Variational Formula Research Articles

Robin Spectral Rigidity of the Ellipse

In this paper, we investigate $$C^1$$ isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. If the deformation is in fact smooth, reparametrizing allows us to show that the first variation actually vanishes to infinite order. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard’s variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we in fact construct an explicit parametrix for the wave propagator in the interior, microlocally near geodesic loops.

Local variation formulas of solutions for nonlinear controlled functional differential equations with constant delays and the discontinuous initial condition

Abstract For the nonlinear controlled functional differential equations with several constant delays, the local variation formulas of solutions are proved, in which the effects of the discontinuous initial condition and perturbations of delays and the initial moment are detected.

Study on tool orientation feasible region with constraint of non-linear error for high-precision five-axis machining

Five-axis machine is one of the most versatile machine tools in complex surface machining while the machining interference occurs easily. To avoid the adverse consequences caused by interference, interference detection, and feasible region, solution are proposed. However, with the required processing accuracy of the complex surface parts increasing, the common feasible region solution is generally difficult to achieve. The error caused by the tool orientation variation along the tool path, just the non-linear error, may exceed the specified tolerance. Considering that both the interference problem and the non-linear problem could be solved by modifying the tool orientation, this study proposes a feasible region solution for tool orientation with the constraint of non-linear error, which can avoid tool interference and achieve high geometric accuracy for tool path planning in the extreme high-precision machining. The detection region that corresponds to the tool orientation at cutter contact (CC) point is calculated to avoid the collision firstly, and the feasible region of tool orientation for a certain CC point is established. After building the tool path equations between two adjacent CC points, the relationship between the movement angles of rotation axis and the machining error is built. By considering the non-linear error of the different rotatory axes respectively, the calculation formula for the movement angle variation of the rotation axis is deduced. Finally, according to the single point non-interference feasible region of the tool angle, the non-interference feasible region for the extreme high-precision processing is constructed with the constraint of non-linear error. A comparative experiment is carried out to verify the proposed feasible region construction method. The experimental results show that with the proposed method, the profile arithmetic average error and the maximum of profile deviation are 53 μm and 71 μm, respectively, which decrease by 30.13% and 32.31% compared with the conventional non-interference feasible region. The achievements have good applicability and are significant for the improvement of processing quality, which provide guidance for the extreme high-precision machining of complex surface parts.

A formula for the anisotropic total variation of SBV functions

The purpose of this paper is to present the relation between certain BMO–type seminorms and the total variation of SBV functions. Following some ideas of [2], we give a representation formula of the total variation of SBV functions which does not make use of the distributional derivatives. We consider an anisotropic variant of the BMO–type seminorm introduced in [4], by using, instead of cubes, covering families made by translations of a given open bounded set with Lipschitz boundary.

A Variational Characterization of the Risk-Sensitive Average Reward for Controlled Diffusions on $\mathbb{R}^d$

We address the variational formulation of the risk-sensitive reward problem for non-degenerate diffusions on $\mathbb{R}^d$ controlled through the drift. We establish a variational formula on the whole space and also show that the risk-sensitive value equals the generalized principal eigenvalue of the semilinear operator. This can be viewed as a controlled version of the variational formulas for principal eigenvalues of diffusion operators arising in large deviations. We also revisit the average risk-sensitive minimization problem and by employing a gradient estimate developed in this paper, we extend earlier results to unbounded drifts and running costs.

Uncertainty Quantification for Markov Processes via Variational Principles and Functional Inequalities

Information-theory based variational principles have proven effective at providing scalable uncertainty quantification (i.e., robustness) bounds for quantities of interest in the presence of nonparametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincaré, $\log$-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds for time-averaged observables, comparing a Markov process to a second (not necessarily Markov) process. These bounds are well behaved in the infinite-time limit and apply to steady-states of both discrete and continuous-time Markov processes.

Volumes of quasifuchsian manifolds

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil-Petersson distance between the conformal metrics at infinity.

Singular extremals inL1-optimal control problems: sufficient optimality conditions

In this paper we are concerned with generalisedL1-minimisation problems,i.e.Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proved by means of Hamiltonian methods. As a by-product of the result, we provide an explicit invariant formula for the second variation along the singular arc.

On the general dual Orlicz-Minkowski problem

For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $\phi:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_{\phi})$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_\phi(K)=\int_{\mathbb{R}^n \setminus K} \phi(x)\,dx.$$ Under certain conditions on $\phi$, we prove a variational formula for the general dual ($L_{\phi})$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{\phi,\mathcal{V}}(K, \cdot)$, the general dual ($L_{\phi})$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski problem: {\it Given a nonzero finite Borel measure $\mu$ defined on $S^{n-1}$ and a continuous function $\phi: \mathbb{R}^n\setminus\{o\}\rightarrow (0, \infty)$, can one find a constant $\tau>0$ and a convex body $K$ (ideally, containing $o$ in its interior), such that,} $$\mu=\tau\widetilde{C}_{\phi,\mathcal{V}}(K,\cdot)? $$ Based on the method of Lagrange multipliers and the established variational formula for the general dual ($L_{\phi})$ Orlicz quermassintegral, a solution to the general dual Orlicz-Minkowski problem is provided. In some special cases, the uniqueness of solutions is proved and the solution for $\mu$ being a discrete measure is characterized.

One-sided reflected Brownian motions and the KPZ fixed point

AbstractWe consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling-invariant Markov process defined in [MQR17] and believed to govern the long-time, large-scale fluctuations for all models in the KPZ universality class. Brownian last-passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.

Generalization of Schläfli formula to the volume of a spherically faced simplex

The purpose of this paper is to present a variational formula of Schläfli type for the volume of a spherically faced simplex in the Euclidean space. It is described in terms of Cayley–Menger determinants and their differentials involved with hypersphere arrangements. We derive it as a limit of fundamental identities for hypergeometric integrals associated with hypersphere arrangements obtained by the authors in the preceding article.

Strong Converse Using Change of Measure Arguments

The strong converse for a coding theorem shows that the optimal asymptotic rate possible with vanishing error cannot be improved by allowing a fixed error. Building on a method introduced by Gu and Effros for centralized coding problems, we develop a general and simple recipe for proving strong converse that is applicable for distributed problems as well. Heuristically, our proof of strong converse mimics the standard steps for proving a weak converse, except that we apply those steps to a modified distribution obtained by conditioning the original distribution on the event that no error occurs. A key component of our recipe is the replacement of the hard Markov constraints implied by the distributed nature of the problem with a soft information cost using a variational formula introduced by Oohama. We illustrate our method by providing a short proof of the strong converse for the Wyner-Ziv problem and strong converse theorems for interactive function computation, common randomness and secret key agreement, and the wiretap channel; the latter three strong converse problems were open prior to this work.

F-Minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

AbstractThe aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind including minimal Lagrangians and soliton solutions for Lagrangian mean curvature flow (LMCF). We derive 2nd variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh’s formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

A limit theorem for the survival probability of a simple random walk among power-law renewal obstacles

We consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1−e −β , where β is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent γ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is γ/(γ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter β that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case β corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5.

Solution of Scalar Riccati Differential Equation of Fractional Order

The main objective of this paper is to generalize the scalar Riccati differential equation for factional order derivatives using Caputo definition, and then to find its approximate solution using the variational iteration method. The present work consists of the statement and the proof of the variational iteration formula and then prove of its convergent to the exact solution. Some illustrative examples are considered and simulated using Mathcad 15 computer program.

Stability of regime-switching jump diffusion processes

This work studies the stability of regime-switching jump diffusion processes in a finite or a countably infinite state space. Some criteria with sufficient conditions for stability and instability are provided based on characterizing the stability property of the processes in any fixed state through constants under common measurements. Also, some variational formula of these constants are given. Moreover, some examples of nonlinear regime-switching jump diffusion processes are provided to show the usefulness and sharpness of these criteria.

A numerical study on the ventilation coefficients of falling lobed hailstones

The ventilation coefficients that represent the enhancement of mass transfer rate due to the falling motion of spherical hailstones in an atmosphere of 460hPa and 248K are computed by numerically solving the unsteady Navier‐Stokes equation for airflow past hailstones and the convective diffusion equation for water vapor diffusion aroundthe fallinghailstones.The diameters of the hailstonesinvestigated are from 1to 10cm, corresponding to Reynolds number from 5935 to 177148. The calculated ventilation coefficients vary approximately linearly with the hailstone diameter, from about 19 for a 1-cm hailstone to about 208 for a 10-cm hailstone. Empirical formulas for ventilation coefficient variation with hailstone diameter as well as Reynolds and Schmidt numbers are given. Implications of these ventilation coefficients are discussed.

Effects of infinite occurrence of hybrid impulses with quasi-synchronization of parameter mismatched neural networks

This article is deeply concerned with the effects of hybrid impulses on quasi-synchronization of neural networks with mixed time-varying delays and parameter mismatches. Hybrid impulses allow synchronizing as well as desynchronizing impulses in one impulsive sequence, so their infinite time occurrence with the system may destroy the synchronization process. Therefore, the effective hybrid impulsive controller has been designed to deal with the difficulties in achieving the quasi-synchronization under the effects of hybrid impulses, which occur all the time, but their density of occurrence gradually decrease. In addition, the new concepts of average impulsive interval and average impulsive gain have been applied to cope with the simultaneous existence of synchronizing and desynchronizing impulses. Based on the Lyapunov method together with the extended comparison principle and the formula of variation of parameters for mixed time-varying delayed impulsive system, the delay-dependent sufficient criteria of quasi-synchronization have been derived for two separate cases, viz., Ta<∞ and Ta=∞. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples.

Affine isoperimetric inequalities for intersection mean ellipsoids

A variational formula for Lutwak’s dual affine quermassintegrals $${\widetilde{\Lambda }}_{j}$$ of convex bodies in $${\mathbb {R}}^n$$ is established when $$1\le j\le n-1.$$ Using new ellipsoids associated with the intersection functions of convex bodies, we prove several sharp affine isoperimetric inequalities.

Effects of the initial moment and several delays perturbations in the variation formulas for a solution of a functional differential equation with the continuous initial condition

Abstract For the nonlinear functional differential equation with several constant delays, the variation formulas for its solution are proved, in which the effects of perturbations of delays and the initial moment are detected.