Convex Potential Research Articles

Analysis of a general dynamic history-dependent variational–hemivariational inequality

This paper is devoted to the study of a general dynamic variational–hemivariational inequality with history-dependent operators. These operators appear in a convex potential and in a locally Lipschitz superpotential. The existence and uniqueness of a solution to the inequality problem is explored through a result on a class of nonlinear evolutionary abstract inclusions involving a nonmonotone multivalued term described by the Clarke generalized gradient. The result presented in this paper is new and general. It can be applied to study various dynamic contact problems. As an illustrative example, we apply the theory on a dynamic frictional viscoelastic contact problem in which the contact is modeled by a nonmonotone Clarke subdifferential boundary condition and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the total slip.

Predictions of the quantum landscape multiverse

The 2015 Planck data release has placed tight constraints on the class of inflationary models allowed. The current best fit region favors concave downwards inflationary potentials, since they produce a suppressed tensor to scalar index ratio r. Concave downward potentials have a negative curvature , therefore a tachyonic mass square that drives fluctuations. Furthermore, their use can become problematic if the field rolls in a part of the potential away from the extrema, since the semiclassical approximation of quantum cosmology, used for deriving the most probable wavefunction of the universe from the landscape and for addressing the quantum to classical transition, breaks down away from the steepest descent region. We here propose a way of dealing with such potentials by inverting the metric signature and solving for the wavefunction of the universe in the Euclidean sector. This method allows us to extend our theory of the origin of the universe from a quantum multiverse, to a more general class of concave inflationary potentials where a straightforward application of the semiclassical approximation fails. The work here completes the derivation of modifications to the Newtonian potential and to the inflationary potential, which originate from the quantum entanglement of our universe with all others in the quantum landscape multiverse, leading to predictions of observational signatures for both types of inflationary models, concave and convex potentials.

A Dissipation of Relative Entropy by Diffusion Flows

Given a probability measure, we consider the diffusion flows of probability measures associated with the partial differential equation (PDE) of Fokker–Planck. Our flows of the probability measures are defined as the solution of the Fokker–Planck equation for the same strictly convex potential, which means that the flows have the same equilibrium. Then, we shall investigate the time derivative for the relative entropy in the case where the object and the reference measures are moving according to the above diffusion flows, from which we can obtain a certain dissipation formula and also an integral representation of the relative entropy.

Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces

We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.

Constraints on single-field inflation

Many alternatives to canonical slow-roll inflation have been proposed over the years, one of the main motivations being to have a model, capable of generating observable values of non-Gaussianity. In this work, we (re-)explore the physical implications of a great majority of such models within a single, effective field theory framework (including novel models with large non-Gaussianity discussed for the first time below). The constraints we apply—both theoretical and experimental—are found to be rather robust, determined to a great extent by just three parameters: the coefficients of the quadratic EFT operators (δN)2 and δNδE, and the slow-roll parameter ε. This allows to significantly limit the majority of single-field alternatives to canonical slow-roll inflation. While the existing data still leaves some room for most of the considered models, the situation would change dramatically if the current upper limit on the tensor-to-scalar ratio decreased down to r < 10−2. Apart from inflationary models driven by plateau-like potentials, the single-field model that would have a chance of surviving this bound is the recently proposed slow-roll inflation with weakly-broken galileon symmetry. In contrast to canonical slow-roll inflation, the latter model can support r < 10−2 even if driven by a convex potential, as well as generate observable values for the amplitude of non-Gaussianity.

Inflation model constraints from data released in 2015

We provide the latest constraints on the power spectra of both scalar and tensor perturbations from the CMB data (including \textit{Planck}~2015, BICEP2 \& \textit{Keck Array} experiments) and the new BAO scales from SDSS-III BOSS observation. We find that the inflation model with a convex potential is not favored and both the inflation model with a monomial potential and the natural inflation model are marginally disfavored at around $95\%$ confidence level. But both the Brane inflation model and the Starobinsky inflation model fit the data quite well.

