Variational Formula Research Articles

SIF formula based on exact SIF distribution for semi‐elliptical surface cracks subjected to mode I, II, III uniform loading

AbstractThe stress intensity factor (SIF) distribution of a semi‐elliptical surface crack is often used for evaluating the fatigue strength of structures. Virtually exact distributions of SIFs can be provided along the crack front by solving the hypersingular integral equation of the body force method. In this paper, to create a very accurate SIF variation formula, the elliptical crack SIF solutions , , are used and the SIF ratios , , are mainly focused. Paying attention to the corner point singularity, by applying the least squares method to the ratio , in the whole range of parametric angle , and formulas can be proposed. Instead, formula can be proposed by applying the method of least squares to the ratio of in the range but applying directly to in the range . In this way, all of formulas proposed in this paper provide the SIFs with better than 1.0% accuracy.

Gromov–Hausdorff variational principles and measure stability * *KL and CAM were partially supported by Basic Science Research Program through the NRF funded by the Ministry of Education of the Republic of Korea (Grant Number: 2022R1l1A3053628).

Variational formulae for the Gromov–Hausdorff distances of compact metric spaces and their continuous maps are obtained. Also, a notion of µ-topological GH-stability on metric-measure spaces with probability µ is given. We prove that every expansive map with the weak µ-shadowing property (Lee and Rojas 2022 Russ. J. Nonlinear Dyn. 18 297–307) is µ-topologically GH-stable.

On the self-overlap in vector spin glasses

We consider vector spin glass models with self-overlap correction. Since the limit of free energy is an infimum, we use arguments analogous to those for generic models to show the following: (1) the averaged self-overlap converges; (2) the self-overlap concentrates; (3) the infimum optimizes over paths whose right endpoints are the limit of self-overlap. Lastly, using these, we directly verify the equivalence between the variational formula obtained in Chen [Electron. J. Probab. 28, 1 (2023)] and Panchenko’s generalized Parisi formula in Panchenko [Ann. Probab. 46(2), 865 (2018)].

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.

Matrix Measure-Based Event-Triggered Impulsive Quasi-Synchronization on Coupled Neural Networks.

In this article, the quasi-synchronization for a kind of coupled neural networks with time-varying delays is investigated via a novel event-triggered impulsive control approach. In view of the randomly occurring uncertainties (ROUs) in the communication channels, the global quasi-synchronization for the coupled neural networks within a given error bound is considered instead of discussing the complete synchronization. A kind of distributed event-triggered impulsive controllers is presented with considering the Bernoulli stochastic variables based on ROUs, which works at each event-triggered impulsive instant. According to the matrix measure method and the Lyapunov stability theorem, several sufficient conditions for the realization of the quasi-synchronization are successfully derived. Combining with the mathematical methodology with the formula of variation of parameters and the comparison principle for the impulsive systems with time-varying delays, the convergence rate and the synchronization error bound are precisely estimated. Meanwhile, the Zeno behaviors could be eliminated in the coupled neural network with the proposed event-triggered function. Finally, a numerical example is presented to prove the results of theoretical analysis.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

Entanglement Monogamy via Multivariate Trace Inequalities

Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.

Empirical Measures, Geodesic Lengths, and a Variational Formula in First-Passage Percolation

This monograph resolves—in a dense class of cases—several open problems concerning geodesics in i.i.d. first-passage percolation on Z d \mathbb {Z}^d . Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 0 to n ξ n{\boldsymbol \xi } , where ξ {\boldsymbol \xi } is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as n → ∞ n\to \infty , answering a question of Hoffman. These families include arbitrarily small L ∞ L^\infty -perturbations of any given distribution, almost every finitely supported distribution, uncountable collections of continuous distributions, and certain discrete distributions whose atoms can have any prescribed sequence of probabilities. Moreover, the constructions are explicit enough to guarantee examples possessing certain features, for instance: both continuous and discrete distributions whose support is all of [ 0 , ∞ ) [0,\infty ) , and distributions given by a density function that is k k -times differentiable. All results also hold for ξ {\boldsymbol \xi } -directed infinite geodesics. In comparison, we show that if Z d \mathbb {Z}^d is replaced by the infinite d d -ary tree, then any distribution for the weights admits a unique limiting empirical measure along geodesics. In both the lattice and tree cases, our methodology is driven by a new variational formula for the time constant, which requires no assumptions on the edge-weight distribution. Incidentally, this variational approach also allows us to obtain new convergence results for geodesic lengths, which have been unimproved in the subcritical regime since the seminal 1965 manuscript of Hammersley and Welsh.

