-reduced order method. Based upon differential inequality techniques and Lyapunov function method, a new sufficient condition is presented to ensure all solutions of the addressed model and their derivatives converge to zero vector, which refines

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated

For the problem of inconsistent quantitative standards for running status analysis of rolling bearings, this paper uses principal component analysis (PCA) to extract a new index F, which is the joint parameters of time domain and frequency domain, and by establishing the value of F to analyze the running states of the rolling bearings. Firstly, the acceleration sensors are used to collect the vibration

In this research paper, we solve a new n-variable mixed type additive-quadratic functional equation and prove the Ulam stability of the new n-variable mixed type additive-quadratic functional equation in probabilistic modular spaces by using fixed point method.

In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of Hölder’s inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions

In this paper, we consider the existence of an reversed S -shaped connected component in the set of positive solutions for second order periodic boundary value problem with a sign-changing weight function. By bifurcation technique, we identify the interval of bifurcation parameter in which the periodic boundary value problem has one or two or three positive solutions according to the asymptotic behavior

In the article, we establish a Simpson-type generalized identity containing multi-parameters and derive some new estimates for the generalized Simpson’s quadrature rule via the Raina fractional integrals. As applications, we provide several inequalities for the f -divergence measures and probability density functions.

Gamma function and its generalizations always have played a basic role in various disciplines. The aim of present study is to investigate a new representation of the λ-generalized gamma function. This representation is developed by using different modified forms of delta function. This development explores their extended use as generalized functions (distributions), which are meaningful to exist over

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics

A novel fractional-order jerk equation with resonant boundary value conditions is proposed. Using coincidence degree theory, we obtain the existence of solutions of nonlinear fractional jerk equation with two-point boundary conditions. This paper enriches some existing literatures. Finally, an example is given to demonstrate the effectiveness of our main result.

In this paper, we investigate the existence of $W_0^{1,1}(\Omega)$ solutions to the following elliptic equation with principal part having noncoercivity and singular quadratic term \begin{equation*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{\nabla u}{(1+|u|)^{\gamma}}\right)+\frac{|\nabla u|^2}{u^{\theta}}=f,&x\in\Omega,\\ u=0,&x\in\partial\Omega, \end{array} \right. \end{equation*} where $\Omega$

Let $G(V,E)$ be a simple connected graph of order $n$. A graph of order $n$ is called pancyclic if it contains all the cycles $C_k$ for $k\in \{3,4,\cdot\cdot\cdot,n\}$. In this paper, some new spectral sufficient conditions for the graph to be pancyclic are established in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.

Taking into accounting time-varying delays and anti-periodic environments, this paper deals with the global convergence dynamics on a class of anti-periodic high-order inertial Hopfield neural networks. First of all, with the help of Lyapunov function method, we prove that the global solutions are exponentially attractive to each other. Secondly, by using analytical techniques in uniform convergence

In this study, the characterization of position vectors belonging to non-lightlike Bertrand W curve mate with constant curvature are obtained depending on differentiable functions. The position vector of Bertrand W curve is stated by a linear combination of its Frenet frame with differentiable functions. There exist also different cases for the curve depending on the value of curvature and torsion

In this paper, the new version of the celebrated Montgomery identity is determined via quantum integral operators. By using it, certain quantum integral inequalities of Ostrowski type are established. Moreover, the relevant connection of the obtained results of this work with the derived results in previously published works is discussed.

The aim of the current manuscript is to prove the Hyers-Ulam stability of supremum, infimum and multiplication preserving functional equations in Banach f -algebras. In fact, by using the direct method and the fixed point method, the Hyers-Ulam stability of the functional equations is proved.

In this paper we consider a non-autonomous Navier-Stokes-Voigt model including a variety of delay terms in a unified formulation. Firstly, we prove the existence and uniqueness of solutions by using a Galerkin scheme. Next, we prove the existence and eventual uniqueness of stationary solutions, as well as their exponential stability by using three methods: first, a Lyapunov function which requires

In this study, we introduce a new general identity for n th order differentiable functions. Also, we establish some new inequalities regarding general perturbed trapezoid inequality for the functions whose the absolute values of n th derivatives are convex. Finally, some applications for special means are provided.

