canonical p dimension of algebraic groups we describe a way to compute the p relative version of the berhuy reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic p is any prime integer the canonical p dimension is computed for all split simple groups of classical types

fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

the u invariant of the function fields of p adic curves the u invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field it is an open question whether the u invariant of function fields of p aidc curves is 8 in this paper we answer this question in the affirmative for function fields of nondyadic p adic curves

on the first witt index of quadratic forms we prove hoffmann s conjecture determining the possible values of the first witt index of anisotropic quadratic forms of any given dimension the proof makes use of the steenrod type operations on the modulo 2 chow groups constructed by p brosnan

the index of a brauer class on a brauer severi variety let d and e be central division algebras over k let k be the generic splitting field of e we show that the index of d k k is the minimum of the indices of d e i as i varies we use this to calculate the index of d under related central extensions and to construct division algebras with special properties

acoustic pulse spreading in a random fractal fractal medium models are used to model for instance the heterogeneous earth and the turbulent atmosphere a wave pulse propagating through such a medium will be affected by multiscale medium fluctuations for a class of one dimensional fractal random media defined in terms of fractional brownian motion we show how the wave interacts with the medium fluctuations the modification in the pulse shape depends on the roughness of the medium and can be described in a deterministic way when the pulse is observed at its random arrival time for very rough media the coherent wave is confined to a surface layer

wild pfister forms over henselian fields k theory and conic division algebras the epicenter of this paper concerns pfister quadratic forms over a field f with a henselian discrete valuation all characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but f does not we also prove results about round quadratic forms composition algebras generalizations of composition algebras we call conic algebras and central simple associative symbol algebras finally we give relationships between these objects and kato s filtration on the milnor k groups of f

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

upper motives of algebraic groups and incompressibility of severi brauer varieties let g be a semisimple affine algebraic group of inner type over a field f we write xg for the class of all finite direct products of projective g homogeneous f varieties we determine the structure of the chow motives with coefficients in a finite field of the varieties in xg more precisely it is known that the motive of any variety in xg decomposes in a unique way into a sum of indecomposable motives and we describe the indecomposable summands which appear in the decompositions in the case where g is the group pgla of automorphisms of a given central simple f algebra a for any variety in the class xg which includes the generalized severi brauer varieties of the algebra a we determine its canonical dimension at any prime p in particular we find out which varieties in xg are p incompressible if a is a division algebra of degree p for some n 0 then the list of p incompressible varieties includes the generalized severi brauer variety x p a of ideals of reduced dimension p for m 0 1 n

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

computation of generalized equivariant cohomologies of kac moody flag varieties abstract in 1998 goresky kottwitz and macpherson showed that for certain projective varieties x equipped with an algebraic action of a complex torus t the equivariant cohomology ring h t x can be described by combinatorial data obtained from its orbit decomposition in this paper we generalize their theorem in three different ways first our group g need not be a torus second our space x is an equivariant stratified space along with some additional hypotheses on the attaching maps third and most important we allow for generalized equivariant cohomology theories e g instead of h t for these spaces we give a combinatorial description of e g x as a subring of e g f i where the f i are certain invariant subspaces of x our main examples are the flag varieties g p of kac moody groups g with the action of the torus of g in this context the f i are the t fixed points and e g is a t equivariant complex oriented cohomology theory such as h t k t or mu t we detail several explicit examples

strongly inner anisotropic forms of simple algebraic groups the definition of the index of a semisimple algebraic group over a field will be recalled in 1 55 more precisely we show in proposition 2 b cf 3 1 that if over a given field k there exists a central division algebra of degree 4 i e of dimension 16 and of order 4 in the brauer group then there exists a group of type ei t over a purely transcendental extension of degree 1 k t of k this will be deduced in two different ways from theorem 1 of section 2 whose precise statement is too technical to be given in this introduction but whose spirit is that it enables one from the existence of groups of a given type with preassigned properties of a geometric and cohomological nature over a field k to infer the existence of groups of a more complicated type meeting similar requirements over k t that theorem has other interesting applications to classification problems for instance we also deduce from it again in two different ways proposition 3 b in 3 2 that the existence of a division algebra of degree 3 over k implies the existence of groups with index

frequency content of randomly scattered signals the statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it the asymptotic analy

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

wave propagation in micro heterogeneous porous media a model based on an integro differential wave equation a model hyperbolic partial differential equation with singular convolution operators and infinitely smooth solutions is studied it is shown that short pulses including finite bandwidth pulses propagate with a delay with respect to the wavefront for a two parameter family of such equations green s functions are obtained in a simple self similar form as an application it is demonstrated that the gurevich lopatnikov dispersion law for a thin layered porous medium can be approximated by a hyperbolic equation with singular memory

acoustic waves in long range random media we consider waves propagating through multiscale media much is known about waves propagating through a medium that satisfies a scale separation assumption with random fluctuations on a microscale here we go beyond this situation and consider waves propagating through a medium defined in terms of a long range process such a medium can for instance be modeled in terms of a one dimensional fractional brownian motion with variations on a continuum of scales fractal medium models are used to model for example the heterogeneous earth and the turbulent atmosphere we set forth a framework using the theory of rough paths in which propagation problems of this nature can be analyzed in the case with anticipative medium fluctuations with a hurst exponent h 1 2 we show how the wave interacts with the medium fluctuations in this case and that the interaction is qualitatively different from the situation where the medium satisfies a separation of scales assumption in the long range case considered here the

withdrawn parameterization of rainfall models with 1 emphasis on extremes temporal rainfall is usually represented as a series of storms with different duration d and average intensity id separated by dry periods inside each storm rainfall intensity fluctuates producing different average values id in different intervals of duration d d we consider storm models of this type to predict the distribution of the annual maximum of id of special importance is the dependence of the upper tails of id and id on d and d d id respectively from the analysis of 23 hourly rainfall records we conclude that all marginal and conditional upper tails of id and id have lognormal shape the upper tail of id is largely independent of d and is the same as the upper tail of id for d d both tail distributions scale in a multifractal way with duration these findings greatly simplify the parameterization of the model and the estimation of rainfall extremes we compare extreme rainfall results from this model with those produced by methods that use only the marginal upper tail of id and the number of wet d intervals in one year and with the empirical distributions from the historical annual maxima

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

shells of twisted flag varieties and the rost invariant we introduce two new general methods to compute the chow motives of twisted flag varieties and settle a 20 year old conjecture of markus rost about the rost invariant for groups of type e 7

motivic decomposition of isotropic projective homogeneous varieties we give a decomposition of the chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties as an application we prove a generalization of rost s nilpotence theorem

frequency domain wave equation and its time domain solutions in attenuating media our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law to achieve this a frequency domain wave equation was developed using previously derived causal dispersion relations an inverse space and time fourier transform of the solution to this algebraic equation results in a time domain solution it is shown that this solution satisfies the convolutional time domain wave equation proposed by szabo j acoust soc am 96 491 500 1994 the form of the convolutional loss operator contained in this wave equation is obtained solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution using our solution comparisons are made for transient one dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of szabo s equation these show good agreement

dynamic poroelasticity of thinly layered structures abstract compressional seismic p waves propagating in poroelastic fluid saturated laminated sediments are strongly affected by the medium heterogeneity here simple analytical expressions for the p wave phase velocity and attenuation coefficient are derived both are functions of frequency and statistical medium parameters such as correlation lengths and variances the theoretical results are compared with results from numerical simulations and show good agreement in heterogeneous media impedance fluctuations lead to poroelastic scattering variations of the layer compressibilities cause inter layer flow a 1 d macroscopic local flow from the seismic frequency range 10 100 hz up to ultrasonic frequencies attenuation due to heterogeneity is strongly enhanced compared to homogeneous biot models the new theory automatically includes different asymptotic approximations such as poroelastic backus averaging in the quasi static and the no flow limit geometrical optics and intermediate frequency ranges

isotropy of unitary involutions we prove the so called unitary isotropy theorem a result on isotropy of a unitary involution the analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal symplectic and unitary involutions are formal consequences of this theorem a component of the proof is a detailed study of the quasi split unitary grassmannians

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations

effective fractional acoustic wave equations in one dimensional random multiscale media this paper considers multiple scattering of waves propagating in a non lossy one dimensional random medium with short or long range correlations using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation which corresponds to an effective medium with a frequency dependent attenuation that obeys a power law with an exponent between 0 and 2 the exponent is related to the hurst parameter of the medium which is a characteristic parameter of the correlation properties of the fluctuations of the random medium moreover the frequency dependent attenuation is associated with a special frequency dependent phase which ensures that causality and kramers kronig relations are satisfied in the time domain the effective wave equation has the form of a linear integro differential equation with a fractional derivative

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko