wave speed dispersion associated with an attenuation obeying a frequency power law an attenuation scaling as a power of frequency over an infinite bandwidth is neither analytic nor square integrable thus calling into question the application of the kramers kronig dispersion relations for determining the frequency dependence of the associated phase speed in this paper three different approaches are developed all of which return the dispersion formula for the wavenumber k the first analysis relies on the properties of generalized functions and the causality requirement that the impulse response k t the inverse fourier transform of ik must vanish for t 0 second a wave equation is introduced that yields the phase speed dispersion associated with a frequency power law attenuation finally it is shown that with minor modification the kramers kronig dispersion relations with no subtractions the plemelj formulas do in fact hold for an attenuation scaling as yielding the same dispersion formula as the other two derivations from this dispersion formula admissible values of the exponent are established physically the inadmissible values of which include all the integers correspond to attenuation dispersion pairs whose fourier components cannot combine in such a way as to make the impulse response k t vanish for t 0 there is no upper or lower limit on the value that may take

analytical time domain green s functions for power law media frequency dependent loss and dispersion are typically modeled with a power law attenuation coefficient where the power law exponent ranges from 0 to 2 to facilitate analytical solution a fractional partial differential equation is derived that exactly describes power law attenuation and the szabo wave equation time domain wave equations for lossy media obeying a frequency power law j acoust soc am 96 491 500 1994 is an approximation to this equation this paper derives analytical time domain green s functions in power law media for exponents in this range to construct solutions stable law probability distributions are utilized for exponents equal to 0 1 3 1 2 2 3 3 2 and 2 the green s function is expressed in terms of dirac delta exponential airy hypergeometric and gaussian functions for exponents strictly less than 1 the green s functions are expressed as fox functions and are causal for exponents greater than or equal than 1 the green s functions are expressed as fox and

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

asymptotic distributions of continuous time random walks a probabilistic approach we provide a systematic analysis of the possible asymptotic distributions o one dimensional continuous time random walks ctrws by applying the limit theorems of probability theory biased and unbiased walks of coupled and decoupled memory are considered in contrast to previous work concerning decoupled memory and levy walks we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean obtaining asymmetric levy stable limits suprisingly it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution we discuss interesting properties of walks with an infinite mean waiting time between successive jumps

finite bandwidth kramers kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions kramers kronig kk analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of kk integrals for a complete infinite spectrum of input information for data exhibiting localized extrema kk relations can provide accurate transforms over finite bandwidths due to the local weighting properties of the kk kernel recently acoustic kk relations have been derived for the determination of the group velocity cg and the derivative of the attenuation coefficient components of the derivative of the acoustic complex wave number these relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters in contrast to twice subtracted finite bandwidth kk predictions for phase velocity and attenuation coefficient components of the undifferentiated wave number these more recently derived relations for cg and provide stricter tests of causal consistency because the resulting shapes are inv

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

finite bandwidth effects on the causal prediction of ultrasonic attenuation of the power law form kramers kronig k k relations exist as a consequence of causality placing nonlocal constraints on the relationship between dispersion and absorption the finite bandwidth method of applying these relations is examined where the k k integrals are restricted to the spectrum of the experimental data these finite bandwidth k k relations are known to work with resonant type data and here are applied to dispersion data consistent with a power law attenuation coefficient exponent from 1 to 2 bandwidth restricted forms of the zero and once subtracted k k relations are used to determine the attenuation coefficient from phase velocity analytically it is shown that these transforms produce the proper power law form of the attenuation coefficient as a stand alone term summed with artifacts that are dependent on the integration limits calculations are performed to demonstrate how these finite bandwidth artifacts affect the k k predictions under a variety of conditions the predictions are studied in a local context as a function of subtraction frequency bandwidth and power law exponent the k k predictions of the power law exponent within various decades of the spectrum are also examined in general the agreement between finite bandwidth k k predictions and exact values grows as the power law exponent approaches 1 and with increasing bandwidth

causal theories and data for acoustic attenuation obeying a frequency power law this study compares causal theories based on kramers kronig relations fractional calculus and on those derived from new time domain causal relationships to diverse data all these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number the data for lossy media with attenuation having a power law frequency dependence with an exponent y include cases for both liquids and solids ranging from acoustic to ultrasound frequencies data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer experimental results and supporting evidence show that the classical case of frequency squared attenuation is dispersionless an approximate nearly local kramers kronig theory is in agreement with the time causal theory when the exponent is close to one but deviates for other values the comprehensive time causal theory is shown to be equivalent to two other theories derived from exact kramers kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches attenuation dispersion relations are presented in two forms one for a general frequency range and another for a finite range it is demonstrated that complete velocity dispersion within a signal bandwidth can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency y 1 and or at a high frequency limit 0 y 1 in addition attenuation data can be predicted from velocity data

fractional biharmonic operator equation model for arbitrary frequency dependent scattering attenuation in acoustic wave propagation this paper proposes a fractional biharmonic operator equation model in the time space domain to describe scattering attenuation of acoustic waves in heterogeneous media compared with the existing models the proposed fractional model is able to describe arbitrary frequency dependent scattering attenuation which typically obeys an empirical power law with an exponent ranging from 0 to 4 in stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature little has been reported on frequency dependent scattering attenuation this is largely because the order of the fractional laplacian is from 0 to 2 and is infeasible for scattering attenuation in this study the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power law dependency an

surface acoustic impedance and causality several models that are commonly used to calculate the surface acoustic impedance of porous materials are examined in light of the causality requirement in its nonlocal and its approximate local forms strict nonlocal causality requires that the imaginary part of the surface acoustic impedance be the hilbert transform of the real part it is found that the local approximation is not a good test of causality for these models and that even in its general form the equation describing the surface acoustic impedance of these porous materials is not strictly causal at high frequencies

on a time domain representation of the kramers kronig dispersion relations the development of kramers kronig dispersion relations is typically carried out in the frequency domain an alternative approach known as the time causal theory develops dispersion relations for media with attenuation obeying a frequency power law through analysis in the time domain t l szabo j acoust soc am 96 491 500 1994 although both approaches predict identical dispersion relations it is perceived that these two approaches are distinct from each other it is shown however that the time causal theory is in essence a time domain formulation of the kramers kronig dispersion relations for the special case of media with attenuation obeying a frequency power law additionally it is shown that time domain representations of the kramers kronig dispersion relations are available for a broader class of media than simply those with power law attenuation the time causal theory and the kramers kronig dispersion relations can be viewed as two complementary yet equivalent approaches to the study

verification of kramers kronig relationship in porous materials having a rigid frame abstract the propagation of acoustic waves in porous materials having a rigid frame is well described by several models a doubt about the causality of these models has been raised recently in the literature a verification of the causality of these models is studied in this paper using the kramers kronig dispersion relations adapted to the frequency power law dependence of the attenuation it is shown that these models are causal in the high and low frequency range a time domain wave equation and time causal theory have been treated

uncoupled continuous time random walk finite jump length probability density function in this work the uncoupled continuous time random walk model with the exponential jump length probability density function pdf is considered exact solutions for the pdf and second moment are obtained and investigated for different kinds of waiting time pdf

fourier coefficients of automorphic forms and integrable discrete series abstract let g be the group of r points of a semisimple algebraic group g defined over q assume that g is connected and noncompact we study fourier coefficients of poincare series attached to matrix coefficients of integrable discrete series we use these results to construct explicit automorphic cuspidal realizations which have appropriate fourier coefficients 0 of integrable discrete series in families of congruence subgroups in the case of g sp 2 n r we relate our work to that of li 14 for g quasi split over q we relate our work to the result about poincare series due to khare larsen and savin 12

