the beta model maximum likelihood cramer rao bounds and hypothesis testing we study the maximum likelihood estimator in a setting where the dependent variable is a random graph and covariates are available on a graph level the model generalizes the well known beta model for random graphs by replacing the constant model parameters with regression functions cramer rao bounds are derived for special cases of the undirected beta model the directed beta model and the covariate based beta model the corresponding maximum likelihood estimators are compared with the bounds by means of simulations moreover examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance finally the applicability of the model is demonstrated using temporal social network data describing communication among healthcare workers

a generalized one dimensional fast multipole method with application to filtering of spherical harmonics the need to filter functions defined on the sphere arises in a number of applications such as climate modeling electromagnetic and acoustic scattering and several other areas recently it has been observed that the problem of uniform resolution filtering on the sphere can be performed efficiently via the fast multipole method fmm in one dimension in this paper we introduce a generalization of the fmm that leads to an accelerated version of the filtering process instead of multipole expansions the scheme uses special purpose bases constructed via the singular value decomposition of appropriately chosen submatrices of the filtering matrix the algorithm is applicable to a fairly wide class of projection operators its performance is illustrated with several numerical examples

joint estimation of porosity and saturation using stochastic rock physics modeling sediment porosity and saturation affect bulk modulus shearmodulus anddensity consequently estimatinghydrocarbon saturation and reservoir porosity from seismic data is a joint estimation problem uncertainty in porosity will lead toerrorsinsaturationprediction andviceversa porosityand saturation can be jointly estimated using stochastic rockphysicsmodelingandformalbayesianestimationmethodology knowledge of shear impedance reduces the uncertainty inporosityandthusalsoreducesuncertaintyinsaturationestimation this study investigates joint estimation of porosity and saturation by using rock physics stochastic modeling and bayesian estimation theory to derive saturation and porosity maps of expected pay sands in the field example the uncertainty in porosity quantified by the standard deviation stdassociatedwiththeposteriorprobabilitydensityfunctionpdf derivedfrominversionofseismicdataismuchless than the uncertainty in the derived saturation for a typical case the std associated with saturation is 24 while porositystdisabout1 34porosityunitsgivenseismic derived inversionattributeswithreasonableaccuracy comparisonof these numbers with prior estimates showed that inversion of seismic data decreased the uncertainty in porosity to 15 of the prior uncertainty while saturation uncertainty was only reduced to 92 of the prior uncertainty although these resultsmayvaryfromonelocationtoanother themethodology isgeneralandcanbeappliedtootherlocations

limits of sensing temporal concentration changes by single cells berg and purcell biophys j 20 193 1977 calculated how the accuracy of concentration sensing by single celled organisms is limited by noise from the small number of counted molecules here we generalize their results to the sensing of concentration ramps which is often the biologically relevant situation e g during bacterial chemotaxis we calculate lower bounds on the uncertainty of ramp sensing by three measurement devices a single receptor an absorbing sphere and a monitoring sphere we contrast two strategies simple linear regression of the input signal versus maximum likelihood estimation and show that the latter can be twice as accurate as the former finally we consider biological implementations of these two strategies and identify possible signatures that maximum likelihood estimation is implemented by real biological systems

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

relationship between minimum gap and success probability in adiabatic quantum computing we explore the relationship between two figures of merit for an adiabatic quantum computation process the success probability p and the minimum gap min between the ground and first excited states investigating to what extent the success probability for an ensemble of problem hamiltonians can be fitted by a function of min and the computation time t we study a generic adiabatic algorithm and show that a rich structure exists in the distribution of p and min in the case of two qubits p is to a good approximation a function of min of the stage in the evolution at which the minimum occurs and of t this structure persists in examples of larger systems

