amplitude effect of the free surface in elastic reverse time extrapolation we evaluate the physical validity of surface boundary conditions of the computational model in reverse time extrapolation of 3d three component 3 c elastic seismic data acquired at the earth s free surface by using mathematical derivations and numerical simulations reverse time extrapolation of elastic data assumes that only the incident p or s waves are reconstructed during extrapolation into the computational grid however superposition of the upgoing incident waves and the downgoing reflected and converted waves generated at the free surface also is recorded in data acquisition and is input into reverse time extrapolation in elastic reverse time extrapolation the computational model needs to have an absorbing top boundary when the 3d 3 c elastic data are inserted into the computational model during reverse time extrapolation the originally incident p or s wave is reconstructed in addition the free surface p to p reflected and p to s converted waves recombine to reconstruct a second in

reciprocity theorems for two way and one way wave vectors a comparison for acoustic applications in which there is a preferred direction of propagation the axial direction it is useful to arrange the two way and one way wave equations into the same matrix vector formalism in this formalism axial variations of the wave vector are expressed in terms of lateral variations of the same wave vector the two way wave vector contains the field quantities pressure and velocity axial component only whereas the one way wave vector contains waves propagating in the positive and negative axial direction by exploiting the equivalent form of the two way and one way matrix vector equations it appears to be possible to derive two way and one way reciprocity theorems that have an equivalent form but a different interpretation the main differences appear in the boundary integrals for unbounded media in the contrast terms and for the correlation type theorems in the handling of evanescent waves

wave propagation in randomly stratified media and the law of large numbers abstract the propagation of scalar harmonic waves through a randomly stratified slab is investigated the transition matrix method is applied for the analysis of the problem to obtain the effective parameters of the material of the slab the law of large numbers for non commuting products is applied the results are illustrated with the numerical example of the propagation of elastic waves in steel titanium laminate in an aluminium environment

extreme points in triangular uhf algebras we examine the strongly extreme point structure of the unit balls of triangular uhf algebras the semisimple triangular uhf algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal in contrast to this for the class of full nest algebras we prove a krein milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points results concerning the geometry of the unit ball have a long history both in banach space theory and in the theory of operator algebras this geometry can be affiliated with structural and algebraic properties moreover differences in geometric properties can prove useful in classification problems two fundamental results are the russo dye theorem and kadison s theorem on isometries more recently there has been interest in the unit balls of nonselfadjoint operator algebras especially nest algebras 1 2 3 4 15 16 this paper concerns the unit balls of triangular uhf algebras these and the larger class of triangular af algebras are nonselfadjoint analogues of the uhf and af c algebras studied by glimm and bratteli their theory has grown rapidly cf 8 9 18 19 20 we focus on the extreme point structure and our results have a different flavor than those for nest algebras specifically we study the strongly extreme points those boundary points whose stable character with respect to approximations makes them behave well under direct limits triangular uhf algebras are direct limits of full upper triangular matrix algebras unit balls embed into unit balls in the direct limit scheme and some types of embeddings respect the extreme point structure while others do not this leads to structural differences in the limit algebras the geometric structures of the unit balls of different triangular uhf algebras can be very dissimilar the convex hull of the strongly extreme points even without closure always contains the unit ball of the diagonal theorem 7 shows that the two coincide if and only if the algebra is semisimple this is a characterization of a purely geometric property in terms of a purely algebraic one in contrast to received by the editors january 11 1996 and in revised form march 28 1996 1991 mathematics subject classification primary 47d25 46k50 46b20

flux normalized wavefield decomposition and migration of seismic data separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix in conventional pressurenormalized wavefield decomposition the resulting one way wave equations contain an interaction term which depends on the reflectivity function applying directional wavefield decomposition using flux normalized eigenvalue decomposition and disregarding interaction between up and downgoing wavefields these interaction terms were absent by also applying a correction term for transmission loss the result was an improved estimate of the up and downgoing wavefields in the wave equation angle transform a crosscorrelation function in local offset coordinates was fourier transformed to produce an estimate of reflectivity as a function of slowness or angle we normalized this wave equation angle transform with an estimate of the planewave reflection coefficient the flux normalized oneway wave propagation scheme was applied to imaging and to the normalized wave equation angle transform on synthetic and field data this proved the effectiveness of the new methods

estimates for p harmonic functions vanishing on a flat we study p harmonic functions in a domain crn near an m dimensional plane an m flat m where 0 m n 1 in particular let u be a positive p harmonic function with n p vanishing on a porti

concentration inequalities for random matrix products abstract suppose x k k z is a sequence of bounded independent random matrices with common dimension d d and common expectation e x k x under these general assumptions the normalized random matrix product z n i d 1 n x n i d 1 n x n 1 i d 1 n x 1 converges to e x as n normalized random matrix products of this form arise naturally in stochastic iterative algorithms such as oja s algorithm for streaming principal component analysis here we derive nonasymptotic concentration inequalities for such random matrix products in particular we show that the spectral norm error satisfies z n e x o log n 2 log d n with probability exceeding 1 this rate is sharp in n d and up to logarithmic factors the proof relies on two key points of theory the matrix bernstein inequality concerning the concentration of sums of random matrices and baranyai s theorem from combinatorial mathematics concentration bounds for general classes of random matrix products are hard to come by in the literature and we hope that our result will inspire further work in this direction

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

infinity laplacian equation with strong absorptions abstract we study regularity properties of solutions to reaction diffusion equations ruled by the infinity laplacian operator we focus our analysis in models presenting plateaus i e regions where a non negative solution vanishes identically we obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets in particular we show that the n hausdorff measure of the plateaus boundary is finite for a universal number 0

weak fubini property and infinity harmonic functions in riemannian and sub riemannian manifolds we examine the relationship between infinity harmonic functions absolutely minimizing lipschitz extensions strong absolutely minimizing lipschitz extensions and absolutely gradient minimizing extensions in carnot carath eodory spaces using the weak fubini property we show that absolutely minimizing lipschitz extensions are infinity harmonic in any sub riemannian manifold

c1 regularity for infinity harmonic functions in two dimensions a continuous function open image in new window is said to be infinity harmonic if it satisfies the pde

reciprocity properties of one way propagators acoustic reciprocity is a fundamental property of the wave equation for the total acoustic wavefield rayleigh 1878 in its most elementary form the reciprocity principle states that an acoustic response remains the same when the source and receiver are interchanged in terms of green s functions this reciprocity principle reads g r2 r1 g r1 r2 with r x y z for more general expressions and their seismic applications see fokkema and van den berg 1993

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

migration of primaries and multiples using an imaging condition for amplitude normalized separated wavefields abstractmigration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution a joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders in recent works primaries and multiples have therefore been imaged separately using the upgoing and downgoing pressure wavefields obtained by decomposing dual sensor streamer data the matches between the corresponding depth images are still not found to be sufficiently accurate so new and more appropriate imaging conditions need to be found we reviewed the classical imaging approach used in one way wave equation migration with the aim of extending it for simultaneous migration of primaries and all orders of multiples based on rayleigh s reciprocity theorem and well known theoretical developments we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical velocity wavefield t

