on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

three transmit antenna space time codes based on su 3 fully diverse constellations i e a set of unitary matrices whose pairwise differences are nonsingular are useful in multiantenna communications especially in multiantenna differential modulation since they have good pairwise error properties recently group theoretic ideas especially fixed point free fpf groups have been used to design fully diverse constellations of unitary matrices here we give systematic design methods of space time codes which are appropriate for three transmit antenna differential modulation the structures of the codes are motivated by the special unitary lie group su 3 one of the codes which is called the ab code has a fast maximum likelihood ml decoding algorithm using complex sphere decoding diversity products of the codes can be easily calculated and simulated performance shows that they are better than group based codes especially at high rates and as good as the elaborately designed nongroup code

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

cyclic algebras for noncoherent differential space time coding we investigate cyclic algebras for coding over the differential noncoherent channel cyclic algebras are an algebraic object that became popular for coherent space time coding since it naturally yields linear families of matrices with full diversity coding for the differential noncoherent channel has a similar flavor in the sense that it asks for matrices that achieve full diversity except that these matrices furthermore have to be unitary in this work we give a systematic way to find infinitely many unitary matrices inside cyclic algebras which holds for all dimensions we show how cyclic algebras generalize previous families of unitary matrices obtained using the representation of fixed point free groups as an application of our technique we present families of codes for three and four antennas that achieve high coding gain

cayley differential unitary space time codes one method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter and to decode differentially without knowing the channel coefficients at the receiver since channel knowledge is not required at the receiver differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible either because of rapid changes in the channel characteristics or because of limited system resources although this basic principle is well understood it is not known how to generate good performing constellations of unitary matrices for any number of transmit and receive antennas and for any rate this is especially true at high rates where the constellations must be rapidly encoded and decoded we propose a class of cayley codes that works with any number of antennas and has efficient encoding and decoding at any rate the codes are named for their use of the cayley transform which maps the highly nonlinear stiefel manifold of unitary matrices to the linear space of skew hermitian matrices this transformation leads to a simple linear constellation structure in the cayley transform domain and to an information theoretic design criterion based on emulating a cauchy random matrix moreover the resulting cayley codes allow polynomial time near maximum likelihood ml decoding based on either successive nulling canceling or sphere decoding simulations show that the cayley codes allow efficient and effective high rate data transmission in multiantenna communication systems without knowing the channel

algebraic cayley differential space time codes cayley space time codes have been proposed as a solution for coding over noncoherent differential multiple input multiple output mimo channels based on the cayley transform that maps the space of hermitian matrices to the manifold of unitary matrices cayley codes are particularly suitable for high data rate since they have an easy encoding and can be decoded using a sphere decoder algorithm however at high rate the problem of evaluating if a cayley code is fully diverse may become intractable and previous work has focused instead on maximizing a mutual information criterion the drawback of this approach is that it requires heavy optimization which depends on the number of antennas and rate in this work we study cayley codes in the context of division algebras an algebraic tool that allows to get fully diverse codes we present an algebraic construction of fully diverse cayley codes and show that this approach naturally yields without further optimization codes that perform similarly or closely to previous unitary differential codes including previous cayley codes and codes built from lie groups

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

differential space time modulation space time coding and modulation exploit the presence of multiple transmit antennas to improve the performance on multipath radio channels thus far most work on space time coding has assumed that perfect channel estimates are available at the receiver in certain situations however it may be difficult or costly to estimate the channel accurately in which case it is natural to consider the design of modulation techniques that do not require channel estimates at the transmitter or receiver we propose a general approach to differential modulation for multiple transmit antennas based on group codes this approach ran be applied to any number of transmit and receive antennas and any signal constellation we also derive low complexity differential receivers error bounds and modulator design criteria which we use to construct optimal differential modulation schemes for two transmit antennas these schemes can be demodulated with or without channel estimates this permits the receiver to exploit channel estimates when they are available the performance degrades by approximately 3 db when estimates are not available

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

representation theory for high rate multiple antenna code design multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment but the practical success of using multiple antennas depends crucially on our ability to design high rate space time constellations with low encoding and decoding complexity it has been shown that full transmitter diversity where the constellation is a set of unitary matrices whose differences have nonzero determinant is a desirable property for good performance we use the powerful theory of fixed point free groups and their representations to design high rate constellations with full diversity furthermore we thereby classify all full diversity constellations that form a group for all rates and numbers of transmitter antennas the group structure makes the constellations especially suitable for differential modulation and low complexity decoding algorithms the classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd we therefore also consider extensions of the constellation designs to nongroups we conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

kinetic approach to long time behavior of linearized fast diffusion equations we show that the rate of convergence towards the self similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self similar solution at a fixed time as a consequence we find an improved rate of convergence to self similarity in terms of a fourier based distance between two solutions the results are based on the asymptotic equivalence of a collisional kinetic model of boltzmann type with a linear fokker planck equation with nonconstant coefficients and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous boltzmann equation for maxwell molecules

self similarity and power like tails in nonconservative kinetic models in this paper we discuss the large time behavior of solution of a simple kinetic model of boltzmann maxwell type such that the temperature is time decreasing and or time increasing we show that under the combined effects of the nonlinearity and of the time monotonicity of the temperature the kinetic model has non trivial quasi stationary states with power law tails in order to do this we consider a suitable asymptotic limit of the model yielding a fokker planck equation for the distribution the same idea is applied to investigate the large time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and or decreasing of the mean wealth in this last case the large time behavior of the solution shows a pareto power law tail numerical results confirm the previous analysis

optimal space time constellations from groups we consider the design of space time constellations based on group codes for fading channels with multiple transmit and receive antennas these codes can be viewed as multiantenna extensions of phase shift keying psk in the sense that all codewords have equal energy all are rotations of a fixed codeword and there is a simple differential transmission rule that allows data to be sent without channel estimates at the transmitter or receiver for coherent detection we show that all optimal full rank space time group codes are unitary each code matrix has equal energy orthogonal rows this leads to a simpler code design criterion and suggests that unitary codes may play an important role in coherent as well as noncoherent communication for any number of transmit antennas t we then use the design criterion to characterize all full rank unitary space time group codes of minimum block length also t which have 2 sup p codewords these results allow us to characterize all optimal 2 sup p ary unitary group codes with square code matrices this restricted class of block codes matches the class proposed for differential modulation by hughes see ieee trans inform theory vol 46 p 2567 78 nov 2000 and by hochwald and sweldens see ieee trans commun vol 48 p 2041 2052 dec 2000

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

on the conjectures of su 3 and ab unitary space time codes proofs to the conjectures made by jing and hassibi on having fully diverse 3times3 su 3 and ab unitary space time codes are presented in this correspondence we first prove that the su 3 codes are fully diverse if and only if the design parameters p q r and s are all odd integers and in addition are relatively prime for the type i ab codes it is shown that full diversity can be achieved if and only if the integers p q r and s are relatively prime finally we show that such condition is also sufficient for having fully diverse type ii ab codes

unitary nongroup stbc from cyclic algebras space time block codes stbcs are designed for multiple input multiple output mimo channels in order to avoid errors single input single output siso fading channels require long coding blocks and interleavers that result in high delays if one wishes to increase the data rate it is necessary to take advantage of space diversity early stbc that where developed by alamouti for known channels and by tarokh for unknown channels have been proven to increase the performance of channels characterized by rayleigh fading codes that are based on division algebras have by definition nonzero diversity and therefore are suitable for stbc in order to achieve high rates at low symbol to noise ratio snr this work presents new high diversity group based stbcs with improved performance both in known and unknown channels we describe two new sets of codes for multiple antenna communication the first set is a set of superquaternions and improves considerably on the alamouti codes it is based on the mathematical fact that normalized integral quaternions are very well distributed over the unit sphere in four dimensional 4 d euclidean space the second set of codes gives arrays of 3times3 unitary matrices with full diversity here the idea is to use cosets of finite subgroups of division algebras that are nine dimensional 9 d over their center which is a finite cyclotomic extension of the field of rational numbers it is shown that these codes outperform alamouti and g mr