robust linear channel estimation methods for per subcarrier transmit antenna selection the linear minimum mean squared error mse channel estimator for systems employing per subcarrier transmit antenna selection is developed frequency domain correlations after the selection process are shown to be approximated well using a simple function which makes near optimal channel estimation practically possible however the resultant estimators are not robust to errors in the assumed model in terms of the antenna to subcarrier assignment used at transmission as we motivate both analytically and via simulations this is an issue when the channels for conveying this information to the receiver are severely constrained we present windowed channel estimations and new robust estimation algorithms either individually or in combination as solutions to counter this sensitivity the robust estimators are developed based on the principles of 1 minimising the worst case mse over the set of possible models and 2 minimising the expected mse over the set of possible models the latter estimator is preferred due to the lower implementation complexity and better mse performance we also prove theoretical asymptotic in signal to noise ratio performances of the two estimators used with the proposed correlation model simulation results illustrate the performance gains and improved robustness offered by the developed estimators

active learning applied to rcs computations with nonuniform sampling using different objective functions an active learning framework is introduced to reduce the number of frequencies and angles one must consider for wideband monostatic scattering computations or measurements this method is used to optimally select those frequencies and angles that would be most informative resulting in nonuniform sampling and often a reduced number of points vis agrave vis uniform sampling in this paper we focus on jointly two dimensional optimal sampling in frequency and incident angle thetas for monostatic scattering the method consists of two basic steps one step involves estimation of model parameters using a least square ls algorithm the next step is to optimally choose the next point frequency and thetas for analysis by the computational model or experiment this new point is selected with the goal of reducing uncertainty in the parametric model quantified via the fisher information matrix iterating these two steps a sequence of numerical computations or measurements are performed each at the most informative point for learning the parameters of the associated simpler parametric model this idea is demonstrated here in the context of reducing the number of points frequencies and orientations at which a computational model must be employed and in order to avoid repeatedly gathering samplings at the edge of the input space an alternative objective function is applied which makes the actively selected points closer to a region of interest

closed form crlbs for snr estimation from turbo coded bpsk msk and square qam modulated signals in this contribution we derive for the first time the closed form expressions for the cramer rao lower bounds crlbs of the signal to noise ratio snr estimates from bpsk msk and square qam modulated signals over turbo coded transmissions these crlbs relatively easy to derive from bpsk msk and qpsk transmissions become extremely challenging with higher order square qam modulated signals in the latter by exploiting the structure of the gray mapping we are able to factorize the likelihood function thereby linearizing all the derivation steps for the fisher information matrix fim elements we also propose another approach that allows the evaluation of the considered bounds using extensive monte carlo computer simulations the analytical crlbs coincide exactly with their empirical counterparts validating thereby our new analytical expressions numerical results suggest that the clrbs for code aided ca snr estimates range between the crlbs for non data aided nda snr estimates and those for data aided da ones thereby highlighting the expected potential in snr estimation improvement from the coding gain indeed the ca crlbs improve by decreasing the overall coding rate due to enhanced decoding capabilities however they do increase with the modulation order for a given code rate finally the derived bounds are also valid for ldpc coded systems and they can be evaluated when the latter are decoded using the turbo principal

adiabatic quantum computation is equivalent to standard quantum computation the model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities we describe an efficient adiabatic simulation of any given quantum circuit this implies that the adiabatic computation model and the standard circuit based quantum computation model are polynomially equivalent our result can be extended to the physically realistic setting of particles arranged on a two dimensional grid with nearest neighbor interactions the equivalence between the models allows one to state the main open problems in quantum computation using well studied mathematical objects such as eigenvectors and spectral gaps of hamiltonians

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

fast spectral projection algorithms for density matrix computations we present a fast algorithm for the construction of a spectral projector this algorithm allows us to compute the density matrix as used in e g the kohn sham iteration and so obtain the electron density we compute the spectral projector by constructing the matrix sign function through a simple polynomial recursion we present several matrix representations for fast computation within this recursion using bases with controlled space spatial frequency localization in particular we consider wavelet and local cosine bases since spectral projectors appear in many contexts we expect many additional applications of our approach

