closed subsets in duals of commutative table algebras a lower bound on the number of elements outside a closed subset of a c basis of a commutative c algebra with dual a table algebra is derived as is an equivalent condition for when the lower bound is met as corollaries lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative imprimitive association scheme and for a given normal subgroup n of a finite group g on the number of irreducible characters of g whose kernels do not contain n and on the number of conjugacy classes of g not contained in n also found are equivalent conditions for when these lower bounds are met

orthogonalization process by recurrence relations an orthogonalization process is proposed applicable to spaces which are realizations of abstract hilbert space it is simpler than the gram schmidt process a recurrence relation which orthogonalizes a physical space is proposed and it is shown that the generalized langevin equation is contained therein this process serves as a basis for understanding the nature of the dynamic many body formalism

on wreath products of c algebras table algebras and association schemes abstract in this paper we will first present characterizations of wreath products of c algebras and show that the dual of the wreath product of c algebras is isomorphic to the wreath product of the duals of those c algebras in the reverse order then we prove the krull schmidt type theorems for the wreath products of table algebras as well as for the wreath products of association schemes

reality based algebras with a two dimensional representation abstract it is shown that a two dimensional irreducible representation of a reality based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti automorphism on the distinguished basis this generalizes known results for algebras of dimensions 5 6 7 and 8 explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

noncommutative standard table algebras with at most one nontrivial multiplicity abstract standard table algebras that have an irreducible character of degree two and with all other irreducible characters of multiplicity one are classified to exact isomorphism

the generalized terwilliger algebra and its finite dimensional modules when d 2 terwilliger j algebraic combin 1 1992 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations are often called the triple product relations they determine much of the structure of t though not all of it in general to illuminate the role these relations play the current author introduced j algebra233 2000 213 252 a generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c the dimensions of c and c are equal let d 1 denote this common dimension intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s triple product relations t is infinite dimensional and noncommutative in general in this paper we study t and its finite dimensional modules when d 2 and t has no extra vanishing intersection numbers or dual intersection numbers in this case we show t is c algebra isomorphic to m3 c a where m3 c denotes the c algebra consisting of all 3 by 3 matrices with entries in c and a denotes the associative c algebra with identity generated by the symbols e and f subject to the relations e2 e and f2 f we find a basis for a and we determine the center of a we classify the finite dimensional indecomposable a modules up to isomorphism there are four such a modules in every odd dimension and in every even dimension these modules are parameterized by a single complex number we also classify the finite dimensional irreducible a modules up to isomorphism using our results concerning a we find a basis for t we describe the center of t and we classify both the finite dimensional indecomposable and the finite dimensional irreducible t modules up to isomorphism

full waveform inversion and the truncated newton method full waveform inversion fwi is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield this method consists in minimizing the distance between

model based inversion of amplitude variations with offset data using a genetic algorithm i cast the inversion of amplitude variation with offset avo data into the framework of bayesian statistics under such a framework the model parameters and the physics of the forward problem are used to generate synthetic data these synthetic data are then matched with the observed data to obtain an a posteriori probability density ppd function in the model space the genetic algorithm ga uses a directed random search technique to estimate the shape of the ppd unlike the classical inversion methods ga does not depend upon the choice of an initial model and is well suited for the avo inversion for the single layer avo inversion where the amplitudes from a single reflection event are inverted ga estimates the normal incidence reflection coefficient r0 and the contrast of the poisson s ratio to reasonable accuracy even when the signal to noise ratio is poor comparisons of single layer amplitude inversion using synthetic data demonstrate that ga inversion obtains more accurate results tha

