Leibniz Formula Research Articles

Doubly Exponential Growth and Decay for a Semilinear Heat Equation With Logarithmic Nonlinearity

ABSTRACTIn this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. and Han. We firstly estimate the upper bound function with doubly exponential property for the weak solutions and establish the doubly exponential decay and at most doubly exponential growth criteria for the weak solutions by the logarithmic Sobolev inequality, the derivative formula for the product, the Newton–Leibniz formula, and the logarithmic properties. We secondly establish a newly infinity blow‐up criterion such that the lower bound function of weak solutions is at least doubly exponential growth by the nonconcavity method and the nonincreasing property for the energy functional. We finally establish a sufficient condition such that the upper bound function with doubly exponential growth is greater than or equal to the lower bound function with doubly exponential growth by the quotient comparison principle. In addition, we also find the threshold of the doubly exponential growth and the doubly exponential decay and obtain for any as These research results improve previous results from both decay and growth.

A new formula for the determinant and bounds on its tensor and Waring ranks

AbstractWe present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$ -th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$ . Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$ . Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$ , and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$ .

The Euler-Lagrange equations of nabla derivatives for variational approach to optimization problems on time scales

This paper investigates the variational approach using nabla (denoted as ∇) within the framework of time scales. By employing two different methods, we derive the Euler-Lagrange equations for first-order variational approach to optimization problems involving exponential functions, as well as for those with both exponential functions and their ∇-derivatives. To establish the high-order variational approach to optimization problem, we present the Leibniz Formula for ∇-derivatives along with its proof. Additionally, we determine the high-order variational approach to optimization problem incorporating ∇-derivatives of exponential functions. Through these analyses, we aim to contribute to the understanding and application of the variational calculus on time scales, offering insights into the behavior of dynamic systems governed by exponential functions and their derivatives.

Gröbner bases for bipartite determinantal ideals

Nakajima’s graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals. We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia–Guccione–Guccione. As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics.

Differential Game-Based Deep Reinforcement Learning in Underwater Target Hunting Task.

To meet requirements for real-time trajectory scheduling and distributed coordination, underwater target hunting task is challenging in terms of turbulent ocean environments and dynamic adversarial environment. Despite the existing research in game-based target hunting area, few approaches have considered dynamic environmental factors, such as sea currents, winds, and communication delay. In this article, we focus on a target hunting system consisted of multiple unmanned underwater vehicles (UUVs) and a target with high maneuverability. Besides, differential game theory is leveraged to analyze adversarial behaviors between hunters and the escapee. However, it is intractable that UUVs have to deploy an adaptive scheme to guarantee the consistency and avoid the escape of the target without collision. Therefore, we conceive the Hamiltonian function with Leibniz's formula to obtain feedback control policies. In addition, it proves that the target hunting system is asymptotically stable in the mean, and the system can satisfy Nash equilibrium relying on the proposed control policies. Furthermore, we design a modified multiagent reinforcement learning (MARL) to facilitate the underwater target hunting task under the constraints of energetic flows and acoustic propagation delay. Simulation results show that the proposed scheme is superior to the typical MARL algorithm in terms of reward and success rate.

Pseudo-Stieltjes calculus: $ \alpha- $pseudo-differentiability, the pseudo-Stieltjes integrability and applications

<p>In this paper, the concepts of the $ \alpha- $pseudo-differentiability and the pseudo-Stieltjes integrability are proposed, and the corresponding transformation theorems and Newton–Leibniz formula are established. The obtained results provide a framework for analyzing nonlinear differential equations.</p>

Synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay

The topic of synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay is addressed in this paper. By combining graph theory, the Lyapunov method, and the Razumikhin method, the synchronization criteria are obtained by designing a new time-varying graph-theoretical Lyapunov function. Different from the typical quadratic Lyapunov function, the time-varying graph-theoretical Lyapunov function contains the product of three-term functions and poses the difficulty of fractional derivative due to the unavailability of the general Leibniz formula and the chain rule. A new inequality about the fractional derivative rule of the time-varying graph-theoretical Lyapunov function is established based on the convex function creatively. In addition, compared with relevant literatures, the synchronization criteria are related to the proportionality coefficient and the order of fractional derivative. Then, the synchronization of fractional-order power system established on fuzzy complex networks with time-varying couplings and proportional delay is studied for the first time by applying the feedback control. Lastly, some simulations are exhibited to demonstrate the effectiveness of the theoretical results.