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

power spectral density of the heterogeneous fracture compliance from scattered elastic wavefields using the scattered elastic wavefield a method to derive the power spectral density psd of the heterogeneous compliance distribution along the plane of a single fracture is formulated the method involves estimation of the stress field at the fracture depth from the scattered wavefield followed by least squares optimization of the psd of the stress field we consider a 2d geometry and incident plane waves the derivations are made in the frequency wavenumber domain to derive the relationship between the scattered wavefield and the psd of the heterogeneous compliance perturbation theory is used in the forward modeling the scattering response of the heterogeneous compliance is estimated assuming a stationary random process numerical tests of the proposed method of psd estimation offer important insights the estimated psd of the compliance given by the standard deviation and the correlation length lc assuming a gaussian distribution as a statistical measure of the heterogeneous fracture compliance is robust against fracture depth uncertainties furthermore the sparse spatial sampling of the wavefield is not problematic as long as the data contain those wavenumber components which represent the slope of the psd of the stress field however when the sampling interval is too large compared to the correlation length of the heterogeneous compliance distribution it is difficult to estimate accurately the psd of the compliance due to spatial aliasing and small amplitude variation within the available wavenumber for a given spatial sampling the use of higher frequencies results in a stronger amplitude variation in the scattered wavefield this leads to better estimates of the psd of the compliance via a least squares optimization in the presence of white noise a change in the psd of the stress field at the fracture depth significantly affects the psd estimates in this case a data driven amplitude correction improves the estimates of and lc

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

matrix propagator method for layered porous media analytical expressions and stability criteria we fully expand the thomson haskell propagator matrix method to obtain closed form analytical expressions for the reflectivity and transmittivity of a horizontally stratified poroelastic biot type medium that is bounded from above and below by fluid half spaces stable and unstable regions of the propagator matrix method are determined by means of comparison with the inherently stable reflectivity method it was found that the stability is mainly determined by the imaginary part of the vertical wavenumber of the biot slow wave

the russo dye theorem in nest subalgebras of factors abstract let n be a nest in a factor r in this paper we introduce the concept of the admissible nest for r and prove that the weak operator topology closure of the convex hull of all the unitary elements in the nest subalgebra of the factor is the unit ball if and only if n is an admissible nest in r

inversion versus migration a new perspective to an old discussion seismic imaging techniques can be subdivided into inversion and migration the object functions for inversion and migration are respectively the medium contrast parameters and reflectivity in this paper the relationship between inversion and migration is approached by analyzing the underlying representations the forward models it appears that the two way representation which underlies inversion as well as the one way representation which underlies migration can both be expressed in terms of a volume integral over the appropriate object function in their linearized form these representations account for primaries only in this case the one way representation in terms of reflectivity is the most accurate of the two which implies that proper migration is more accurate than linearized inversion internal multiples can be taken into account by the nonlinear representations as an alternative however the generalized primary representation is introduced in its explicit form this one way

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

spectral statistics for the difference of two wishart matrices in this work we consider the weighted difference of two independent complex wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite dimension scenario using two distinct approaches the first derivation involves the use of unitary group integral while the second one relies on applying the derivative principle the latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with monte carlo simulation results analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived additionally we provide a large dimension asymptotic result for the spectral density using the stieltjes transform approach for algebraic random matrices finally we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean

a non selfadjoint russo dye theorem one of the well known results in the theory of c algebras is the russo dye theorem 19 given a c algebra r the closed convex hull of the unitary elements in r equals the closed unit ball of r this result was later refined by gardner and reached its final form by kadison and pedersen today it is known that every operator in a c algebra r whose norm is less than 1 is the average of unitaries from a the russo dye theorem initiated the theory of unitary rank in selfadjoint operator algebras if r is an operator algebra the unitary rank of an element a e r is defined as the smallest number for which there is a convex combination of unitaries from r of length u a and equaling a if no such decomposition exists in particular if liaii 1 we define u a oo the literature on unitary rank is vast the earliest result is due to murray and yon neumann who proved that any selfadjoint operator of norm i or less is the mean of two unitary operators 12 p 239 1937 the first systematic study was given by r kadison and g pedersen 8 in 1984 previous work in the field included contributions by popa 15 robertson 17 gardner 6 and others in 1986 c olsen and g pedersen 14 characterized all elements in a factor von neumarm algebra with finite unitary rank in the general case of a c algebra a characterization was obtained by rordam in his important paper 18 for more details and further information on the theory of unitary rank we refer to the excellent articles of u haagerup 7 and m rordam 18 in the first section of the present paper we prove a russo dye type theorem for infinite multiplicity nest algebras the techniques employed in the proof of our result are different from that of gardner and kadison pedersen to our knowledge this is the first result of this type for non selfadjoint operator algebras and clearly initiates the unitary rank theory for such algebras