Long time behavior of telegraph processes under convex potentials

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift towards the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.

Stability of solutions to stochastic partial differential equations

We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models.

Aliasing and oblique dual pair designs for consistent sampling

In this paper we study some aspects of oblique duality between finite sequences of vectors F and G lying in finite dimensional subspaces W and V, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to F lying in V. We compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for F with norm restrictions; as an application, we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible, among oblique duals with norm restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces V and W has in oblique duality. We apply this analysis to compute those rigid rotations U for W such that the canonical oblique dual of U⋅F minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations U for W such that the canonical oblique dual pair associated to U⋅F minimize the aliasing. We point out that these two last problems are intrinsic to oblique duality, within the context of consistent sampling.

The incremental Cauchy Problem in elastoplasticity: General solution method and semi-analytic formulae for the pressurised hollow sphere

A general solution method to the Cauchy Problem (CP) formulated for incremental elastoplasticity is designed. The method extends previous works of the authors on the solution to Cauchy Problems for linear operators and convex nonlinear elasticity in small strain to the case of generalised standard materials defined by two convex potentials. The CP is transformed into the minimisation of an error between the solutions to two well-posed elastoplastic evolution problems. A one-parameter family of errors in the constitutive equation is derived based on Legendre–Fenchel residuals. The method is illustrated by the simple example of a pressurised thick-spherical reservoir made of elastic, linear strain-hardening plastic material. The identification of inner pressure and plasticity evolution has been carried-out using semi-analytical solutions to the elastoplastic behaviours to build the error functional.

Universality of the Stochastic Airy Operator

AbstractWe introduce a new method for studying universality of random matrices. Let Tn be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, Tn converges to the stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, these ideas lead to conjectured operator limits for the entire family of soft edge distributions.© 2015 Wiley Periodicals, Inc.

Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential

We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically small dissipation and convex potential. We prove that if the damping term behaves like Ktα as t→+∞, for some K>0 and α∈]0,1[, then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper of Cabot and Frankel (2012) [6].

A comparative analysis of two formulations for non linear hardening plasticity models: Application to shakedown analysis

The paper is mainly devoted to a comparative analysis of two plasticity models with a nonlinear hardening law, namely the Armstrong and Frederick model, and a recent modification proposed in literature in the framework of Generalized Standard Materials (GSM). We first provide a detailed mathematical analysis of the two models by appropriately resorting to the bipotential theory. This delivers for the GSM model a closed form expression of a bipotential. Moreover, it is demonstrated for the first time that the Armstrong and Frederick model does not admit a convex potential; this result confirms the necessary requirement of a non associated framework for this model. Then, for the modified model, making use of the above bipotential-based tools, we carry out a shakedown analysis of a thin walled tube under constant tension and alternating cyclic torsion. The accuracy of the obtained results is checked by comparing them to data obtained by numerical solving the corresponding shakedown bounds problems. Finally, the predictions of the two models are compared and illustrated their significant differences.

Periodic solutions of second-order systems with subquadratic convex potential

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: ( ¨ u(t) + m 2 w 2 u(t) +rF(t, u(t)) = 0 a.e. t2 (0, T), u(0) u(T) = ˙ u(0) ˙ u(T) = 0, where m > 0, the potential F(t, x) is convex in x and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem (Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989).

10108 液滴部のボルタポテンシャルプロファイルとウォーターマーク形状

To reveal the mechanism of watermark formation on a Cu film, the Volta potential in a droplet area on Cu was evaluated using the scanning Kelvin-probe method. The droplet area on Cu exhibited an upward convex potential profile, indicating that the electrochemical reactions were more active in the areas around the droplet than at the center of the droplet. On the other hand, Si exhibited a profile converse to that of Cu: electrochemical reactions were more active at the center of the droplet area than in the areas around it. These evaluations revealed that the pinning and ring-shaped watermark formation at the droplet area on a Cu film resulted from the said Volta potential profile.

Multiplicative Lidskii's inequalities and optimal perturbations of frames

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame F={fj}j∈In for Cd we compute those dual frames G of F that are optimal perturbations of the canonical dual frame for F under certain restrictions on the norms of the elements of G. On the other hand, we compute those V⋅F={Vfj}j∈In – for invertible operators V which are close to the identity – that are optimal perturbations of F. That is, we compute the optimal perturbations of F among frames G={gj}j∈In that have the same linear relations as F. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.

Finite Cycle Gibbs Measures on Permutations of $${{\mathbb Z}^d}$$ Z d

We consider Gibbs distributions on the set of permutations of $${\mathbb Z}^d$$ associated to the Hamiltonian $$H(\sigma ):=\sum _{x} {V}(\sigma (x)-x)$$ , where $$\sigma $$ is a permutation and $${V}:{\mathbb Z}^d\rightarrow {\mathbb R}$$ is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on $${V}$$ ensuring that for large enough temperature $$\alpha >0$$ there exists a unique infinite volume ergodic Gibbs measure $$\mu ^\alpha $$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $$\mu ^{\alpha }$$ as the unique invariant measure of a Markov process on the set of finite-cycle permutations that can be seen as a loss-network, a continuous-time birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define $$\tau _v$$ as the shift permutation $$\tau _v(x)=x+v$$ . In the Gaussian case $${V}=\Vert \cdot \Vert ^2$$ , we show that for each $$v\in {\mathbb Z}^d$$ , $$\mu ^\alpha _v$$ given by $$\mu ^\alpha _v(f)=\mu ^\alpha [f(\tau _v\cdot )]$$ is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $$\tau _v$$ boundary conditions. For a general potential $${V}$$ , we prove the existence of Gibbs measures $$\mu ^\alpha _v$$ when $$\alpha $$ is bigger than some $$v$$ -dependent value.

The Dirichlet–Neumann problem revisited after modelling a new class of non-smooth phenomena

Generalized Standard Materials are governed by maximal cyclically monotone operators and modelled by convex potentials. Gery de Saxce’s Implicit Standard Materials are modelled by biconvex bipotentials. Analyzing the intermediate class of n-monotone materials governed by maximal n-monotone operators and modelled by Fitzpatrick’s functions, we find out that n-monotonicity is a relevant criterion for the materials characterisation and classification. Additionally, the Fitzpatrick’s functions allow to describe the thermal or mechanical equilibrium equations of n-monotone materials by primal–dual two-fields variational principles. In doing so, we are led to Dirichlet–Neumann problems that we solve by Uzawa-type algorithms.

Feedback control for spacecraft reorientation under attitude constraints via convex potentials

A novel guidance algorithm is proposed for the attitude reorientation of a rigid body spacecraft in the presence of multiple types of attitude-constrained zones. In this direction, two types of attitude-constrained zones are first developed using unit quaternions, namely, the attitude-forbidden and -mandatory zones. The paper then utilizes a convex parameterization of forbidden and mandatory zones for constructing a strictly convex logarithmic barrier potential that is subsequently used for the synthesis of feedback attitude control laws while the inevitable unwinding phenomenon is given a simple and effective remedy. Model-independent and model-dependent control laws are then implemented by using the Lyapunov direct method and the modified integrator backstepping method. The paper concludes with a set of simulation results to evaluate the effectiveness and demonstrate the viability of the proposed methodology.

Spectral gap for spherically symmetric log-concave probability measures, and beyond

Let μ be a probability measure on R n ( n ≥ 2 ) with Lebesgue density proportional to e − V ( ‖ x ‖ ) , where V : R + → R is a smooth convex potential. We show that the associated spectral gap in L 2 ( μ ) lies between ( n − 1 ) / ∫ R n ‖ x ‖ 2 μ ( d x ) and n / ∫ R n ‖ x ‖ 2 μ ( d x ) , improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying operator in the energy, leading to weighted Poincaré inequalities. All our results are illustrated by some classical and less classical examples.