Variational formulae for conformal capacity

Analogues of Hadamard's classical variational formula are presented for the Dirichlet integral of a normalized harmonic function under deformations of its domain of definition. Variational formulae for quadratic forms with coefficients depending on the inner radii, Robin radii, Green's functions and Robin functions of the domains under consideration are also presented. Bibliography: 17 titles.

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Stability of minimal surfaces in the sub-Riemannian manifold $\widetilde{E(2)}$

In the paper we study smooth oriented surfaces in the universal covering space of the group of orientation-preserving Euclidean plane isometries, which has a three-dimensional sub-Riemannian manifold structure. This structure is constructed as a restriction of the Euclidean metric on the group to some completely non-integrable left invariant distribution. The sub-Riemannian area of a surface is then defined as the integral of the length of its unit normal field projected orthogonally onto this distribution. We calculate the first variation formula of the sub-Riemannian surface area and derive the minimality criterion from it. Here we call a surface minimal if it is a critical point of the sub-Riemannian area functional under normal variations with compact support. We show that the minimality in this case is not equivalent to the vanishing of the sub-Riemannian mean curvature. We then prove that a Euclidean plane is minimal if and only if it is parallel or orthogonal to the $z$-axis (where the $z$-coordinate corresponds to the rotation angle of an isometry). Also we obtain the minimality condition for a graph and give examples of minimal graphs. The examples considered in the paper demonstrate, in particular, that the minimality of a surface in the Riemannian (in this case Euclidean) sense does not imply its sub-Riemannian minimality, and vice versa. Next, we consider the stability of minimal surfaces. For this purpose, we derive the second variation formula of the sub-Riemannian area and show with it that minimal Euclidean planes are stable. We introduce a class of surfaces for which the tangent planes are perpendicular to the planes of the sub-Riemannian structure, and call them vertical surfaces. In particular, for such surfaces the second variation formula is simplified significantly. Then we prove that complete connected vertical minimal surfaces are either Euclidean planes or helicoids and that helicoids are unstable. This implies a following Bernstein type result: a complete connected vertical minimal surface is stable if and only if it is a Euclidean plane orthogonal to the $z$-axis.

Stability and moduli space of generalized Ricci solitons

The generalized Einstein–Hilbert action is an extension of the classic scalar curvature energy and Perelman’s F-functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application we show that all Bismut-flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso [1–4] and Podesta, Spiro, Kröncke [5–8] to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.

Grid entropy in last passage percolation — A superadditive critical exponent approach

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we arrive at the notion of grid entropy — a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a 2020 paper by Bates. We partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates.

A Fractal Prediction Method for Contact Stiffness of Helical Gear Considering Asperity Lateral Contact and Interaction

The normal contact stiffness (NCS) on rough surfaces has a significant impact on the dynamic characteristics of helical gear. Aiming at the problem of inaccurate calculation of the NCS model under the traditional Hertz theory of smooth surfaces, a fractal prediction model of helical gear contact stiffness considering asperity lateral contact and interaction between asperities is proposed in this paper. The variation formula of asperity and the correction coefficient of a tooth contact surface under asperity lateral contact and interaction are derived, and the influence of micro-elements on normal load and NCS is qualitatively analyzed. The results show that the NCS of considering the interaction and lateral contact of asperity is closer to the experimental results; the contact surface correction coefficient increases with the increase of curvature radius and load. The NCS of a tooth surface increases with the increase in fractal dimension D or the decrease in roughness amplitude G. The influence of asperity lateral contact and interaction decreases with the increase in D and the decrease in G. The NCS of the helical gear decreases under the lateral contact and interaction of the asperity, which is critical for exact estimation of the NCS of contact surfaces in gear.

