Constant Formula Research Articles

On the asymptotic behavior of solutions to bilinear Caputo stochastic fractional differential equations

In this paper, we focus on investigating the asymptotic behavior of solutions in a mean square sense to bilinear Caputo stochastic fractional differential equations (CSFDEs). The main tools in the proof include a variation of the constant formula for CSFDEs, the Jordan normal form of a matrix, the summation formula of Djrbashian type, and constructing a weighted norm in the associated Banach space.

Analysis of Strong Coupling Constant with Machine Learning and Its Application

We investigate the nature of the strong coupling constant and related physics. Through the analysis of accumulated experimental data around the world, we employ the ability of machine learning to unravel its physical laws. The result of our efforts is a formula that captures the expansive panorama of the distribution of the strong coupling constant across the entire energy range. Importantly, this newly derived expression is very similar to the formula derived from the Dyson–Schwinger equations based on the framework of Yang–Mills theory. By introducing the Euler number e into the functional formula of the strong coupling constant at high energies, we successfully solve the puzzle of the infrared divergence, which allows for a seamless transition of the strong coupling constant from the perturbative to the non-perturbative energy regime. Moreover, the obtained ghost and gluon dressing function distribution results confirm that the obtained strong coupling constant formula can well describe the physical properties of the non-perturbed regime. In addition, we study the quantum-chromodynamics strong coupling constant result of the Bjorken sum rule Γ1p−n and the quark–quark static energy E 0(r), and find that the global energy scale can effectively interpret the experimental data. The present results shed light on the puzzling properties of quantum chromodynamics and the intricate interplay of strong coupling constants at both low and high energy scales.

Global exponential convergence and synchronization for exponential numerical competitive neural networks with different time scales and fuzzy logic

For nonlocal fuzzy delayed competitive neural networks, a novel difference model is first developed in this article using the Mittag-Leffler Euler difference method. Secondly, the existence of a sole globally bounded solution and globally exponential stability to the discrete-time networks are addressed based on Banach contractive mapping principle and the constant variation formula for sequences. Besides, exponential synchronization of the discrete-time networks is investigated and a feedback controller is designed from the viewpoint of the theory of controls. It is the first time to consider nonlocal fuzzy delayed competitive neural networks by using Mittag-Leffler Euler differences. In contrast to nonlocal discrete approaches, such difference competitive neural networks could preferably figure the features of continuous networks. The difference method in this paper extends and improves the traditional difference approaches, for example, Adams-Bashforth Euler differences. Moreover, this article explores some avenue for the research of discrete-time fractional-order systems and develops a set of theories and methodologies for the future investigations.

Studies over the existence of a certain impulse-based fuzzy integrodifferential equations of the Sobolev type

The present investigation employs impulses and a non-local constraint to prove the existence are some various types of abstract differential and integrodifferential equations related to the Sobolev type. Semigroup theory, specifically variants of constant formula, is utilized to get the analytical results for those equations. Furthermore, findings using the Banach fixed point approach were examined using fuzzy numbers with values spanning the En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {E}_n$$\\end{document} range, which includes the normal, convex, upper semi-continuous, and compactly supported interval. A description is given for each situation to illustrate the principle.

Riemann Solver for Hyperbolic Equations with Discontinuous Coefficients: A Mathematical Proof of the Constant State Formula

Riemann Solver for Hyperbolic Equations with Discontinuous Coefficients: A Mathematical Proof of the Constant State Formula

Random periodic oscillations and global mean-square exponential stability of discrete-space and discrete-time stochastic competitive neural networks with Dirichlet boundary condition

The current article explores the affects of space-time discrete stochastic competitive neural networks. In line with a discrete-space and discrete-time constant variation formula, boundedness and stability are addressed to the space-time discrete stochastic competitive neural networks. Notably, the best convergence speed can be computed by a non-linear optimization problem. In the end, random periodic sequences with respect to time variable of the discrete-space and discrete-time stochastic competitive neural networks are discussed. The results indicate that spatial diffusion with non-negative density factors has no effect on the global mean square boundedness and stability and random periodicity of the network model. The current article is precursory in consideration of space-time discrete competitive neural networks.

