Potential Theory

A general theory for nonadiabatic electron-transferreactions at high temperatureinvolving three Marcus parabolic potential surfaces is presented. The theory can be applied to a threecomponent system with a donor, a bridging intermediateand an acceptor as well as to a system with charge separation from a photo-excited state followed by c"bargerecombinationto a third or groundstate. Using the nonperturbative stochastic LiouviUeapproach, analytical expressions are derived for the superexchange and the sequential electron-transferrate constants coveting all three conditions: the "nondegenerate", the ndegenerate"and the "quasi-degenerate" regimes.

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (= SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P (φ) 1 -time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's 2 -method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.

The main purpose of this paper is to establish the parabolic Harnack inequality for the transition semigroup associated with the time dependent Ginzburg-Landau type stochastic partial differential equation (= SPDE, in abbreviation). In view of quantum field theory, this dynamics is called a P (φ) 1 -time evolution. We prove the main result by adopting a stochastic approach which is different from Bakry-Emery's 2 -method. As an application of our result, we study some estimates on the transition probability for our dynamics. We also discuss the Varadhan type asymptotics.

The problem of boundedness of the Riesz potential in local Morreytype spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values The research of V.

We introduce a novel class of fractals, termed Irwin Fractals, generated by perturbing the classical quadratic map with bounded complex sequences. Specifically, we study the iterative system: z n+1 = z 2 n + c + g(n), n ≥ 0, z 0 = 0, where c ∈ C and g : N → C is a bounded sequence. We demonstrate the Principle of Sequence Sensitivity, a generalized butterfly effect stating that arbitrarily small changes in g(n) can induce exponentially divergent orbits and produce measurably distinct fractal sets: F (g) = {c ∈ C : |z n | remains bounded as n → ∞}. We quantify this sensitivity using Lyapunov exponents, Hausdorff distances, and escape-time metrics, providing both rigorous proofs and visualizations of the resulting fractal structures.

Let Ω be a simply-connected domain in the complex plane, let Ω and let K( z, ζ) denote the Bergman kernel function of Ω with respect to ζ. Also, let K ζ ∈ n (z , ζ) denote the n th degree polynomial approximation to K{ z , ζ ), given by the classical Bergman kernel method, and let n π denote the corresponding n th degree Bieberbach polynomial approximation to the conformal map f of Ω onto a disc. Finally, let B be any subdomain of Ω. In this paper we investigate the following two local errors , || || , || ) , (. ) , (. ||

In this paper, we devise a layer stripping algorithm for any dynamical isotropic elasticity system of equations in three-dimensional space. We give an explicit reconstruction of both Lamé moduli and the density, as well as their derivatives, at the boundary, from the full symbol of the dynamical Dirichlet to Neumann map.

For a nice Markov process such as Brownian motion on a bounded domain, we introduce a non-linear potential operator defined in terms of running suprema, and we prove a non-linear Riesz representation of a given function as the sum of a harmonic function and a non-linear potential. The proof involves a family of optimal stopping problems in analogy to the general construction of Bank and El Karoui [3], but here the analysis is carried out in terms of probabilistic potential theory.

We obtain a maximum principle, and "a priori" upper estimates for solutions of a class of non linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.

The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice which do not belong to the space.

We resolve two problems in Mathematical Physics. First, we prove that any L^∞ connection Γ on the tangent bundle of an arbitrary differentiable manifold with L^∞ Riemann curvature can be smoothed by coordinate transformation to optimal regularity, Γ∈ W^1,p, any p<∞, (i.e., one derivative smoother than the curvature). For Lorentzian metrics in GR this implies that shock wave solutions of the Einstein-Euler equations are non-singular—geodesic curves, locally inertial coordinates and the Newtonian limit exist. The proof is based on extending authors' existence theory for the RT-equations, a system of nonlinear elliptic partial differential equations, to the level of L^∞ connections, and for this we introduce the reduced RT-equations. Secondly, we prove that this existence theory suffices to extend Uhlenbeckcompactness from the case of connections on vector bundles over Riemannian manifolds (2019 Abel Prize and 2007 Steele Prize), to the case of connections on the tangent bundle ...