Sampled-data exponential consensus of multi-agent systems with Lipschitz nonlinearities

In this paper, the exponential consensus of leaderless and leader-following multi-agent systems with Lipschitz nonlinear dynamics is illustrated with aperiodic sampled-data control using a two-sided loop-based Lyapunov functional (LBLF). Firstly, applying input delay approach to reformulate the resulting sampled-data system as a continuous system with time-varying delay in the control input. A two-sided LBLF which captures the information on sampled-data pattern is constructed and the symmetry of the Laplacian matrix together with Newton–Leibniz formula have been employed to obtain reduced number of decision variables and decreased LMI dimensions for the exponential sampled-data consensus problem. Subsequently, an aperiodic sampled-data controller was designed to simplify and enhance stability conditions for computation and optimization purposes in the proposed approach. Finally, based on the controller design, simulation examples including the power system are proposed to illustrate the theoretical analysis, moreover, a larger sampled-data interval can be acquired by this method than other literature, thereby conserving bandwidth and reducing communication resources.

A Direct Reinforcement Learning Approach for Nonautonomous Thermoacoustic Generator

For nonautonomous nonlinear systems, the optimal control design is affected by the terms of partial derivative. If a reinforcement learning (RL) strategy is developed to approximate the optimal control scheme in nonautonomous nonlinear systems, then the closed control system might be unstabilizing. Therefore, in this article, the approach of direct RL law for a nonautonomous thermoacoustic generator (TAG) is investigated. We establish the mathematical model of TAG by partial differential equations (PDEs) and then transforming them into time varying nonlinear systems. The direct RL technique with Newton–Leibniz formula is implemented to consider the partial derivative term from classical policy iteration (PI) method by modifying the computation using data collection between the two sampling times. Finally, several simulation studies with some comparisons are conducted to validate the theoretical analyses.

New results on robust exponential stability of Takagi–Sugeno fuzzy for neutral differential systems with mixed time-varying delays

This study aims to analyse the robust exponential stability for neutral differential equations of the uncertain Takagi–Sugeno fuzzy system with distributed and discrete time interval delays by applying the Newton–Leibniz formula, the Lyapunov–Krasovskii functional, and the zero equations. By these methods, the less conservative exponential stability criteria are obtained for a special case of the generalized fuzzy of the neutral differential system. By applying Matlab LMI toolbox, the new conditions for exponential stability are obtained in the forms of linear matrix inequalities (LMIs), which can verify the numerical performance by the LMI toolbox of MATLAB. Finally, numerical examples are provided to manifest the efficiency and reduce the conservatism of the obtained results rather than the results which exist in the literature.

On the Exponential Stability of Neutral Linear Systems with Variable Delays

We investigate the exponential stability of a linear system of neutral-type with variable time lags. By using the Newton–Leibniz formula and a Lyapunov–Krasovskii functional, we prove two results on the exponential stability of solutions. The stability criteria are stated in the form of linear matrix inequalities. By using the MATLAB-Simulink software, we present two numerical examples illustrating the applicability of our assumptions. The obtained results extend and generalize the already known results available from the related literature.

A tool for the Analytic Hierarchy Process based on Leibniz's formula for determinants computation

A tool for the Analytic Hierarchy Process based on Leibniz's formula for determinants computation

Formalizing Calculus without Limit Theory in Coq

Formal verification of mathematical theory has received widespread concern and grown rapidly. The formalization of the fundamental theory will contribute to the development of large projects. In this paper, we present the formalization in Coq of calculus without limit theory. The theory aims to found a new form of calculus more easily but rigorously. This theory as an innovation differs from traditional calculus but is equivalent and more comprehensible. First, the definition of the difference-quotient control function is given intuitively from the physical facts. Further, conditions are added to it to get the derivative, and define the integral by the axiomatization. Then some important conclusions in calculus such as the Newton–Leibniz formula and the Taylor formula can be formally verified. This shows that this theory can be independent of limit theory, and any proof does not involve real number completeness. This work can help learners to study calculus and lay the foundation for many applications.

A Fuzzy Lyapunov Function Method to Stability Analysis of Fractional-Order T–S Fuzzy Systems

This article investigates the stability analysis and stabilization problems for fractional-order T–S fuzzy systems via fuzzy Lyapunov function method. A membership-function-dependent fuzzy Lyapunov function instead of the general quadratic Lyapunov function is employed to obtain the stability and stabilization criteria. Different from the general quadratic Lyapunov function, the fuzzy Lyapunov functions contain the product of three term functions. Since the general Leibniz formula cannot be satisfied for fractional derivative, the current results on the fractional derivative for the quadratic Lyapunov functions cannot be extended to the fuzzy Lyapunov functions. Therefore, to estimate the fractional derivative of fuzzy Lyapunov functions, the fractional derivative rule for the product of three term functions is proposed. Based on the proposed fractional derivative rule, the corresponding stability and stabilization criteria are established, which extend the existing results. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed theoretical analysis.

Fast and Accurate Computation of Nonadiabatic Coupling Matrix Elements Using the Truncated Leibniz Formula and Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory.

We present a fast and accurate numerical algorithm for computing the first-order nonadiabatic coupling matrix element (NACME). The algorithm employs the truncated Leibniz formula (TLF) approximation within the finite-difference method, which makes it easily applicable in connection with any wave function-based methodology. In this work, we used the algorithm in connection with the recently developed mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT, MRSF for brevity). The accuracy is assessed for NACME between the singlet electronic states of a dissociating hydrogen molecule. It is demonstrated that an intermediate approximation, TLF(1), affords a negligible numeric error on the order of ∼10-10 a.u. while enabling a fast computation of NACME. As the MRSF method yields the correct description of the dissociation curves of H2 for all the electronic states involved, the numeric TLF(1)/MRSF NACME values are in excellent agreement with the reference analytical values obtained by the full configuration interaction. For polyatomic molecules, the MRSF NAC vectors agree very closely with the MRCISD NAC vectors. Hence, the proposed protocol is a promising tool for the evaluation of NACMEs.

Derivation of Commutative Rings and the Leibniz Formula for Power of Derivation

Summary In this article we formalize in Mizar [1], [2] a derivation of commutative rings, its definition and some properties. The details are to be referred to [5], [7]. A derivation of a ring, say D, is defined generally as a map from a commutative ring A to A-Module M with specific conditions. However we start with simpler case, namely dom D = rng D. This allows to define a derivation in other rings such as a polynomial ring. A derivation is a map D : A → A satisfying the following conditions: (i) D(x + y) = Dx + Dy, (ii) D(xy) = xDy + yDx, ∀x, y ∈ A. Typical properties are formalized such as: D ( ∑ i = 1 n x i ) = ∑ i = 1 n D x i D\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) = \sum\limits_{i = 1}^n {D{x_i}} and D ( ∏ i = 1 n x i ) = ∑ i = 1 n x 1 x 2 ⋯ D x i ⋯ x n ( ∀ x i ∈ A ) . D\left( {\prod\limits_{i = 1}^n {{x_i}} } \right) = \sum\limits_{i = 1}^n {{x_1}{x_2} \cdots D{x_i} \cdots {x_n}} \left( {\forall {x_i} \in A} \right). We also formalized the Leibniz Formula for power of derivation D : D n ( x y ) = ∑ i = 0 n ( i n ) D i x D n - i y . {D^n}\left( {xy} \right) = \sum\limits_{i = 0}^n {\left( {_i^n} \right){D^i}x{D^{n - i}}y.} Lastly applying the definition to the polynomial ring of A and a derivation of polynomial ring was formalized. We mentioned a justification about compatibility of the derivation in this article to the same object that has treated as differentiations of polynomial functions [3].

