Heat Kernel

With the increasing amount of 3D data and the ability of capture devices to produce low-cost multimedia data, the capability to select relevant information has become an interesting research field. In 3D objects, the aim is to detect a few salient structures which can be used, instead of the whole object, for applications like object registration, retrieval, and mesh simplification. In this paper, we present an interest points detector for 3D objects based on Harris operator, which has been used with good results in computer vision applications. We propose an adaptive technique to determine the neighborhood of a vertex, over which the Harris response on that vertex is calculated. Our method is robust to several transformations, which can be seen in the high repeatability values obtained using the SHREC feature detection and description benchmark. In addition, we show that Harris 3D outperforms the results obtained by recent effective techniques such as Heat Kernel Signatures.

This survey summarizes briefly results obtained recently in the Casimir energy studies devoted to the following subjects: i) account of the material characteristics of the media in calculations of the vacuum energy (for example, Casimir energy of a dilute dielectric ball); ii) application of the spectral geometry methods for investigating the vacuum energy of quantized fields with the goal to

We prove Gaussian upper bounds for kernels associated with non -symmetric, non -autonomous second order parabolic operators of divergence form subject to various boundary conditions. The growth of the kernel in time is determined by the boundary conditions and the geometric properties of the domain. The theory gives a unified treatment for Dirichlet, Neumann and Robin boundary conditions, and the existence of a Gaussian type bound is essentially reduced to verifying some properties of the Hilbert space in the weak formulation of the problem.

We develop a manifestly covariant technique for heat kernel calculation in the presence of arbitrary background fields in a curved space.The four lowest-order coefficients of Schwinger-De Witt asymptotic expansion are explicitly computed.We calculate also the heat kernel asymptotic expansion up to the terms of third order in the rapidly varying background fields (curvatures). This approximate series is summed up and the covariant nonlocal expressions for heat kernel,ζ-function and one-loop effective action are obtained.Other related problems are discussed. † The permanent address after

A short informal overview about recent progress in the calculation of the effective action in quantum gravity is given. I describe briefly the standard heat kernel approach to the calculation of the effective action and discuss the applicability of the Schwinger - De Witt asymptotic expansion in the case of strong background fields. I propose a new ansatz for the heat kernel that generalizes the Schwinger - De Witt one and is always valid. Then I discuss the general structure of the asymptotic expansion and put forward some approximate explicitly covariant methods for calculating the heat kernel, namely, the high-energy approximation as well as the low-energy one. In both cases the explicit formulae for the heat kernel are given.

The heat-kernel expansion and ζ-regularization techniques for quantum field theory and extended objects on curved space-times are reviewed. In particular, ultrastatic space-times with spatial section consisting in manifold with constant curvature are discussed in detail. Several mathematical results, relevant to physical applications are presented, including exact solutions of the heat-kernel equation, a simple exposition of hyperbolic geometry and an elementary derivation of the Selberg trace formula. With regards to the physical applications, the vacuum energy for scalar fields, the one-loop renormalization of a self-interacting scalar field theory on a hyperbolic space-time, with a discussion on the topological symmetry breaking, the finite temperature effects and the Bose-Einstein condensation, are considered. Some attempts to generalize the results to extended objects are also presented, including some remarks on path integral quantization, asymptotic properties of extended objects and a novel representation for the one-loop (super)string free energy.

This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of Psi_{H}DO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providing us with sharp regularity estimates for hypoelliptic operators on Heisenberg manifolds; (ii) Criterions on the principal symbol of $P$ to invert the heat operator $P+\partial_{t}$ and to derive the small time heat kernel asymptotics for $P$; (iii) Weyl asymptotics for hypoelliptic operators which can be reformulated geometrically for the main geometric operators on CR and contact manifolds, that is, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers, as well as the contact Laplacian. For dealing with complex powers of hypoelliptic operators we cannot make use of the standard approach of Seeley, so we rely on a new approach based on the pseudodifferential approach representation of the heat kernel. This is especially suitable for dealing with positive hypoelliptic operators. We will deal with more general operator in a forthcoming paper using another new approach.