on the true and the modified cramer rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters we consider the cramer rao bound crb for the estimation of a scalar parameter in the presence of nuisance parameters whereas this true crb is hard to evaluate in general we present a simple analytical expression for its high snr asymptote i e the asymptotic crb acrb we show that this acrb is related to the crb for the joint estimation of the scalar parameter and the nuisance parameters previously a modified crb mcrb for the estimation of a scalar parameter in the presence of nuisance parameters has been derived this mcrb is also simple to evaluate and is related to the crb for the estimation of the scalar parameter assuming that the value of the nuisance parameters is a priori known we show that the mcrb can be quite loose at high snr when the scalar parameter is coupled with the nuisance parameters in the case of synchronization parameter estimation we find that the acrb equals the mcrb whether or not some of the nuisance parameters are a priori known

iterative cdma multiuser receiver with soft decision directed channel estimation we develop an iterative code division multiple access cdma receiver for multipath block fading channels in the uplink the mean square error mse of the channel estimates is reduced iteratively by using soft data estimates as additional pilots the multiuser detector performs parallel interference cancellation pic with subsequent linear filtering we consider a single user matched filter sumf and develop a low complexity linear minimum mse lmmse filter for intersymbol interference isi channels that allows for 33 more users than the sumf we investigate two choices of training sequences for initial channel estimation the first is a random pilot symbol sequence modulated by a short spreading sequence rand and the second is a perfect root of unity sequence prus used as long sounding chip sequence we observe that the rand scheme approaches the single user bound up to 2 db and in case of prus up to 1 db at a target bit error rate ber of 10 3 further several channel estimators are developed that use various combinations of soft decision symbols and statistical a priori knowledge about the channel these are the approximated least squares als approximated linear mmse almmse and linear mmse lmmse estimators the iterative estimators show an improvement of 2 5 and 5 db with prus and rand pilots respectively over estimators that do not exploit soft feedback symbols at a ber of 10 3 the als is sacrificing less than 0 6 db to obtain a ber of 10 3 in a system at load 1 5 when compared with the almmse and lmmse schemes the load is understood as the ratio of the number of users to the spreading factor this paper shows that significant improvements in system capacity can be achieved in multipath scenarios with low complexity choices of multiuser detectors and channel estimators

boundary layer resolving pseudospectral methods for singular perturbation problems pseudospectral methods are investigated for singularly perturbed boundary value problems for ordinary differential equations odes which possess boundary layers it is well known that if the boundary layer is very small then a very large number of spectral collocation points is required to obtain accurate solutions we introduce here a new effective procedure based on coordinate stretching and the chebyshev pseudospectral method to resolve the boundary layers stable and accurate results are obtained for very thin boundary layers with a fairly small number of spectral collocation points

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations

accuracy enhancement for higher derivatives using chebyshev collocation and a mapping technique a new method is investigated to reduce the roundoff error in computing derivatives using chebyshev collocation methods by using a grid mapping derived by kosloff and tal ezer and the proper choice of the parameter alpha the roundoff error of the k th derivative can be reduced from o n 2k to o n lge k where epsilon is the machine precision and n is the number of collocation points this drastic reduction of roundoff error makes mapped chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large n several other aspects of the mapped chebyshev differentiation matrix are also studied revealing that the mapped chebyshev methods require much less than pi points to resolve a wave the eigenvalues are less sensitive to perturbation by roundoff error and larger time steps can be used for solving pdes all these advantages of the mapped chebyshev methods can be achieved while maintaining spectral accuracy

on minimum variance unbiased estimation of clock offset in a two way message exchange mechanism for many applications distributed networks require the local clocks of the constituent nodes to run close to an agreed upon notion of time most of the widely used clock synchronization algorithms in such systems employ the sender receiver protocol based on a two way timing message exchange paradigm maximum likelihood estimator mle of the clock offset based on the timing message exchanges between two clocks was derived in d r jeske on maximum likelihood estimation of clock offset ieee trans commun vol 53 pp 53 54 jan 2005 when the fixed delays are symmetric and the variable delays in each direction assume an exponential distribution with an unknown mean herein the best linear unbiased estimate using order statistics blue os of the clock offset between two nodes is derived assuming both symmetric and asymmetric exponential network delays respectively the rao blackwell lehmann scheffe theorem is then exploited to obtain the minimum variance unbiased estimate mvue for the clock offset which it is shown to coincide with the blue os in addition it is found that the mvue of the clock offset in the presence of symmetric network delays also coincides with the mle finally in the presence of asymmetric network delays although the mle is biased it is shown to achieve lesser mean square error mse than the mvue in the region around the point where the bidirectional network link delays are symmetric and hence its merit as the most versatile estimator is fairly justified