the subconstituent algebra of a distance regular graph thin modules with endpoint one abstract we consider a distance regular graph with diameter d 3 intersection numbers ai bi ci and eigenvalues 0 1 d let x denote the vertex set of and fix x x let t t x denote the subalgebra of mat x c generated by a e 0 e 1 e d where a denotes the adjacency matrix of and e i denotes the projection onto the ith subconstituent of with respect to x t is called the subconstituent algebra or terwilliger algebra of with respect to x an irreducible t module w is said to be thin whenever dim e i w 1 for 0 i d by the endpoint of w we mean min i e i w 0 we describe the thin irreducible t modules with endpoint 1 let w denote a thin irreducible t module with endpoint 1 observe e 1w is a one dimensional eigenspace for e 1ae 1 let denote the corresponding eigenvalue it is known 1 d where 1 1 b 1 1 1 1 and d 1 b 1 1 d 1 for 1 and d the structure of w was worked out by go and the present author tight distance regular graphs and the subconstituent algebra preprint for 1 d we obtain the following results we show the dimension of w is d let v denote a nonzero vector in e 1w we show w has a basis eiv 1 i d where ei denotes the primitive idempotent of a associated with i we show this basis is orthogonal with respect to the hermitian dot product and we compute the square norm of each basis vector we show w has a basis e i 1aiv 0 i d 1 where ai denotes the ith distance matrix for we find the matrix representing a with respect to this basis we show this basis is orthogonal and we compute the square norm of each basis vector we find the transition matrix relating our two bases for w

on structures of quasi thin table algebras and applications to association schemes abstract in this paper we develop a structure theory for a class of table algebras i e the so called quasi thin table algebras they are direct generalizations of quasi thin association schemes although this class of table algebras is rather close to finite groups its structural properties are quite different from those of finite groups the thin radical and thin residue give rise to two group structures for every table algebra they play the central role in our discussions applications to association schemes are also discussed and some new results about quasi thin association schemes are obtained in particular we will give necessary and sufficient conditions under which a commutative quasi thin association scheme has multiplicities 1 or 2 we will need these results in the study of the characterization and classification of double quasi thin association schemes in 22

a generalization of sylow s theorems on finite groups to association schemes let x g denote an association scheme or briefly a scheme in the sense of 5 and let p denote a prime number an element g in g will be called p valenced if ng is a power of p a subset of g will be called p valenced if each of its elements is p valenced a p valenced subset f of g will be called a p subset if nf is a power of p let us assume x g to be finite then by definition ng is finite we shall denote by sylp g the set of all closed p subsets h of g such that p does not divide ng h it is the purpose of the present note to prove the following theorem

on morphisms of association schemes abstract in this paper we present a necessary and sufficient condition under which a combinatorial morphism between association schemes induces an algebra homomorphism between their scheme rings we will also discuss scheme ring homomorphisms of naturally valenced association schemes

varietal terwilliger algebras arising from wreath products of rank two association schemes abstract as abstractions of terwilliger algebras generalized terwilliger algebras and varietal terwilliger algebras have been studied in 5 6 17 in general it is difficult to determine the structure of a varietal terwilliger algebra as proposed in the concluding remarks in 4 in this paper we study the structures of those varietal terwilliger algebras arising from the wreath products of rank two association schemes in particular we give the explicit formulas for their central primitive idempotents applications to terwilliger algebras of association schemes are also discussed and an affirmative answer to the terwilliger conjecture mention in 4 is given

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

on a class of integral table algebras abstract in this paper we give the classification for a class of integral table algebras and their duals as well then we give an application of this classification to association schemes

irreducible characters of wreath products in reality based algebras and applications to association schemes abstract wreath products in reality based algebras are generalizations of wreath products of table algebras and generalized camina frobenius pairs of c algebras in this paper we present characterizations of the wreath product in a reality based algebra by its irreducible characters and by the size of the zero submatrix of its character table applications to finite groups table algebras and association schemes are also discussed in particular we will show that the wreath product of one class association schemes is characterized by the zeros in its first eigenmatrix

target oriented linear least squares and nonlinear trust region newton inversions of plane waves using ava and pva data for elastic model parameters abstractvariations in elastic properties directly impact observed seismic amplitude and phase as a function of incident angle amplitude variation with angle ava and with phase pva conventional simultaneous prestack inversion uses precritical pp reflections or joint pp and ps reflections data to invert for elastic properties via linearized least squares ls methods incorporating measured phase information as additional constraints for joint pp ps and joint ava pva inversions helps to stabilize the convergence path and improves the accuracy of the inverted parameters combining these dual joint constraints in a nonlinear trust region reflective newton trn solver produces a robust inversion with improved convergence rates partial derivatives for the jacobian and hessian matrices are computed numerically two weighting functions one for pp ps and one for ava pva data are developed for minimizing the global objective function during inversion numerical results for data from a salt interface r