unitary space time constellation designs from group codes in this correspondence we propose new unitary space time constellation designs with high diversity products our hamiltonian and product constellations are based on slepian s group codes and can be used for any number of antennas and any data rate our hamiltonian constellations achieve the theoretical upper bound of diversity product when the number of transmitter antennas is even and the cardinality of the signal constellation is less than or equal to 5 many of our hamiltonian and product constellations outperform and have higher diversity products than the best known designs in the literature these include orthogonal designs dicyclic groups cyclic groups parametric codes numerical approaches nongroup designs cayley codes tast codes and some constellations obtained from fixed point free groups

omnidirectional precoding for massive mimo with uniform rectangular array part ii numerical optimization based schemes in part ii of this two part paper we develop novel numerical optimization approaches to design omnidirectional precoding schemes for a massive multiple input multiple output system equipped with a uniform rectangular array ura to this end we first formulate the omnidirectional precoding design as a semidefinite program sdp with rank constraint an iterative rank reduction algorithm is then proposed to produce omnidirectional transmission solution with a minimum number of precoding vectors it is shown that the proposed algorithm can always obtain three precoding vectors i e a rank 3 precoding matrix to generate a perfectly omnidirectional power radiation pattern for any p times q ura with min lbrace p q rbrace 2 or yield an omnidirectional transmission solution with theoretically minimum two precoding vectors for any p times q ura with min lbrace p q rbrace leq 2 in addition to facilitate analog precoding design we further impose constant modulus constraints for every entries of the precoding matrix through judicious re formulation we develop an efficient newton s method which can compute four constant modulus precoding vectors to generate an omnidirectional power radiation pattern for any ura configuration the numerical results demonstrate the merits of the proposed schemes

strong convergence towards self similarity for one dimensional dissipative maxwell models we prove the propagation of regularity uniformly in time for the scaled solutions of the one dimensional dissipative maxwell models introduced in d ben avraham e ben naim k lindenberg a rosas self similarity in random collision processes phys rev e 68 2003 r050103 this result together with the weak convergence towards the stationary state proven in l pareschi g toscani self similarity and power like tails in nonconservative kinetic models j stat phys 124 2 4 2006 747 779 implies the strong convergence in sobolev norms and in the l1 norm towards it depending on the regularity of the initial data as a consequence the original nonscaled solutions are also proved to be convergent in l1 towards the corresponding self similar homogeneous cooling state the proof is based on the uniform in time control of the tails of the fourier transform of the solution and it holds for a large range of values of the mixing parameters in particular in the case of the one dimensional inelastic boltzmann equation the result does not depend of the degree of inelasticity

convex sobolev inequalities derived from entropy dissipation we study families of convex sobolev inequalities which arise as entropy dissipation relations for certain linear fokker planck equations extending the ideas recently developed by the first two authors a refinement of the bakry emery method is established which allows us to prove non trivial inequalities even in situations where the classical bakry emery criterion fails the main application of our theory concerns the linearized fast diffusion equation in dimensions d 1 which admits a poincare but no logarithmic sobolev inequality we calculate bounds on the constants in the interpolating convex sobolev inequalities and prove that these bounds are sharp on a specified range in dimension d 1 our estimates improve the corresponding results that can be obtained by the measure theoretic techniques of barthe and roberto as a by product we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by denzler and mccann in further applications of our method we prove convex sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy and the lasota model for blood cell production

on the absolute mahler measure of polynomials having all zeros in a sector ii kklet be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z in the paper on the absolute mahler measure of polynomials having all zeros in a sector g rhin and c smyth compute the greatest lower bound c of the absolute mahler measure d i 1 max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in this paper we improve the result to thirteen subintervals of 0 and extend some existing subintervals

decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model we quantify the long time behavior of solutions to the nonlinear boltzmann equation for spatially uniform freely cooling inelastic maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures existence uniqueness and precise estimates of overpopulated high energy tails of the self similar profile proved in ref 9 are revisited and derived from this new liapunov functional for general initial conditions the solutions of the boltzmann equation are then proved to converge with computable rate as t to the self similar solution in this distance which metrizes the weak convergence of measures moreover we can relate this fourier distance to the euclidean wasserstein distance or tanaka functional proving also its exponential convergence towards the homogeneous cooling states the findings are relevant in the understanding of the conjecture formulated by ernst and brito in refs 15 16 and complement and improve recent studies on the same problem of bobylev and cercignani 9 and bobylev cercignani and one of the authors 11

a study on neural learning on manifold foliations the case of the lie group su 3 learning on differential manifolds may involve the optimization of a function of many parameters in this letter we deal with riemannian gradient based optimization on a lie group namely the group of unitary unimodular matrices su 3 in this special case subalgebras of the associated lie algebra su 3 may be individuated by computing pair wise gell mann matrices commutators subalgebras generate subgroups of a lie group as well as manifold foliation we show that the riemannian gradient may be projected over tangent structures to foliation giving rise to foliation gradients exponentiations of foliation gradients may be computed in closed forms which closely resemble rodriguez forms for the special orthogonal group so 3 we thus compare optimization by riemannian gradient and foliation gradients

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

differential unitary space time modulation we present a framework for differential modulation with multiple antennas across a continuously fading channel where neither the transmitter nor the receiver knows the fading coefficients the framework can be seen as a natural extension of standard differential phase shift keying commonly used in single antenna unknown channel systems we show how our differential framework links the unknown channel system with a known channel system and we develop performance design criteria as a special ease we introduce a class of diagonal signals where only one antenna is active at any time and demonstrate how these signals may be used to achieve full transmitter diversity and low probability of error

size distribution of cities a kinetic explanation abstract we present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration in most cases the boltzmann type kinetic description allows to obtain within an asymptotic procedure a fokker planck equation with variable coefficients of diffusion and drift which describes the evolution in time of some probability density of the city size it is shown that in dependence of the microscopic rules of migration the equilibrium density can follow both a power law for large values of the size variable which contains as particular case a zipf s law behavior and a lognormal law for middle and low values of the size variable in particular connections between the value of pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined the theoretical findings are tested with recent data of the populations of italy and switzerland

design of fully diverse multiple antenna codes based on sp 2 fully diverse constellations i e sets of unitary matrices whose pairwise differences are nonsingular are useful in multiple antenna communications especially in multiple antenna differential modulation since they have good pairwise error properties recently group theoretic ideas especially fixed point free fpf groups have been used to design fully diverse constellations of unitary matrices here we construct four transmit antenna constellations appropriate for differential modulation based on the symplectic group sp 2 they can be regarded as extensions of alamouti s celebrated two transmit antenna orthogonal design which can be constructed from the group sp 1 we further show that the structure of sp 2 codes lends itself to efficient maximum likelihood ml decoding via the sphere decoding algorithm finally the performance of sp 2 codes is compared with that of other existing codes including alamouti s orthogonal design a 4 spl times 4 complex orthogonal design cayley differential unitary space time codes and group based codes

spectrally efficient differential space time coding using non full diverse constellations in this paper a method is proposed to construct spectral efficient unitary space time codes for high rate differential communications over multiple antenna channels unlike most of the known methods which are designed to maximize the diversity product the minimum determinant distance our objective is to increase the spectral efficiency the simulation results indicate that for high spectral efficiency and for more than one receive antenna the new method significantly outperforms the existing alternatives in the special case of two transmit antennas which is the main focus of this paper the relation between the proposed code and the alamouti scheme helps us to provide an efficient maximum likelihood decoding algorithm also we demonstrate that similar ideas can be applied for designing codes for more than two transmit antennas as an example we present a construction for a 4 times 4 unitary constellations which has a good performance as compared with the other known codes

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule