schur subalgebras and an application to the symmetric group abstract let k be an infinite field of prime characteristic p and let d r be positive integers of the same parity satisfying a certain congruence condition we prove that the schur algebra s 2 d is isomorphic to a subalgebra of the form es 2 r e where e is a certain idempotent of s 2 r translating this result via ringel duality to the symmetric groups d and r we obtain lattice isomorphisms between specht modules between young modules and between permutation modules here modules labelled by the partitions r k k correspond to modules labelled by d k k this provides a representation theoretical interpretation for part of the fractal structures observed for the decomposition numbers of the symmetric groups corresponding to two part partitions

renormalizing curvature integrals on poincare einstein manifolds abstract after analyzing renormalization schemes on a poincare einstein manifold we study the renormalized integrals of scalar riemannian invariants the behavior of the renormalized volume is well known and we show any scalar riemannian invariant renormalizes similarly we consider characteristic forms and their behavior under a variation of the poincare einstein structure and obtain from the renormalized integral of the pfaffian an extension of the gauss bonnet theorem

boundary regularity of conformally compact einstein metrics we show that c conformally compact riemannian einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions and polyhomogeneous in odd dimensions greater than 3

covariance matrix functions of vector χ 2 random fields in space and time this paper introduces vector multivariate or multiple 2 and rayleigh random functions or random fields on a spatial temporal or spatio temporal index domain and explores their basic properties formulated as a sum of squares of independent gaussian random functions a vector 2 random function has an interesting feature that its finite dimensional laplace transforms are not determined by its own covariance matrix but by that of the underlying gaussian one with the conditionally negative definite matrix as an important building block this paper constructs a class of vector 2 random functions from whose covariance matrices one can easily identify those of the underlying gaussian one so that the resulting vector 2 random function can be easily simulated and analyzed

double forms curvature structures and the p q curvatures we introduce a natural extension of the metric tensor and the hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra these properties are then used to study new riemannian curvature invariants called the p q curvatures they are a generalization of the p curvature obtained by substituting the gauss kronecker tensor to the riemann curvature tensor in particular for p 0 the 0 q curvatures coincide with the h weyl curvature invariants for p 1 the 1 q curvatures are the curvatures of generalized einstein tensors and for q 1 the p 1 curvatures coincide with the p curvatures also we prove that the second h weyl curvature invariant is nonnegative for an einstein manifold of dimension n 4 and it is nonpositive for a conformally flat manifold with zero scalar curvature a similar result is proved for the higher h weyl curvature invariants

ramified partition algebras for each natural number n poset t and t tuple of scalars q we introduce the ramified partition algebra q which is a physically motivated and natural generalization of the partition algebra 24 25 the partition algebra coincides with case t 1 for fixed n and t these algebras like the partition algebra have a basis independent of q we investigate their representation theory in case t 1 2 leq we show that q is quasi hereditary over field k when q 1 q 2 is invertible in k and k is such that certain finite group algebras over k are semisimple e g when k is algebraically closed characteristic zero under these conditions we determine an index set for simple modules of q and construct standard modules with this index set we show that there are unboundedly many choices of q such that q is not semisimple for sufficiently large n but that it is generically semisimple for all n we construct tensor space representations of certain non semisimple specializations of q and show how to use these to build clock model transfer matrices 24 in arbitrary physical dimensions

insights into head related transfer function spatial dimensionality and continuous representation this paper studies head related transfer function hrtf sampling and synthesis in a three dimensional auditory scene based on a general modal decomposition of the hrtf in all frequency range angle domains the main finding is that the hrtf decomposition with the derived spatial basis function modes can be well approximated by a finite number which is defined as the spatial dimensionality of the hrtf the dimensionality determines the minimum number of parameters to represent the hrtf corresponding to all directions and also the required spatial resolution in hrtf measurement the general model is further developed to a continuous hrtf representation in which the normalized spatial modes can achieve hrtf near field and far field representations in one formulation the remaining hrtf spectral components are compactly represented using a fourier spherical bessel series where the aim is to generate the hrtf with much higher spectral resolution in fewer parameters from typical measurements which usually have limited spectral resolution constrained by sampling conditions a low computation algorithm is developed to obtain the model coefficients from the existing measurements the hrtf synthesis using the proposed model is validated by three sets of data i synthetic hrtfs from the spherical head model ii the mit kemar knowles electronics mannequin for acoustics research data and iii 45 subject cipic hrtf measurements

meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature abstract we give a detailed description of the schwartz kernel of the resolvent of the laplacian on a certain class of complete riemannian manifolds with negative sectional curvature near infinity these manifolds have compactifications as c manifolds with boundary on which the metric is conformai to a c metric up to the boundary with conformal factor 2 where is a defining function for the boundary the modified resolvent r s 2 n 1 defined and analytic in r n is shown to extend to be meromorphic in the whole complex plane when s 2 is equal to the sectional curvature at infinity for hyperbolic spaces obtained as quotients hn 1 where is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity s is constant we then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the eisenstein series the basic method used to produce a parametrix for the laplacian is part of a more general construction by g mendoza and the second author but since this is not yet available the present paper has been made essentially self contained

generalized sampling expansion for functions on the sphere functions on the sphere appear in several applications including geodesics imaging and acoustics sampling of these functions may result in aliasing if the sampling condition is not met the generalized sampling expansion introduced by papoulis enables the reconstruction of a band limited function sampled at a frequency lower than the nyquist frequency using the outputs of several linear time invariant systems this paper formulates the generalized sampling expansion for functions on the sphere using spherical harmonics decomposition facilitating sub nyquit sampling without aliasing error an analysis of linear systems on the sphere and the aliasing phenomenon in the spherical harmonics domain is presented examples demonstrating the performance of the method and its limitations are studied

degrees of freedom of a communication channel using dof singular values a fundamental problem in any communication system is given a communication channel between a transmitter and a receiver how many independent signals can be exchanged between them arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper the well known notions of degrees of freedom dof at level and essential dimension of such channels are developed in this general setting we argue that the dof at level and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver we also generalize the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a hilbert space structure we show how these generalized singular values which we call dof singular values can be used to calculate the dof at level and the essential dimension of compact operators that describe communication channels we describe physically realistic channels that require such general channel models

signal recovery with cost constrained measurements we are concerned with the problem of optimally measuring an accessible signal under a total cost constraint in order to estimate a signal which is not directly accessible an important aspect of our formulation is the inclusion of a measurement device model where each device has a cost depending on the number of amplitude levels that the device can reliably distinguish we also assume that there is a cost budget so that it is not possible to make a high amplitude resolution measurement at every point we investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error this problem differs from standard estimation problems in that we are allowed to design the number and noise levels of the measurement devices subject to the cost constraint our main results are presented in the form of tradeoff curves between the estimation error and the cost budget although our primary motivation and numerical examples come from wave propagation problems our formulation is also valid for other measurement problems with similar budget limitations where the observed variables are related to the unknown variables through a linear relation we discuss the effects of signal to noise ratio distance of propagation and the degree of coherence correlation of the waves on these tradeoffs and the optimum cost allocation our conclusions not only yield practical strategies for designing optimal measurement systems under cost constraints but also provide insights into measurement aspects of certain inverse problems

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

poincare lovelock metrics on conformally compact manifolds abstract an important tool in the study of conformal geometry and the ads cft correspondence in physics is the fefferman graham expansion of conformally compact einstein metrics we show that conformally compact metrics satisfying a generalization of the einstein equation poincare lovelock metrics also have fefferman graham expansions moreover we show that conformal classes of metrics that are near that of the round metric on the n sphere have fillings into the ball satisfying the lovelock equation extending the existence result of graham lee for einstein metrics

low complexity estimators for distributed sources in antenna array applications the propagation environment is often more complicated than the ordinarily assumed model of plane wavefronts here a low complexity algorithm is suggested for estimating both the doa and the spread angle of a source subject to local scattering using a uniform linear array the parameters are calculated from the estimates obtained using a standard algorithm such as root music to fit a two ray model to the data the algorithm is shown to give consistent estimates and the statistical performance is studied analytically and through simulations

