Green Kernel Research Articles

Interior Capacities of Condensers in Locally Compact Spaces

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a finite collection of arbitrary sets with sign + 1 or − 1 prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in \(\mathbb R^n\) and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying Fuglede’s condition of consistency between the strong and vague (= weak*) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We describe those solutions and the corresponding equilibrium constants, analyze their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in \(\mathbb R^n\), which is important in applications.

Estimées des noyaux de Green et de la chaleur sur les espaces symétriques

On majore les noyaux de Green et de la chaleur au dehors de la diagonale pour des operateurs de type laplacien sur les espaces symetriques. We provide an upper bound for the off-diagonal entries of the Green and heat kernel for Laplace-type operators on symmetric spaces.

The resolvent kernel for PCF self-similar fractals

For the Laplacian Δ \Delta defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions and also with Neumann boundary conditions. That is, we construct a symmetric function G ( λ ) G^{(\lambda )} which solves ( λ I − Δ ) − 1 f ( x ) = ∫ G ( λ ) ( x , y ) f ( y ) d μ ( y ) (\lambda \mathbb {I} - \Delta )^{-1} f(x) = \int G^{(\lambda )}(x,y) f(y) \, d\mu (y) . The method is similar to Kigami’s construction of the Green kernel and G ( λ ) G^{(\lambda )} is expressed as a sum of scaled and “translated” copies of a certain function ψ ( λ ) \psi ^{(\lambda )} which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket S G 3 SG_3 .

A Singularity-Free Boundary Equation Method for Wave Scattering

Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting Maxwell's system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green's functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method. It can be generalized to 3D problems and to other classes of linear problems, including acoustics and elasticity.

Knowledge-Based Green's Kernel for Support Vector Regression

This paper presents a novel prior knowledge-based Green's kernel for support vector regression (SVR). After reviewing the correspondence between support vector kernels used in support vector machines (SVMs) and regularization operators used in regularization networks and the use of Green's function of their corresponding regularization operators to construct support vector kernels, a mathematical framework is presented to obtain the domain knowledge about magnitude of the Fourier transform of the function to be predicted and design a prior knowledge-based Green's kernel that exhibits optimal regularization properties by using the concept of matched filters. The matched filter behavior of the proposed kernel function makes it suitable for signals corrupted with noise that includes many real world systems. We conduct several experiments mostly using benchmark datasets to compare the performance of our proposed technique with the results already published in literature for other existing support vector kernel over a variety of settings including different noise levels, noise models, loss functions, and SVM variations. Experimental results indicate that knowledge-based Green's kernel could be seen as a good choice among the other candidate kernel functions.

Generalized inverses of symmetric [formula omitted]-matrices

In this work we carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M -matrices. The key idea in our approach is to identify any symmetric M -matrix with a positive semi-definite Schrödinger operator on a connected network whose conductances are given by the off-diagonal elements of the M -matrix. Moreover, the potential of the operator is determined by the positive eigenvector of the M -matrix. We prove that any generalized inverse can be obtained throughout a Green kernel plus some projection operators related to the positive eigenfunction. Moreover, we use the discrete Potential Theory associated with any positive semi-definite Schrödinger operator to get an explicit expression for any generalized inverse, in terms of equilibrium measures. Finally, we particularize the obtained result to the cases of tridiagonal matrices and circulant matrices.

A FRACTIONAL POISSON EQUATION: EXISTENCE, REGULARITY AND APPROXIMATIONS OF THE SOLUTION

We consider a stochastic boundary value elliptic problem on a bounded domain D ⊂ ℝk, driven by a fractional Brownian field with Hurst parameter H = (H1,…,Hk) ∈ [½, 1[k. First, we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove the existence and uniqueness of solution along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.

Existence and Comparison of Green Kernels for Some Elliptic Operators on (0, ∞)

In this paper we give necessary and sufficient conditions for the existence and the comparability of Green kernels for second order elliptic operators in the one dimensional case.