We review the status of covariant methods in quantum field theory and quantum gravity, in particular, some recent progress in the calculation of the effective action via the heat kernel method. We study the heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold without boundary. We develop a manifestly covariant method for computation of the heat kernel asymptotic expansion as well as new algebraic methods for calculation of the heat kernel for covariantly constant background, in particular, on homogeneous bundles over symmetric spaces, which enables one to compute the low-energy non-perturbative effective action.

We construct an explicit solution of the Cauchy initial value problem for certain diffusiontype equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding non-homogeneous equation is also found.

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion co...

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.

In this series of lectures a method is developed to compute one-loop shifts to classical masses of kinks, multi-component kinks, and self-dual vortices. Canonical quantization is used to show that the mass shift induced by one-loop quantum fluctuations is the trace of the square root of the differential operator governing these fluctuations. Standard mathematical techniques are used to deal with

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods.

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of ${\overline{\partial}}$ -operators on the Teichmüller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space ${\mathcal{M}_{g,n}}$ in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

We consider the stochastic heat equation with multiplicative noise $u_{t}=\frac{1}{2}\Delta u+u\dot{W}$ in ℝ+×ℝd , whose solution is interpreted in the mild sense. The noise $\dot{W}$ is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.

The spectral analysis of the electromagnetic field on the background of a infinitely thin flat plasma layer is carried out. This model is loosely imitating a single base plane from graphite and it is of interest for theoretical studies of fullerenes. By making use of the Hertz potentials the solutions to Maxwell equations with the appropriate matching conditions at the plasma layer are derived and on this basis the spectrum of electromagnetic oscillations is determined. The model is naturally split into the TE-sector and TM-sector. Both the sectors have positive continuous spectra, but the TM-modes have in addition a bound state, namely, the surface plasmon. This analysis relies on the consideration of the scattering problem in the TE-and TM-sectors. The spectral zeta function and integrated heat kernel are constructed for different branches of the spectrum in an explicit form. As a preliminary, the rigorous procedure of integration over the continuous spectra is formulated by introducing the spectral density in terms of the scattering phase shifts. The asymptotic expansion of the integrated heat kernel at small values of the evolution parameter is derived. By making use of the technique of integral equations, developed earlier by the same authors, the local heat kernel (Green's function or fundamental solution) is constructed also. As a by-product, a new method is demonstrated for deriving the fundamental solution to the heat conduction equation (or to the Schrödinger equation) on an infinite line with the δ-like source. In particular, for the heat conduction equation on an infinite line with the δ-source a nontrivial counterpart is found, namely, a spectral problem with point interaction, that possesses the same integrated heat kernel while the local heat kernels (fundamental solutions) in these spectral problems are different.

In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph in a vector space. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat kernel of the graph is found by exponentiating the Laplacian eigensystem over time. In this paper, we explore how graphs can be characterized in a geometric manner using embeddings into a vector space obtained from the heat kernel. We explore two different embedding strategies. The first of these is a direct method in which the matrix of embedding coordinates is obtained by performing a Young-Householder decomposition on the heat kernel. The second method is indirect and involves performing a low-distortion embedding by applying multidimensional scaling to the geodesic distances between nodes. We show how the required geodesic distances can be computed using parametrix expansion of the heat kernel. Once the nodes of the graph are embedded using one of the two alternative methods, we can characterize them in a geometric manner using the distribution of the node coordinates. We investigate several alternative methods of characterization, including spatial moments for the embedded points, the Laplacian spectrum for the Euclidean distance matrix and scalar curvatures computed from the difference in geodesic and Euclidean distances. We experiment with the resulting algorithms on the COIL database.

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.

This paper establishes a heat semigroup version of Bernstein's theorem, applicable to any unimodular Lie group. The result has an intrinsic geometric content, involving estimates for the norms of the heat kernels for small time and large time. The theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. The geometric content is further underlined by showing that, in a certain sence, the theorem is best-possible.

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on (or near) a low dimensional manifold M d embedded in R p , we prove the relation between VDM and the connection-Laplacian operator for vector fields over the manifold.