The unified integral operator presented in Definition 4 produces several kinds of known fractional and conformable integral operators. The goal of this paper is to obtain bounds of this unified integral operator by using the definition of (s,m)-convexity. The resulting inequalities in specific cases represent the bounds of many known fractional and conformable fractional integral operators in a compact

The q-gradient is the generalization of the gradient based on the q-derivative. The q-version of the steepest descent method for unconstrained multiobjective optimization problems is constructed and recovered to the classical one as q equals 1. In this method, the search process moves step by step from global at the beginning to particularly neighborhood at last. This method does not depend upon a

In this paper, we study topologies generated by simple undirected graphs without isolated vertices and their properties. We generate firstly a topology using a simple undirected graph without isolated vertices. Moreover, we investigate properties of the topologies generated by certain graphs. Finally,we present continuity and opennes of functions defined from one graph to another via the topologies

The main purpose of this paper is using the mean value of Dirichlet L-functions and character sums to study a kind of mean value of the Hardy sums over short intervals, and give some interesting formulae.

We investigate some relationships between two vastly studied parameters of a simple graph $G$. These parameters include mixed domination number (denoted by $\gamma_m(G)$) and 2-independence number ($\beta_2(G)$). For a tree $T$, we obtain $\frac{3}{4}\beta_2(T)\ge \gamma_m(T)$ and characterized all those trees which attain the equality.

The numerical method is proposed in this paper to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients and the time-dependent matrices are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities of the involved functions, a methodology for not equally

This paper provides an answer if the nonuniform behavior can destroy the structural stability of nonlinear systems. We show that if the linear system $\dot{x}(t)=A(t)x(t)$ admits a nonuniform exponential dichotomy, then the perturbed nonautonomous system $\dot{x}(t)=A(t)x(t)+f(t,x)$ is structurally stable under suitable conditions.

It is well known that the approximation operators are always primitive concepts in kinds of general rough set theories. In this paper, considering L to be a completely distributive lattice, we introduce a notion of L-fuzzy upper approximation operator based on L-generalized fuzzy remote neighborhood systems of L-fuzzy points. It is shown that the new approximation operator is a fuzzification of the

In this article, we use elementary methods and the estimate for character sums to study the properties of a certain primitive roots modulo p (an odd prime), and prove that the generalized Golomb’s conjecture is correct in a reduced residue system modulo p. This solved an open problem proposed by W. P. Zhang and T. T. Wang in [3].

For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb

In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients $\binom{p^a n}{n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that $\binom{pn}{n}_F$

As an extension of neutrosophic set, interval complex neutrosophic set is a new research topic in the field of neutrosophic set theory, which can handle the uncertain, inconsistent and incomplete information in periodic data. Distance measure is an important tool to solve some problems in engineering and science. Hence, this paper presents some interval complex neutrosophic distance measures to deal

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form $\int_{0}^{\infty}\ln^k(\alpha y)\ln(R(y))dy$ in terms of a special function, where $R(y)$ is a general function and $k$ and $\alpha$ are arbitrary complex numbers.

In this paper, we introduce the notion of an orthogonal (F, ψ)-contraction of Hardy-Rogers-type mapping and prove some fixed point theorem for such contraction mappings in orthogonally metric spaces. Our result extend and improve the main result of the paper by Sawangsup et al. [Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces, J. Fixed Point Theory Appl. (2020)

In this paper, we deal with a class of fractional Hénon equation and by using the Lyapunov-Schmidt reduction method, under some suitable assumptions, we derive the existence of infinitely many solutions, whose energy can be made arbitrarily large. Compared to the previous works, we encounter some new challenges because of the nonlocal property for fractional Laplacian. But by doing some delicate estimates

The objective of this paper is to study the ordered $h$-regular semirings by the properties of their ordered $h$-ideals. It is proved that each $h$-regular ordered semiring is an ordered $h$-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and $h$-regular semirings are given. It is also proved that each regular ordered semiring