Unique Minimum Research Articles

Positive Lyapunov Exponent and Minimality for the Continuous 1-d Quasi-Periodic Schrödinger Equation with Two Basic Frequencies

We consider the time-independent quasi-periodic Schrodinger equation $$-u^{\prime\prime}(x) + K^{2}V (x, \theta + \omega x)u(x) = Eu(x), \,\,\,\, x \in {\mathbb{R}}$$ , with a potential function \(V : {\mathbb{T}}^{2} \rightarrow {\mathbb{R}}\) of class C2 with a unique non-degenerate global minimum, large coupling constants K2 and energies E in the bottom of the spectrum of the associated Schrodinger operator. We obtain estimates on the Lyapunov exponents and the Lebesgue measure of the spectrum, as well as localization results. Moreover, we show that the projective flow on \({\mathbb{T}}^{2} \times {\mathbb{P}}^{1}\) induced by the Schrodinger equation often is minimal.

Harnack’s principle for quasiminimizers

We study Harnack type properties of quasiminimizers of the $p\mspace{1mu}$ -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincare inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the $p\mspace{1mu}$ -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincare inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27

A first principles study of the optical properties of B xC y single wall nanotubes

The optical properties of small radius (<1 nm) single wall carbon nanotubes (SWCNTs) alloyed with boron were examined using relaxed C–C bond length ab initio calculations in the long wavelength limit. The magnitude of the static dielectric constant essentially depends on the B concentration as well as the direction of polarization. The maximum value of the absorption coefficient is shown to strongly depend on the concentration of B in a non-linear way with a minimum at a critical concentration of 0.40 for both the parallel polarization and the un-polarized cases and of 0.29 for perpendicular polarization of the electromagnetic field. The peak of the loss function in parallel polarization and unpolarized cases shifts to a lower frequency with increasing concentration up to 50% but then shifts to a higher frequency. The non-linear fits to the plasma resonance frequency variation with B concentration indicate the existence of a unique minimum. All these factors may shed light on the nature of collective excitations in B-alloyed SWCNTs.

The problem of minimizing locally a 𝐶² functional around non-critical points is well-posed

In this paper, we prove the following general result: Let X X be a real Hilbert space and J : X → R J:X\to \textbf {R} a C 1 C^1 functional, with locally Lipschitzian derivative. Then, for each x 0 ∈ X x_0\in X with J ′ ( x 0 ) ≠ 0 J’(x_0)\neq 0 , there exists δ &gt; 0 \delta &gt;0 such that, for every r ∈ ] 0 , δ [ r\in ]0,\delta [ , the restriction of J J to the sphere { x ∈ X : ‖ x − x 0 ‖ = r } \{x\in X : \|x-x_0\|=r\} has a unique global minimum toward which every minimizing sequence strongly converges.

On a pulsating Brownian motor and its characterization

As a model of Brownian motor we consider the jump diffusion motion of a particle in the presence of an asymmetric periodic potential with a unique minimum and subject to half-period space shifts at the instants of occurrence of two Poisson processes. The relevant quantities, i.e., probability current, effective driving force, stall force, power and efficiency of the motor are explicitly calculated as averages of certain functions of the random variable representing the particle position.

Robust Array Beamforming With Sidelobe Control Using Support Vector Machines

Robust beamforming is a challenging task in a number of applications (radar, sonar, wireless communications, etc.) due to strict restrictions on the number of available snapshots, signal mismatches, or calibration errors. We present a new approach to adaptive beamforming that provides increased robustness against the mismatch problem as well as additional control over the sidelobe level. We generalize the conventional linearly constrained minimum variance cost function by including a regularization term that penalizes differences between the actual and the target (ideal) array responses. By using the so-called epsi-insensitive loss function for the penalty term, the final cost function adopts the form of a support vector machine (SVM) for regression. In particular, the resulting cost function is convex with a unique global minimum that has traditionally been found using quadratic programming (QP) techniques. To alleviate the computational cost of conventional QP techniques, we use an iterative reweighted least-squares (IRWLS) procedure, which also converges to the SVM solution. Computer simulations demonstrate an improved performance of the proposed SVM-based beamformer, in comparison with other recently proposed robust beamforming techniques

“Bottom of the well” semi-classical trace invariants

Let $\hat H$ be an $\h$-admissible pseudodifferential operator whose principal symbol, $H$, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues of $\hat H$ corresponding to the ``bottom of the well determine the Birkhoff normal form of $H$ at the minimum. We treat both the resonant and the non-resonant cases.

Some inverse spectral results for semi-classical Schrödinger operators

We show that the Birkhoff normal form of a classical Hamiltonian $H(x,\xi) = \norm{\xi}^2+V(x)$ at a non-degenerate minimum $x_0$ of the potential determines the Taylor series of the potential at $x_0$, provided the eigenvalues of the Hessian are linearly independent over $\bbQ$ and $V$ satisfies a symmetry condition near $x_0$. As a consequence, if $x_0$ is the unique global minimum of $V$, the low-lying eigenvalues of the semi-classical Schr\odinger operator, $-\h^2\Delta + V(x)$, determine the Taylor series of the potential at $x_0$.

Dichotomy in a scaling limit under Wiener measure with density

In general, if the large deviation principle holds for a sequence of probability measures and its rate functional admits a unique minimizer, then the measures asymptotically concentrate in its neighborhood so that the law of large numbers follows. This paper discusses the situation that the rate functional has two distinct minimizers, for a simple model described by the pinned Wiener measures with certain densities involving a scaling. We study their asymptotic behavior and determine to which minimizers they converge based on a more precise investigation than the large deviation's level.

Potential field method to navigate several mobile robots

Navigation of mobile robots remains one of the most challenging functions to carry out. Potential Field Method (PFM) is rapidly gaining popularity in navigation and obstacle avoidance applications for mobile robots because of its elegance. Here a modified potential field method for robots navigation has been described. The developed potential field function takes care of both obstacles and targets. The final aim of the robots is to reach some pre-defined targets. The new potential function can configure a free space, which is free from any local minima irrespective of number of repulsive nodes (obstacles) in the configured space. There is a unique global minimum for an attractive node (target) whose region of attraction extends over the whole free space. Simulation results show that the proposed potential field method is suitable for navigation of several mobile robots in complex and unknown environments.

The reconstruction of groundwater parameters from head data in an unconfined aquifer

The estimation of groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system. In an unconfined aquifer, the problem is more complex because the governing equation for the well heads, the Boussinesq equation, is non-linear. We consider here a new method that allows for the simultaneous computation of the unconfined groundwater parameters as the unique minimum of a convex functional.

Existence and concentration of ground states of coupled nonlinear Schrödinger equations

In this paper we consider the existence and concentration of ground states of coupled nonlinear Schrödinger equations with trap potentials. When the interaction between two states is repulsive, we prove the existence of ground states. Then concentration phenomenon of these ground states is studied as the small perturbed parameter (Planck constant) approaches zero. Roughly speaking, we prove that components of the ground states concentrate at the unique global minimum points of their potentials. Moreover, we prove the existence of ground states when the interaction is attractive.

An iterative algorithm for approximating convex minimization problem

We suggest and analyze an iterative algorithm without the assumption of any type of commutativity on an infinite family of nonexpansive mappings. We show that the proposed iterative algorithm converges to the unique minimizer of some quadratic function over the common fixed point sets of an infinite family of nonexpansive mappings. Our result extend and improve many results announced by many authors.

Navigation technique to control several mobile robots

This paper describes potential field navigation method optimized by simulated annealing technique. In the current analysis a new potential field function is developed so as to take care of unknown obstacles and targets during navigation. The final aims of the robots are to reach targets. Local minimum obtained due to potential field being avoided by simulated annealing technique. After utilizing simulated annealing technique for optimizing potential field method, it is observed that, there is a unique global minimum for an attractive node i.e., target. Afterwards potential field method being optimized by simulated annealing technique is hybridized with neuro-fuzzy technique to improve the navigation of robots in a highly cluttered environment. Simulation results show that the proposed potential-field-simulated-annealing-neuro-fuzzy technique is most efficient against potential-field and potential-field-simulated-annealing techniques.

Statistical analysis of a dynamical multifragmentation path

A microcanonical multifragmentation model (MMM) is used for investigating whether equilibration really occurs in the dynamical evolution of two heavy ion collisions simulated via a stochastic mean field approach. The standard deviation function between the dynamically obtained freeze-out fragment distributions corresponding to the reaction $^{129}\mathrm{Xe}+^{119}\mathrm{Sn}$ at 32 MeV/nucleon and the MMM ones corresponding to a wide range of mass, excitation energy, freeze-out volume, and nuclear level density cutoff parameter shows a unique minimum. A distinct statistically equilibrated stage is identified in the dynamical evolution of the system.

Low-temperature resistivity minimum in dilute Mn-dopingNa0.7Co1−xMnxO2: A Kondo-like oxide with strong disorder characteristics

We present the results of low-temperature transport behavior and magnetic properties for dilute Mn-doped ${\mathrm{Na}}_{0.7}{\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}{\mathrm{O}}_{2}$ system with $x=0$, 0.03, 0.07, 0.1. The results show that the properties are very sensitive to Mn doping at low temperatures. A unique resistivity minimum is found with a characteristic of metallic-semiconductor transition $(&lt;50\phantom{\rule{0.3em}{0ex}}\mathrm{K})$ for all the doped samples. Combining with the resistivity data in applied field, it can be understood within the theory of Kondo scattering induced by magnetic impurities and electron-electron (e-e) interaction enhanced by disorder. The best fitting was made in the framework of the two scattering mechanisms in a wide temperature range of 2--$100\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The results prove that the layer ${\mathrm{Na}}_{0.7}{\mathrm{Co}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}{\mathrm{O}}_{2}$, a kind of typical Kondo-like oxide like the dilute convention alloys, is a strong disorder system in which exists enhanced e-e interaction, which reflects a typical characteristic of strong correlation systems.

Criteria for global minimum of sum of squares in nonlinear regression

Demidenko [2000. Is this the least squares estimate? Biometrika 87, 437–452] has established the relationship between the curvature of nonlinear regression and the local convexity of a sum of squares: the Hessian matrix is positive definite if the sum of squares is less than the minimum squared radius of the full curvature. In this paper, we continue developing the criteria for the global minimum of the sum of squares in nonlinear regression. In particular, the concept of the local unimodality is introduced; a function is called locally unimodal on a level set if it has a unique local minimum in each component of that level set. We show that the level of the local unimodality of the sum of squares is equal to the minimum squared radius of the intrinsic curvature of the nonlinear regression function. A new class of unidirected nonlinear regression models is introduced with an interpretation in terms of differential geometry. The criteria are illustrated by several popular nonlinear regression models.

The mean‐field approximation in quantum electrodynamics: The no‐photon case

AbstractWe study the mean‐field approximation of quantum electrodynamics (QED) by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal ordering or choice of bare electron/positron subspaces. Neglecting photons, we properly define this Hamiltonian in a finite box [−L/2; L/2)3, with periodic boundary conditions and an ultraviolet cutoff λ. We then study the limit of the ground state (i.e., the vacuum) energy and of the minimizers as L goes to infinity, in the Hartree‐Fock approximation.In the case with no external field, we prove that the energy per volume converges and obtain in the limit a translation‐invariant projector describing the free Hartree‐Fock vacuum. We also define the energy per unit volume of translation‐invariant states and prove that the free vacuum is the unique minimizer of this energy.In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as L goes to infinity. We obtain in the limit the so‐called Bogoliubov‐Dirac‐Fock functional. The Hartree‐Fock (polarized) vacuum is a Hilbert‐Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov‐Dirac‐Fock energy. © 2006 Wiley Periodicals, Inc.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CONSERVATION LAWS WITH PERIODIC FORCING

We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.

Monte Carlo Estimation of a Joint Density Using Malliavin Calculus, and Application to American Options

We use the Malliavin integration by parts formula in order to provide a family of representations of the joint density (which does not involve Dirac measures) of (X_?, X ? + ?), where X is a d-dimensional Markov diffusion (d ? 1), ? > 0 and ? > 0. Following Bouchard et al. (2004), the different representations are determined by a pair of localizing functions. We discuss the problem of variance reduction within the family of separable localizing functions: We characterize a pair of exponential functions as the unique integrated-variance minimizer among this class of separable localizing functions. We test our method on the d-dimensional Brownian motion and provide an application to the problem of American options valuation by the quantization tree method introduced by Bally et al. (2002).