Boundary Element Calculation of Eddy Currents in Cylindrical Structures Containing Cracks

Electromagnetic fields due to excitation coils in boreholes containing cracks have been calculated using a boundary element method formulated in terms of a scalar decomposition of the field. By using a Green's kernel that ensures continuity of the tangential electric and magnetic field at the cylindrical surface, it is only necessary to render the flaw region in a discrete form. The effect of an ideal crack, one with zero opening but acting as a perfect barrier to eddy currents, is here represented by a current dipole layer orientated normal to the crack surface. An integral equation determines the source density on the crack and a numerical solution is found using boundary elements. The coil impedance change due to the crack is then computed using an expression based on a reciprocity principle.

Thin-Layer Drying of Green Peas and Selection of a Suitable Thin-Layer Drying Model

The thin-layer drying of three varieties of green peas was carried out in hot air-drying chamber using an automatic weighing system at five temperatures (55–75°C) and air velocity of 100 m/min. The green peas were blanched and sulphited before drying. The variety Pb-87 dried at 60°C was judged to be best for quality on the basis of sensory evaluation and rehydration ratio. The Thomson model was found to represent thin-layer drying kinetics within 99.9% accuracy. The effective diffusivity was determined to be 3.95 × 10−10 to 6.23 × 10−10 m2/s in the temperature range of 55 to 75°C. The activation energy for diffusion was calculated to be 22.48 kJ/mol. The variation in shrinkage exhibited a linear relationship with moisture content of the product during drying. The Dincer number at drying air temperature 60°C and drying air velocity 100 m/min was determined to be 2,838,087. The difference between temperatures of drying air and that of green pea kernels was found to decrease with drying time for all the drying temperatures taken for investigation.

Correlation asymptotics for non‐translation invariant lattice spin systems

AbstractWe obtain asymptotic expressions for the Green kernels of certain non‐translation invariant transition matrices using methods of semiclassical and microlocal analysis. Combined with a result by Bach and Møller this yields asymptotic formulas for the truncated two‐point correlation functions of certain non‐translation invariant lattice models of real‐valued spins. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Uniform asymptotic approximations of Green's functions in a long rod

AbstractAsymptotic approximations for Green's function for the operator −Δ in a long rod are derived. These approximations are uniformly valid over the whole domain including the end regions of the rod. Connections are established between the asymptotic approximations in a long rod and the asymptotics in thin domains. Overview of asymptotic approximations of Green's kernels in a domain with a small hole and domains with singularly perturbed smooth or conical boundaries is also given. Copyright © 2008 John Wiley & Sons, Ltd.

Mapping QTLs for milling yield and grain characteristics in a tropical japonica long grain cross

Percent milling yield is an economically important trait of commercial rice because it largely determines the price that farmers receive for their crop. Analyzing 22 trait variables including milling yield, grain dimensions, chemistry and appearance, we identified 43 quantitative trait loci (QTLs) in a long grain japonica by long grain japonica cross. We report one QTL explaining 20% of the variation in brown rice recovery; two QTLs explaining 14% and 13% of the variation in milled rice recovery; and one QTL explaining 14% of the variation in head rice (HR) recovery. QTLs for the proportion of pre-broken brown rice kernels, seed density, amylose content, and kernel whiteness and chalkiness were found in the same region as the HR QTL. QTLs explaining up to 54% of the variation in grain shape measurements were identified and mapped to areas independent from those identified for milling yield. Analyses of grain appearance traits identified two QTLs for chalk in brown rice and one in head rice, and a QTL explaining up to 33% of the variance in green kernel area. Our results confirm previous findings on the multigenically complex nature of milling yield.

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Ψ H DO operators introduced by Beals–Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Ψ H DOs induced by the analytic extension of the usual trace to non-integer order Ψ H DOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding Ψ H DO. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order Ψ H DOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators, logarithmic metric estimates for Green kernels of hypoelliptic operators, and the extension of the Dixmier trace to the whole algebra of integer order Ψ H DOs. In the third part, we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact Laplacian on contact manifolds. In the fourth part, we present two applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower-dimensional volumes in pseudohermitian geometry, e.g., we can give sense to the area of any 3-dimensional CR manifold endowed with a pseudohermitian structure. On the way we obtain a spectral interpretation of the Einstein–Hilbert action in pseudohermitian geometry.

Disparity estimation through Green’s functions of matching equations

Binocular disparities arise from positional differences of scene features projected in the two retinae, and constitute the primary sensory cue for stereo vision. Here we introduce a new computational model for disparity estimation, based on the Green's function of an image matching equation. When filtering a Gabor-function-modulated signal, the considered Green's function yields a similarly modulated but shifted version of the original signal. Since a Gabor function models the receptive field of a cortical simple cell, the Green's kernel thus allows the simulation of relative shifts between the cell's left and right binocular inputs. A measure of the local degree of matching of such shifted inputs can then be introduced which affords disparity estimation in a similar manner to the energy model of the complex cortical cells. We have therefore effectively reformulated, in physiologically plausible terms, an image matching approach to disparity estimation. Our experiments show that the Green's function method allows the detection of disparities both from random-dot and real-world stereograms.

Kato class measures of symmetric Markov processes under heat kernel estimates

We establish the coincidence of two classes of Kato class measures in the framework of symmetric Markov processes admitting upper and lower estimates of heat kernel under mild conditions. One class of Kato class measures is defined by way of the heat kernel, another is defined in terms of the Green kernel depending on some exponents related to the heat kernel estimates. We also prove that pth integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of Kato class if p is greater than a constant related to the estimate under the same conditions. These are complete extensions of some results for the Brownian motion on Euclidean space by Aizenman and Simon. Our result can be applicable to many examples, for instance, symmetric (relativistic) stable processes, jump processes on d-sets, Brownian motions on Riemannian manifolds, diffusions on fractals and so on.

Optimal stopping of Hunt and Lévy processes

Infinite horizon (perpetual) optimal stopping problems for Hunt processes on R are studied via the representation theory of excessive functions. In particular, we focus on problems with one-sided structure, that is, there exists a point x* such that the stopping region is of the form . The main result states that if it is possible to find a Radon measure such that the excessive function induced by this measure via the spectral representation has some very intuitive properties then the constructed excessive function coincides with the value function of the problem. Corresponding results for two-sided problems are also indicated. Specializing to Lévy processes, we obtain, by applying the Wiener–Hopf factorization, a general representation of the value function in terms of the maximum of the Lévy process. To illustrate the results, an explicit expression for the Green kernel of Brownian motion with exponential jumps is computed and some optimal stopping problems for Poisson process with positive exponential jumps and negative drift are solved.

A Low Dimensional Fluid Motion Estimator

In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a vorticity free component. The objective is to provide a low-dimensional parametric representation of optical flows by depicting them as deformations generated by a reduced number of vortex and source particles. Both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measures. The resulting so called irrotational and solenoidal fields consist of linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on synthetic examples and real world sequences.

Complex counterpart of Chern-Simons-Witten theory and holomorphic linking

In this paper we are begining to explore the complex counterpart of the Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking number for complex curves embedded in a Calabi-Yau 3-fold using the formal path integral that leads to a rigorous mathematical expression. We give an analytic and geometric interpretation of our holomorphic linking following the parallel with the real case. We show in particular that the Green kernel that appears in the explicit integral for the Gauss linking number is replaced by the Bochner-Martinelli kernel. We also find canonical expressions of the holomorphic linking using the Grothendieck-Serre duality in local cohomology, the latter admits a generalization for an arbitrary field.

Shelf Life Extension of Roasted Peanuts by Surface Lipid Removal

Removal of lipids on the surface or from a layer adjacent to the surface of a roasted peanut kernel was investigated as a means to extend the oxidative stability or shelf life of roasted peanuts. Georgia green runner-type peanut kernels were roasted at 178C in an electric oven for 15 min. Roasted peanuts were subjected to lipid extraction in hexane by mechanical shaking and sonication at room temperature for various durations. Peanut oil was recovered from hexane using a Soxtec lipid extraction system. The surface conditions of the peanuts before and after lipid removal were examined using a florescence microscope. Results showed that surface lipids of the roasted peanuts used in this study existed at a level of approximately 0.56% of the total peanut mass. Florescence micrographs revealed that the surface of sonicated peanut kernels was clean and almost free of oil stains, as opposed to that of freshly roasted peanut kernels, where a considerable amount of lipids remained on the surface of roasted peanuts. Statistical analysis indicated that lipid removal could significantly improve the oxidative stability of roasted peanuts, and it was possible to extend the shelf life of roasted peanuts by removing partial lipids from the roasted peanuts. The results also showed that power ultrasound could remove the lipids more effectively than mechanical shaking.