improved hardy and rellich inequalities on riemannian manifolds in this paper we establish improved hardy and rellich type inequalities on a riemannian manifold m furthermore we also obtain sharp constants for improved hardy and rellich type inequalities on the hyperbolic space n

on optimal multilayer cyclotomic space time code designs high rate and large diversity product or coding advantage or coding gain or determinant distance or minimum product distance are two of the most important criteria often used for good space time code designs in recent linear lattice based space time code designs more attention is paid to the high rate criterion but less to the large diversity product criterion in this paper we consider these two criteria together for multilayer cyclotomic space time code designs in a previous paper we recently proposed a systematic cyclotomic diagonal space time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas where diagonal codes correspond to single layer codes in this paper in this paper we first propose a general multilayer cyclotomic space time codes we present a general optimality theorem for these infinitely many cyclotomic diagonal or single layer space time codes over general cyclotomic number rings for a general number of transmit antennas we then present optimal multilayer full rate cyclotomic space time code designs for two and three transmit antennas we also present an optimal two layer cyclotomic space time code design for three and four transmit antennas the optimality here is in the sense that for a fixed mean transmission signal power its diversity product is maximized or equivalently for a fixed diversity product its mean transmission signal power is minimized it should be emphasized that all the optimal multilayer cyclotomic space time codes presented in this paper have the nonvanishing determinant property

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

hardy and uncertainty inequalities on stratified lie groups abstract we prove various hardy type and uncertainty inequalities on a stratified lie group g in particular we show that the operators t f l 2 f where is a homogeneous norm 0 q p and l is the sub laplacian are bounded on the lebesgue space l p g as consequences we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality we also deduce a general version of the heisenberg pauli weyl inequality relating the l p norm of a function f to the l q norm of f and the l r norm of l 2 f

sticky spheres and related systems some properties of a system of hard core particles with attractive wells in the baxter sticky sphere limit and a related limit are considered as is the approach to these limits a demonstration of the result of stell and williams that sticky spheres of equal diameter in the baxter limit are not thermodynamically stable is given and the way in which size polydispersity can be expected to restore thermodynamic stability is discussed the implications of these results for the py sticky sphere approximation and recent sticky sphere computer simulations are then examined it is concluded that the baxter py sticky sphere approximation for a monodisperse system may well be a reasonable one for a slightly polydisperse system of sticky spheres and that existing simulation results may also be relevant to such a system how polydisperse a system must be in quantitative terms in order for the py approximation to be useful remains to be seen however the question of whether the py sticky sphere approximation may prove to be useful and appropriate in describing monodisperse systems with pair potentials for which the attractive wells arenot extremely narrow is also considered it is noted that firm evidence concerning this question also appears to be lacking implications for systems near but not in the limit of zero attractive well width are also considered especially in terms of the relative size of the well width and the degree of size polydispersity in the repulsive cores the possible pertinence of such considerations to colloidal systems is observed the importance of taking into consideration the extremely long equilibration times that can be expected for systems with very narrow attractive wells is also pointed out in connection both with real colloidal systems and in computer simulations it is further observed that in the baxter limit sticky spheres described quantum mechanically are indistinguishable from hard spheres so described near the zero well width limit the quantal behavior hinges on the number of bound states and thus the well depth as well as the relative size of the de broglie thermal wavelength and the well width related results and investigations relevant to the issues described above are cited

a geometrical approach to computing free energy landscapes from short ranged potentials particles interacting with short ranged potentials have attracted increasing interest partly for their ability to model mesoscale systems such as colloids interacting via dna or depletion we consider the free energy landscape of such systems as the range of the potential goes to zero in this limit the landscape is entirely defined by geometrical manifolds plus a single control parameter these manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system equilibrium or nonequilibrium can be computed for arbitrary potentials to consider dynamical quantities we compute the asymptotic limit of the fokker planck equation and show that it becomes restricted to the low dimensional manifolds connected by sticky boundary conditions to illustrate our theory we compute the low dimensional manifolds for identical particles providing a complete description of the lowest energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom the results can be directly tested on colloidal clusters this limit is a unique approach for understanding energy landscapes and our hope is that it can also provide insight into finite range potentials

