random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

optimum sequence multisets for synchronous code division multiple access channels it is shown that the sum capacity of the symbol synchronous code division multiple access channel with equal average input energy constraints is maximized precisely by those spreading sequence multisets that meet welch s lower bound on total squared correlation it is further shown that the symmetric capacity of the channel determined by these same sequence multisets is equal to the sum capacity

a functional approach to the stationary waiting time and idle period distributions of the gi g 1 queue the gi g 1 queueing model is regarded as a functional which maps the service and interarrival time distributions onto output quantities of interest such as the stationary waiting time distribution for the case where the input distributions have densities techniques from infinite dimensional analysis are used to obtain derivatives and taylor series expansions for the functionals these yield approximations to the output distributions which can be viewed as nonparametric alternatives to parametric approximations such as those provided by infinitesimal perturbation analysis or the phase method

group detection for synchronous gaussian code division multiple access channels the concept of group detection is introduced to address the design of suboptimum multiuser detectors for code division multiple access cdma channels a group detection scheme consists of a bank of p group detectors one each for detecting the information symbols of users in each group of a p group partition of the k simultaneously transmitting users in a parallel group detection scheme these group detectors operate independently whereas in a sequential scheme each group detector uses the decisions of the previous group detectors to successively cancel the interference from those users group detectors based on the generalized likelihood ratio test glrt are obtained for the synchronous gaussian cdma channel the complexity of these detectors is exponential in the group size whereas that of the optimum detector is exponential in k since the partition of users is a design parameter group sizes can be chosen to satisfy a wide range of complexity constraints a key performance result is that the glrt group detectors are optimally group near far resistant furthermore upper and lower bounds on the asymptotic efficiency of the sequential group detectors are derived these bounds reveal that the sequential group detectors can under certain conditions perform as well as glrt group detectors of much larger group sizes group detection provides a unifying approach to multiuser detection when the users are partitioned into k single user groups the glrt a modified form of glrt and the sequential group detectors reduce to previously proposed suboptimal detectors namely the decorrelator the two stage detector and the decorrelating decision feedback detector respectively for the other nontrivial partitions the group detectors are new and have a performance that is commensurate with their complexity

the cassels tate pairing on polarized abelian varieties let a e be a principally polarized abelian variety defined over a global field k and let a be its shafarevich tate group let a nd denote the quotient of a by its maximal divisible subgroup cassels and tate constructed a nondegenerate pairing a nd a nd q z if a is an elliptic curve then by a result of cassels the pairing is alternating but in general it is only antisymmetric using some new but equivalent definitions of the pairing we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on a nd these criteria are expressed in terms of an element c a nd that is canonically associated to the polarization e in the case that a is the jacobian of some curve a down to earth version of the result allows us to determine effectively whether a if finite is a square or twice a square we then apply this to prove that a positive proportion in some precise sense of all hyperelliptic curves of even genus g 2 over q have a jacobian with nonsquare if finite for example it appears that this density is about 13 for curves of genus 2 the proof makes use of a general result relating global and local densities this result can be applied in other situations

outage theorems for mimo block fading channels the connection between the average codeword or frame error probability fep of space time codes and the outage probability over general block fading multiple input multiple output mimo channels is established three archetypal problems are considered under general fading distributions in a single framework wherein the receiver has channel state information whereas the transmitter knows a the fading distribution but not the channel realization b the channel realization but must follow a short term per codeword average power constraint and c the channel realization but is constrained only by a long term average power constraint three telescoping sets of space time codes are defined for a given rate and it is shown that average feps arbitrarily close to the respective outage probabilities for each of the three cases a c can be achieved by codes in each set for sufficiently large frame lengths for the smallest set among the three which contains codes with a spectral norm constraint that is stricter than the average or maximum energy constraints commonly assumed firm sphere packing lower bounds on the fep are obtained and consequently strong converse theorems are proved which assert that the respective outage probabilities also represent the best achievable fep in the large frame length limit moreover the set of spectral norm constrained codes are also shown to be large enough to contain universal codes that can communicate reliably over any channel realization for which the mutual information exceeds the information rate of the code

on harmonic renewal measures let be a probability and nu h sum limits n 1 infty frac 1 n mu n the corresponding harmonic renewal measure complementing earlier results where is concentrated on a halfline we investigate the behaviour ofvh x x 1 and the harmonic renewal functiong x vh x asx ifm1 x dx 0 we also consider the casem1 0

two coverings of jacobians of curves of genus 2 given a curve c of genus 2 defined over a field k of characteristic different from 2 with a jacobian variety j we show that the two coverings corresponding to elements of a large subgroup of h1 gal ks k j 2 ks containing the selmer group when k is a global field can be embedded as an intersection of 72 quadrics in 15k just as the jacobian j itself moreover we actually give explicit equations for the models of these twists in the generic case extending the work of gordon and grant which applied only to the case when all weierstrass points are rational in addition we describe elegant equations of the jacobian itself and answer a question of cassels and flynn concerning a map from the kummer surface in 3 to the desingularized kummer surface in 5

capacity of root mean square bandlimited gaussian multiuser channels continuous time additive white gaussian noise channels with strictly time limited and root mean square rms bandlimited inputs are studied the capacity of the single user and two user rms bandlimited channels are found in easy to compute parametric forms and are compared to the classical formulas for the capacity of strictly bandlimited channels in addition channels are considered where the inputs are further constrained to be pulse amplitude modulated waveforms the capacity of the single user rms bandlimited pam channel is shown to coincide with shannon s capacity formula for the strictly bandlimited channel this shows that the laxer bandwidth constraints precisely offsets the pam structural constraint and illustrates a tradeoff between the time domain and frequency domain constraints in the synchronous two user channel the pair of pulses that achieves the boundary of the capacity region is derived and it is shown that the shapes of the optimal pulses depend not only on the bandwidth but also on the respective signal to noise ratios

the generalized diversity multiplexing tradeoff of the mimo z interference channel the generalized diversity multiplexing tradeoff gdmt of the two user quasi static fading multi input multi output mimo z interference channel z ic is established for the general case with an arbitrary number of antennas at each node under the assumptions of full channel state information at the transmitters csit and a short term average power constraint in the gdmt framework which captures the rate versus reliability tradeoff in the high signal to noise ratio snr regime the direct link snr and cross link interference to noise ratio inr are allowed to be disparate so that their ratios relative to a nominal snr in the decibel scale i e the snr and inr exponents are arbitrary and fixed it is shown that a simple han kobayashi message splitting partial interference decoding scheme that uses only partial csit in which the second transmitter s signal depends only on its cross link channel matrix and the first user s transmit signal does not need any csit can achieve the full csit gdmt of the mimo z ic the gdmt of the mimo z ic under the no csit assumption is also obtained for some range of multiplexing gains the size of this range depends on the numbers of antennas at the four nodes and the snr and inr exponents of the direct and cross links respectively for certain classes of channels including those in which the interfered receiver has more antennas than do the other nodes or when the inr exponent is greater than a certain threshold the gdmt of the mimo z ic under the no csit assumption is also completely characterized

on subordinated distributions and generalized renewal measures where an n eno is some sequence of nonnegative numbers sn neno is the sequence of partial sums s0 0 sn xflxk of another sequence xk ken of i i d random variables and a c r is a fixed borel set such as 0 1 or 0 oo examples of such convolution series are subordinated distributions f 0oan 1 which arise as distributions of random sums and harmonic and ordinary renewal measures a0 0 an 1 n for all n c n in the first an 1 for all n c no in the second case these examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes they are also of interest in connection with representation theorems such as the levy representation of infinitely divisible distributions a traditional approach to such problems is via regular variation if the underlying random variables are nonnegative we can use laplace transforms and the related abelian and tauberian theorems see e g stam 1973 in the context of subordination and feller 1971 xiv 3 in connection with renewal theory embrechts maejima and omey 1984 is a recent treatment of generalized renewal measures along these lines the approach of the present paper is based on the wiener levy gel fand theorem and has occasionally been called the banach algebra method in gruibel 1983 we gave a new variant of this method for the special case of lattice distributions showing that by using the appropriate banach algebras of sequences arbitrarily fine expansions are possible under certain assumptions on the higher order differences of p x1 n fnen here we give a corresponding treatment of nonlattice distributions we restrict ourselves to an analogue of first order differences and obtain a number of theorems which perhaps are described best as next term results to explain this let us consider a special case in more detail

transcendental obstructions to weak approximation on general k3 surfaces abstract we construct an explicit k3 surface over the field of rational numbers that has geometric picard rank one and for which there is a transcendental brauer manin obstruction to weak approximation to do so we exploit the relationship between polarized k3 surfaces endowed with particular kinds of brauer classes and cubic fourfolds

a strong renewal theorem for generalized renewal functions in the infinite mean case letf x be a nonarithmetic c d f on 0 such that 1 f x x l x wherel x is slowly varying and 0 1 leta x be regularly varying with exponent 1 a strong renewal theorem of blackwell type for generalized renewal functions of the form g t equiv sum limits n 0 infty a n f n t is proved here thus extending the recent work of embrechts maejima and omey 1 and that of erickson 4

decision feedback multiuser detection a systematic approach a systematic approach to decision feedback multiuser detection is introduced for the joint detection of symbols of k simultaneously transmitting users of a synchronous correlated waveform multiple access cwma channel with gaussian noise a new performance criterion called symmetric energy is defined which is a low noise indicator of the joint error rate that at least one user is detected erroneously even the best linear detectors can perform poorly in terms of symmetric energy compared to the maximum likelihood detector a general class of decision feedback detectors dfds is defined with o k implementational complexity per user the symmetric energy of arbitrary dfd and bounds on their asymptotic effective energy aee performance are obtained along with an exact bit error rate and aee analysis for the decorrelating dfd the optimum dfd that maximizes symmetric energy is obtained each one of the k optimum decorrelating and conventional dfds that correspond to the k orders in which the users can be detected are shown to outperform the linear optimum decorrelating and conventional detectors respectively in terms of symmetric energy moreover algorithms are obtained for determining the choice of order of detection for the three dfds which guarantee that they uniformly user wise outperform their linear counterparts in addition to optimality in symmetric energy it is also shown that under certain conditions the optimum dfd achieves the aee performance of the exponentially complex maximum likelihood detector for all users simultaneously none of the results of this paper make the perfect feedback assumption the implications of our work on power control for multiuser detection are also discussed

harmonic renewal measures and bivariate domains of attraction in fluctuation theory each probability measure c on a first orthant is associated with a harmonic renewal measure g specifically we consider n s n the ladder time place of a random walk sn using bivariate g we show that when s1 is in a domain of attraction so is n sn this unifies and generalizes results of sinai rogosin

nontrivial elements of sha explained through k3 surfaces we present a new method to show that a principal homogeneous space of the jacobian of a curve of genus two is nontrivial the idea is to exhibit a brauer manin obstruction to the existence of rational points on a quotient of this principal homogeneous space in an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose jacobians have nontrivial tate shafarevich group

tate shafarevich groups and k3 surfaces this paper explores a topic taken up recently by logan and van luijk finding nontrivial 2 torsion elements of the tate shafarevich group of the jacobian of a genus 2 curve by exhibiting brauer manin obstructions to rational points on certain quotients of principal homogeneous spaces of the jacobian whose desingularizations are explicit k3 surfaces the main difference between the methods used in this paper and those of logan and van luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras rather than elliptic fibrations

galois sections for abelianized fundamental groups given a smooth projective curve x of genus at least 2 over a number field k grothendieck s section conjecture predicts that the canonical projection from the etale fundamental group of x onto the absolute galois group of k has a section if and only if the curve has a rational point we show that there exist curves where the above map has a section over each completion of k but not over k in the appendix victor flynn gives explicit examples in genus 2 our result is a consequence of a more general investigation of the existence of sections for the projection of the etale fundamental group with abelianized geometric part onto the galois group we also point out the relation to the elementary obstruction of colliot thelene and sansuc

two cover descent on hyperelliptic curves we describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve with the property that any rational point will lift to one of the covers in particular if the algorithm returns an empty set then the hyperelliptic curve has no rational points this provides a relatively efficiently tested criterion for solvability of hyperelliptic curves we also discuss applications of this algorithm to curves of genus 1 and to curves with rational points

on the brauer group of a surface artin and tate 15 5 1 have shown in 1966 the existence of a canonical skew symmetric pairing on the non p part of br x whose kernel is exactly the set of divisible elements it follows from this fact that if the non p part of br x is finite then its order is a square or twice a square a few years later manin published examples of rational surfaces that is surfaces birational over k to the projective plane with brauer groups equal to z 2z it is only in 1996 that the examples of manin were revisited by urabe who found a mistake in them for rational surfaces the brauer group is relatively easy to understand and urabe 16 remark after 1 17 showed improving on a result of milne 9 that the brauer group of a rational surface has order a square in 17 0 3 urabe then proves in full generality that when p 2 the 2 part of br x modulo its divisible subgroup has order a square thus to complete the proof of theorem 1 it remains to treat the case where p 2 and x k is not rational as we remarked above it was wrongly assumed for about 30 years that br x was not always a square our method of proof provides for any p that the 2 part of br x has order a square via the knowledge of the 2 part

functions of discrete probability measures rates of convergence in the renewal theorem let and be finite measures on the set of integers such that the fourier transform of is an analytic function of the transform the central result shows how may be approximated by linear combinations of convolution powers of applications are given to renewal theory infinitely divisible measures and age dependent branching processes

optimal successive group decoders for mimo multiple access channels we consider a slow fading narrowband multiple input multiple output mimo multiple access channel mac in which multiple users each equipped with multiple transmit antennas communicate to a receiver equipped with multiple receive antennas the users are unaware of the channel state information csi whereas the receiver has perfect csi and employs a successive group decoder sgd we obtain achievable outage probabilities for the case where an outage must be declared simultaneously for all users common outage as well as the case where outages can be declared individually for each user individual outage we then derive the optimum successive group decoder osgd that simultaneously minimizes the common outage probability and the individual outage probability of each user over all sgds of permissible decoding complexity for each channel realization the osgd is also shown to maximize the error exponent of the decodable set of users an adaptive sgd is derived which not only retains the outage optimality of the osgd but also minimizes the expected decoding complexity asymptotically tight in the limit of high signal to noise ratio snr affine approximations are then obtained for the weighted sum common and individual outage capacities and the symmetric outage capacity yielded by the osgd limiting expressions for the relevant capacities as the number of users and the number of receive antennas approach infinity are also obtained and it is shown that the osgd yields symmetric capacity gains commensurate with the decoding complexity allowed simulation results with practical low density parity check ldpc outer codes show that the osgd offers significantly improved performance at low decoding complexity

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

error exponents for maximum likelihood and successive decoders for the gaussian cdma channel random coding error exponents are derived for the gaussian code division multiple access cdma channel for the maximum likelihood and optimum successive decoders error exponents not only specify the capacity region of the channel which is known but also give lower bounds on the rate of exponential decay of the average probability of error as a function of the block length of random codes a comparison of the two decoders in terms of their error exponents is included

on generalized renewal measures and certain first passage times by relating it to the family x t infln 1 sn t t 0 one of the major results is that up t oo for all t e r iff 0 t u0 t 1 a 1 ua i1as t a oo iff e xj 2 x1 oo provided 4 is ultimately increasing a 2 0 a related result is proved for u t h u t and up t e n 2 o4 n p mn t where mn max0 i n s our results form extensions of earlier ones by heyde kalma gut and others who either considered more specific functions 4 or used stronger moment assumptions the proofs are based on a regeneration technique from renewal theory and two martingale inequalities by burkholder davis and gundy