Dwell‐time‐dependent conditions for exponential stability and hybrid L2 × l2‐gain of linear neutral time‐delay systems with impulsive effects

AbstractThis article considers the problems of achieving exponential stability and finite hybrid L2 × l2‐gain with respect to continuous and discrete disturbances for linear neutral time‐delay systems with impulsive effects. In the framework of descriptor system representation, a piecewise timer‐dependent Lyapunov functional is introduced to analyze the double impacts of state‐delay and impulse‐dwell‐time on stability of the underlying system. This functional depends on the state over an impulse interval and the state over a delay interval. Its construction is based on a delay‐fractioning scheme which ensures all delay subintervals contains at most one impulse instant. The relations among the augmented state variables are explored by using the impulse‐timer functions and the Newton–Leibniz formula. It is proved that the positive definiteness property of the functional at nonimpulse instants is not necessary for ensuring stability. The hybrid L2 × l2‐gain analysis is carried out by jointly using the introduced functional and a discrete‐time Lyapunov function. The obtain criteria for exponential stability and hybrid L2 × l2‐gain are expressed in terms of linear matrix inequalities. Their effectiveness is verified through two numerical examples.

"Efficiency Comparison of Algorithms for Finding Determinants Via Permutations "

"Determiners have been used extensively in a selection of applications throughout history. It also biased many areas of mathematics such as linear algebra. There are algorithms commonly used for computing a matrix determinant such as: Laplace expansion, LDU decomposition, Cofactor algorithm, and permutation algorithms. The determinants of a quadratic matrix can be found using a diversity of these methods, including the well-known methods of the Leibniz formula and the Laplace expansion and permutation algorithms that computes the determinant of any n×n matrix in O(n!). In this paper, we first discuss three algorithms for finding determinants using permutations. Then we make out the algorithms in pseudo code and finally, we analyze the complexity and nature of the algorithms and compare them with each other. We present permutations algorithms and then analyze and compare them in terms of runtime, acceleration and competence, as the presented algorithms presented different results.

Exponential stabilization and non-fragile sampled-date dissipative control for uncertain time-varying delay T-S fuzzy systems with state quantization

In this paper, the problem of exponential dissipation stability of T-S fuzzy system with state quantization is studied by using non-fragile sampled-data control. First, A Lyapunov-Krasovskii function containing all sampled-data and quantization information is constructed. Second, the better results can be obtained by using the integral inequality and the Newton Leibniz formula. In addition, a dissipative controller for non-fragile sampled-data is designed. Finally, The reasonable examples illustrate the significant improvement and value of this paper.

Synchronization Analysis for Stochastic T-S Fuzzy Complex Networks with Markovian Jumping Parameters and Mixed Time-Varying Delays via Impulsive Control

In this paper, we propose and explore the synchronization examination for fuzzy stochastic complex networks’ Markovian jumping parameters portrayed by Takagi-Sugeno (T-S) fuzzy model with mixed time-varying coupling delays via impulsive control. The hybrid coupling includes time-varying discrete and distributed delays. Based on appropriate Lyapunov–Krasovskii functional (LKF) approach, Newton–Leibniz formula, and Jensen’s inequality, the stochastic examination systems and Kronecker product to create delay-dependent synchronization criteria that guarantee stochastically synchronous of the proposed T-S fuzzy stochastic complex networks with mixed time-varying delays. Adequate conditions for the synchronization criteria for the frameworks are established in terms of linear matrix inequalities (LMIs). At long last, numerical examples and simulations are given to demonstrate the correctness of the hypothetical outcomes.