We use theζ-function regularization and an integral representation of the complex power of a pseudo differential operator to give an unambiguous definition of the determinant of the Dirac operator. We bring this definition to a workable form by making use of an asymmetric Wigner representation. The formalism is amenable to the study of several problems of which we consider in detail two, the inverse mass expansion and the gradient expansion. We obtain explicit closed-form expressions for the corresponding Seeley–DeWitt coefficients to all orders. The determinant is shown to be vector gauge invariant and to posses the correct axial and scale anomalies. The main virtue of our approach is that it is conceptually simple and systematic, and can be extended naturally to more general problems (bosonic operators, gravitational fields, etc). In particular, it avoids defining the real and imaginary parts of the effective action separately. In addition, it does not reduce the problem to a bosonic one in order to apply heat kernel expansions, nor performs further analytical rotations of the fields in order to make the Dirac operator Hermitian. We illustrate the flexibility of the method by studying some interesting cases.

We study coercive inequalities on finite dimensional metric spaces with probability measures which do not have volume doubling property. This class of inequalities includes Poincar\'e and Log-Sobolev inequality. Our main result is proof of Log-Sobolev inequality on Heisenberg group equipped with either heat kernel measure or "gaussian" density build from optimal control distance. As intermediate results we prove so called U-bounds.

In a paper by Krylov [6], a parabolic Littlewood-Paley inequality and its application to an L p -estimate of the gradient of the heat kernel are proved. These estimates are crucial tools in the development of a theory of parabolic stochastic partial differential equations constructed by Krylov [7]. We generalize these inequalities so that they can be applied to stochastic integrodifferential equations. ISBN 978-951-22-9441-1 (print) ISBN 978-951-22-9442-8 (PDF)

The demands of gauge field theories, quantum theory in external gauge and gravitational backgrounds, and quantum gravity have spurred intense study of the asymptotic heat kernel expansion for nonminimal differential operators on curved manifolds in the presence of a gauge field. There are several computer programs for calculating the coefficients in an expansion of the heat kernel. When applied to the nonminimal operator these programs appeared to be able to compute no more than the second order coefficient. We report the first complete calculation of the fourth coefficient for nonminimal operator. The calculation has been carried out on PC with the help of a program written in C. Some other new results have been also obtained by this program.

We establish a representation formula for the transition probability density of a di usion perturbed by a vector ÿeld, which takes a form of Cameron-Martin's formula for pinned di usions. As an application, by carefully estimating the mixed moments of a Gaussian process, we deduce explicit, strong lower and upper estimates for the transition probability function of Brownian motion with drift of linear growth.

A proper understanding of boundary-value problems is essential in the attempt of developing a quantum theory of gravity and of the birth of the universe. The present paper reviews these topics in light of recent developments in spectral geometry, i.e.

Let L be a non-negative self adjoint operator acting on L 2 (X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernels p t (x, y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted L p -norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L 2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.

We introduce a new method for obtaining heat kernel on-diagonal lower bounds on noncompact Lie groups and on infinite discrete groups. By using this method, we are able to recover the previously known results for unimodular amenable Lie groups as well as for certain classes of discrete groups including the polycyclic groups, and to give them a geometric interpretation. We also obtain new results for some discrete groups which admit the structure of a semi-direct product or of a wreath product. These include the two-generators groups of affine transformations of the real line x → x + 1, x → λx with λ algebraic, as well as lamplighter groups with nilpotent base.

We establish several versions of Hardy's theorem for the Fourier transform on the Heisenberg group. Let \(\hat f (\lambda)\) be the Fourier transform of a function f on \(H^n\) and assume \(\hat f (\lambda)^\ast \hat f (\lambda) \leq c \hat p_{2b} (\lambda)\) where \(p_s\) is the heat kernel associated to the sublaplacian. We show that if \(|f(z,t)| \leq c p_a (z,t)\) then \(f=0\) whenever \(a < b\) . When \(a \geq b\) we replace the condition on f by \(|f^\lambda (z)| \leq c p_a^\lambda (z)\) where \(f^\lambda (z)\) is the Fourier transform of f in the t-variable. Under suitable assumptions on the ‘spherical harmonic coefficients’ of \(\hat f (\lambda)\) we prove: (i) \(f^\lambda (z)=c(\lambda) p^\lambda_a (z)\) when a=b; (ii) when a > b there are infinitely many linearly independent functions f satisfying both conditions on \(f^\lambda\) and \(\hat f (\lambda)\) .

A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the Lie group of isometries. The heat kernel diagonal is obtained in form of an integral over the isotropy subgroup.

We calculate and discuss the one-loop corrections to the photon sector of QED interacting to a background gravitational field. At high energies the fermion field can be taken as massless and the quantum terms can be obtained by integrating conformal anomaly. We present a covariant local expression for the corresponding effective action, similar to the one obtained earlier for the gravitational sector. At the moderate energies the quantum terms can be obtained through the heat-kernel method. In this way we derive the exact one-loop β-function for the electric charge in the momentum subtraction scheme and explore both massless and large-mass limits. The relation between the two approaches is shown and the difference discussed in view of the possible applications to cosmology and astrophysics.

The application of Fourier series to the heat conduction on a circular ring is considered. The temperature distribution function u(θ;t)and so u(θ;t)=u(θ+2πn;t), where n is an integer, positive or negative.  It is periodic as after integer number of revolutions, we return to the same position on the circle.  u(θ;t) satisfies diffusion equation. This is given by the convolution of the initial distribution at time t=0,  viz.,u(x;0) and heat kernel  G(x-x^';t). This kernel is also called propagator or Green’s function in different context. This heat kernel is identified as Jacobi ϑ_3 (πx:2πt),  which is periodic in x and t. The physical significance of the kernel is explained clearly.  The extraction of heat kernel as the impulse response for an initial ‘delta heating’ on a very small part of the ring and how it can be measured are clearly explained.

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceḂ s p,q in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesḂ s p,q , with 1 ≤ p, q < ∞ and s ∈ R.

In this article, we study the equation @ @t u(x,t) = c 2 k B u(x,t) with the initial condition u(x,0) = f(x) for x 2 R + . The operator k is named the Bessel ultra-hyperbolic operator iterated k times and is defined by given generalized function and c is a positive constant. We obtain the solution of such equation which is related to the spectrum and the kernel which is so called the Bessel ultra-hyperbolic heat kernel. Moreover, such the Bessel ultra-hyperbolic heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

This work consists essentially of two parts. The first part is an analysis of the one-loop effective action using the zeta-function approach. This gives a simple expression for the effective action in terms of the background field propagator. The next-of-kin to the zetafunction, the heat kernel, is given in terms of B. Dewitt's proper time expansion (also known as P. B. Gilkey's theorem). It is calculated in the second part for fermions interacting with an external electromagnetic field to first nonvanishing order in the variations of the gauge field.

Edge detection plays a fundamental role on image processing. The detected edges describe an object contour that greatly improves the pattern recognition process. Many edge detectors have been proposed. Most of them apply smooth filters to minimize the noise and the image derivative or gradient to enhance the edges. However, smooth filters produce ramp edges with the same gradient magnitude as those produced by noise. This work presents an algorithm that enhances the gradient correspondent to ramp edges without amplifying the noisy ones. Moreover, an efficient method for edge detection without set a threshold value is proposed. The experimental results show that the proposed algorithm enhances the gradient of ramp edges, improving the gradient magnitude without shifting the edge location. Further, we are testing the implementation of the proposed algorithm in hardware for real time vision applications.

In this paper we present a new method in order to transfer boundedness results for operators associated with Hermite functions to boundedness results for operators associated with Laguerre functions. The technique relies on an exact point-wise identity relating the heat kernels of both systems. The method that we present here has the novelty that can be used backwards, that is, boundedness results for Laguerre systems can be also transfered to boundedness results for Hermite systems. We apply our method in order to get new properties of some operators in the Laguerre setting. Among others, we mention the description of Riesz transforms as principal value operators. As an application of the reversibility of the method we characterize the class of Banach spaces B for which the Riesz transforms (in the Laguerre setting) are bounded from L p B into itself. It is shown that this class coincides with the UMD class.

We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.

We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R d , whose solution is interpreted in the mild sense. The noiseẆ is fractional in time (with Hurst index H ≥ 1/2), and colored in space (with spatial covariance kernel f ). When H > 1/2, the equation generalizes the Itô-sense equation for H = 1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α < d, then the sufficient condition for the existence of the solution is d ≤ 2 + α (if H > 1/2), respectively d < 2 + α (if H = 1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the k-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of k independent d-dimensional Brownian motions.