Variational Formula Research Articles

First‐order asymptotic perturbation theory for extensions of symmetric operators

AbstractThis work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

Hadamard’s Variational Formula for Simple Eigenvalues

Here, we study Hadamard’s variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for a two-dimensional domain, which implies several new inequalities.

Attributes of residual neural networks for modeling fractional differential equations

This paper offers a pioneering in-depth exploration of applying residual neural networks to approximate Erdélyi-Kober fractional derivatives and establishes a parameter upper bound for these networks. We validate this method using the variational iteration formula to obtain the exact solution of a differential equation. The resulting structure from the variational iteration method serves as a basis for showcasing how residual neural networks can effectively estimate these equations. Furthermore, we provide illustrative examples to elucidate the application of residual neural networks in solving equations involving Erdélyi-Kober fractional derivatives.

Flows of geometric structures

Abstract We develop an abstract theory of flows of geometric 𝐻-structures, i.e., flows of tensor fields defining 𝐻-reductions of the frame bundle, for a closed and connected subgroup H ⊂ SO ⁢ ( n ) H\subset\mathrm{SO}(n) , on any connected and oriented 𝑛-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of 𝐻-structures, by way of the natural infinitesimal action of GL ⁢ ( n , R ) \mathrm{GL}(n,\mathbb{R}) on tensors combined with various bundle decompositions induced by 𝐻-structures. We compute evolution equations for the intrinsic torsion under general flows of 𝐻-structures and, as applications, we obtain general Bianchi-type identities for 𝐻-structures, and, for closed manifolds, a general first variation formula for the L 2 L^{2} -Dirichlet energy functional ℰ on the space of 𝐻-structures. We then specialise the theory to the negative gradient flow of ℰ over isometric 𝐻-structures, i.e., their harmonic flow. The core result is an almost-monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulas by Chen–Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 3, 485–502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 1, 83–103] for the harmonic map heat flow. This yields an 𝜀-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the L ∞ L^{\infty} -norm of initial torsion, in the spirit of Chen–Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 3, 567–578]. Moreover, below a certain energy level, the absence of a torsion-free isometric 𝐻-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat 𝑛-tori, so long as the set [ S n , SO ⁢ ( n ) / H ] [\mathbb{S}^{n},\mathrm{SO}(n)/H] of homotopy classes of maps S n → SO ⁢ ( n ) / H \mathbb{S}^{n}\to\mathrm{SO}(n)/H contains more than one element and the universal cover of SO ⁢ ( n ) / H \mathrm{SO}(n)/H is a sphere, e.g. when n = 7 n=7 and H = G 2 H=\mathrm{G}_{2} , or n = 8 n=8 and H = Spin ⁢ ( 7 ) H=\mathrm{Spin}(7) .

A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application

A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application

Some potential theorems for countable transient jump processes

. This article extends the potential theory to countable transient jump processes based on the works of Kemeny, Snell, and Knapp (1976) and Syski (1979). First, it presents the construction of excessive measures. Using excessive measures, this article defines elementary concepts in potential theory. Thus most conclusions for Markov chains can be generalized to jump processes. One of the most important results is the variational formulas for the energy of equilibrium potentials, which holds for asymmetric processes.

Magneto-thermoelastic nonlinear dynamic modeling of a rotating functionally graded shell

Research is conducted on the dynamic modeling of a rotating functionally graded (FG) cylindrical shell subjected to magnetic and temperature fields. Based on the elasticity theory and generalized Hooke's law on the physical neutral surface, nonlinear geometric equations and thermoelastic constitutive relations are determined. According to the Kirchhoff-Love theory, variational formulas of strain energies for deformation, temperature, and centrifugal force are obtained. Considering the rotational effect, the kinetic energy and its variational formula are derived. The electromagnetic force model incorporating magnetization effect of the ferromagnetic FG shell is established by utilizing the electromagnetic theory. Subsequently, the magneto-thermoelastic dynamic model of the rotating FG shell is developed by adopting the Hamilton's principle. The model can reveal the coupling mechanisms of the interaction and superposition of multi-physical fields. Finally, taking the primary resonance as example, detailed numerical analyses are performed to investigate the effects of different parameters on vibration response and dynamical stability.

A new method to calculate LED’s thermal impedance and a new voltage-current formula of LED

The thermal impedance and voltage-current relation of LEDs can be obtained using a photodetector response to pulsed light emitted by LEDs. In this study, LEDs have been driven by pulsed high currents. Photodetector voltage is proportional to the radiant power of LED. A triangle has been drawn on the top of the photodetector voltage figure, and some geometrical magnitudes have been measured. The junction temperature can be calculated using these magnitudes and the LED’s relative radiant power-junction temperature coefficient. Thermal impedance can be calculated if junction temperature and ambient temperature difference are divided by thermal power dissipated from the LED. In this study, the thermal impedance of LEDs and their error have been calculated and a new voltage-current formula for LEDs has been given. The formulas for LED junction temperature-time and thermal impedance-time variations have been obtained. Besides these, the maximum pulse duration of the current pulse that the LED can work without breaking down has been calculated.