a double angular propagation model with cluster scattering a statistical multipath propagation model is developed in the double angular domain to include effects of both angle of departure and angle of arrival the model is characterized by a discrete stochastic double angular spectrum of the green s function between two volumes that communicate with each other in the presence of cluster scattering the whole modeling process is a combination of theories assumptions and numerical verification by rigorous full wave electromagnetic simulations notably an uncorrelated scattering assumption for received fields is extended to the double angular domain and the transport theory plays a key role defining the statistical properties of the spectrum of the green s function the spectrum is eventually modeled as a random matrix whose entries are uncorrelated zero mean complex laplacian random variables the real and imaginary parts of the matrix entries are independent and identically distributed and the variance profile follows the reduced incident intensity of the waves through random media the parameters of the model are completely obtained from bulk statistical knowledge of the scattering environments and the resulting model demonstrates good performance on em dof and mimo capacity estimation

degrees of freedom in multiple antenna channels a signal space approach multiple antenna systems that are limited by the area and geometry of antenna arrays are considered given these physical constraints the limit on the available number of spatial degrees of freedom is derived the commonly used statistical multiple input multiple output mimo model is inadequate antenna theory is applied to take into account the area and geometry constraints and to define the spatial signal space so as to interpret experimental channel measurements in an array independent but manageable description of the physical environment based on these modeling strategies for a spherical array of effective aperture a in a physical environment of angular spread spl omega in solid angle the number of spatial degrees of freedom is shown to be a spl omega for uni polarized antennas and 2a spl omega for tri polarized antennas together with the 2wt degrees of freedom for a system of bandwidth w transmitting in an interval t the total degrees of freedom of a multiple antenna channel is therefore 4wta spl omega

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

quadratic variational framework for signal design on the 2 sphere this paper introduces a quadratic variational framework for solving a broad class of signal design problems on the 2 sphere the functional to be extremized combines energy concentration measures using a weighting function in the spatial domain multiplicative weights in the spectral domain and a total energy constraint this leads to two formulations of the signal design problem on the 2 sphere one a fredholm integral equation in the spatial domain and the other an infinite matrix equation in the spectral domain the framework is illustrated by deriving the key equations for the two classical spatio spectral concentration problems on the 2 sphere and for an isotropic filter design that maximizes the filtered energy in addition using the proposed framework we formulate a joint 3 d beamforming application which achieves optimal directivity and spatial resolution simultaneously

intrinsic limits of dimensionality and richness in random multipath fields we study the dimensions or degrees of freedom of farfield multipath that is observed in a limited source free region of space the multipath fields are studied as solutions to the wave equation in an infinite dimensional vector space we prove two universal upper bounds on the truncation error of fixed and random multipath fields a direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension for circular and spherical spatial regions we show that this finite dimension is proportional to the radius and area of the region respectively we use the karhunen loegraveve kl expansion of random multipath fields to quantify the notion of multipath richness the multipath richness is defined as the number of significant eigenvalues in the kl expansion that achieve 99 of the total multipath energy we establish a lower bound on the largest eigenvalue this lower bound quantifies to some extent the well known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum

sublinear capacity scaling laws for sparse mimo channels recent attention on performance analysis of single user multiple input multiple output mimo systems has been on understanding the impact of the spatial correlation model on ergodic capacity in most of these works it is assumed that the statistical degrees of freedom dof in the channel can be captured by decomposing it along a suitable eigenbasis and that the transmitter has perfect knowledge of the statistical dof with an increased interest in large antenna systems in state of the art technologies these implicit channel modeling assumptions in the literature have to be revisited in particular multiantenna measurements have showed that large antenna systems are sparse where only a few dof are dominant enough to contribute towards capacity thus in this work it is assumed that the transmitter can only afford to learn the dominant statistical dof in the channel the focus is on understanding ergodic capacity scaling laws in sparse channels unlike classical results where linear capacity scaling is implicit sparsity of mimo channels coupled with a knowledge of only the dominant dof is shown to result in a new paradigm of sublinear capacity scaling that is consistent with experimental results and physical arguments it is also shown that uniform power signaling over all the antenna dimensions is wasteful and could result in a significant penalty over optimally adapting the antenna spacings in response to the sparsity level of the channel and transmit snr

a generalized spatial correlation model for 3d mimo channels based on the fourier coefficients of power spectrums previous studies have confirmed the adverse impact of fading correlation on the mutual information mi of two dimensional 2d multiple input multiple output mimo systems more recently the trend is to enhance the system performance by exploiting the channels degrees of freedom in the elevation which necessitates the derivation and characterization of three dimensional 3d channels in the presence of spatial correlation in this paper an exact closed form expression for the spatial correlation function scf is derived for 3d mimo channels the proposed method resorts to the spherical harmonic expansion she of plane waves and the trigonometric expansion of legendre and associated legendre polynomials the resulting expression depends on the underlying arbitrary angular distributions and antenna patterns through the fourier series fs coefficients of power azimuth and elevation spectrums the novelty of the proposed method lies in the scf being valid for any 3d propagation environment the developed scf determines the covariance matrices at the transmitter and the receiver that form the kronecker channel model in order to quantify the effects of correlation on system performance the information theoretic deterministic equivalents of the mi for the kronecker model are utilized in both mono user and multi user cases numerical results validate the proposed analytical expressions and elucidate the dependence of the system performance on azimuth and elevation angular spreads and antenna patterns some useful insights into the behavior of mi as a function of downtilt angles are provided the derived model will help evaluate the performance of correlated 3d mimo channels in the future

on landau s eigenvalue theorem and information cut sets a variation of landau s eigenvalue theorem describing the phase transition of the eigenvalues of a time frequency limiting self adjoint operator is presented the total number of degrees of freedom of square integrable multidimensional bandlimited functions is defined in terms of kolmogorov s n width and computed in some limiting regimes where the original theorem cannot be directly applied results are used to characterize up to order the total amount of information that can be transported in time and space by multiple scattered electromagnetic waves rigorously addressing a question originally posed in the early works of toraldo di francia and gabor applications in the context of wireless communication and electromagnetic sensing are discussed

spherical coordinate descriptions of cylindrical and spherical bessel beams this paper derives a generalized spherical harmonic description of bessel beams the spherical harmonic description of the well known cylindrical bessel beams is reviewed and a family of spherical bessel beams are introduced which can provide a number of azimuthal phase variations for a single beam radial amplitude the results are verified by numerical simulations

parametric localization of distributed sources most array processing algorithms are based on the assumption that the signals are generated by point sources this is a mathematical constraint that is not satisfied in many applications in this paper we consider situations where the sources are distributed in space with a parametric angular cross correlation kernel we propose an algorithm that estimates the parameters of this model using a generalization of the music algorithm the method involves maximizing a cost function that depends on a matrix array manifold and the noise eigenvectors we study two particular cases coherent and incoherent spatial source distributions the spatial correlation function for a uniformly distributed signal is derived from this we find the array gain and show that in contrast to point sources it does not increase linearly with the number of sources we compare our method to the conventional point source music algorithm the simulation studies show that the new method outperforms the music algorithm by reducing the estimation bias and the standard deviation for scenarios with distributed sources it is also shown that the threshold signal to noise ratio required for resolving two closely spaced distributed sources is considerably smaller for the new method

new aspects of electromagnetic information theory for wireless and antenna systems this paper investigates information theoretic characterization via shannon s information capacity and number of degrees of freedom of wave radiation antenna and wireless propagation systems specifically the paper derives from the fundamental physical point of view of maxwell s equations describing electromagnetic fields the shannon information capacity of space time wireless channels formed by electromagnetic sources and receivers in a known background medium the theory is developed first for the case of sources working at a fixed frequency time harmonic case and is expanded later to the more general case of temporally bandlimited systems time domain fields in the bandlimited case we consider separately the two cases of time limited and essentially bandlimited systems and of purely bandlimited systems the developments take into account the physical radiated power constraint in addition to a constraint in the source l 2 norm which acts to avoid antenna superdirectivity based on such radiated power and current l 2 norm constraints we derive the shannon information capacity of canonical wireless and antenna systems in free space for a given additive gaussian noise level as well as an associated number of degrees of freedom resulting from such capacity calculations the derived results also illustrate from a new information theoretic point of view the transition from near to far fields