3 d prestack full wavefield inversion a full wavefield inversion algorithm for direct imaging of a 3 d compressional wave velocity distribution is based on the full 3 d scalar wave equation and operates on common source data recorded by areal arrays for each source the method involves reverse time extrapolation of the residual wavefield application of the image condition by crosscorrelation with the source wavefield at each time step produces a 3 d image whose amplitude at each point is proportional to the required velocity update at that point convergence to local minima is mitigated against by gradually increasing the wavenumber bandwidth in the estimated 3 d velocity distribution as iterations proceed starting from the smallest wavenumber the algorithm is illustrated by successful application to synthetic data for a multilayered monocline and for a multilayered structure with the geometry of the standard french model the latter demonstrates good performance with noisy unequally spaced data with significant elevation statics

fluctuation dissipation dispersion relation and quality factor for slow processes a generalization of the callen welton formula for systems with slowly varying parameters is given using the momentum method and the time multiscale technique it is shown that not only the dissipation but also the dispersive contributions determine the spectral function of the fluctuations in these systems the general formalism is illustrated for an oscillating electrical circuit and the influence of the dispersion contributions on the quality factor of the system is discussed

minimization of a large scale quadratic function subject to a spherical constraint an important problem in linear algebra and optimization is the trust region subproblem minimize a quadratic function subject to an ellipsoidal or spherical constraint this basic problem has several important large scale applications including seismic inversion and forcing convergence in optimization methods existing methods to solve the trust region subproblem require matrix factorizations which are not feasible in the large scale setting this paper presents an algorithm for solving the large scale trust region subproblem that requires a fixed size limited storage proportional to the order of the quadratic and that relies only on matrix vector products the algorithm recasts the trust region subproblem in terms of a parameterized eigenvalue problem and adjusts the parameter with a superlinearly convergent iteration to find the optimal solution from the eigenvector of the parameterized problem only the smallest eigenvalue and corresponding eigenvector of the parameterized problem needs to be computed the implicitly restarted lanczos method is well suited to this subproblem

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

on nakayama automorphisms of double frobenius algebras abstract double frobenius algebras or df algebras were recently introduced by the author the concept generalizes finite dimensional hopf algebras adjacency algebras of non commutative association schemes and c algebras or character algebras this paper studies basic properties of various nakayama automorphisms in a df algebra as an application the theorem of d e radford stating that the antipode s of a finite dimensional hopf algebra is of finite order is generalized to df algebras and so is his formula for s 4 n in terms of certain group like elements

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

fusion algebras with negative structure constants abstract we introduce fusion algebras with not necessarily positive structure constants and without identity element we prove that they are semisimple when tensored with c and that their characters satisfy orthogonality relations then we define the proper notion of subrings and factor rings for such algebras for certain algebras r we prove the existence of a ring r with nonnegative structure constants such that r is a factor ring of r we give some examples of interesting factor rings of the representation ring of the quantum double of a finite group then we investigate the algebras associated to hadamard matrices for an n n matrix the corresponding algebra is a factor ring of a subalgebra of z z 2 z n 2

on thin residues and basis digraphs of nilpotent table algebras and applications to nilpotent groups abstract in this paper we study characterizations of nilpotent table algebras in terms of thin residues thin residue matrices thin residue digraphs and basis digraphs we will also discuss applications to nilpotent groups in particular we give a conceptual explanation of the results in takegahara and yoshida 2008 ty

