estimation of the eigenvalues and the smallest singular value of matrices abstract the purpose of this paper is to discuss the estimation of eigenvalues and the smallest singular value of matrices firstly we prove that all the eigenvalues of arbitrarily complex matrix are located in one closed disk around trm n of radius n 1 n m f 2 trm 2 n max 1 k n 1 b k n k f c n k k f 2 secondly we present a lower bound for the smallest singular value of matrices finally we give some numerical examples which will show the effectiveness of our results

superiority of superposition multiaccess with single user decoding over tdma in the low snr regime this paper studies the gaussian multiaccess channel with multiantenna basestation in the low signal to noise ratio snr regime we compare the spectral efficiencies of the optimal superposition channel sharing scheme and two simple alternatives the time division multiaccess tdma scheme and superposition multiaccess with single user decoding ssd due to the fact that ssd but not tdma exploits the multiuser multiplexing gain the relative spectral efficiency of ssd over tdma grows drastically as the number of antennas at the basestation increases the results suggest that in the low snr regime with multiple antenna basestation tdma s suboptimality can no longer be offset by its simplicity since ssd can achieve much higher spectral efficiency while the simplicities of the two channel sharing schemes are similar

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

some lower bounds for the spectral radius of matrices using traces abstract let a be an n n matrix with eigenvalues 1 2 n and let m be an integer satisfying rank a m n if a is real the best possible lower bound for its spectral radius in terms of m tr a and tr a 2 is obtained if a is any complex matrix two lower bounds for j 1 n j 2 are compared and furthermore a new lower bound for the spectral radius is given only in terms of tr a tr a 2 a a a aa n and m

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

lower bounds for the spectral radius of a matrix abstract we develop lower bounds for the spectral radius of symmetric skew symmetric and arbitrary real matrices our approach utilizes the well known leverrier faddeev algorithm for calculating the coefficients of the characteristic polynomial of a matrix in conjunction with a theorem by lucas which states that the critical points of a polynomial lie within the convex hull of its roots our results generalize and simplify a proof recently published by tarazaga for a lower bound on the spectral radius of a symmetric positive definite matrix in addition we provide new lower bounds for the spectral radius of skew symmetric matrices we apply these results to a problem involving the stability of fixed points in recurrent neural networks

error estimates for the davenport heilbronn theorems we obtain the first known power saving remainder terms for the theorems of davenport and heilbronn on the density of discriminants of cubic fields and the mean number of 3 torsion elements in the class groups of quadratic fields in addition we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2 torsion elements in the class groups of cubic fields these results prove analytic continuation of the related dirichlet series to the left of the line r s 1

optimum power control at finite blocklength this paper investigates the maximal channel coding rate achievable at a given blocklength n and error probability epsilon when the codewords are subjected to a long term i e averaged over all codeword power constraint the second order term in the large n expansion of the maximal channel coding rate is characterized both for additive white gaussian noise awgn channels and for quasi static fading channels with perfect channel state information available at both the transmitter and the receiver it is shown that in both the cases the second order term is proportional to n 1 ln n 1 2 for the quasi static fading case this second order term is achieved by truncated channel inversion namely by concatenating a dispersion optimal code for an awgn channel subject to a short term power constraint with a power controller that inverts the channel whenever the fading gain is above a certain threshold easy to evaluate approximations of the maximal channel coding rate are developed for both the awgn and the quasi static fading case

an algorithm to compute relative cubic fields let k be an imaginary quadratic number field with class number 1 we describe a new essentially linear time algorithm to list all isomorphism classes of cubic extensions l k up to a bound x on the norm of the relative discriminant ideal the main tools are taniguchi s generalization of davenport heilbronn parametrisation of cubic extensions and reduction theory for binary cubic forms over imaginary quadratic fields finally we give numerical data for k q i and we compare our results with ray class field algorithm ones and with asymptotic heuristics based on a generalization of roberts conjecture