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

treatment of singularity in the method of boundary integral equations for 2 5d electromagnetic modeling abstractfast and accurate simulation of responses of electromagnetic logging while drilling tools in 2d and 3d formations is important for further development of the proactive geosteering service which is currently based on the 1d layered model inversion in search of an optimal method of forward simulation we have developed boundary integral equations in a 2d formation model with planar boundaries a peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy and hence the computation time increases as the transmitter approaches the boundary to retain efficiency in the case of transmitter boundary proximity we use the following modification of the method the main term of the calculated electromagnetic field namely the solution for the two layered model corresponding to the closest boundary is calculated by the well known explicit formulas whereas the anomalous field is found from the system of integral equations the modified method is implemented and tes

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

3d finite difference frequency domain modeling of controlled source electromagnetic data direct solution and optimization for high accuracy three dimensional modeling of marine controlled source electromagnetic csem data is vital to improve the understanding of electromagnetic em responses collected in increasingly complex geologic settings a modeling tool for simulating 3d marine csem surveys based on a finite difference discretization of the helmholtz equation for the electric fields has been developed optimizations for csem simulations include the use of a frequency domain technique a staggering scheme that reduces inaccuracies especially for horizontal electric dipole sources located near the seafloor and a new interpolation technique that provides highly accurate em field values for receivers located in the immediate vicinity of the seafloor source singularities are eliminated through a secondary field approach in which the primary fields are computed analytically for a homogeneous or a 1d layered background the secondary fields are computed using the finite difference technique exploiting recent advances in computer technology and algorithmic developments the system of finite difference equations is solved using the mumps direct matrix solver in combination with the other optimizations this allows accurate em field computations for moderately sized models on small computer clusters the explicit availability of matrix factorizations is advantageous for multisource modeling and makes the algorithm well suited for future use within an inversion scheme comparisons of simulated data for 1 1d models to data generated using a 1d reflectivity technique and 2 3d models to published 3d data demonstrate the accuracy and benefits of the approach

shot profile true amplitude crosscorrelation imaging condition the u d imaging condition for shot profile migration can be used to estimate the angle dependent reflection coefficient but is difficult to implement numerically because of the spectral division involved most techniques for stabilizing the division require a damping factor which might be difficult to estimate and which also introduces bias into the final result a stable result can be achieved by approximating the imaging condition with a crosscorrelation of the up and downgoing wavefields at zero time lag but this will lead to incorrect amplitudeversus angle ava behavior of the estimated reflection coefficient we use a simple model for wave propagation of primary reflections in the wavenumber frequency domain and invert the model with respect to the reflection coefficient by using the properties of wavefield extrapolators it can then be shown that the reflection coefficients can be estimated by crosscorrelation of the upgoing wavefield and a downgoing wavefield where the initial wavefield is the inverse of the wavefield generated by a point source the new imaging condition gives the correct ava behavior for horizontal reflectors for dipping reflectors it is shown that a postmigration correction factor can be used to recover the correct angle behavior of the reflection coefficient the new imaging condition is numerically stable does not involve damping factors is simple to implement numerically and is a simple modification of the classical crosscorrelation imaging condition numerical examples confirm the correct ava behavior of the new imaging condition

unified double and single sided homogeneous green s function representations in wave theory the homogeneous green s function consists of the impulse response to a point source minus its time reversal it can be represented by a closed boundary integral in many practical situations the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only the inherent approximations are acceptable as long as the effects of multiple scattering are negligible however in case of strongly inhomogeneous media the effects of multiple scattering can be severe we derive double and single sided homogeneous green s function representations the single sided representation applies to situations where the medium can be accessed from one side only it correctly handles multiple scattering it employs a focusing function instead of the backward propagating green s function in the classical double sided representation when reflection measurements are available at the accessible boundary of the medium the focusing function can be retrieved from these measurements throughout the paper we use a unified notation which applies to acoustic quantum mechanical electromagnetic and elastodynamic waves we foresee many interesting applications of the unified single sided homogeneous green s function representation in holographic imaging and inverse scattering time reversed wave field propagation and interferometric green s function retrieval

accounting for free surface multiples in marchenko imaging imagine placing a receiver at any location in the earth and recording the response at that location to sources on the surface in such a world we could place receivers around our reservoir to better image the reservoir and understand its properties realistically this is not a feasible approach for understanding the subsurface we have developed an alternative and realizable approach to obtain the response of a buried virtual receiver for sources at the surface our method is capable of retrieving the green s function for a virtual point in the subsurface to the acquisition surface in our case a physical receiver is not required at the subsurface point instead we require the reflection measurements for sources and receivers at the surface of the earth and a macromodel of the velocity no small scale details of the model are necessary we can interpret the retrieved green s function as the response to sources at the surface for a virtual receiver in the subsurface we obtain this green s function by solving the marchenko equation an integral equation pertinent to inverse scattering problems our derivation of the marchenko equation for the green s function retrieval takes into account the free surface reflections present in the reflection response previous work considered a response without free surface multiples we decompose the marchenko equation into up and downgoing fields and solve for these fields iteratively the retrieved green s function not only includes primaries and internal multiples as do previous methods but it also includes freesurface multiples we use these up and downgoing fields to obtain a 2d image of our area of interest in this case below a synclinal structure

reaction diffusion equations for the infinity laplacian abstract we derive sharp regularity for viscosity solutions of an inhomogeneous infinity laplace equation across the free boundary when the right hand side does not change sign and satisfies a certain growth condition we prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function then the free boundary is a porous set and hence has zero lebesgue measure additionally we derive a liouville type theorem when near the origin the right hand side grows not faster than third degree homogeneous function we show that if a non negative viscosity solution vanishes at a point then it has to vanish everywhere

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

regularity for degenerate evolution equations with strong absorption abstract in this manuscript we study geometric regularity estimates for degenerate parabolic equations of p laplacian type 2 p under a strong absorption condition p u u t 0 u q in t 0 t where 0 q 1 this model is interesting because it yields the formation of dead core sets i e regions where non negative solutions vanish identically we shall prove sharp and improved parabolic c regularity estimates along the set f 0 u t u 0 t the free boundary where p p 1 q 1 1 p 1 some weak geometric and measure theoretical properties as non degeneracy positive density porosity and finite speed of propagation are proved as an application we prove a liouville type result for entire solutions a specific analysis for blow up type solutions will be done as well the results are new even for dead core problems driven by the heat operator

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

convex combinations of unitary operators in von neumann algebras the unitary rank of an element t in a von neumann algebra u is defined as the smallest number u t for which there is a convex combination of unitaries from u of length u t and equalling t we determine the unitary rank for every element t in the closed unit ball of u in terms of the index i t of t and the distance t from t to the group of invertible elements in u if i t 0 then u t 2 if i t 0 then u t n when n 1 2 1 t 1 n and u t when t 1 we also determine precisely which asymmetric convex decompositions of t can be realized we show that only an approximate version of these results holds in a general c algebra

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

exact evaluations of some meijer g functions and probability of all eigenvalues real for the product of two gaussian matrices we provide a proof to a recent conjecture by forrester 2014 j phys a math theor 47 065202 regarding the algebraic and arithmetic structure of meijer g functions which appear in the expression for probability of all eigenvalues real for the product of two real gaussian matrices in the process we come across several interesting identities involving meijer g functions

universal moduli of continuity for solutions to fully nonlinear elliptic equations this paper provides universal optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations f x d2u f x based on the weakest and borderline integrability properties of the source function f in different scenarios the primary result established in this work is a sharp log lipschitz estimate on u based on the ln norm of f which corresponds to optimal regularity bounds for the critical threshold case optimal c1 regularity estimates are also delivered when f in l n varepsilon the limiting upper borderline case f in l infty also has transcendental importance to elliptic regularity theory and its applications in this paper we show under the convexity assumption on f that u in c 1 rm log lip provided f has bounded mean oscillation once more such an estimate is optimal for the lower borderline integrability condition allowed by the theory we establish interior a priori estimates on the c 0 frac n 2 varepsilon n varepsilon norm of u based on the ln e norm of f where is the escauriaza universal constant the exponent frac n 2 varepsilon n varepsilon is optimal when the source function f lies in lq n q n e we also obtain the exact improved sharp holder exponent of continuity

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

mixed phase deconvolution we describe a new algorithm for mixed phase deconvolution it is valid only for pulses whose z transform has no zeros on the unit circle that is the amplitude spectrum cannot be zero for any frequency using the z transform of a discrete time signal and assuming that the signal has zeros inside the unit circle the inverse of its minimum delay component may be estimated by solving the extended yule walker eyw system of equations with the lag of the autocorrelation function acf on diagonal of the coefficient matrix this property of the solution of eyw equations is exploited to derive mixed phase inverse filters and their corresponding mixed phase pulses for different values of a suite of inverse filters is generated using the same acf to choose the best decomposition and its corresponding mixed phase inverse filter we have used the value of which gives the maximum value of the lp norm of the filtered signal the optimal value of does not seem to be very sensitive to the choice of norm

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

review of elastic and electromagnetic wave propagation in horizontally layered media the objective of this paper is to provide a unified treatment of elastic and electromagnetic em wave propagation in horizontally layered media for which the parameters in the partial differential equations are piece wise continuous functions of only one spatial variable by applying a combination of fourier laplace and bessel transforms to the partial differential equations describing the elastic or em wave propagation i obtain a system of 2n linear ordinary differential equations the 2n 2n coefficient matrix is partitioned into 4n n submatrices by a proper choice of variables the diagonal submatrices are zero and the off diagonal submatrices are symmetric all the results in the paper are derived from the symmetry properties of this general equation in the appendices it is shown that three dimensional elastic waves cylindrical p sv waves acoustic waves and electromagnetic waves in isotropic layered media can all be represented by an equation with the same properties the symmetry properties of

low frequency layer induced anisotropy abstractit is often useful to compare velocities estimated from seismic experiments with those measured by well logs at much smaller scales and higher frequencies in the presence of fine layering seismic velocities and anisotropy are scale and frequency dependent and upscaling of well logs is necessary to allow comparison when upscaling well log data measured at sonic frequencies we assumed a layered medium with layer thickness given by logging step standard upscaling gives the exact solution for effective properties of a layered medium assuming that all constituents of the medium are linearly elastic and there are no anelastic energy losses this method is static meaning that the solution is obtained in the zero frequency limit we expand upscaling to low frequencies and propose to use a seismic wavelet estimate for weighted averaging of effective properties by expanding the logarithm of the propagator matrix computed from the stack of horizontal transversely isotropic layers with a vertical symm

central limit theorem for products of random matrices using the semigroup product formula of p chernoff a central limit theorem is derived for products of random matrices applications are presented for representations of solutions to linear systems of stochastic differential equations and to the corresponding partial differential evolution equations included is a discussion of stochastic semigroups and a stochastic version of the lie trotter product formula

product formulas for solutions of initial value partial differential equations i abstract families of operators which approximate semi groups or evolution systems generated by partial differential operators are constructed product formulas are used to recover these semi groups or evolution systems through product integrals conditions on generators are provided under which its semi group or evolution system can be approximated in this way by families of specific types of operators

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations