Basilis Gidas

Symmetry and related properties via the maximum principle

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.

A priori bounds for positive solutions of nonlinear elliptic equations

We derive a priori bounds for positive solutions of the non-linear elliptic boundary value problem where Ω is a bounded domain in R n. Our proof is by contradiction and uses a scaling (“blow up”) argument reminiscent to that used in the theory of Minimal Surfaces. This procedure reduces the problem of a priori bounds to global results of Liouville type.

Global optimization via the Langevin equation

We provide a simple proof of the convergence of the cooling algorithms, i.e., the annealing algorithm and the Langevin equation. The convergence is established for temperature schedules which are very near to optimal ones. Our methods are based on Differential Equations techniques.

The Langevin Equation as a Global Minimization Algorithm

During the past two years a great deal of attention has been given to simulated annealing as a global minimization algorithm in combinatorial optimization problems [11], image processing problems [2], and other problems [9]. The first rigorous result concerning the convergence of the annealing algorithm was obtained in [2]. In [4], the annealing algorithm was treated as a special case of non-stationary Markov chains, and some optimal convergence estimates and an ergodic theorem were established. Optimal estimates for the annealing algorithm have recently been obtained by nice intuitive arguments in [7].

Soliton mass and surface tension in the (λ|Ø|4)2 quantum field model

The spectrum of the mass operator on the soliton sectors of the anisotropic (λ|ø|4)2—and the (λø4)2—quantum field models in the two phase region is analyzed. It is proven that, for small enough λ>0, the mass gapms(λ) on the soliton sector is positive, andms(λ)=0(λ−1). This involves estimatingms(λ) from below by a quantity τ(λ) analogous to the surface tension in the statistical mechanics of two dimensional, classical spin systems and then estimating τ(λ) by methods of Euclidean field theory. In principle, our methods apply to any two dimensional quantum field model with a spontaneously broken, internal symmetry group.

Motion detection and tracking using deformable templates

We propose an object-based framework for detection and tracking of moving objects in a sequence of images. Two key ingredients of the approach are appropriate object models based on Grenanders (see General Pattern Theory, 1993) deformable templates and spatio-temporal data models. Detection and tracking problems are formulated as optimization problems. Detection employs a Metropolis-type procedure starting from a random initial configuration, while tracking involves a deterministic nonlinear Gauss-Seidel algorithm. We present experimental results with real data on a highway traffic sequence.<<ETX>>

Stop Consonants Discrimination and Clustering Using Nonlinear Transformations and Wavelets

We present a new algorithmic procedure for the classification and clustering of the English six stop consonants /p, t, k, b, d, g/ on the basis of C V (Consonant-Vowel) or VC syllables. The central difficulties of the stop consonant problem lie in the nonstationary and nonlinear statistical structures of the acoustic signal in the burst and transition regions, Nonstationarity renders the application of Fourier transform methods questionable, and points to time-domain methods, Nonlinearities point to nonparametric statistical methods. We deal with the nonstationary and nonlinear effects by combining two powerful tools: (a) a wavelet transform of the acoustic signal and the associated “waveletogram”; (b) nonparametric transformations of the “waveletogram” and a nonlinear classification rule based on these transformations. An important byproduct of our method is the construction of certain two-dimensional clustering plots for stop consonants as well as for vowels. We know of no other method in the literature that yields such clustering plots for consonants. We present two experiments; in both experiments, the correct classification rates for stop consonants is over 93% when 50 to 60 ms of speech signal are used, and over 95% when 90 to 100 ms are used. For the classification of both the stop consonants and the vowels, the optimal trade-off occurs when 140 to 160 ms of speech signal are used, with correct classification rates for stop consonants over 95% and nearly 100% for vowels.

A Multilevel-Multiresolution Technique For Computer Vision Via Renormalization Group Ideas

A multilevel-multiresolution method for image processing tasks and computer vision in general, is presented. The method is based on a combination of probabilistic models, Monte Carlo type algorithms, and renormalization group ideas. The method is suitable for implementation on massively parallel computers. It also yields a new global optimization algorithm potentially applicable to any cost function, but especially efficient for problems which are governed by local spatial relations.

Simulations and Global Optimization

Let Zd be the usual d-dimensional cubic lattice. To each site i e Zd, we associate a random variable (“spin”) xi with values in a spin stable space X.

Deformations and spectral properties of merons

We consider a meron–antimeron pair located at a, b, ∈ R4, and show that the spectrum of its stability operator is not bounded below [in precise mathematical terms: The stability operator defined on C∞0(R4−{a,b}) has a self‐adjoint extension, possibly many, all of which are unbounded below]. We regularize a single meron located at the origin by replacing it inside a sphere of radius R0 and outside a sphere of radius R by ’’half instantons,’’ and show that for R≫R0 the regularized configuration continues to be unstable. For R0 finite and R=∞, we show that the spectrum of the stability operator continues to extend to −∞. We employ a singular transformation to embed R4 into S3×R where the meron pair takes a simple form and its stability operator L becomes L=−d2/dτ2+V, where τ∈R, and the potential V can be diagonalized in terms of the angular momenta, spin, and isospin of the vector field. The spectrum of L is continuous and extends from −2 to +∞. We determine the number of (generalized) zero eigenmodes of L, ...