Conservation laws for systems of non-standard Birkhoffians with fractional derivatives

Conservation laws for systems of exponential, power-law and logarithmic non-standard Birkhoffian with fractional derivatives are investigated respectively, and their relationships with Noether symmetries are revealed. Corresponding to the three forms of non-standard Birkhoffians, the Pfaff–Birkhoff principles are proposed and non-standard Birkhoff’s equations with fractional derivatives are derived. The variation of Pfaff action is investigated in depth, and the formulas of total variation are deduced, and on this basis the criterion equations of Noether symmetry are set up. Then, Noether’s theorems for systems of non-standard Birkhoffian with fractional derivatives are proved, which reveals the relationship between symmetry and conserved quantity. Finally, three examples are given to illustrate how to calculate the symmetry and find out the conserved quantity, and the correctness of the obtained conserved quantities is shown by numerical calculation.

Matrix measure-based distributed impulsive consensus on nonlinear multi-agent systems with mixed time-varying delays

This paper concentrates on researching the global and exponential leader-following consensus issue for an array of nonlinear multi-agent systems with the system time-varying delay and the distributed time-varying delay. An innovative distributed impulsive controller is proposed to force the states of all agents to track the trajectories of leader agent. By jointly introducing the matrix measure protocol, the comparison principle for impulsive systems, and the average impulsive interval, sufficient criteria for the realization of leader-following consensus are derived. In addition, considering different functions of impulsive signal, two different convergence rates are precisely estimated by utilizing the parameter variation formula, respectively. Finally, two numerical examples are given to demonstrate the effectiveness of proposed control strategy and the rightness of theoretical analysis.

The maximal curves and heat flow in general-affine geometry

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space R2 must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term “affine geometry” refers to “equi-affine geometry”.) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in R2? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in R2, and show the general-affine maximal curves in R2 are much more abundant and include the explicit curves y=xα(αis a constant andα∉{0,1,12,2}) and y=xlog⁡x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA(n)=GL(n)⋉Rn. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved.

Unfitted generalized finite element methods for Dirichlet problems without penalty or stabilization

AbstractUnfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM.

Cumulant GAN.

In this article, we propose a novel loss function for training generative adversarial networks (GANs) aiming toward deeper theoretical understanding as well as improved stability and performance for the underlying optimization problem. The new loss function is based on cumulant generating functions (CGFs) giving rise to Cumulant GAN. Relying on a recently derived variational formula, we show that the corresponding optimization problem is equivalent to Rényi divergence minimization, thus offering a (partially) unified perspective of GAN losses: the Rényi family encompasses Kullback-Leibler divergence (KLD), reverse KLD, Hellinger distance, and χ2 -divergence. Wasserstein GAN is also a member of cumulant GAN. In terms of stability, we rigorously prove the linear convergence of cumulant GAN to the Nash equilibrium for a linear discriminator, Gaussian distributions, and the standard gradient descent ascent algorithm. Finally, we experimentally demonstrate that image generation is more robust relative to Wasserstein GAN and it is substantially improved in terms of both inception score (IS) and Fréchet inception distance (FID) when both weaker and stronger discriminators are considered.

Influence of Social Entrepreneurship Training on Job Creation in Ilorin's Small Business Sector

Small and medium enterprises, particularly micro-businesses have been the rescue platform for the survival and sustainability of the poor, unemployed youths and women as well as being the nursery-bed of entrepreneurship development in the nation. The study analyzes the influence of social entrepreneurship training on job creation in Ilorin's small business sector. The study focuses on understanding the impact of social entrepreneurship training programs on the ability of small business owners in Ilorin to create employment opportunities. The study adopted a descriptive survey research design. The study sampled 226 small business enterprises out of 520 using the Taro Yamane Coefficient of Variation formula. The researcher's instrument for data collection was a well-structured five-point Likert scale questionnaire which was administered to entrepreneurs from each selected entrepreneurial business type comprised of poultry farming, snail rearing, Rabbit rearing and 58 fish farming. The study adopted regression analysis and Pearson product moment correlation coefficient to test the hypotheses. The findings reveal that social entrepreneurship training has a significant positive influence on job creation in Ilorin's small business sector. Small business owners who have received training are more likely to create employment opportunities compared to those who have not undergone any training. It was concluded from the findings of the study that the social entrepreneurship development programs have greatly contributed to the rate of growth of small business enterprises in Ilorin.