Weighted pseudo [formula omitted]-almost periodic sequence solution and guaranteed cost control for discrete-time and discrete-space stochastic inertial neural networks

This article considers the dual hybrid influences of both discrete spatial diffusions and discrete time in a stochastic inertial neural networks via the methods of exponential Euler difference and second order central finite difference. Based on a non-decomposed constant variation formula, and the theories of semi-flow and metric dynamical systems, the existence of a unique weighted pseudo θ-almost periodic sequence solution is addressed to the discrete space and time stochastic inertial neural networks. Further, a guaranteed cost controller is designed to complete a global exponential stabilization for this discrete networks by establishing a framework of drive, response and error networks. Meanwhile, global exponential stability in the sense of mean square is achieved as well. This discussion is pioneering in considering discrete spatial diffusions in inertial neural networks and offers the studying bases for future researches.

On Arthur's unitarity conjecture for split real groups

Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the Langlands element (i.e., the one specified by Arthur) of all unipotent Arthur packets for split real groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet L-functions, to reduce to a more combinatorial problem about intertwining operators.

Grassmann 공간에서 2-입력, 2-출력, 4차 시스템의 정적출력궤한 극배치

In this paper, a parametric study of pole placement of two-input, two-output, 4SUPth/SUP order systems with strictly proper transfer functions is investigated in real Grassmann space. To characterize the features of pole placement, the well-known Plücker matrix formula is introduced as a real constant matrix formula constructed from the static output feedback (SOF) variable’s coefficient functions of characteristic polynomial under Grassmann coordinates. It is shown that the SOF pole placement entirely depends upon the columns’ condition of a real Grassmann parameter. The results presented in this paper shows that this real Grassmannian parametrization method can generalize the existing pole placement methods for two-input, two-output, 4SUPth/SUP order systems.

Linear quaternion-valued difference equations: Representation of solutions, controllability, and observability

In this paper, we present the fundamental theory of linear quaternion-valued difference equations. Firstly, we derive general solutions for linear homogeneous equations and give the algorithm for calculating the fundamental matrix in the case of the diagonalizable form and Jordan form. Secondly, we apply the variation of the constant formula and Z transformation to study general solutions of linear nonhomogeneous equations. We obtain the representation of solutions in the case of quaternion and complex numbers. Thirdly, we adopt the ideas from the Gram matrix and the rank of the criteria to establish sufficient and necessary conditions to guarantee that linear quaternion-valued difference equations are controllable and observable in the sense of quaternion-valued and complex numbers, respectively. In addition, a direct method to solve the control function and duality is also given. Finally, we illustrate our theoretical results with some examples.

Constant Depth Formula and Partial Function Versions of MCSP Are Hard

Attempts to prove the intractability of the Minimum Circuit Size Problem (MCSP) date as far back as the 1950s and are well-motivated by connections to cryptography, learning theory, and average-case complexity. In this work, we make progress, on two fronts, towards showing MCSP is intractable under worst-case assumptions. While Masek showed in the late 1970s that the version of MCSP for DNF formulas is NP-hard, extending this result to the case of depth-3 AND/OR formulas was open. We show that determining the minimum size of a depth- $d$ formula computing a given Boolean function is N P-hard under quasipolynomial-time randomized reductions for all constant $d\geq 2$ . Our approach is based on a method to “lift” depth- $d$ formula lower bounds to depth-( $d+1$ ). This method also implies the existence of a function with a $2^{\Omega_{d}(n^{1/5}}$ ) additive gap between its depth-d and depth-( $d+1$ ) formula complexity. We also make progress in the case of general, unrestricted circuits. We show that the version of MCSP where the input is a partial function (represented by a string in $\{0,1, ?\}^{\ast}$ ) is not in P under the Exponential Time Hypothesis (ETH). Intriguingly, we formulate a notion of lower bound statements being (P/poly)-recognizable that is closely related to Razborov and Rudich's definition of being (P/poly)-constructive. We show that unless there are subexponential-sized circuits computing SAT, the lower bound statements used to prove the correctness of our reductions cannot be (P/poly)-recognizable.