some structure theory of table algebras and applications to association schemes abstract in this paper we prove some basic structure theorems for arbitrary table algebras in particular we prove a krull schmidt theorem for arbitrary table algebras and give applications to association schemes

nilpotency property in table algebras abstract table algebras form an important class of c algebras the dual of a table algebra may not be a table algebra but just a c algebra it is not known under what conditions the dual of a table algebra is also a table algebra in this paper we prove that if a table algebra has nilpotency property then its dual also is a table algebra finally we conclude that if a b is a nilpotent table algebra then its dual also is a nilpotent table algebra

quotient structures homomorphisms and partial wreath products in reality based algebras abstract homomorphisms and quotient structures of reality based algebras are presented and described related central idempotents and quotient sets are characterized and the extent to which a homomorphism preserves the given anti automorphisms of the algebras is determined the notion of a partial wreath product an algebraic abstraction of the wedge product of association schemes is developed prototypical results previously obtained for various special cases including table algebras and commutative c algebras are thus uniformly generalized

the terwilliger algebra of a distance regular graph of negative type abstract let denote a distance regular graph with diameter d 3 assume has classical parameters d b with b 1 let x denote the vertex set of and let a mat x c denote the adjacency matrix of fix x x and let a mat x c denote the corresponding dual adjacency matrix let t denote the subalgebra of mat x c generated by a a we call t the terwilliger algebra of with respect to x we show that up to isomorphism there exist exactly two irreducible t modules with endpoint 1 their dimensions are d and 2 d 2 for these t modules we display a basis consisting of eigenvectors for a and for each basis we give the action of a

terwilliger algebras of direct and wreath products of association schemes abstract we will consider structures of terwilliger algebras of direct and wreath products of association schemes in general it is difficult to determine the structure of the terwilliger algebras though they are known to be semisimple c algebras but we get the structure of terwilliger algebras of these cases under some assumptions

the recognition problem for table algebras and reality based algebras given a finite dimensional noncommutative semisimple algebra a over c with involution we show that a always has a basis b for which a b is a reality based algebra for algebras that have a one dimensional representation we show that there always exists an rba basis for which is a positive degree map we characterize all rba bases of the 5 dimensional noncommutative semisimple algebra for which the algebra has a positive degree map and give examples of rba bases of c mn c for which the rba has a positive degree map for all n 2

bayesian linearized avo inversion a new linearized avo inversion technique is developed in a bayesian framework the objective is to obtain posterior distributions for p wave velocity s wave velocity and density distributions for other elastic parameters can also be assessed for example acoustic impedance shear impedance and p wave to s wave velocity ratio the inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the zoeppritz equation the solution is represented by a gaussian posterior distribution with explicit expressions for the posterior expectation and covariance hence exact prediction intervals for the inverted parameters can be computed under the specified model the explicit analytical form of the posterior distribution provides a computationally fast inversion method tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero with realistic noise levels acoustic impedance was the best determined parameter

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

long offset avo inversion of pp reflections from plane interfaces using effective reflection coefficients a conventional amplitude variation with offset avo inversion is based on geometrical seismics which exploit plane wave reflection coefficients to describe the reflection phenomenon widely exploited linearizations of plane wave coefficients are mostly valid at pre critical offsets for media with almost flat and weak contrast interfaces existing linearizations do not account for the seismic frequency range by ignoring the frequency content of the wavelet which is a strong assumption planewave reflection coefficients do not fully describe the reflection of seismic waves at near critical and post critical offsets because reflected seismic waves are typically generated by point sources we propose an improved approach to avo inversion which is based on effective reflection coefficients ercs ercs generalize plane wave coefficients for seismic waves generated by point sources and therefore more accurately describe nearcritical and post critical reflections where head waves are generated moreover they are frequency dependent and incorporate the local curvatures of the wavefront and the reflecting interface in our study we neglect the effect of interface curvature and demonstrate the advantages of our approach on synthetic data for a simple model with a plane interface separating two isotropic half spaces a comparison of the inversion results obtained with our approach and the results from an avo inversion method based on the exact plane wave reflection coefficient suggests that our method is superior in particular for long offset ranges which extend to and beyond the critical angle we thus propose that long offsets can be successfully exploited in an avo inversion under the correct assumption about the reflection coefficient such long offset avo inversion shows the potential of outperforming a conventional moderate offset avo inversion in the accuracy of estimated model parameters

terwilliger algebras of wreath products of association schemes abstract the terwilliger algebra of an association scheme of order n introduced in 13 is a subalgebra of the matrix algebra of all n n matrices terwilliger algebras of wreath products of special association schemes are studied in several papers in this paper we study the terwilliger algebra of the wreath product t s of two arbitrary association schemes s and t we will express the terwilliger algebra of t s and its primitive central idempotents in terms of the terwilliger algebras of s and t and their primitive central idempotents the known results of hanaki kim etc cf 7 10 are special cases of our results

on semisimple varietal terwilliger algebras whose non primary ideals are 1 dimensional abstract as an abstraction of the terwilliger algebra of a commutative association scheme the notion of a varietal terwilliger algebra was introduced in 11 in this paper we first prove that there exists a varietal terwilliger algebra defined by any given standard c algebra and a generalized terwilliger algebra 6 is also a varietal terwilliger algebra then we present a necessary and sufficient condition under which a quotient algebra of a varietal terwilliger algebra is also a varietal terwilliger algebra we will also give a characterization of a semisimple varietal terwilliger algebra whose non primary ideals are all 1 dimensional and prove that a semisimple regular varietal terwilliger algebra with 1 dimensional non primary ideals is characterized by the structure of its base c subalgebra as direct consequences we obtain some results in 18 for finite association schemes and a conceptual explanation of the results in 9

fusion algebras for imprimitive complex reflection groups abstract we prove that the fourier matrices for the imprimitive complex reflection groups introduced by malle in gunter malle unipotente grade imprimitiver komplexer spiegelungsgruppen j algebra 177 3 1995 768 826 define fusion algebras with not necessarily positive but integer structure constants hence they define z algebras as a result we obtain that all known fourier matrices belonging to spetses define algebras with integer structure constants

accurate estimation of density from the inversion of multicomponent prestack seismic waveform data using a nondominated sorting genetic algorithm estimation of density and other elastic parameters is important in characterizing conventional unconventional and carbon dioxide co2 sequestrated reservoirs while it is possible to estimate velocities from the conventional p wave seismic data it is difficult to accurately estimate density from such data because the converted wave p sv reflections are more sensitive to density compared to the primary pp reflections using both pp and p sv data in a multicomponent inversion has been shown to accurately estimate density compared to using p wave data alone the problem of such multicomponent inversion is however nonlinear requires simultaneous optimization of multiple sets of data and has multiple optimal solutions such simultaneous optimization is better handled by a nondominated sorting genetic algorithm in comparison to the classical approach applying the nondominated sorting genetic algorithm to multicomponent synthetic seismic data here we demonstrate that this method can potentially extr

elastic inversion of near and postcritical reflections using phase variation with angle abstractnear and postcritical wide angle reflections provide the potential for velocity and density inversion because of their large amplitudes and phase shifted waveforms we tested using phase variation with angle pva data in addition to or instead of amplitude variation with angle ava data for elastic inversion accurate pva test data were generated using the reflectivity method two other forward modeling methods were also investigated including plane wave and spherical wave reflection coefficients for a two half space model linearized least squares was used to invert pva and ava data for the p wave velocity s wave velocity and the density of the lower space and the s wave velocity of the upper space inversion tests showed the feasibility and robustness of pva inversion a reverse time migration test demonstrated better preservation of pva information than ava information during wavefield propagation through a layered overburden phases of deeper reflections were less affected than ampl

why irreversibility is not a sufficient condition for ergodicity khinchin s theorem of ergodicity is examined by means of linear response theory the resulting ergodic condition shows that contrary to the theorem irreversibility is not a sufficient condition for ergodicity by the recurrence relations method we prove that irreversibility is broader in scope than ergodicity showing why it can only be a necessary condition for ergodicity

ergodic theory infinite products and long time behavior in hermitian models ergodic theory has been approached from the side of the time averages with correlation functions from many body models the condition for ergodic behavior is formulated in terms of infinite products of certain numbers associated with time evolution physical properties that make a model ergodic are identified

on quasi thin association schemes with odd number of points abstract let x r be an association scheme in the sense of p h zieschang 1996 an algebraic approach to association schemes lecture notes in mathematics vol 1628 springer new york berlin where x is a finite set and r is a partition of x x we say that x r is quasi thin if each element of r has a valency of at most two in this paper we focus on quasi thin association schemes with an odd number of points and obtain that x r has a regular automorphism group when n o r is square free

polynomial table algebras and their covering numbers abstract in this article we introduce the notion of polynomial table algebras and discuss their covering numbers in particular we prove that the real table algebras a b with cn b 2 b 2 are polynomial table algebras such that by a suitable reordering of x i b if necessary the first intersection matrices are tridiagonal as follows formula where b i 0 c j 0 a k 0

pseudo direct sums and wreath products of loose coherent algebras with applications to coherent configurations abstract we introduce the notion of a loose coherent algebra which is a special semisimple subalgebra of the matrix algebra and define two operations to obtain new loose coherent algebras from the old ones the pseudo direct sum and the wreath product for two arbitrary coherent configurations c d and their wreath product c d it is difficult to express the terwilliger algebra t x y c d in terms of the terwilliger algebras t x c and t y d by using the concept and operations of loose coherent algebras we find a very simple such expression as a direct consequence of this expression we obtain the central primitive idempotents of t x y c d in terms of the central primitive idempotents of t x c and t y d many results in 4 6 10 12 are special cases of the results in this paper

effective reflection coefficients for curved interfaces in transversely isotropic media plane wave reflection coefficients pwrcs are routinely used in amplitude variation with offset analysis and for generating boundary data in kirchhoff modeling however the geometrical seismics approximation based on pwrcs becomes inadequate in describing reflected wavefields at near and postcritical incidence angles also pwrcs are derived for plane interfaces and break down in the presence of significant reflector curvature here we discuss effective reflection coefficients ercs designed to overcome the limitations of pwrcs for multicomponent data from heterogeneous anisotropic media we represent the reflected wavefield in the immediate vicinity of a curved interface by a generalized plane wave decomposition which approximately reduces to the conventional weyl type integral computed for an apparent source location the erc then is obtained as the ratio of the reflected and incident wavefields at each point of the interface to conduct diffraction modeling we combine ercs with the tip wave superposition methodtwsm extended to elastic media this methodology is implemented for curved interfaces that separate an isotropic incidence half space and a transversely isotropic ti medium with the symmetry axis orthogonal to the reflector if the interface is plane ercs generally are close to the exact solution sensitive to the anisotropy parameters and source receiver geometry numerical tests demonstrate that the difference between ercs and pwrcs for typical ti models can be significant especially at low frequencies and in the postcritical domain for curved interfaces ercs provide a practical approximate tool to compute the reflected wavefield we analyze the dependence of ercs on reflector shape and demonstrate their advantages over pwrcs in 3d diffraction modeling of pp and ps reflection data

testing boltzmann s ergodic hypothesis with electron gas models kubo proposed a physical approach to proving the validity of boltzmann s ergodic hypothesis it is to perform the time averages on dynamical functions thereby avoiding the difficulties of measure theory inherent in the classical approach to perform time averaging properly one must have a general solution for the heisenberg equation of motion such as by the recurrence relations method a time averaging carried over with a recurrence relations solution is found to yield an ergodic condition in the form of an infinite product it is linked to the energy transfer mechanisms hence to the ergodicity itself the electron gas models are fertile ground for testing ergodicity by this approach for several static domains we have evaluated the infinite product and drawn from them a general physical picture that underlies the ergodic hypothesis