a construction of a space time code based on number theory we construct a full data rate space time st block code over m 2 transmit antennas and t 2 symbol periods and we prove that it achieves a transmit diversity of 2 over all constellations carved from z i sup 4 further we optimize the coding gain of the proposed code and then compare it to the alamouti code it is shown that the new code outperforms the alamouti see ieee j select areas commun vol 16 p 1451 58 1998 code at low and high signal to noise ratio snr when the number of receive antennas n 1 the performance improvement is further enhanced when n or the size of the constellation increases we relate the problem of st diversity gain to algebraic number theory and the coding gain optimization to the theory of simultaneous diophantine approximation in the geometry of numbers we find that the coding gain optimization is equivalent to finding irrational numbers the furthest from any simultaneous rational approximations

trace orthonormal full diversity cyclotomic space time codes in this paper we consider the design of full diversity space time codes for a coherent multiple input multiple output mimo communication system starting from both the information theoretic and detection error viewpoints we first establish that a desirable property for general linear dispersion ld codes is to have an interunitary as well as an intraunitary structure a structure we call trace orthonormality by imposing the trace orthonormal structure on an ld code and applying cyclotomic number theory we establish for an arbitrary number of transmitter and receiver antennas a systematic and simple method to jointly design a unitary cyclotomic matrix the diophantine number and the corresponding constellation for an ld code as a result this enables us to construct full diversity rectangular cyclotomic ld codes with any symbol transmission rate less than or equal to the number of transmitter antennas in addition for the case when the number of transmitter antennas is greater than the number of receiver antennas by taking advantage of the delay we also arrive at the design of a special trace orthonormal full diversity cyclotomic space time block code which for the number of transmitter antenna being equal to 2m can be proved to minimize the worst case pairwise error probability of a maximum likelihood ml detector for a q ary quadrature amplitude modulation qam signal constellation and therefore achieves optimal coding gain computer simulations show that these codes have bit error performance advantages over currently available codes

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

sharp poincare hardy and poincare rellich inequalities on the hyperbolic space abstract we study hardy type inequalities associated to the quadratic form of the shifted laplacian h n n 1 2 4 on the hyperbolic space h n n 1 2 4 being as it is well known the bottom of the l 2 spectrum of h n we find the optimal constant in a resulting poincare hardy inequality which includes a further remainder term which makes it sharp also locally the resulting operator is in fact critical in the sense of 17 a related improved hardy inequality on more general manifolds under suitable curvature assumption and allowing for the curvature to be possibly unbounded below is also shown it involves an explicit curvature dependent and typically unbounded potential and is again optimal in a suitable sense furthermore with a different approach we prove rellich type inequalities associated with the shifted laplacian which are again sharp in suitable senses

some trace hardy type inequalities and trace hardy sobolev maz ya type inequalities abstract we prove a trace hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of alvino et al and of tzirakis on the upper half space we also prove some trace hardy sobolev maz ya type inequalities which generalize the recent results of filippas et al in applications we derive some hardy type inequalities and hardy sobolev maz ya type inequalities for fractional laplacian finally we prove the logarithmic sobolev trace inequalities and logarithmic hardy trace inequalities on the upper half spaces the best constants in these inequalities are explicitly computed in the radial case

on some strong poincare inequalities on riemannian models and their improvements abstract we prove second and fourth order improved poincare type inequalities on the hyperbolic space involving hardy type remainder terms since theirs l h s only involve the radial part of the gradient or of the laplacian they can be seen as stronger versions of the classical poincare inequality we show that such inequalities hold true on model manifolds as well under suitable curvature assumptions and sharpness of some constants is also discussed

sharp constants in the hardy rellich inequalities optimal constants are found in hardy rellich inequalities containing derivatives of arbitrary not necessarily integer order l some new inequalities of this type are also obtained

optimizing improved hardy inequalities let be a bounded domain in rn n 3 containing the origin motivated by a question of brezis and vazquez we consider an improved hardy inequality with best constant b that we formally write as n 22 21 x 2 bv x we first give necessary conditions on the potential v under which the previous inequality can or cannot be further improved we show that the best constant b is never achieved in h01 and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in h01 our analysis reveals that the original inequality can be repeatedly improved by adding on the right hand side specific potentials this leads to an infinite series expansion of hardy s inequality the series obtained is in some sense optimal in establishing these results we derive various sharp improved hardy sobolev inequalities

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

stress adapted numerical form finding of pre stressed surfaces by the updated reference strategy in this paper we present the updated reference strategy for numerical formfinding of pre stressed membranes which is based on standard finite elementdiscretization the singularities of the inverse problem are regularizedby a homotopy mapping a projection scheme is proposed where anisotropicpre stress is defined with respect to an additional reference plane whichreflects the initially developable surface of membrane strips in the productionprocess physically problematic combinations of edge geometry and surfacestress are solved by a self adaptive stress correction scheme the algorithmis based on a local criterion derived from differential geometry severalexamples illustrate the success of each idea and implementation

the natural force density method for the shape finding of taut structures this work presents with the aid of the natural approach an extension of the force density method for the initial shape finding of cable and membrane structures which leads to the solution of a system of linear equations this method here called the natural force density method preserves the linearity which characterizes the original force density method at the same time it overcomes the difficulties that the original procedure presents to cope with irregular triangular finite element meshes furthermore if this method is applied iteratively in the lines prescribed herewith it leads to a viable initial configuration with a uniform isotropic plane cauchy stress state this means that a minimal surface for the membrane can be achieved through a succession of equilibrated configurations several numerical examples illustrate the simplicity and robustness of the method

remarks on the rellich inequality we study the rellich inequalities in the framework of equalities we present equalities which imply the rellich inequalities by dropping remainders this provides a simple and direct understanding of the rellich inequalities as well as the nonexistence of nontrivial extremisers

mmse analysis of certain large isometric random precoded systems linear precoding consists in multiplying by an n spl times k matrix a k dimensional vector obtained by serial to parallel conversion of a symbol sequence to be transmitted in this paper new tools borrowed from the so called free probability theory are introduced for the purpose of analyzing the performance of minimum mean square error mmse receivers for certain large random isometric precoded systems on fading channels the isometric condition represents the case of precoding matrices with orthonormal columns it is shown in this contribution that the signal to interference plus noise ratio sinr at the equalizer output converges almost surely to a deterministic value depending on the probability distribution of the channel coefficients when n spl rarr spl infin and k n spl rarr spl alpha spl les 1 these asymptotic results are used to analyze the impact of orthogonal spreading as well as to optimally balance the redundancy introduced between linear precoding versus classical convolutional coding while preserving a simple mmse equalization scheme at the receiver

a noncommutative analogue of the odlyzko bounds and bounds on performance for space time lattice codes this paper considers space time coding over several independently rayleigh faded blocks in particular we will concentrate on giving upper bounds for the coding gain of lattice space time codes as the number of blocks grow this problem was previously considered in the single antenna case by bayer fluckiger et al in 2006 crucial to their work was odlyzko s bound on the discriminant of an algebraic number field as this provides an upper bound for the normalized coding gain of number field codes in the mimo context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding noncommutative algebra in this paper we will develop analogues of the odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space time codes these bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases

almost universal codes for mimo wiretap channels despite several works on secrecy coding for fading and mimo wiretap channels from an error probability perspective the construction of information theoretically secure codes over such channels remains an open problem in this paper we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information our channel model includes static channels i i d block fading channels and ergodic stationary fading with fast decay of large deviations for the eavesdropper s channel we extend the flatness factor criterion from the gaussian wiretap channel to fading and mimo wiretap channels and establish a simple design criterion where the normalized product distance minimum determinant of the lattice and its dual should be maximized simultaneously moreover we propose concrete lattice codes satisfying this design criterion which are built from algebraic number fields with constant root discriminant in the single antenna case and from division algebras centered at such number fields in the multiple antenna case the proposed lattice codes achieve strong secrecy and semantic security for all rates r where c b and c e are bob and eve s channel capacities respectively and kappa is an explicit constant gap furthermore these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models finally we consider a compound wiretap model with a more restricted uncertainty set and show that rates r are achievable where bar c b is a lower bound for bob s capacity and bar c e is an upper bound for eve s capacity for all the channels in the set

low complexity adaptive design for full diversity full rate space time codes full diversity full rate fdfr space time codes for open loop multiple antenna systems achieve both high data rate and good performance but come with very high decoding complexity in this paper we propose a low complexity adaptive fdfr design for closed loop multiple input multiple output systems with only partial channel subspace knowledge at the transmitter we adapt the open loop fdfr code to the channel to maintain the special layer structure of the code at the receiver that special layer structure enables us to decouple the joint detection over dimension cnt2 into nt individual decoders of dimension cnt where nt is the number of transmit antennas this can also be seen as combining the channel diagonalization with signal diversity rotation the performance of the proposed scheme is analyzed and it is shown that the full diversity property is maintained in the adaptive design adaptive power loading is also incorporated to further exploit channel state information in term of the knowledge of the singular values of the channel the optimal loading schemes are derived for systems with linear receivers

the amplify and forward half duplex cooperative system pairwise error probability and precoder design in this paper an exact asymptotic pairwise error probability pep is derived for a half duplex cooperative system employing an amplify and forward af protocol when compared with the pep of a traditional multiple input multiple output mimo system the diversity gain for the cooperative system is no longer just a simple exponential function of the signal to noise ratio snr rather it involves the logarithm of the snr the term diversity gain function is used to designate this characteristic of the pep the coding gain on the other hand is found similar to that for the mimo system and is proportional to the determinant of the autocorrelation of the error matrix based on our analysis and observations we propose a design of unitary precoder for the cooperative system to achieve the full diversity gain function for the case of a 4 qam signal being transmitted we further optimize the coding gain and arrive at a closed form optimum precoder simulations indicate that our proposed precoder designs greatly improve the performance of the cooperative system

algebraic tools to build modulation schemes for fading channels a unified framework is presented in order to build lattice constellations matched to both the rayleigh fading channel and the gaussian channel the method encompasses the situations where the interleaving is done on the real components or on two dimensional signals in the latter case a simple construction of lattices congruent to the densest binary lattices with respect to the euclidean distance is proposed it generalizes in a sense to be clarified later the structural construction proposed by forney 1991 these constellations are next combined with coset codes the partitioning rules and the gain formula are similar to those used for the gaussian channel

diagonal algebraic space time block codes we construct a new family of linear space time st block codes by the combination of rotated constellations and the hadamard transform and we prove them to achieve the full transmit diversity over a quasi static or fast fading channels the proposed codes transmit at a normalized rate of 1 symbol s when the number of transmit antennas n 1 2 or n is a multiple of four we spread a rotated version of the information symbol vector by the hadamard transform and send it over n transmit antennas and n time periods for other values of n we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n spl times n st code matrix the codes maintain their rate diversity and coding gains for all real and complex constellations carved from the complex integers ring z i and they outperform the codes from orthogonal design when using complex constellations for n 2 the maximum likelihood ml decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity it is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity

diagonal lattice space time codes from number fields and asymptotic bounds in this paper we reformulate some constructions of real and complex diagonal lattice space time codes from number fields which have been given explicitly or implicitly by other researchers these constructions establish a connection between good diagonal lattice space time codes and number fields with small absolute values of discriminants we present two tables for diversity products of some lattice space time codes from these constructions the maximal rank of diagonal lattice space time codes with positive diversity product is determined we also discuss the asymptotic problem of lattice space time codes by using an infinite tower of hilbert class field and a tamely ramified class field tower we obtain asymptotically good sequences of lattice space time codes some asymptotic upper bounds are given in the paper as well

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

space time block code designs based on quadratic field extension for two transmitter antennas space time block code designs based on algebraic field extension for full rate large diversity product and nonvanishing minimum determinant of codewords have received great attention there are many different types of codes available for two transmitter antennas such as cyclotomic space time block codes the golden space time block code and rotation based space time block codes in this paper a more general space time block code design scheme which is called quadratic space time block coding is proposed for the two transmitter antennas using quadratic field extension the optimal design of the quadratic space time block codes in terms of a diversity product criterion is also presented it is shown that the optimal quadratic space time block codes designed in this paper do not belong to the existing space time block code family such as the cyclotomic golden and rotation based space time block codes the simulation results demonstrate that the average codeword error rate of the optimal quadratic space time block code attains about 0 5 db signal to noise ratio gain over those of the optimal cyclotomic and golden space time block codes

optimal hardy rellich inequalities maximum principle and related eigenvalue problem in this paper we deal with three types of problems concerning the hardy rellich s embedding for a bilaplacian operator first we obtain the hardy rellich inequalities in the critical dimension n 4 then we derive a maximum principle for fourth order operators with singular terms then we study the existence non existence simplicity and asymptotic behavior of the first eigenvalue of the hardy rellich operator 2 n 2 n 4 2 16 q x x 4 under various assumptions on the perturbation q

new algebraic constructions of rotated z sup n lattice constellations for the rayleigh fading channel in this correspondence we present various families of full diversity rotated z sup n lattice constellations based on algebraic number theory constructions we are able to give closed form expressions of their minimum product distance using the corresponding algebraic properties

underconstrained jammed packings of nonspherical hard particles ellipses and ellipsoids continuing on recent computational and experimental work on jammed packings of hard ellipsoids donev et al science 303 990 2004 we consider jamming in packings of smooth strictly convex nonspherical hard particles we explain why an isocounting conjecture which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle overline z 2 d f does not apply to nonspherical particles we develop first and second order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained hypoconstrained overline z l2 d f we apply an algorithm using these conditions to computer generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings even close to the sphere point we also consider packings that are nearly jammed and draw connections to packings of deformable but stiff particles finally we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point

adaptive force density method for form finding problem of tensegrity structures a numerical method is presented for form finding of tensegrity structures eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates the equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation a unique and non degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates a simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non degenerate structure several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

an overview and comparison of structural form finding methods for general networks abstract this paper discusses and compares existing form finding methods for discrete networks well known methods such as the force density method dynamic relaxation updated reference strategy and others are discussed by mathematically structuring and presenting them in the same way using the same notation and combining terminology based on this a single computational framework using a sparse branch node data structure is presented it is shown how each method approaches the initial equilibrium problem defines and linearizes the equilibrium equations applied to linear elements and uses particular solving strategies this framework marginalizes any differences related to operating platforms programming language and style offering a better baseline for independent comparison of performance and results as a consequence it is possible to more clearly relate distinguish and compare existing methods allow for hybrid methods and identify new avenues for research

precoding and decoding paradigms for distributed vector data compression in this paper we consider the problem of lossy coding of correlated vector sources with uncoded side information available at the decoder in particular we consider lossy coding of vector source xisinrn which is correlated with vector source yisinrn known at the decoder we propose two compression schemes namely distributed adaptive compression dac and distributed universal compression duc schemes the dac algorithm is inspired by the optimal solution for gaussian sources and requires computation of the conditional karhunen loegraveve transform cklt of the data at the encoder the duc algorithm however does not require knowledge of the cklt at the encoder the duc algorithms are based on the approximation of the correlation model between the sources y and x through a linear model y hx n in which h is a matrix and n is a random vector and independent of x this model can be viewed as a fictitious communication channel with input x and output y utilizing channel equalization at the receiver we convert the original vector source coding problem into a set of manageable scalar source coding problems furthermore inspired by bit loading strategies employed in wireless communication systems we propose for both compression schemes a rate allocation policy which minimizes the decoding error rate under a total rate constraint equalization and bit loading are paired with a quantization scheme for each vector source entry a slightly simplified version of the so called discus scheme the merits of our work are as follows 1 it provides a simple yet optimized implementation of wyner ziv quantizers for correlated vector sources by using the insight gained in the design of communication systems 2 it provides encoding schemes that with or without the knowledge of the correlation model at the encoder enjoy distributed compression gains

systematic construction of full diversity algebraic constellations a simple and systematic approach for constructing full diversity m dimensional constellations carved from lattices over a number ring r is proposed for an arbitrary dimension m when r z wn the nth cyclotomic number ring all the possible dimensions that allow for achieving the optimal minimum product distances using the proposed approach are determined it turns out that one can construct optimal unitary transformations using our construction if and only if m factors into a power of 2 and powers of the primes dividing n for m not satisfying these conditions a method based on diophantine approximation theory is proposed to optimize the minimum product distance a lower bound on the product distance is given in this case thus ensuring full diversity with good minimum product distances furthermore the proposed approach subsumes the optimal unitary transformations proposed by giraud et al over r z w4 and r z w3 while giving optimal unitary transformations for infinitely many new values of n and m

the hardy inequality and nonlinear parabolic equations on carnot groups abstract in this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation u t g p u v x u p 1 in 0 t 1 p 2 u x 0 u 0 x 0 in u x t 0 on 0 t where g p is the p sub laplacian on a carnot group g and v l loc 1

simple iterative methods to exploit the signal space diversity signal space diversity is a power and bandwidth efficient diversity technique to exploit the signal space diversity joint maximum likelihood ml detection at the receiver is usually needed where the complexity grows exponentially with the dimension of the lattice in this letter we propose a serial concatenated scheme and two simple iterative methods to exploit the signal space diversity the simple iterative methods are based on the idea of soft interference cancellation the first iterative method is based on a vector gaussian approximation while the second one is based on a scalar gaussian approximation the complexity of the first iterative method grows cubically with the dimension of the lattice and the simulations show that its performance approaches that of the optimal maximum a posteriori detection method the complexity of the second iterative method grows linearly with the dimension of the lattice and the simulations show that when the dimension of the lattice n 32 at bit error rate 10 sup 5 the performance gap between the rayleigh fading channel and the gaussian channel is only 0 3 db

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

hardy lieb thirring inequalities for fractional schrodinger operators we show that the lieb thirring inequalities on moments of negative eigenvalues of schrodinger like operators remain true with possibly different constants when the critical hardy weight c x 2 is subtracted from the laplace operator we do so by first establishing a sobolev inequality for such operators similar results are true for fractional powers of the laplacian and the hardy weight and in particular for relativistic schrodinger operators we also allow for the inclusion of magnetic vector potentials as an application we extend for the first time the proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge z 2 for less than some critical value

sharp trace hardy sobolev maz ya inequalities and the fractional laplacian in this work we establish trace hardy and trace hardy sobolev maz ya inequalities with best hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity we then use them to produce fractional hardy sobolev maz ya inequalities with best hardy constants for various fractional laplacians in the case where the domain is the half space our results cover the full range of the exponent s in 0 1 of the fractional laplacians in particular we give a complete answer in the l2 setting to an open problem raised by frank and seiringer sharp fractional hardy inequalities in half spaces in around the research of vladimir maz ya international mathematical series 2010

on the nonexistence of rate one generalized complex orthogonal designs orthogonal space time block coding proposed recently by alamouti 1998 and tarokh et al 1999 is a promising scheme for information transmission over rayleigh fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum likelihood ml decoding algorithm tarokh et al extended the theory of classical orthogonal designs to the theory of generalized real or complex linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space time block codes stbc with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas it has been known that the stbc constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation contrary to this in this correspondence we prove that there does not exist rate one stbc from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation

hardy and rellich inequalities identities and sharp remainders on homogeneous groups abstract we give sharp remainder terms of l p and weighted hardy and rellich inequalities on one of most general subclasses of nilpotent lie groups namely the class of homogeneous groups as consequences we obtain analogues of the generalised classical hardy and rellich inequalities and the uncertainty principle on homogeneous groups we also prove higher order inequalities of hardy rellich type all with sharp constants a number of identities are derived including weighted and higher order types

algebraic lattice constellations bounds on performance in this work we give a bound on performance of any full diversity lattice constellation constructed from algebraic number fields we show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain furthermore we discuss constructions minimum product distance and bounds for full diversity complex rotated z i sup n lattices for any dimension n which avoid the need of component interleaving

improved lp poincare inequalities on the hyperbolic space abstract we investigate the possibility of improving the p poincare inequality h n u p p p u p p on the hyperbolic space where p 1 and p n 1 p p is the best constant for which such inequality holds we prove several different and independent improved inequalities one of which is a poincare hardy inequality namely an improvement of the best p poincare inequality in terms of the hardy weight r p r being geodesic distance from a given pole certain hardy maz ya type inequalities in the euclidean half space are also obtained

densest local sphere packing diversity ii application to three dimensions the densest local packings of n three dimensional identical nonoverlapping spheres within a radius r mathrm min n of a fixed central sphere of the same size are obtained for selected values of n up to n 1054 in the predecessor to this paper a b hopkins f h stillinger and s torquato phys rev e 81 041305 2010 we described our method for finding the putative densest packings of n spheres in d dimensional euclidean space mathbb r d and presented those packings in mathbb r 2 for values of n up to n 348 here we analyze the properties and characteristics of the densest local packings in mathbb r 3 and employ knowledge of the r mathrm min n using methods applicable in any d to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings in mathbb r 3 we find wide variability in the densest local packings including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry we compare the densest local packings of n spheres near a central sphere to minimal energy configurations of n 1 points interacting with short range repulsive and long range attractive pair potentials e g 12 6 lennard jones and find that they are in general completely different a result that has possible implications for nucleation theory we also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in mathbb r 3 the barlow packings and find that the densest local packings are almost always most similar as measured by a similarity metric to the subsets of barlow packings with the smallest number of coordination shells measured about a single central sphere e g a subset of the fcc barlow packing additionally we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance r mathrm min n in particular we find two maracas packings at n 77 and n 93 each consisting of a few unjammed spheres free to rattle within a husk composed of the maximal number of spheres that can be packed with centers at respective r mathrm min n

non linear ground state representations and sharp hardy inequalities we determine the sharp constant in the hardy inequality for fractional sobolev spaces to do so we develop a non linear and non local version of the ground state representation which even yields a remainder term from the sharp hardy inequality we deduce the sharp constant in a sobolev embedding which is optimal in the lorentz scale in the appendix we characterize the cases of equality in the rearrangement inequality in fractional sobolev spaces

mimo channel modeling and the principle of maximum entropy we devise theoretical grounds for constructing channel models for multiple input multiple output mimo systems based on information theoretic tools the paper provides a general method to derive a channel model which is consistent with one s state of knowledge the framework we give here has already been fruitfully explored with success in the context of bayesian spectrum analysis and parameter estimation for each channel model we conduct an asymptotic analysis in the number of antennas of the achievable transmission rate using tools from random matrix theory a central limit theorem is provided on the asymptotic behavior of the mutual information and validated in the finite case by simulations the results are useful both in terms of designing a system based on criteria such as quality of service and in optimizing transmissions in multiuser networks

structure of finite sphere packings via exact enumeration implications for colloidal crystal nucleation we analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration the number of nonisomorphic isostatic packings grows exponentially with the number of spheres n and their diversity of structure and symmetry increases with increasing n and decreases with increasing hyperstaticity h n c n iso where n c is the number of pair contacts and n iso 3n 6 maximally contacting packings are in general neither the densest nor the most symmetric analyses of local structure show that the fraction f of nuclei with order compatible with the bulk rhcp crystal decreases sharply with increasing n due to a high propensity for stacking faults five and near fivefold symmetric structures and other motifs that preclude rhcp order while f increases with increasing h a significant fraction of hyperstatic nuclei for n as small as 11 retain non rhcp structure classical theories of nucleation that consider only spherical nuclei or only nuclei with the same ordering as the bulk crystal cannot capture such effects our results provide an explanation for the failure of classical nucleation theory for hard sphere systems of n 10 particles we argue that in this size regime it is essential to consider nuclei of unconstrained geometry our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard core like repulsive and short ranged attractive interactions

systematic and optimal cyclotomic lattices and diagonal space time block code designs in this correspondence a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory this design provides infinitely many full diversity cyclotomic lattices for a given lattice size based on the packing theory and the concrete form of the design optimal cyclotomic lattices are presented by minimizing the mean transmission signal power for a given minimum diversity product or equivalently maximizing the minimum product for a given mean transmission signal power the newly proposed cyclotomic lattices can be applied to both space time code designs for multi antenna systems and linear precode design for signal space diversity in single antenna systems over fast rayleigh fading channels although there are some cyclotomic lattices space time codes existing in the literature most of them are not optimal

free energy of singular sticky sphere clusters networks of particles connected by springs model many condensed matter systems from colloids interacting with a short range potential and complex fluids near jamming to self assembled lattices and various metamaterials under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero frequency vibrational modes yet such singular modes are at the epicenter of many interesting behaviors in the systems above we consider a system of n spherical particles and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged although the contribution to the partition function from singular clusters diverges in the limit its asymptotic value can be calculated and depends on only two parameters characterizing the depth and range of the potential the result holds for systems that are second order rigid a geometric characterization that describes all known ground state rigid sticky clusters to illustrate the applications of our theory we address the question of emergence how does crystalline order arise in large systems when it is strongly disfavored in small ones we calculate the partition functions of all known rigid clusters up to n 21 and show the cluster landscape is dominated by hyperstatic clusters those with more than 3n 6 contacts singular and isostatic clusters are far less frequent despite their extra vibrational and configurational entropies since the most hyperstatic clusters are close to fragments of a close packed lattice this underlies the emergence of order in sticky sphere systems even those as small as n 10

constellations matched to the rayleigh fading channel we introduce a new technique for designing signal sets matched to the rayleigh fading channel in particular we look for n dimensional n spl ges 2 lattices whose structure provides nth order diversity our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number geometric approach specifically a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body s x x sub 1 spl middot spl middot spl middot x sub n spl isin r sup n spl pi sub i 1 sup n x sub i spl les 1 the lattices among which we search for an optimal solution are the standard embeddings in r sup n of the number ring of some totally real number field of degree n over q simulation results confirm that this approach yields lattices with considerable coding gains

an algebraic family of complex lattices for fading channels with application to space time codes a new approach is presented for the design of full modulation diversity fmd complex lattices for the rayleigh fading channel the fmd lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance d sub p sup 2 of the finite signal set carved out of the lattice we approach the problem of maximizing d sub p sup 2 by minimizing the average energy of the signal constellation obtained from a new family of fmd lattices the unnormalized minimum product distance for every lattice in the proposed family is lower bounded by a nonzero constant minimizing the average energy of the signal set translates to minimizing the frobenius norm of the generator matrices within the proposed family the two strategies proposed for the frobenius norm reduction are based on the concepts of successive minima sm and basis reduction of an equivalent real lattice the lattice constructions in this paper provide significantly larger normalized minimum product distances compared to the existing lattices in certain dimensions the proposed construction is general and works for any dimension as long as a list of number fields of the same degree is available

investigation of minimal forms with conjugate gradient method abstract amongst the numerous calculation methods available to investigate minimal forms an approach based on the mechanical consideration of uniform tension in the domain leads to the writing of innovative relationships indeed by establishing the equivalence between the vector of nodal internal forces due to the prestressed domain and the gradient of the function to be minimized this study proposes to use the conjugate gradient method as a minimal area shape form finding tool the determination of descent directions may refer to fletcher reeves suggestion or to an optimized value based on the polak ribiere formula moreover the steplength calculation is envisaged in accordance with the more and thuente s line search algorithm or with a modified approach especially adapted to the studied configurations numerical experiments illustrate the use of the conjugate gradient method and thus point out the efficiency of the suggested modifications related to required computation times in a second time the conjugate gradient method is compared with density methods and a mixed formulation is therefore put forward numerical tests enable the comparison between these different approaches

almost universal codes achieving ergodic mimo capacity within a constant gap this paper addresses the question of achieving capacity with lattice codes in multi antenna block fading channels when the number of fading blocks tends to infinity a design criterion based on the normalized minimum determinant is proposed for division algebra multi block space time codes over fading channels this plays a similar role to the hermite invariant for gaussian channels under maximum likelihood decoding it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels moreover if the number of receive antennas is greater than or equal to the number of transmit antennas the same constant gap is achieved under naive lattice decoding as well in the case of independent identically distributed rayleigh fading the error probability vanishes exponentially fast in contrast to the standard approach in the literature which employs random lattice ensembles the existence results in this paper are derived from the number theory first the gap to capacity is shown to depend on the discriminant of the chosen division algebra then class field theory is applied to build families of algebras with small discriminants the key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants

full diversity blind space time block codes in this paper the systematic design of a space time block code is considered for a wireless communication system with multiple transmitter receiver antennas and flat fading in which channel state information is completely unknown from the viewpoint of blind signal processing a necessary and sufficient condition is given for the unique identification of the multi input multi output flat fading channel and transmitted signal then some novel unique factorizations for a pair of coprime p ary and q ary phase shift keying psk constellations are established with this and currently available coherent space time block code designs a method is developed to systematically construct full diversity blind nonunitary space time block codes as well as unitary codes by just performing the qr decomposition of the nonunitary codes it is proved that under a noise free environment the code design proposed in this paper can guarantee that the transmitted signals and the channel coefficients are uniquely identified and that under a complex gaussian noise environment in which pth order and q th order statistics p and q are co prime of the received signals are available the channel coefficients can be still uniquely identified in addition a closed form solution to determine the channel coefficients is obtained

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free