Extending results of Davies and of Keicher on p we show that the peripheral point spectrum of the generator of a positive bounded C0-semigroup of kernel operators on L p is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators.

A unified view is given to recent developments about a systematic method of constructing rational mappings as ergodic transformations with non-uniform invariant measures on the unit interval $I=[0,1]$ . All of the rational ergodic mappings of $I$ with explicit non-uniform invariant densities can be obtained by addition theorems of elliptic functions. It is shown here that the class of the rational ergodic map- pings $Iarrow I$ are essentially same as the permutable rational functions obtained by J. F. Ritt.

New nonlinear evolution equations are derived that generalize the system by Matsuno and a terrain-following Boussinesq system by Nachbin . The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters .

In this paper, we consider the propagation of water waves in a long-wave asymptotic regime, when the bottom topography is periodic on a short length scale. We perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced Boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated. We obtain explicit expressions for the coefficients of the resulting effective Korteweg-de Vries ͑KdV͒ equations. We show that it is possible to choose the free parameter of the reduced model so as to match the KdV limits of the full and reduced models. Hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem, which has a microstructure. We also discuss the impact of the rough bottom on the effective wave propagation. In particular, nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case.

Intermediate depth, Boussinesq-type modeling is used to generalize previously known results for surface water waves propagating over arbitrarly shaped topographies. The improved reduced wave model is obtained after studying how small changes in the linear dispersion relation (over a flat bottom) can become dramatically important in the presence of a highly-fluctuating topography. Numerical validation of the dispersive properties, regarding several possible truncations for the reduced models, are compared with the complete (non-truncated) linear potential theory model. Moreover, linear L 2 estimates are extended from the analysis of KdV-type models to include the improved Boussinesq systems in contrast with potential theory. Discrepancies observed among the different possible reduced models become even more important in the waveform inversion problem. The time reversal technique is used for recompressing a long fluctuating signal, representing a highly scattered wave that has propagated for very long distances. When properly backpropagated (through a numerical model), the scattered signal refocuses into a smooth profile representing the onset of the ocean's surface disturbance. Previous Boussinesq models underestimate the original disturbance's amplitude. The improved Boussinesq system agrees very well with the full potential theory predictions.

Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) X ∼ uo of a Banach lattice X and identify it as the order continuous part of the order continuous dual X ∼ n . The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel-Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.

An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class U (V, M) of the functions u : M → [0, π/4[ which are plurisubharmonic in M and such that u(p) = 0 for each p ∈ V . If U (V, M) admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E V,M on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampère models introduced by Lempert and Szöke in .

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{\mu\nu}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{\mu\nu}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equati...

Maryam Mirzakhani's Harvard PhD dissertation under Curt McMullen was widely acclaimed and contained already the seeds of what would become her first three major papers. All three of these results-a new proof of Witten's conjecture, a computation of the volume of the moduli space of curves, and an asymptotic count of the number of simple closed geodesics on a hyperbolic surface-were deep, beautiful, and unexpected. Subsequent papers revealed the geometry and dynamics of the moduli space of hyperbolic surfaces, culminating in the rigidity result achieved jointly with Alex Eskin. Her work

Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel.

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin δ \delta for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the H r ( Ω ) H^{r}(\Omega ) , 0 ≤ r ≤ 2 0\le r \...

The class of multianalytic functions are defined. For this class the notions of essential and nonessential isolated singularities and of exceptional values are introduced. It is then shown that a multianalytic function has at most one exceptional value at an essential isolated singularity.

A real valued function f defined on a convex K is an approximately convex function iff it satisfies A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function E vanishing on the vertices of a simplex. A set A in a normed space is an approximately convex set iff for all a, b ∈ A the distance of the midpoint (a + b)/2 to A is ≤ 1. The bounds on approximately convex functions are used to show that in R n with the Euclidean norm, for any approximately convex set A, any point z of the convex hull of A is at a distance of at most [log 2 (n -1)] + 1 + (n -1)/2 [log 2 (n-1)] from A. Examples are given to show this is the sharp bound. Bounds for general norms on R n are also given.

Starting from band structures of the constituent materials, the electronic band structure of the semiconducting alloys Ge.~Sn,_, and Si~Sn,_x are calculated by the empirical pseudopotential method using a corrected virtual crystal approximation (VCA) which incorporates compositional disorder as an effective potential. Various quantities, including the bowing parameter of the fundamental band gap, the energies of several optical gaps, and the crossover of the band gaps are predicted.

This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.

In 1906, Clark defined and studied the set of p-adic Liouville numbers and, in 1985, Schikhof also studied this set in his book Ultrametric Calculus. In this paper, we introduce the set of weak p-adic Liouville numbers, which is a set of transcendental p-adic numbers that contains the p-adic Liouville numbers, and we show some properties about these numbers. In particular, we shall prove an analogous result to a classic theorem of Maillet about Liouville numbers.

We investigate various boundary decay estimates for p(•)-harmonic functions. For domains in R n , n ≥ 2 satisfying the ball condition (C 1,1 -domains) we show the boundary Harnack inequality for p(•)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for p(•)harmonic functions in NTA domains in R n and provide lower-and upper-growth estimates and a doubling property for a p(•)-harmonic measure.

In this paper we give a new compactness criterion in the Lebesgue spaces L p ((0, T ) × Ω) and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.

The following problem is considered: a transversely isotropic elasttc half-space has at its boundary a circular domain, where tangential displacements are prescribed, while the rest of the boundary is stressfree. The normal stress vanishes all over the boundary of the half-space. The problem of finding the tangential stress distribution over the circular area is solved exactly and in closed form. The author has used his new method of continuation of solutions. Several examples are presented.

We provide a detailed analysis of atomic * -representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal * -dilations of defect free representations modelled on finite groups of rank 2.

We prove explicit upper and lower bounds for the L 1 -moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P m in ambient Riemannian spaces N n . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N n themselves.

In this paper, we introduce the concept of a w * -compatible mappings to obtain coupled coincidence point and coupled common fixed points of nonlinear contractive mappings in partially ordered metric spaces. Our results generalize, extend, unify, enrich and complement various comparable results in the existing literature.

We consider the application of semi-iterative methods (SIM) to the standard SOR method with complex relaxation parameter ω, under the following two assumptions: (i) the associated Jacobi matrix J is consistently ordered and weakly cyclic of index 2 (ii) the spectrum σ(J) of J belongs to a compact subset Σ of the complex plane C, which is symmetric with respect to the origin. By using results from potential theory, we determine the region of optimal choice of ω ∈ C for the combination SIM-SOR and settle, for a large class of compact sets Σ, the classical problem of characterizing completely all the cases for which the use of the SIM-SOR is advantageous over the sole use of SOR, in view of the hypothesis that σ(J) ⊂ Σ. In particular, our results show that, unless the outer boundary of Σ is an ellipse, SIM-SOR is always better and, furthermore, one of the best possible choices is an asymptotically optimal SIM applied to the Gauss-Seidel method. In addition, we derive the optimal complex SOR parameters for all ellipses which are symmetric with respect to the origin. Our work was motivated by recent results of M. Eiermann and R.S. Varga.

Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P n , n ≥ 0, be polynomials of respective degrees n = 0, 1, . . . that are orthonormal in G with respect to the area measure (the socalled Bergman polynomials). Let ϕ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P n 's, the case when ϕ has a singularity on L. To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P n 's for such lens domains is supported on a Jordan arc joining the two vertices of G. We determine this arc along with the distribution function.

In this paper, we prove two-sided pointwise estimates for the Green function of a parabolic operator with singular first order term on a C 1,1 -cylindrical domain . Basing on these estimates, we establish the equivalence of the parabolic measure, the adjoint parabolic measure and the surface measure on the lateral boundary of . These results are first studied by some authors for certain elliptic and less general parabolic operators.

In this paper, we study the boundary behaviour of positive solutions of certain parabolic operators with lower order terms in a half-space. Basing on these results, we characterize the Martin boundary and show that any positive solution has an integral representation. We thus generalize to a large class of parabolic operators some results proved by Kaufman, Wu, Mair, and Taylor for the heat equation.