structural properties of optimal transmission policies over a randomly varying channel we consider the problem of transmitting packets over a randomly varying point to point channel with the objective of minimizing the expected power consumption subject to a constraint on the average packet delay by casting it as a constrained markov decision process in discrete time with time averaged costs we prove structural results about the dependence of the optimal policy on buffer occupancy number of packet arrivals in the previous slot and the channel fading state for both i i d and markov arrivals and channel fading the techniques we use to establish such results convexity stochastic dominance decreasing differences are among the standard ones for the purpose our main contribution however is the passage to the average cost case a notoriously difficult problem for which rather limited results are available the novel proof techniques used here are likely to have utility in other stochastic control problems well beyond their immediate application considered here

new eigenvalue estimates for complex matrices abstract in a recent work 1 we have proved that all the eigenvalues j of a complex matrix a of order n lie in a disk with center at tr a n and radius n 1 n k 1 n k 2 tr a 2 n 1 2 in this paper we prove that the eigenvalues of a are contained in a rectangle with vertices on the boundary circle of the mentioned disk improving in this way the result in 1 we recall some inequalities which alloww us to bound the radius of the disk and the rectangle the bounds can be computed without knowing the eigenvalues of a

counting cubic extensions with given quadratic resolvent abstract given a number field k and a quadratic extension k 2 we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose galois closure contains k 2 as quadratic subextension ordered by the norm of their relative discriminant ideal the main tool is kummer theory we also study in detail the error term of the asymptotics and show that it is o x for an explicit 1

the mean number of 3 torsion elements in the class groups and ideal groups of quadratic orders we determine the mean number of 3 torsion elements in the class groups of quadratic orders where the quadratic orders are ordered by their absolute discriminants moreover for a quadratic order o we distinguish between the two groups cl3 o the group of ideal classes of order 3 and i3 o the group of ideals of order 3 we determine the mean values of both cl3 o and i3 o as o ranges over any family of orders defined by finitely many or in suitable cases even infinitely many local conditions as a consequence we prove the surprising fact that the mean value of the difference cl3 o i3 o is equal to 1 regardless of whether one averages over the maximal orders in complex quadratic fields or over all orders in such fields or indeed over any family of complex quadratic orders defined by local conditions for any family of real quadratic orders defined by local conditions we prove similarly that the mean value of the difference cl3 o 1 3 i3 o is equal to 1 independent of the family

localization of eigenvalues in elliptic regions abstract we prove that the eigenvalues j of an n by n complex matrix a with its characteristic polynomial having real coefficients lie in the elliptic region defined by 2 tr a n 2 2 y 2 2 2 where n 1 n n 1 n re k 2 re te a 2 n 1 2 and n 1 n k 1 n im k 2 1 2 this region is intersected with the strip y 1 2 k 1 u im k 2 1 2 to obtain an improved eigenvalue localization region we also give bounds for the semiaxes which can be computed without knowing the eigenvalues of a when a has r n nonzero eigenvalues we obtain a smaller elliptic region containing such nonzero eigenvalues

adaptive training for correlated fading channels with feedback we consider data transmission through a time selective correlated first order markov rayleigh fading channel subject to an average power constraint the channel is estimated at the receiver with a pilot signal and the estimate is fed back to the transmitter the estimate is used for coherent demodulation and to adapt the data and pilot powers we derive the hamilton jacobi bellman hjb equation for the optimal policy in a continuous time limit where the channel state evolves as an ornstein uhlenbeck diffusion process and is estimated by a kalman filter at the receiver finding an explicit solution to the hjb equation as well as proving that a twice differentiable solution exists appears to be quite challenging however assuming that such a solution does exist we explicitly determine the optimal pilot and data power control policies the optimal pilot policy switches between zero and the maximum peak constrained value bang bang control and approximates the optimal discrete time policy at low signal to noise ratios snrs equivalently large bandwidths the switching boundary is defined in terms of the system state estimated channel mean and associated error variance and can be explicitly computed under the optimal policy the transmitter conserves power by decreasing the training power when the channel is faded thereby increasing the data rate numerical results show a significant increase in achievable rate due to the adaptive training scheme with feedback relative to constant nonadaptive training which does not require feedback the gain is more pronounced at relatively low snrs and with fast fading results are further verified through monte carlo simulations

power minimization of central wishart mimo block fading channels in this correspondence a central wishart multiple input multiple output mimo k block fading channel with an information outage probability constraint is studied under this block limited channel condition we aim to minimize the transmit power required for attaining a given outage probability based on the statistical channel information at the transmitter scit using gaussian approximation to express the probability density function pdf of the instantaneous mutual information and by deriving analytically the mean and variance of the mutual information of the mimo channel the optimal power allocation can be obtained numerically by a simple one dimensional sampling method such as dividing rectange direct

more estimates for eigenvalues and singular values abstract there is an interesting relation between the angle that a matrix forms with the identity and its eigenvalues then we show for symmetric matrices that some information about the matrix can be known if the angle is computed a lower bound for the maximum eigenvalue is obtained for symmetric positive semidefinite matrices this inequality and the upper bound for eigenvalues given in 5 provide us an interval in which must lie the maximum eigenvalue finally some results from 5 and the first part of this paper are generalized for the non symmetric case

approaching the zero outage capacity of mimo ofdm without instantaneous water filling orthogonal frequency division multiplexing ofdm transforms a frequency selective multiple input multiple output mimo fading channel into a mimo ofdm channel that has a well defined outage capacity a transmitter with channel knowledge can achieve this capacity by a combination of eigenbeamforming and water filling the eigenbeamforming transforms the mimo ofdm channel into a parallel bank of scalar channels and the water filling procedure optimally allocates rate and energy to the scalar channels a form of adaptive modulation this paper shows that the water filling procedure is not necessary to approach the zero outage capacity of the mimo ofdm channel it is sufficient instead to use a combination of eigenbeamforming and a fixed nonadaptive rate allocation the fixed allocation depends only on the statistics of the channel and is independent of the particular channel realization this paper proves that the capacity penalty incurred by the fixed allocation approaches zero as the number of antennas grows large numerical results indicate that the convergence is fast for example the fixed allocation suffers an snr penalty of less than 0 2 db for a 6 input 6 output rayleigh fading mimo ofdm channel at 8 bits per signaling interval when the channel is assumed to be uncorrelated between antennas and between channel taps a main conclusion is that eigenbeamforming is the most valuable way to exploit knowledge of the channel at the transmitter and that any subsequent adaptive modulation has minimal relative value

low complexity delay constrained beamforming for multi user mimo systems with imperfect csit we consider the delay constrained beamforming control for downlink multi user mimo mu mimo systems with imperfect channel state information at the transmitter csit the delay constrained control problem is formulated as an infinite horizon average cost partially observed markov decision process to deal with the curse of dimensionality we introduce a virtual continuous time system and derive a closed form approximate value function using perturbation analysis w r t the csit errors to deal with the challenge of the conditional packet error rate per we build a tractable closed form approximation using a bernstein type inequality based on the closed form approximations of the relative value function and the conditional per we propose a conservative formulation of the original beamforming control problem the conservative problem is non convex and we transform it into a convex problem using the semidefinite relaxation sdr technique we then propose an alternating iterative algorithm to solve the sdr problem finally we show through simulations that the proposed scheme has significant performance gain compared with various baselines

optimal power delay tradeoffs in fading channels small delay asymptotics when transmitting stochastically arriving data over fading channels there is an inherent tradeoff between the required average transmission power and the average queueing delay experienced by the data this tradeoff can be exploited by appropriately scheduling the transmission of data over time in this paper we study the behavior of the optimal power delay tradeoff for a single user in the regime of asymptotically small delays in this regime we first lower bound how much average power is required as a function of the average queueing delay we show that the rate at which this bound increases as the delay becomes asymptotically small depends on the behavior of the fading distribution near zero as well as the arrival statistics we lower bound this rate for two different classes of fading distributions one class that requires infinite power to minimize the queueing delay and one class that requires only finite power we then show that for both classes the bounds can essentially be achieved by a sequence of simple channel threshold policies which only transmit when the channel gain is greater than a given threshold we also consider several other transmission scheduling policies and characterize their convergence behavior in the small delay regime

tabulation of cubic function fields via polynomial binary cubic forms we present a method for tabulating all cubic function fields over mathbb f q t whose discriminant d has either odd degree or even degree and the leading coefficient of 3d is a non square in mathbb q up to a given bound x on d q deg d our method is based on a generalization of belabas method for tabulating cubic number fields the main theoretical ingredient is a generalization of a theorem of davenport and heilbronn to cubic function fields along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms the algorithm requires o q 4 x 1 epsilon field operations when d has odd degree and o q 5 x 1 epsilon when d has even degree it performs quite well in practice the algorithm examples and numerical data for q 5 7 11 13 are included

the best possible lower bound for the perron root using traces abstract let a be an n n matrix with real eigenvalues wolkowicz and styan presented bounds for the eigenvalues using only n tr a and tr a 2 we show that their lower bound for the largest eigenvalue works also as a lower bound for the perron root of a if a is nonnegative and its eigenvalues are not necessarily real we also show that this bound is optimal under certain conditions finally we solve completely the problem to find the optimal lower bound for the perron root using only n tr a and tr a 2

channels with block interference a new class of channel models with memory is presented in order to study various kinds of interference phenomena it is shown among other things that when all other parameters are held fixed channel capacity c is an em increasing function of the memory length while the cutoff rate r 0 generally is a em decreasing function calculations with various explicit coding schemes indicate that c is better than r 0 as a performance measure for these channel models as a partial resolution of this c versus r 0 paradox the conjecture is offered that r 0 is more properly a measure of coding delay rather than of coding complexity

communication over fading channels with delay constraints we consider a user communicating over a fading channel with perfect channel state information data are assumed to arrive from some higher layer application and are stored in a buffer until transmitted we study adapting the user s transmission rate and power based on the channel state information as well as the buffer occupancy the objectives are to regulate both the long term average transmission power and the average buffer delay incurred by the traffic two models for this situation are discussed one corresponding to fixed length variable rate codewords and one corresponding to variable length codewords the tradeoff between the average delay and the average transmission power required for reliable communication is analyzed a dynamic programming formulation is given to find all pareto optimal power delay operating points we then quantify the behavior of this tradeoff in the regime of asymptotically large delay in this regime we characterize simple buffer control policies which exhibit optimal characteristics connections to the delay limited capacity and the expected capacity of fading channels are also discussed

on the energy delay tradeoff in streaming data finite blocklength analysis this paper investigates basic trade offs between energy and delay in wireless communication systems using finite blocklength theory we first assume that data arrive in constant stream of bits which are put into packets and transmitted over a communications link our results show that depending on exactly how energy is measured in general energy depends on sqrt d 1 or sqrt d 1 log d where d is the delay this means that the energy decreases quite slowly with increasing delay furthermore to approach the absolute minimum of 1 59 db on energy bandwidth has to increase very rapidly much more than what is predicted by infinite blocklength theory we then consider the scenario when data arrive stochastically in packets and can be queued we devise a scheduling algorithm based on finite blocklength theory and develop bounds for the energy delay performance our results again show that the energy decreases quite slowly with increasing delay

on lower and upper bounds in smooth and strongly convex optimization we develop a novel framework to study smooth and strongly convex optimization algorithms focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear operators this in turn reveals a powerful connection between a class of optimization algorithms and the analytic theory of polynomials whereby new lower and upper bounds are derived whereas existing lower bounds for this setting are only valid when the dimensionality scales with the number of iterations our lower bound holds in the natural regime where the dimensionality is fixed lastly expressing it as an optimal solution for the corresponding optimization problem over polynomials as formulated by our framework we present a novel systematic derivation of nesterov s well known accelerated gradient descent method this rather natural interpretation of agd contrasts with earlier ones which lacked a simple yet solid motivation

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

twists of the iwasawa tate zeta function the intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by tate 14 and iwasawa 9 in independent efforts to simplify hecke s analysis of his l functions attached to gr6bencharakters our twists of the iwasawa tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups many of the basic problems of this general theory have already been solved by yukie and the author see 20 and 21 1 but still await appearance in final written form thus it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper let k be a global field i e algebraic number field or function field of a curve defined over a finite field let n and a denote the ring of adeles and the group of ideles of k respectively let co be a quasicharacter of the idele group trivial on the embedded group k of nonzero elements of k the group f2 f2 k of all such quasicharacters co has a natural riemann surface structure let l i denote the usual idele norm on a and for s tl2 let co be the principal quasicharacter co t ltl let re co be the real part of co f2 defined as the unique real number a such that l o i co for any reductive linear algebraic group g defined over k let g denote the adelizat ion of g over k as defined in 16 and let gk be the group of k rational points of g embedded as a discrete group in ga for simplicity we shall fix g as g1 now although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k rational character let p g g1 v be an irreducible k rational representation of g in the finite dimensional vector space v of dimension m let i be the identity matrix in g by schur s lemma there is an integer d such that p ti tdp i for all scalars t we shall always assume that d 0 let dg be any convenient left invariant measure on the quotient ga gk let v be the schwartz bruhat space of functions on v for the time being let vk be any gk invariant subset of vk the zeta function associated to the representation p may now be defined

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

the number of extensions of a number field with fixed degree and bounded discriminant we give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant x these bounds improve on work of schmidt we also prove various related results such as lower bounds for the number of extensions and upper bounds for galois extensions

asymptotic bounds on the power delay tradeoff for fading point to point links from geometric bounds on the stationary distribution of the queue length the optimal power delay tradeoff is studied for a time slotted independently and identically distributed fading point to point link with perfect channel state information at both transmitter and receiver and with random packet arrivals to the transmitter queue it is assumed that the transmitter can control the number of packets served by controlling the transmit power in the slot the optimal tradeoff between average power and average delay is analyzed for stationary and monotone transmitter policies for such policies an asymptotic lower bound on the minimum average delay of the packets is obtained when average transmitter power approaches the minimum average power required for transmitter queue stability the asymptotic lower bound on the minimum average delay is obtained from geometric upper bounds on the stationary distribution of the queue length this approach which uses geometric upper bounds also leads to an intuitive explanation of the asymptotic behavior of average delay the asymptotic lower bounds along with previously known asymptotic upper bounds are used to identify three new cases where the order of the asymptotic behavior differs from that obtained from a previously considered approximate model in which the transmit power is a strictly convex function of real valued service batch size for every fade state

finite horizon cal h _ infty control for discrete time varying systems with randomly occurring nonlinearities and fading measurements this technical note deals with the cal h infty control problem for a class of discrete time varying nonlinear systems with both randomly occurring nonlinearities and fading measurements over a finite horizon the system measurements are transmitted through fading channels described by a modified stochastic rice fading model the purpose of the addressed problem is to design a set of time varying controllers such that in the presence of channel fading and randomly occurring nonlinearities the cal h infty performance is guaranteed over a given finite horizon the model transformation technique is first employed to simplify the addressed problem and then the stochastic analysis in combination with the completing squares method are carried out to obtain necessary and sufficient conditions of an auxiliary index which is closely related to the finite horizon cal h infty performance moreover the time varying controller parameters are characterized via solving coupled backward recursive riccati difference equations rdes a simulation example is utilized to illustrate the usefulness of the proposed controller design scheme

a novel unified approach to invariance conditions for a linear dynamical system in this paper we propose a novel simple and unified approach to explore sufficient and necessary conditions i e invariance conditions under which four classic families of convex sets namely polyhedra polyhedral cones ellipsoids and lorenz cones are invariant sets for a linear discrete or continuous dynamical system for discrete dynamical systems we use the theorems of alternatives i e farkas lemma and s lemma to obtain simple and general proofs to derive invariance conditions this novel method establishes a solid connection between optimization theory and dynamical system also using the s lemma allows us to extend invariance conditions to any set represented by a quadratic inequality such sets include nonconvex and unbounded sets for continuous dynamical systems we use the forward or backward euler method to obtain the corresponding discrete dynamical systems while preserves invariance this enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems finally some numerical examples are presented to illustrate these invariance conditions

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

iterative system solvers for the frequency analysis of linear mechanical systems abstract the paper deals with the numerical treatment of the direct frequency domain dfd analysis of linear mechanical systems attention is mainly focused on the solution of the complex system of equations needed for each analyzed frequency strategies for transforming the system into the shifted form t i z d are proposed and discussed where is related to the frequency for two formulations of the shifted system the performance of some krylov sub space iterative solvers is tested and compared to that of a multi frontal direct method advantage is taken of the shifted form in solving simultaneously a large number of systems resulting from different values of the shift frequency numerical experiments on some prototype structural dynamics problems are reported the results shown demonstrate how the devised strategies for the iterative solution can outperform in many cases the direct solver

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

on the density of unnormalized tamagawa numbers of orthogonal groups ii this is the second part of a series of three papers in this series of papers we determine the density of unnormalized tamagawa numbers of projective special orthogonal groups defined over a fixed number field in this part we determine local constants at infinite places which appear in the density theorem and prove the density theorem when the number of variables n is odd for that purpose we make a dirichlet series whose coefficients are unnormalized tamagawa numbers of projective special orthogonal groups and prove that it is rather simple when n is odd

on ip1 p2 antinorms of nonnegative matrices let 0 p p1 p2 1 x xi rn a aik rm n i e the vector x is elementwise positive and so is the matrix a the lp antinorm of x is defined by x p ixpi1p the lp1 p2 antinorm of a by a p1 p2 k iap1ikp2p11p2 and the lp1 p2 operator antinorm of a by a minp1 p2 infx rn ax p1 x p2 several properties of the antinorms p p1 p2 and minp1 p2 especially the supermultiplicativity of p1 p2 are studied and generalized to hold for nonnegative matrices the results are analogous with the well known submultiplicativity properties of lq1 q2 norms and lq1 q2 operator norms q1 q2 1 a generalization is outlined

bounds for eigenvalues of matrix polynomials upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial or matrix the bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices they generalize various existing bounds for scalar polynomials and single matrices a variety of tools are used in the derivations including block companion matrices gershgorin s theorem the numerical radius and associated scalar polynomials numerical experiments show that the bounds can be surprisingly sharp on practical problems

optimal periodic transmission power schedules for remote estimation of arma processes we consider periodic sensor transmission power allocation with an average energy constraint the sensor sends its kalman filter based state estimate to the remote estimator through an unreliable link dropout probabilities depend on the power level used to encompass applications where the estimator needs to attend to multiple tasks we allow for irregular sampling following a periodic pattern using properties of an underlying markov chain model we derive an explicit expression for the estimation error covariance the results are then used to study optimal sensor power scheduling which minimizes the average error covariance

on the broadcast capacity of wireless networks with cooperative relays a fundamental problem in wireless networks is determining the broadcast capacity i e the maximum data transfer rate from a given node to every other node in a relay network this paper studies the scaling of the broadcast capacity for a network with a single source and n destinations of which f n are randomly selected to also act as relays in high density networks i e the node density goes to infinity the network area is fixed it is shown that the broadcast capacity is upper bounded by log f n schemes are provided that achieve i log f n throughput if the channel fading is spatially continuous ii log log f n throughput if the channel fading is spatially i i d for extended networks i e the node density is fixed the network area goes to infinity the broadcast capacity is upper bounded by 1 under channel models with fading and path loss exponent 2 a multistage cooperative broadcasting scheme which achieves 1 broadcast rate for the high density extended networks with pathloss channel model is proposed these results quantifies the gains obtained due to cooperation compared to multihop noncooperative broadcasting which has a maximum rate that scales as 1 for high density and 1 log f n 2 for extended networks

improving bounds for eigenvalues of complex matrices using traces abstract let a be a complex matrix of order n with eigenvalues j j 1 2 n and m be any integer satisfying rank a m n the bound for j 2 by kress de vries and wegmann is strengthened furthermore new bounds are presented to estimate the spectral radius of a using m and traces of a a 2 a a and a a aa we also improve some wolkowicz styan bounds and previous localization of eigenvalues in rectangular or elliptic regions using traces several simple lower bounds for the spectral radius are given involving tr a tr a 2 tr a 3 and m

optimal beamforming for video streaming in multiantenna interference networks via diffusion limit in this paper we consider queue aware beamforming control for video streaming applications in multiantenna interference network using heavy traffic approach we derive the diffusion limit for the discrete time queuing system and obtain an ergodic excess beamforming control problem for the diffusion limit system the ergodic control problem is to minimize the average power costs of the access points subject to the constraints on the playback interruption costs and buffer overflow costs of the mobile users to deal with the queue coupling challenge we utilize the weak interference coupling property in the network using the theories of calculus and perturbation techniques we derive a closed form approximate priority function of the optimality equation and the associated error bound based on this approximation we propose a low complexity queue aware beamforming control algorithm which is asymptotically optimal for sufficiently small cross channel path gain finally the proposed scheme is compared with various baselines through simulations and it is shown that significant performance gain can be achieved