on the applicability of kramers kronig relations for ultrasonic attenuation obeying a frequency power law in the recent literature concern has been raised regarding the validity of kramers kronig relations for media with ultrasonic attenuation obeying a frequency power law it is demonstrated however that the kramers kronig dispersion relations for application to these types of media are available the developed dispersion relations are compared with measurements on several liquids and agreement is found to better than 1 m s over the experimentally available bandwidth a discussion regarding the validity of these dispersion relations in particular how the dispersion relations relate to the so called paley wiener conditions forms the conclusion

strong limit multiplicity for arithmetic hyperbolic surfaces and 3 manifolds we show that every sequence of torsion free arithmetic congruence lattices in mathrm pgl 2 mathbb r or mathrm pgl 2 mathbb c satisfies a strong quantitative version of the limit multiplicity property we deduce that for r 0 in certain range growing linearly in the degree of the invariant trace field the volume of the r thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3 manifold m is of order at most mathrm vol m 11 12 as an application we prove gelander s conjecture on homotopy type of arithmetic hyperbolic 3 manifolds we show that there are constants a b such that every such manifold m is homotopy equivalent to a simplicial complex with at most a mathrm vol m vertices all of degrees bounded by b

characters l 2 betti numbers and an equivariant approximation theorem let g be a group with a finite subgroup h we define the l 2 multiplicity of an irreducible representation of h in the l 2 homology of a proper g cw complex these invariants generalize the l 2 betti numbers our main results are approximation theorems for l 2 multiplicities which extend the approximation theorems for l 2 betti numbers of luck farber and elek szabo respectively the main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups we discuss applications to the cohomology of arithmetic groups

simplified expressions of the subtracted kramers kronig relations using the expanded forms applied to ultrasonic power law systems the kramers kronig kk relations are a large class of integral transformations that exploit the broad principle of simple causality in order to link the physical properties of matter and materials in applications to the complex valued wavenumber for acoustic propagation the method of subtractions is used to form convergent integral relations between the phase velocity and the attenuation coefficient when the method of subtractions is applied in the usual manner the integrands in the relations become unnecessarily complicated in this work an expanded form of the subtracted relations is presented which is essentially a truncated taylor series expansion of the hilbert transforms the implementation of the relations only requires the explicit evaluation of two simply expressed integrals involving the hilbert transform kernel these two integrals determine the values of the other terms in the subtracted relations demonstrating the computational efficiency of the technique the method is illustrated analytically through its application to power law attenuation coefficients and its associated dispersion which are observed in a wide variety of materials this approach explicitly shows the central role of the hilbert transform kernel in the kk relations which can become obscured in other formulations

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

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

integrodifferential diffusion equation for continuous time random walk in this paper we present an integrodifferential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential and a combination of power law and generalized mittag leffler function we show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is gaussian distribution in the case of the combination of a power law and generalized mittag leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times and probability density function is non gaussian distribution

kramers kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles in this work the kramers kronig k k relations are applied to experimental data of resonant nature by limiting the interval of integration to the measurement spectrum the data are from suspensions of encapsulated microbubbles albunex and have the characteristics of an ultrasonic notch filter the goal is to test the consistency of this dispersion and attenuation data with the kramers kronig relations in a strict manner without any parameters from outside the experimental bandwidth entering in to the calculations in the course of reaching the goal the artifacts associated with the truncation of the integrals are identified and it is shown how their impacts on the results can be minimized the problem is first approached analytically by performing the kramers kronig calculations over a restricted spectral band on a specific hilbert transform pair lorentzian curves the resulting closed form solutions illustrate the type of artifacts that can occur due to truncation and also show that accurate res

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

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

alternative approach to the derivation of dispersion relations for optical constants the derivation of dispersion relations for linear optical constants is considered starting from the representation of an optical property as a herglotz function one form of the kramers kronig relations is determined directly and the second is obtained using elementary properties of the hilbert transform application to the complex refractive index is considered

a k space green s function solution for acoustic initial value problems in homogeneous media with power law absorption an efficient green s function solution for acoustic initial value problems in homogeneous media with power law absorption is derived the solution is based on the homogeneous wave equation for lossless media with two additional terms these terms are dependent on the fractional laplacian and separately account for power law absorption and dispersion given initial conditions for the pressure and its temporal derivative the solution allows the pressure field for any time t 0 to be calculated in a single step using the fourier transform and an exact k space time propagator for regularly spaced cartesian grids the former can be computed efficiently using the fast fourier transform because no time stepping is required the solution facilitates the efficient computation of the pressure field in one two or three dimensions without stability constraints several computational aspects of the solution are discussed including the effect of using a truncated fourier series to represent discrete initial condit

approximate analytical time domain green s functions for the caputo fractional wave equation the caputo fractional wave equation geophys j r astron soc 13 529 539 1967 models power law attenuation and dispersion for both viscoelastic and ultrasound wave propagation the caputo model can be derived from an underlying fractional constitutive equation and is causal in this study an approximate analytical time domain green s function is derived for the caputo equation in three dimensions 3d for power law exponents greater than one the green s function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1 4 r the approximate one dimensional 1d and two dimensional 2d green s functions are also computed in terms of stable distributions finally this green s function is decomposed into a loss component and a diffraction component revealing that the caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation

time domain comparisons of power law attenuation in causal and noncausal time fractional wave equations the attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations to demonstrate some of the similarities and differences that are observed in three related time fractional partial differential equations time domain green s functions are calculated numerically for the power law wave equation the szabo wave equation and for the caputo wave equation these green s functions are evaluated for water with a power law exponent of y 2 breast with a power law exponent of y 1 5 and liver with a power law exponent of y 1 139 simulation results show that the noncausal features of the numerically calculated time domain response are only evident very close to the source and that these causal and noncausal time domain green s functions converge to the same result away from the source when noncausal time domain green s functions are convolved with a short pulse no evidence of noncausal behavior remains in the time domain which suggests that these causal and noncausal time fractional models are equally effective for these numerical calculations

models of anomalous diffusion the subdiffusive case abstract the paper discusses a model for anomalous diffusion processes their one point probability density functions p d f are exact solutions of fractional diffusion equations the model reflects the asymptotic behavior of a jump anomalous random walk process with random jump sizes and random inter jump time intervals with infinite means and variances which do not satisfy the law of large numbers in the case when these intervals have a fractional exponential p d f the fractional komogorov feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed finally some statistical properties of solutions of the related langevin equation are studied the subdiffusive case is explored in detail the emphasis is on a rigorous presentation which however would be accessible to the physical sciences audience

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

experimental validation of the use of kramers kronig relations to eliminate the phase sheet ambiguity in broadband phase spectroscopy the technique of broadband phase spectroscopy proposed in 1978 by sachse and pao j appl phys 49 4320 4327 1978 determines the phase velocity as a function of frequency from the fourier transforms of a received reference and through sample signal although quite successful this approach can be influenced by an ambiguity in the phase velocity calculation which stems from the boundedness of the inverse tangent operation used to calculate phase several empirical approaches to resolve the phase ambiguity have been reported an alternative approach that has not previously been considered appeals to the causal nature of the measurements this article experimentally validates a method which uses the causally consistent kramers kronig relations to eliminate the ambiguity in phase spectroscopy derived phase velocity calculations broadband pulse and narrow band tone burst measurements were performed on three gelatin based phantoms containing different concentrations of graphite particles 0 10 and 20 by volume the phantoms were constructed to have attenuation coefficients which vary approximately linear with frequency a dependence exhibited by many soft tissues the narrow band phase velocity measurements do not suffer from a phase ambiguity and thus they serve as a gold standard against which the broadband phase velocity measurements are compared the experimental results illustrate that using the kramers kronig dispersion relations in conjunction with phase spectroscopy derived phase velocity measurements is an effective means by which to resolve the phase sheet ambiguity in broadband phase spectroscopy

bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms let goo be a semisimple real lie group with unitary dual goo we produce new upper bounds for the multiplicities with which representations e of cohomological type appear in certain spaces of cusp forms on goo the main new idea is to apply noncommutative iwasawa theory to certain p adic completions of the cohomology of locally symmetric spaces