Nondegeneracy Assumption Research Articles

Path integral representation of Lorentzian spinfoam model, asymptotics and simplicial geometries

A new path integral representation of Lorentzian Engle–Pereira–Rovelli–Livine spinfoam model is derived by employing the theory of unitary representation of . The path integral representation is taken as a starting point of semiclassical analysis. The relation between the spinfoam model and classical simplicial geometry is studied via the large-spin asymptotic expansion of the spinfoam amplitude with all spins uniformly large. More precisely, in the large-spin regime, there is an equivalence between the spinfoam critical configuration (with certain nondegeneracy assumption) and a classical Lorentzian simplicial geometry. Such an equivalence relation allows us to classify the spinfoam critical configurations by their geometrical interpretations, via two types of solution-generating maps. The equivalence between spinfoam critical configuration and simplical geometry also allows us to define the notion of globally oriented and time-oriented spinfoam critical configuration. It is shown that only at the globally oriented and time-oriented spinfoam critical configuration, the leading-order contribution of spinfoam large-spin asymptotics gives precisely an exponential of Lorentzian Regge action of General Relativity. At all other (unphysical) critical configurations, spinfoam large-spin asymptotics modifies the Regge action at the leading-order approximation.

Two‐Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front

We continue to develop the regularity theory of general two‐phase free boundary problems for parabolic operators. In a 2010 paper, we establish the optimal (Lipschitz) regularity of a viscosity solution under the assumptions that the free boundary is locally a flat Lipschitz graph and a nondegeneracy condition holds. Here, on one side we improve this result by removing the nondegeneracy assumption; on the other side we prove the smoothness of the front. The proof relies in a crucial way on a local stability result stating that, for a certain class of operators, under small perturbations of the coefficients flat free boundaries remain close and flat. © 2013 Wiley Periodicals, Inc.

Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of [formula omitted

This paper is devoted to the analysis of the following two-space dimensional linear parabolic equation ∂tu-∂x12u-∂x22u=f, subject to Robin type conditions ∂x1u+βu=0, on the lateral boundary, where the coefficient β satisfies suitable non-degeneracy assumptions. The right-hand side f of the equation is taken in L2. The problem is set in a domain of the form t,x1∈R2:0<t<T,φ1(t)<x1<φ2(t)×]0,b[, where φ1 and φ2 are smooth functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where φ1(0)=φ2(0) is allowed. For this purpose, the domain decomposition method is employed.

Geometric Generalisations of shake and rattle

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting.In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

An effective trust-region-based approach for symmetric nonlinear systems

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported.

On the Convergence of Finite Element Methods for Hamilton--Jacobi--Bellman Equations

In this note we study the convergence of monotone P1 finite element methods on unstructured meshes for fully non-linear Hamilton-Jacobi-Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretisations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under non-degeneracy assumptions, strong L2 convergence of the gradients.

Comparison between classes of state-quadratic Lyapunov functions for discrete-time linear polytopic and switched systems

The paper deals with the stability properties of linear discrete-time switched systems with polytopic sets of modes. The most classical way of studying the uniform asymptotic stability of such a system is to check for the existence of a quadratic Lyapunov function. It is known from the literature that letting the Lyapunov function depend on the time-varying switching parameter improves the chance that a quadratic Lyapunov function exists. Our objective is to compare different notions of quadratic stability. The contribution of this paper is twofold. In the first part we consider switching systems satisfying a certain non-degeneracy assumption and we prove that, for such systems, no gain in the stability analysis is obtained if we allow the Lyapunov function to depend explicitly also on time. In the second part we consider the case where the non-degeneracy assumption is violated. We prove that in this case allowing the Lyapunov function to depend on time is less conservative. We also show that new LMI conditions can be used in order to characterize the existence of a time-dependent quadratic Lyapunov function. Moreover in the paper we discuss the case where the variation of the switching parameter is bounded by a prescribed constant between two subsequent times.

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form $dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\ldots,V_d)$ satisfy H\"{o}rmander's bracket condition, we demonstrate that $Y_t$ admits a smooth density for any $t\in(0,T]$, provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter $H>1/4$, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time $T$.

On the Orbital Stability of Standing-Wave Solutions to a Coupled Non-Linear Klein-Gordon Equation

AbstractWe show the existence of standing-wave solutions to a coupled non-linear Klein-Gordon equation. Our solutions are obtained as minimizers of the energy under a two-charges constraint. We prove that the ground state is stable and that standing-waves are orbitally stable under a non-degeneracy assumption.

Mathematical programs with vanishing constraints: a new regularization approach with strong convergence properties

Motivated by a recent method introduced by Kanzow and Schwartz [C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, Preprint 296, Institute of Mathematics, University of Würzburg, Würzburg, 2010] for mathematical programs with complementarity constraints (MPCCs), we present a related regularization scheme for the solution of mathematical programs with vanishing constraints (MPVCs). This new regularization method has stronger convergence properties than the existing ones. In particular, it is shown that every limit point is at least M-stationary under a linear independence-type constraint qualification. If, in addition, an asymptotic weak nondegeneracy assumption holds, the limit point is shown to be S-stationary. Second-order conditions are not needed to obtain these results. Furthermore, some results are given which state that the regularized subproblems satisfy suitable standard constraint qualifications such that the existing software can be applied to these regularized problems.

A simple smooth exact penalty function for smooth optimization problem

For smooth optimization problem with equality constraints, new continuously differentiable penalty function is derived. It is proved exact in the sense that local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function under some nondegeneracy assumption. It is simple in the sense that the penalty function only includes the objective function and constrained functions, and it doesn’t include their gradients. This is achieved by augmenting the dimension of the program by a variable that controls the weight of the penalty terms.

Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler’s equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.

Differentiable Exact Penalty Functions for Nonlinear Second-Order Cone Programs

We propose a method for solving nonlinear second-order cone programs (SOCPs), based on a continuously differentiable exact penalty function. The construction of the penalty function is given by incorporating a multipliers estimate in the augmented Lagrangian for SOCPs. Under the nondegeneracy assumption and the strong second-order sufficient condition, we show that a generalized Newton method has global and superlinear convergence. We also present some preliminary numerical experiments.

Repulsion From resonances

We consider slow-fast systems with periodic fast motion and integrable slow motion in the presence of both weak and strong resonances. Assuming that the initial phases are random and that appropriate non-degeneracy assumptions are satisfied we prove that the effective evolution of the adiabatic invariants is given by a Markov process. This Markov process consists of the motion along the trajectories of a vector field with occasional jumps. The generator of the limiting process is computed from the dynamics of the system near strong resonances.

Degeneracy Resolution for Bilinear Utility Functions

Loss-aversion is a phenomenon where investors are particularly sensitive to losses and eager to avoid them. An efficient method to solve the portfolio optimization problem of maximizing the bilinear utility function is given by Best et al. (Loss-Aversion with Kinked Linear Utility Functions, CORR 2010-04, University of Waterloo, 2010). This method is useful because it performs its computations only using asset related quantities rather than much higher dimensional quantities of the LP formulation. However, a difficulty with this method is that it requires a nondegeneracy assumption which may not be satisfied. This paper implements Bland's least-index rules to the method in such a way that the efficiency of the method is retained. Then we describe the numerical results of applying our algorithm to a series of six asset problems in which the degree of loss-aversion is increased.

A BFGS trust-region method for nonlinear equations

In this paper, a new trust-region subproblem combining with the BFGS update is proposed for solving nonlinear equations, where the trust region radius is defined by a new way. The global convergence without the nondegeneracy assumption and the quadratic convergence are obtained under suitable conditions. Numerical results show that this method is more effective than the norm method.

A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).

A constraint-reduced variant of Mehrotra’s predictor-corrector algorithm

Consider linear programs in dual standard form with n constraints and m variables. When typical interior-point algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires $\mathcal{O}(nm^{2})$ operations per iteration. When n?m it is often the case that at each iteration most of the constraints are not very relevant for the construction of a good update and could be ignored to achieve computational savings. This idea was considered in the 1990s by Dantzig and Ye, Tone, Kaliski and Ye, den Hertog et al. and others. More recently, Tits et al. proposed a simple constraint-reduction scheme and proved global and local quadratic convergence for a dual-feasible primal-dual affine-scaling method modified according to that scheme. In the present work, similar convergence results are proved for a dual-feasible constraint-reduced variant of Mehrotra's predictor-corrector algorithm, under less restrictive nondegeneracy assumptions. These stronger results extend to primal-dual affine scaling as a limiting case. Promising numerical results are reported. As a special case, our analysis applies to standard (unreduced) primal-dual affine scaling. While we do not prove polynomial complexity, our algorithm allows for much larger steps than in previous convergence analyses of such algorithms.

Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel'fand problem in two space dimensions

We consider a sequence of solutions $u_n$ of the problem $$-\Delta u=\lambda e^u \quad \text{in $\Omega$}, \qquad u=0 \quad \text{on $\partial\Omega$},$$ with $\lambda=\{\lambda_n\}_{n\in\mathbb N}$ and blowing up at $m$ points $\kappa_1,\dots,\kappa_m$ in $\Omega$. Under some non-degeneracy assumption on some suitable finite-dimensional function (related to $\kappa_1,\dots,\kappa_m$) we show that $u_n$ is non-degenerate for $n$ large enough.

Harmonic morphisms on conformally flat 3-spheres

We show that under some non-degeneracy assumptions the only submersive harmonic morphism on a conformally flat 3-sphere to a surface is the Hopf fibration. The proof involves an appropriate use of the Chern–Simons functional.