Quenched large deviations in renewal theory

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

Optimal allocation of flood prevention storage and dynamic operation of water levels to increase cascade reservoir hydropower generation

The existing trade-off between flood control and water conservation measured by flood limited water level (FLWL) during flood season hinders multi-reservoir system from realizing comprehensive benefits. To overcome this barrier, we reasonably allocated the complementary flood prevention storage based on the aggregation-decomposition technique, theoretically deduced the mathematical formula of power generation variation simplified from the cascade reservoir operation model, and dynamically operated reservoir water levels considering the inflow forecasting information. The Wudongde (WDD) ∼ Baihetan (BHT) ∼ Xiluodu (XLD) ∼ Xiangjiaba (XJB) cascade reservoirs in the downstream Jinsha River in China is selected as the case study. Results demonstrate that: (1) The derived power generation function monotonically increases only with the growth of the pre-discharge water volume. (2) The reduced proportion of the flood prevention storage of the WDD (or XJB) is favorable to boost the hydroelectric benefits of the WDD ∼ BHT (or XLD ∼ XJB) cascade reservoirs. Optimal allocation of flood prevention storage can increase 1.933 billion kW∙h power generation annually (+2.15 %) for the WDD ∼ BHT ∼ XLD ∼ XJB cascade reservoirs. (3) Further cooperated with dynamic operation of water levels, the WDD ∼ BHT ∼ XLD ∼ XJB cascade reservoirs can totally generate 5.925 billion kW∙h more hydropower annually (+6.60 %) using effective 5-day lead-time inflow forecasting information, while maintaining the design flood prevention standards.

Eigenvalue of (p,q)-Biharmonic System along the Ricci Flow

In this paper, we determine the variation formula for the first eigenvalue of (p,q)-biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived.

A global space–time Trefftz DG scheme for the time-dependent isotropic elastic wave equations

In this paper we are concerned with Trefftz discretizations of the time-dependent linear elastic wave equation in arbitrary space dimensional domains Ω⊂Rd(d∈N). We propose a Trefftz discontinuous Galerkin method, construct Trefftz basis functions and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth. A space–time domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the nonhomogeneous model. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy, and the proposed preconditioner is effective.

Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy

We derive the variational formula of the Loewner driving function of a simple chord under infinitesimal quasiconformal deformations with Beltrami coefficients supported away from the chord. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of its driving function. This result gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo, which has the same variational formula. We also deduce the variation of the total mass of SLE8/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {SLE}_{8/3}$$\\end{document} loops touching the Jordan curve under quasiconformal deformations.

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

In the modern world, where efficiency, stability, and precision play a crucial role, the development and application of optimal control strategies in oscillatory systems hold significant importance. The issues related to the numerical solution of control problems associated with damping oscillatory systems consisting of two objects are considered. To numerically solve the discussed problem, the gradient projection method, based on the formula for the first variation of the functional, and the method of successive approximations, associated with the linearity of boundary problems describing oscillatory processes, are applied. The oscillations of one object are described by a wave equation with first-order boundary conditions, while a second-order ordinary differential equation models the oscillations of the other object. Furthermore, the original and the adjoint boundary value problems are solved using direct methods at each iteration step. An algorithm for the numerical solution of the problem is proposed, and based on this algorithm, a software code for implementation is developed. The numerical results obtained in the study demonstrate that there is convergence in terms of functionality, and the approximately optimal controls found in this process are minimizing sequences in the control space. The mechanism of controlling and regulating the operation of the system according to its input constraints is provided by observed feedback, allowing systems with limited excitation to maintain stability and optimal functioning in conditions of changing external or internal circumstances. The obtained results can also be used to forecast the system's behavior in the future, resource planning, prevention of emergencies, or optimization of production processes

Contributions to Diffusion in Complex Materials Quantified with Machine Learning.

Using machine learning with a variational formula for diffusivity, we recast diffusion as a sum of individual contributions to diffusion-called "kinosons"-and compute their statistical distribution to model a complex multicomponent alloy. Calculating kinosons is orders of magnitude more efficient than computing whole trajectories, and it elucidates kinetic mechanisms for diffusion. The density of kinosons with temperature leads to new accurate analytic models for macroscale diffusivity. This combination of machine learning with diffusion theory promises insight into other complex materials.

On stability of subelliptic harmonic maps with potential

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension ≥3.

Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments

The Herglotz-type vakonomic dynamics of nonholonomic constrained systems with delayed arguments and its Noether theory are studied in this paper. First of all, the Herglotz-type equations of time-delayed vakonomic dynamics for nonholonomic systems are established, and the Herglotz-type local extremal equations are given. Secondly, on the basis of derivation of the variational formulas of Hamilton–Herglotz action with time delay, the Herglotz-type Noether symmetry criteria for time-delayed vakonomic dynamics are investigated. Thirdly, the Herglotz-type Noether’s theorems and inverse theorems for time-delayed vakonomic dynamics of nonholonomic systems are deduced. Finally, an example is presented to demonstrate the application of the results.

Parameter variational based adaptive bipartite formation control of second-order multi-agent systems with antagonistic interactions

This article mainly concentrates on researching the leader-following bipartite formation of second-order nonlinear multi-agent systems (MASs) with antagonistic interactions and time-varying delay. A novel distributed feedback control and impulsive control strategy is proposed for realizing the bipartite formation. By establishing an error system, the bipartite formation issue of the MASs is transformed into the stability issue of the error systems. Sufficient conditions for achieving the bipartite formation of second-order nonlinear MASs are obtained by jointly employing the extended parameters variation formula, the comparison principle and the definition of average impulsive interval. In addition, in order to obtain the optimal control gains and control effects, an adaptive control strategy is taken into account by designing the adaptive updating laws. Ultimately, the effectiveness of the theoretical results is demonstrated by two numerical simulations.

Estimation of the Increase Potential of Global Soil CO<sub>2</sub> Efflux by Surface Air and Soil Warming Based on the Mathematical Model Formula for Diurnal and Seasonal Variations in Soil Temperature

Estimation of the Increase Potential of Global Soil CO<sub>2</sub> Efflux by Surface Air and Soil Warming Based on the Mathematical Model Formula for Diurnal and Seasonal Variations in Soil Temperature

Variation formulae for the volume of coassociative submanifolds

Variation formulae for the volume of coassociative submanifolds