Exponential stability and synchronisation of fuzzy Mittag–Leffler discrete-time Cohen–Grossberg neural networks with time delays

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper.

A Closed Structure Constant Formula for the Universal Enveloping Algebra of the Lie Algebra sl2

Gourley and Kennedy gave a recursive formula for the structure constants of the universal enveloping algebra of sl2. Using Kostant's formula we give a closed formula for these structure constants.

A variation of constant formula for Caputo fractional stochastic differential equations with jump–diffusion

In this paper, the existence and uniqueness of solutions for the Caputo fractional stochastic differential equations with jump–diffusion is discussed. Then, a variation of constant formula for the equation is established. The main ingredient of the proof is to use Itoˆ’s isometry of Poisson jumps and martingale representation theorem.

Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

Considerations on the Castrop formula for calculation of intraocular lens power

BackgroundTo explain the concept of the Castrop lens power calculation formula and show the application and results from a large dataset compared to classical formulae.MethodsThe Castrop vergence formula is based on a pseudophakic model eye with 4 refractive surfaces. This was compared against the SRKT, Hoffer-Q, Holladay1, simplified Haigis with 1 optimized constant and Haigis formula with 3 optimized constants. A large dataset of preoperative biometric values, lens power data and postoperative refraction data was split into training and test sets. The training data were used for formula constant optimization, and the test data for cross-validation. Constant optimization was performed for all formulae using nonlinear optimization, minimising root mean squared prediction error.ResultsThe constants for all formulae were derived with the Levenberg-Marquardt algorithm. Applying these constants to the test data, the Castrop formula showed a slightly better performance compared to the classical formulae in terms of prediction error and absolute prediction error. Using the Castrop formula, the standard deviation of the prediction error was lowest at 0.45 dpt, and 95% of all eyes in the test data were within the limit of 0.9 dpt of prediction error.ConclusionThe calculation concept of the Castrop formula and one potential option for optimization of the 3 Castrop formula constants (C, H, and R) are presented. In a large dataset of 1452 data points the performance of the Castrop formula was slightly superior to the respective results of the classical formulae such as SRKT, Hoffer-Q, Holladay1 or Haigis.

Discrete Subdiffusion Equations with Memory

In this paper, we study a discrete subdiffusion equation with memory. Based on the backward operator and the theory of fractional resolvent families, we find a discrete fractional resolvent sequence which allows to write the solution to this discrete subdiffusion equation as a variation of constant formula.

Neutronic modeling of megawatt-class heat pipe reactors

Megawatt-class heat pipe reactors (HPRs) have been identified as a candidate for the decentralized electricity markets, for instance, remote regions, island communities, and military bases. HPRs exhibit some unique designs, such as the encapsulated solid core with fuel pellets and heat pipes, control drums, and irregular reflectors. The neutronic tools are under development to investigate core characteristics and serve multiphysics coupling. This paper proposes a public neutronic benchmark of megawatt HPR of which all modeling parameters are available. This paper uses open-source Monte-Carlo code OpenMC to carry out the preliminary neutronic analysis of this benchmark. The influence of the cross-section library can be >750 pcm. The addition of axial structures zone can lead to >300 pcm reactivity increase. The Doppler effect of fuel exactly matches the oxide-fueled fast reactor Doppler constant formula. The reactivity worth of the control devices, flux distribution, power distribution, and depletion results are presented and accessible for future benchmarks.

Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions

In this paper, we develop some basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we consider a semilinear equation with nonlocal diffusion and nonlocal boundary condition, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of nontrivial steady states, and establish the stability of steady states. Finally we generalize these results to a nonlinear equation with nonlocal diffusion and nonlocal boundary condition.