density of cubic field discriminants in this paper we give a conjectural refinement of the davenport heilbronn theorem on the density of cubic field discriminants we explain how this refinement is plausible theoretically and agrees very well with computa tional data let an be the number of isomorphism classes of abelian cubic fields with discrim inant n let bn be the number of isomorphism classes of non abelian cubic fields with discriminant n the numbers an are very well understood the numbers bn have been the subject of extensive theoretical and computational study for at least sixty years but are less well understood the object of this note is to contribute to the study of these bn by bringing together the theoretical and computational literature for define g x

optimal rate delay tradeoffs and delay mitigating codes for multipath routed and network coded networks via multiterminal information theory a framework is presented for deriving fundamental rate delay tradeoffs that delay mitigating codes must have when utilized over multipath routed and random linear network coded networks the rate delay tradeoff is formulated as a calculus problem on a capacity region of a related abstracted broadcast channel given this general framework for studying such rate delay tradeoffs the extreme case of uniform networks in which each possible received packet arrival order is equally likely is considered for these networks the rate delay calculus problem is simplified to an integer programming problem which for small numbers of packets may be solved explicitly or for larger numbers of packets may be accurately approximated through the calculus of variations by appropriate relaxation of an integer constraint explicit expressions for the rate delay tradeoff in uniform networks are presented in the special cases of i constant packet inter arrival times and ii exponential independent and identically distributed i i d packet arrival times finally the delay mitigating codes achieving these rate delay tradeoffs are discussed

on the numerical solution of lambda 2 a lambda b c x b and application to structural dynamics in this paper we address the numerical solution of a large linear system depending quadratically on a parameter that varies in a wide range we analyze a solution method whose computational cost grows only sublinearly with the number of parameters that relies on the use of an indefinite inner product important implementation aspects are treated in detail the problem arises in various application areas we shall report on our experience with cases in structural dynamics

mimo transmission control in fading channels a constrained markov decision process formulation with monotone randomized policies this paper addresses the optimal power and rate allocation control in multiple input multiple output mimo wireless systems over markovian fading channels the problem is posed as an infinite horizon average cost constrained markov decision process cmdp with the goal of minimizing the average transmission power subject to delay constraints by using a lagrangian formulation of the cmdp we use the concepts of stochastic dominance submodularity and multimodularity to prove that the optimal randomized policies are monotone three important structural results on the nature of the optimal randomized policies are derived first we show that the action space can be exponentially reduced by decomposing the rate allocation problem into bit loading problem across individual antennas and the total rate allocation based on the current buffer occupancy and channel state second we show that the optimal rate allocation policy is a randomized mixture of two pure policies that are monotonically increasing in the buffer occupancy finally we show that the optimal power allocation is piecewise linear in the delay constraint these three structural results can be exploited to devise efficient online reinforcement learning algorithms for optimal rate allocation

monotonicity of constrained optimal transmission policies in correlated fading channels with arq we consider transmission scheduling using an arq protocol with retransmissions given channel state information csi and a correlated fading channel the problem is formulated as a countable state infinite horizon average cost markov decision process mdp with an average delay constraint our main result is to give sufficient conditions on the channel memory and transmission cost so that the optimal transmission scheduling policy is a monotonically increasing function of the buffer occupancy in proving this result we first prove positive recurrence stability of the buffer the monotone structure proof consists of two steps first the constrained mdp cmdp is transformed into an unconstrained mdp using a lagrangian dynamic programming formulation it is proved that the unconstrained optimal policy is pure and monotonically increasing in the buffer occupancy it is then shown that the constrained optimal policy is a randomized mixture of two pure transmission policies that are monotone in the buffer state finally the monotone structure of the optimal transmission policy is exploited to derive a monotone policy q learning algorithm and a stochastic approximation based monotone policy search algorithm for estimating the optimal policy in real time

on the davenport heilbronn theorems and second order terms we give simple proofs of the davenport heilbronn theorems which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3 torsion elements in the class groups of quadratic fields having bounded discriminant we also establish second main terms for these theorems thus proving a conjecture of roberts our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings