analytic formulas for the regulator of a number field letr rk andw wk be the regulator and the number of roots of unity in the number fieldk we determine allk for whichr w 1 8 there are nine such fields sincew 2 we have allk withr 1 4 in particular we find the first three minima ofr over allk there are three main ingredients to the proof i a new analytic formula theorem a below ii a refinement of remak s geometric approach iii a refinement of zimmert s analytic method

lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

bounds on the spectral shift function and the density of states we study spectra of schrodinger operators on d first we consider a pair of operators which differ by a compactly supported potential as well as the corresponding semigroups we prove almost exponential decay of the singular values n of the difference of the semigroups as n and deduce bounds on the spectral shift function of the pair of operators

capacity of compound mimo gaussian channels with additive uncertainty this paper considers reliable communications over a multiple input multiple output mimo gaussian channel where the channel matrix is within a bounded channel uncertainty region around a nominal channel matrix i e an instance of the compound mimo gaussian channel we study the optimal transmit covariance matrix design to achieve the capacity of compound mimo gaussian channels where the channel uncertainty region is characterized by the spectral norm this design problem is a challenging nonconvex optimization problem however in this paper we reveal that this problem has a hidden convexity property which can be exploited to map the problem into a convex optimization problem we first prove that the optimal transmit design is to diagonalize the nominal channel and then show that the duality gap between the capacity of the compound mimo gaussian channel and the min max channel capacity is zero which proves and generalizes a conjecture of loyka and charalambous the key tools for showing these results are a new matrix determinant inequality and some unitarily invariant properties

a new algorithm for contact detection between spherical particle and triangulated mesh boundary in discrete element method simulations summary in discrete element method dem simulations of real scale the spherical particles are commonly employed for increasing the computation speed and the complex boundary models are represented by triangle meshes with controllable accuracy a new contact detection algorithm has been developed to resolve the contacts between the spheres and the triangle mesh boundaries the application of the barycentric coordinates makes this algorithm more efficient to identify contacts in the intersection test as a particle probably collides with several triangles at the same time the multiple contacts would be reported as face contacts edge contacts or vertex contacts moreover the particle embedding in a triangle can be also contact with the edges or vertices of the next triangles these contacts should be considered as invalid for updating contact forces in the dem to exclude invalid records from the multiple contacts the algorithm gives attention to the mesh structure nearby contacts and analyzes all possible collision situations numerical experiments have been conducted to verify this algorithm by using the algorithm in the dem simulation framework the numerical results suggest that the algorithm can resolve all contacts precisely and stably when the spherical particles collide on the complex boundary circumstances copyright 2013 john wiley sons ltd

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

an accurate and robust contact detection algorithm for particle solid interaction in combined finite discrete element analysis when applying the combined finite discrete element method for analysis of dynamic problems contact is often encountered between the finite elements and discrete elements and thus an effective contact treatment is essential in this paper an accurate and robust contact detection algorithm is proposed to resolve contact problems between spherical particles which represent rigid discrete elements and convex quadrilateral mesh facets which represent finite element boundaries of structural components different contact scenarios between particles and mesh facets or edges or vertices have been taken into account for each potential contact pair the contact search is performed in an hierarchical way starting from mesh facets possibly going to edges and even further to vertices the invalid contact pairs can be removed by means of two reasonable priorities defined in terms of geometric primitives and facet identifications this hierarchical contact searching scheme is effective and its implementation is straightforward numerical examples demonstrated the accuracy and robustness of the proposed algorithm copyright 2015 john wiley sons ltd

achieving awgn channel capacity with lattice gaussian coding we propose a new coding scheme using only one lattice that achieves the 1 2 log 1 snr capacity of the additive white gaussian noise awgn channel with lattice decoding which is provable for signal to noise ratio snr e at present the scheme applies a discrete gaussian distribution over an awgn good lattice but otherwise does not require a shaping lattice or dither thus it significantly simplifies the default lattice coding scheme of erez and zamir which involves a quantization good lattice as well as an awgn good lattice using the flatness factor we show that the error probability of the proposed scheme under minimum mean square error lattice decoding is almost the same as that of erez and zamir for any rate up to the awgn channel capacity we introduce the notion of good constellations which carry almost the same mutual information as that of continuous gaussian inputs we also address the implementation of gaussian shaping for the proposed lattice gaussian coding scheme

awgn goodness is enough capacity achieving lattice codes based on dithered probabilistic shaping in this paper we show that any sequence of infinite lattice constellations which is good for the unconstrained gaussian channel can be shaped into a capacity achieving sequence of codes for the power constrained gaussian channel under lattice decoding and non uniform signaling unlike previous results in the literature our scheme holds with no extra condition on the lattices e g quantization goodness or vanishing flatness factor thus establishing a direct implication between awgn goodness in the sense of poltyrev and capacity achieving codes our analysis uses properties of the discrete gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates in particular we obtain a simple characterization of the finite blocklength behavior of the scheme showing that it approaches the optimal dispersion coefficient for high signal to noise ratio we further show that for low signal to noise ratio the discrete gaussian over centered lattice constellations cannot achieve capacity and thus a shift or dither is essentially necessary

on the densest mimo lattices from cyclic division algebras it is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant using results from class field theory a lower bound to the minimum discriminant of a maximal order with a given center and index the number of tx rx antennas is derived also numerous examples of division algebras achieving the bound are given for example a matrix lattice with quadrature amplitude modulation qam coefficients that has 2 5 times as many codewords as the celebrated golden code of the same minimum determinant is constructed also a general algorithm due to ivanyos and ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed

the spectral shift function and the invariance principle abstract the new representation formula for the spectral shift function due to f gesztesy and k a makarov is considered this formula is extended to the case of relatively trace class perturbations the proof is based on the analysis of a certain new unitary invariant for a pair of self adjoint operators

random ensembles of lattices from generalized reductions we propose a general framework to study constructions of euclidean lattices from linear codes over finite fields in particular we prove general conditions for an ensemble constructed using linear codes to contain dense lattices i e with packing density comparable to the minkowski hlawka lower bound specializing to number field lattices we obtain a number of interesting corollaries for instance the best known packing density of ideal lattices and an elementary coding theoretic construction of asymptotically dense hurwitz lattices all results are algorithmically effective in the sense that for any dimension a finite family containing dense lattices is exhibited for suitable constructions based on craig s lattices this family is smaller in terms of alphabet size than previous ensembles in the literature

a method for representing boundaries in discrete element modelling part ii kinematics the application of the dem to engineering problems involving the dynamic behaviour of discontinuous media has necessitated the introduction of moving boundary surfaces in this paper a method is presented for modelling three dimensional moving boundary surfaces within the discrete element framework the surfaces of boundary objects are discretized into triangular planar surfaces using the finite wall method wall elements are grouped and each group is associated with a single discrete boundary object which may move independently movement comprises any combination of translation and rotation of wall element groups subject to a given acceleration and velocity during a calculation cycle the scheme is explicit due to rigidity of the wall elements which are stationary fixed in position and orientation over a time step any in plane velocity is handled as a contact point velocity within a calculation cycle the kinematic conditions at each calculation cycle may be pre defined or returned from a separate calculation of rigid body motion of the boundary object the method provides a means for coupling sphere based particle dynamics with rigid body dynamics and structural analysis of boundary components copyright 2001 john wiley sons ltd

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

a study on universal codes with finite block lengths based on random codes and typical set decoding an alternative proof of root and varaiya s compound channel coding theorem for linear gaussian channels is presented the performance limit of codes with finite block length under a compound channel is studied through error bounds and simulation although the theorem promises uniform convergence of the probability of error as the block length approaches infinity with short block lengths the performance can differ considerably for individual channels simulation results show that universal performance can be a practical goal as the block lengths become large

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

construction a of lattices over number fields and block fading wiretap coding we propose a lattice construction from totally real and complex multiplication fields which naturally generalizes construction a of lattices from p ary codes obtained from the cyclotomic field mathbb q zeta p p a prime which in turn contains the so called construction a of lattices from binary codes as a particular case we focus on the maximal totally real subfield mathbb q zeta p r zeta p r of the cyclotomic field mathbb q zeta p r r geq 1 our construction has applications to coset encoding of algebraic lattice codes for block fading channels and in particular for block fading wiretap channels

the discrete spectrum in the spectral gaps of semibounded operators with non sign definite perturbations abstract given two self adjoint operators a and v v v we study the motion of the eigenvalues of the operator a t a tv as t increases let 0 and let be a regular point for a we consider the quantities n v n v and n 0 v defined as the number of eigenvalues of the operator a t that pass point from the right to the left from the left to the right or change the direction of their motion exactly at point respectively as t increases from 0 to 0 we study asymptotic characteristics of these quantities as in the present paper the results obtained previously o l safronov comm math phys 193 1998 233 243 are extended and given new applications to differential operators

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

universal lattice codes for mimo channels we propose a coding scheme that achieves the capacity of the compound mimo channel with algebraic lattices our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill conditioned channel realizations to shape the constellation a discrete gaussian distribution over the lattice points is applied these techniques along with algebraic properties of the proposed lattices are then used to construct a sub optimal de coupled coding scheme that achieves a constant gap to compound capacity by decoding in a lattice that does not dependent on the channel realization the gap is characterized in terms of algebraic invariants of the code and is shown to be significantly smaller than previous schemes in the literature we also exhibit alternative algebraic constructions that achieve the capacity of ergodic siso fading channels

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

ring compute and forward over block fading channels the compute and forward c f protocol in quasi static channels normally employs lattice codes based on the rational integers mathbb z the gaussian integers mathbb z i or the eisenstein integers mathbb z omega while its extension to more general channels often assumes channel state information at transmitters csit in this paper we propose a novel scheme for c f in block fading channels without csit which is referred to as ring c f because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers owing to the multiplicative closure of the algebraic lattices employed a relay is able to decode an algebraic integer linear combination of lattice codewords we analyze its achievable computation rates and show it outperforms conventional c f based on the mathbb z lattices by investigating the effect of the diophantine approximation by algebraic conjugates we prove that the degrees of freedom dofs of the optimized computation rate are n l where n is the number of blocks and l is the number of users

spectral shift function in the large coupling constant limit given two selfadjoint operators h0 and v v v we study the spectrum of the operator h h0 v 0 we consider the quantity h h0 r which coincides with krein s spectral shift function of the pair h h0 if v is of trace class and study its asymptotic behavior as applications to differential operators are given

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

exponential convergence under mixing we show that for a mixing stationary sequence of bounded random variables the average of the firstn variables converges exponentially fast withn to the mean value of these random variables

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

eigenvalue asymptotics for the schrodinger operator with perturbed periodic potential we consider the schrodinger operatorh w v acting inl 2 m m 2 with periodic potentialw perturbed by a potentialv which decays slowly at infinity we study the asymptotic behaviour of the discrete spectrum ofh near any given boundary point of the essential spectrum

eigenvalues in gaps and decoupling by neumann boundary conditions abstract continuing earlier investigations we show the existence of eigenvalues of schrodinger operators h w r inside a spectral gap of h v under suitable conditions on v and w our method of proof is based on a decoupling of regions in r v by means of dirichlet or neumann boundary conditions where in particular the use of neumann boundary conditions in the decoupling process leads to a new asymptotic results for eigenvalue counting functions

precoded integer forcing universally achieves the mimo capacity to within a constant gap an open loop single user multiple input multiple output mimo communication scheme is considered where a transmitter equipped with multiple antennas encodes the data into independent streams all taken from the same linear code the coded streams are then linearly precoded using the encoding matrix of a perfect linear dispersion space time code at the receiver side integer forcing equalization is applied followed by standard single stream decoding it is shown that this communication architecture achieves the capacity of any gaussian mimo channel up to a gap that depends only on the number of transmit antennas

on the compound capacity of a class of mimo channels subject to normed uncertainty the compound capacity of uncertain multiple input multiple output channels is considered when the channel is modeled by a class described by a known nominal channel and a constrained norm unknown uncertainty within this framework two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint using the singular value decomposition and related singular value inequalities as the main tools the compound capacity is a maxmin mutual information representing the capacity of the class in which the minimization is done over the class of channels while the maximization is done over the transmit covariance closed form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented it is shown that under certain conditions the compound capacity of the class is equal to the worst case channel capacity thus establishing a saddle point property explicit closed form solutions are given for the worst case channel uncertainty and the capacity achieving transmit covariance matrix the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal known channel eigenmodes with the beam power distribution via the water filling at a degraded snr as the uncertainty increases fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large uncertainty regime for which explicit conditions are given using these results upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out the results are extended to compound multiple access and broadcast channels in all considered cases the price to pay for channel uncertainty is an snr loss or equivalently the nominal channel degradation commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel

on the point spectrum of dirac operators abstract we study the decay of eigenfunctions and we give conditions for the absence of eigenvalues embedded in the continuous spectrum for dirac hamiltonians with long range and locally singular potential

concavity of eigenvalue sums and the spectral shift function abstract it is well known that the sum of negative positive eigenvalues of some finite hermitian matrix v is concave convex with respect to v using the theory of the spectral shift function we generalize this property to self adjoint operators on a separable hilbert space with an arbitrary spectrum more precisely we prove that the spectral shift function integrated with respect to the spectral parameter from to from to is concave convex with respect to trace class perturbations the case of relative trace class perturbations is also considered

strong electric field eigenvalue asymptotics for the perturbed magnetic schrodinger operator we consider the schrodinger operator with constant full rank magnetic field perturbed by an electric potential which decays at infinity and has a constant sign we study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

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

averaging bounds for lattices and linear codes general random coding theorems for lattices are derived from the minkowski hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out a new version of the minkowski hlawka theorem itself is obtained as the limit for p spl rarr spl infin of a simple lemma for linear codes over gf p used with p level amplitude modulation the relation between the combinatorial packing of solid bodies and the information theoretic soft packing with arbitrarily small but positive overlap is illuminated the soft packing results are new when specialized to the additive white gaussian noise channel they reduce to a version of the de buda poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 p n

fundamental limits of infinite constellations in mimo fading channels the fundamental and natural connection between the infinite constellation ic dimension and the best diversity order it can achieve is investigated in this paper in the first part of this paper we develop an upper bound on the diversity order of ics for any dimension and any number of transmit and receive antennas by choosing the right dimensions we prove in the second part of this paper that ics in general and lattices in particular can achieve the optimal diversity multiplexing tradeoff of finite constellations this paper gives a framework for designing lattices for multiple antenna channels using lattice decoding