representation of mutual information via input estimates a relationship between information theory and estimation theory was recently shown for the gaussian channel relating the derivative of mutual information with the minimum mean square error this paper generalizes the link between information theory and estimation theory to arbitrary channels giving representations of the derivative of mutual information as a function of the conditional marginal input distributions given the outputs we illustrate the use of this representation in the efficient numerical computation of the mutual information achieved by inputs such as specific codes or natural language

a fast direct algorithm for the solution of the laplace equation on regions with fractal boundaries an algorithm is presented for the rapid direct solution of the laplace equation on regions with fractal boundaries in a typical application the numerical simulation has to be on a very large scale involving at least tens of thousands of equations with as many unknowns in order to obtain any meaningful results attempts to use conventional techniques have encountered insurmountable difficulties due to excessive cpu time requirements of the computations involved indeed conventional direct algorithms for the solution of linear systems require order o n3 operations for the solution of an n n problem while classical iterative methods require order o n2 operations with the constant strongly dependent on the problem in question in either case the computational expense is prohibitive for large scale problems the direct algorithm of the present paper requires o n operations with a constant dependent only on the geometry of the boundary making it considerably more practical for large scale problems encountered in the computation of harmonic measure of fractals complex iteration theory potential theory and growth phenomena such as crystallization electrodeposition viscous fingering and diffusion limited aggregation

fast algorithms for spectral collocation with non periodic boundary conditions we present a method for the numerical solution of partial differential equations using spectral collocation by employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions the underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations we show how different boundary conditions may be applied and illustrate the technique on the allen cahn equation and the diffusion equation

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

the complexity of the local hamiltonian problem the k locham problem is a natural complete problem for the complexity class qma the quantum analogue of np it is similar in spirit to sc max k sat which is np complete for k geq 2 it was known that the problem is qma complete for any k geq 3 on the other hand 1 locham is in p and hence not believed to be qma complete the complexity of the 2 locham problem has long been outstanding here we settle the question and show that it is qma complete we provide two independent proofs our first proof uses only elementary linear algebra our second proof uses a powerful technique for analyzing the sum of two hamiltonians this technique is based on perturbation theory and we believe that it might prove useful elsewhere using our techniques we also show that adiabatic computation with 2 local interactions on qubits is equivalent to standard quantum computation

on the coalescence time of reversible random walks consider a system of coalescing random walks where each individual performs random walk over a finite graph g or more generally evolves according to some reversible markov chain generator q let c be the first time at which all walkers have coalesced into a single cluster c is closely related to the consensus time of the voter model for this g or q we prove that the expected value of c is at most a constant multiple of the largest hitting time of an element in the state space this solves a problem posed by aldous and fill and gives sharp bounds in many examples including all vertex transitive graphs we also obtain results on the expected time until only k 2 clusters remain our proof tools include a new exponential inequality for the meeting time of a reversible markov chain and a deterministic trajectory which we believe to be of independent interest

ground state spaces of frustration free hamiltonians we study the ground state space properties for frustration free hamiltonians we introduce a concept of reduced spaces to characterize local structures of ground state spaces for a many body system we characterize mathematical structures for the set k of all the k particle reduced spaces which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure the smallest nonzero elements in k called atoms are analogs of extreme points we study the properties of atoms in k and discuss its relationship with ground states of k local frustration free hamiltonians for spin 1 2 systems we show that all the atoms in 2 are unique ground states of some 2 local frustration free hamiltonians moreover we show that the elements in k may not be the join of atoms indicating a richer structure for k beyond the convex structure our study of k deepens the understanding of ground state space properties for frustration free hamiltonians from the new perspective of reduced spaces

on the numerical solution of two point boundary value problems ii abstract in a recent paper greengard and rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two point boundary value problems for second order ordinary differential equations in this paper we extend the method to systems of ordinary differential equations after reducing the system of differential equations to a system of second kind integral equations we discretize the latter via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n p squared n cubed operations with n the number of nodes on the interval p the desired order of convergence and n the number of equations in the system thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference of finite element methods we in addition present a newton method for solving boundary value problems for nonlinear first order systems in which each newton iterate is the solution of a second kind integral equation the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems kr

a quantum walk inspired adiabatic algorithm for solving graph isomorphism problems we present a quantum algorithm for solving graph isomorphism problems that is based on an adiabatic protocol we use a collection of continuous time quantum walks each one generated by an xy hamiltonian to visit the configuration space in this way we avoid a diffusion over all the possible configurations and significantly reduce the dimensionality of the accessible hilbert space within this restricted space the graph isomorphism problem can be translated into searching for a satisfying assignment to a 2 sat satisfiable formula and mapped onto a 2 local hamiltonian without resorting to perturbation gadgets or projective techniques we present an analysis of the time for execution of the algorithm on small graph isomorphism problem instances and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware

complexity of stoquastic frustration free hamiltonians we study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation our results are twofold first we identify a large class of quantum hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer these are stoquastic local hamiltonians with a frustration free ground state a hamiltonian belongs to this class iff it can be represented as h sum a h a where 1 every term h a acts nontrivially on a constant number of qubits 2 every term h a has real nonpositive off diagonal matrix elements in the standard basis and 3 the ground state of h is a ground state of every term h a second we generalize the cook levin theorem proving np completeness of the satisfiability problem to the complexity class ma merlin arthur games a probabilistic analogue of np specifically we construct a quantum version of the k sat problem which we call stoquastic k sat such that stoquastic k sat is contained in ma for any constant k and any promise problem in ma is karp reducible to stoquastic 6 sat this result provides the first nontrivial example of a ma complete promise problem

ramsey numbers and adiabatic quantum computing the graph theoretic ramsey numbers are notoriously difficult to calculate in fact for the two color ramsey numbers r m n with m n 3 only nine are currently known we present a quantum algorithm for the computation of the ramsey numbers r m n we show how the computation of r m n can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution we numerically simulate this adiabatic quantum algorithm and show that it correctly determines the ramsey numbers r 3 3 and r 2 s for 5 s 7 we then discuss the algorithm s experimental implementation and close by showing that ramsey number computation belongs to the quantum complexity class quantum merlin arthur

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

performance study of a near maximum likelihood code aided timing recovery technique in this paper we propose a new code aided ca timing recovery algorithm for various linear constant modulus constellations based on the maximum likelihood ml estimator the first contribution is the derivation of a soft estimator expression of the transmitted symbol instead of its true or hard estimated value which is fed into the timing error detector ted equation the proposed expression includes the log likelihood ratios llrs obtained from a turbo decoder our results show that the proposed ca approach achieves almost as good results as the data aided da approach over a large interval of snr values while achieving a higher spectral efficiency we also derive the corresponding ca cramer rao bounds crb for various modulation orders contrarily to former work we develop here the crb analytical expression for different m psk modulation orders and validate them through comparison to empirical crb obtained by monte carlo iterations the proposed ca estimator realizes an important gain over the nondata aided approach nda and achieves a smaller gap when compared to its relative ca crb especially at moderate snr values where modern systems are constrained to work

auditory motivated level crossing approach to instantaneous frequency estimation we address the problem of estimating the instantaneous frequency if of a phase signal using its level crossing lc information based on front end auditory processing motivation we show that the problem of if estimation using lc information can be cast in the framework of estimation from irregularly sampled data the formulation has the generality of estimating different types of if without the need for a quasistationary assumption we consider two types of if polynomial and bandlimited we use polynomial interpolating functions for the former and for the latter we propose a novel line plus sum of sines model the model parameters are estimated by linear regression considering the noisy case lc data for different levels is analyzed and methods for combining different estimators from lcs are discussed theoretical and extensive simulation results show that the performance of the zero crossing zc based if estimator and the level crossing based if estimator with smaller level values is better than those obtained with higher level values or their combinations the new technique reaches the crame spl acute r rao bound crb roughly above 4 db signal to noise ratio snr and its performance does not deteriorate rapidly with mismatch in the if order compared with the other techniques in the literature

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

low complexity lmmse based mimo ofdm channel estimation via angle domain processing in many situations multiple input multiple output with orthogonal frequency division multiplexing mimo ofdm channels tend to be spatially correlated due for example to limited scattering prior knowledge of this channel spatial correlation and the channel frequency correlation can be exploited by using the linear minimum mean square error lmmse technique however the complexity of the 2 d lmmse technique which fully utilizes both the channel spatial and frequency correlation is quite high to solve this problem this paper presents and analyzes several low complexity suboptimal approximated lmmse channel estimation techniques in the angle domain where the channel model lends itself to a physical interpretation the choice of angle domain techniques is largely dependent on the extent of channel stochastic information e g channel correlation or power that is available to the receiver nevertheless all the proposed angle domain techniques have much lower complexity compared to the 2 d lmmse technique further all the angle domain techniques improve over the conventional least square ls technique for all the typical mimo ofdm models under consideration more importantly our simulation results show that the angle domain quasi 1 d ql d lmmse technique can achieve similar performance compared to the 2 d lmmse technique for all typical mimo ofdm models with significantly lower complexity

a hierarchical o n force calculation algorithm a novel code for the approximate computation of long range forces between n mutually interacting bodies is presented the code is based on a hierarchical tree of cubic cells and features mutual cell cell interactions which are calculated via a cartesian taylor expansion in a symmetric way such that total momentum is conserved the code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort for n 104 the computational costs are found empirically to rise sublinearly with n for applications in stellar dynamics this is the first competitive code with complexity o n it is faster than the standard tree code by a factor of 10 or more

the true cramer rao bound for carrier frequency estimation from a psk signal this paper considers the cramer rao bound crb related to estimating the carrier frequency of a noisy phase shift keying signal the following scenarios are discussed 1 carrier frequency estimation irrespective of the carrier phase based on either known or random data and 2 joint carrier phase and frequency estimation based on either known or random data ideal symbol timing is assumed we compare the results obtained from a commonly used simplified observation model against those resulting from the correct model because of the presence of nuisance parameters random data and or random carrier phase the analytical computation of the corresponding crbs is often not feasible here we present results that are based upon a combined analytical numerical approach our results show that the choice of the observation model has essentially no effect on the crbs at moderate and high signal to noise ratios we also show that of the two scenarios considered joint frequency and phase estimation yields the smaller crb the penalty resulting from frequency estimation irrespective of the carrier phase decreases with increasing observation interval

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

quantum computational complexity of the n representability problem qma complete we study the computational complexity of the n representability problem in quantum chemistry we show that this problem is quantum merlin arthur complete which is the quantum generalization of nondeterministic polynomial time complete our proof uses a simple mapping from spin systems to fermionic systems as well as a convex optimization technique that reduces the problem of finding ground states to n representability

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

a fast algorithm for approximating the ground state energy on a quantum computer estimating the ground state energy of a multiparticle system with relative error using deterministic classical algorithms has cost that grows exponentially with the number of particles the problem depends on a number of state variablesd that is proportional to the number of particles and suers from the curse of dimensionality quantum computers can vanquish this curse in particular we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error using a number of qubits c 0 d log 1 with total cost number of queries plus other quantum operations cd 3 where 0 is arbitrarily small and c and c 0 are inde

bayesian and hybrid cramer rao bounds for the carrier recovery under dynamic phase uncertain channels in this paper we study bayesian and hybrid cramer rao bounds bcrb and hcrb for the code aided ca the data aided da and the non data aided nda dynamical phase estimation of qam modulated signals we address the bounds derivation for both the offline scenario for which the whole observation frame is used and the online which only takes into account the current and the previous observations for the ca scenario we show that the computation of the bayesian information matrix bim and of the hybrid information matrix him is np hard we then resort to the belief propagation bp algorithm or to the bahl cocke jelinek raviv bcjr algorithm to obtain some approximate values moreover in order to avoid the calculus of the inverse of the bim and of the him we present some closed form expressions for the various crbs which greatly reduces the computation complexity finally some simulations allow us to compare the possible improvements enabled by the offline and the ca scenarios

new construction for a qma complete three local hamiltonian we present a new way of encoding a quantum computation into a three local hamiltonian our construction is novel in that it does not include any terms that induce legal illegal clock transitions therefore the weights of the terms in the hamiltonian do not scale with the size of the problem as in previous constructions this improves the construction by kempe and regev quantum inf comput 3 258 264 2003 e print quant ph 0302079 who were the first to prove that three local hamiltonian is complete for the complexity class qma the quantum analog of np quantum k sat a restricted version of the local hamiltonian problem using only projector terms was introduced by bravyi e print quant ph 0602108 as an analog of the classical k sat problem bravyi proved that quantum 4 sat is complete for the class qma with one sided error qma1 and that quantum 2 sat is in p we give an encoding of a quantum circuit into a quantum 4 sat hamiltonian using only three local terms as an intermediate step to this th

phase and frequency estimation for psk packets bounds and algorithms new cramer rao lower bounds crbs are derived for the estimation of phase offset and frequency offset from a block of random bpsk or qpsk symbols at low signal to noise snr ratios these bounds are compared to existing bounds which apply to the unmodulated carrier or to known psk symbols and to the performance of some estimation algorithms

cramer rao lower bounds for qam phase and frequency estimation in this paper we present the true cramer rao lower bounds crlbs for the estimation of phase offset for common quadrature amplitude modulation qam psk and pam signals in awgn channels it is shown that the same analysis also applies to the qam fsk and pam crlbs for frequency offset estimation the ratio of the modulated to the unmodulated crlbs is derived for all qam psk and pam signals and calculated for specific cases of interest this is useful to determine the limiting performance of synchronization circuits for coherent receivers without the need to simulate particular algorithms the hounds are compared to the existing true crlbs for an unmodulated carrier wave cw bpsk and qpsk we investigated new and existing qam phase estimation algorithms in order to verify the new phase crlb this showed that new minimum distance estimator performs close to the qam bound and provides a large improvement over the power law estimator at moderate to high signal to noise ratios

analytic and asymptotic analysis of bayesian cramer rao bound for dynamical phase offset estimation in this paper we present a closed form expression of a bayesian cramer rao lower bound for the estimation of a dynamical phase offset in a non data aided bpsk transmitting context this kind of bound is derived considering two different scenarios a first expression is obtained in an offline context and then a second expression in an online context logically follows the snr asymptotic expressions of this bound drive us to introduce a new asymptotic bound namely the asymptotic bayesian cramer rao bound this bound is close to the classical bayesian bound but is easier to evaluate

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

epsilon net method for optimizations over separable states we give algorithms for the optimization problem max q where q is a hermitian matrix and the variable is a bipartite separable quantum state this problem lies at the heart of several problems in quantum computation and information such as the complexity of qma 2 while the problem is np hard our algorithms are better than brute force for several instances of interest in particular they give pspace upper bounds on promise problems admitting a qma 2 protocol in which the verifier performs only a logarithmic number of elementary gates on both proofs as well as the promise problem of deciding if a bipartite local hamiltonian has a large or small ground energy for q 0 our algorithm runs in time exponential in q f while the existence of such an algorithm was first proved recently by brandao christandl and yard 2011 8 our algorithm is conceptually simpler

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

hybrid fm polynomial phase signal modeling parameter estimation and cramer rao bounds parameter estimation for a class of nonstationary signal models is addressed the class contains combination of a polynomial phase signal pps and a frequency modulated fm component of the sinusoidal or hyperbolic type such signals appear in radar and sonar applications involving moving targets with vibrating or rotating components a novel approach is proposed that allows us to decouple estimation of the fm parameters from those of the pps relying on properties of the multilag high order ambiguity function ml haf the accuracy achievable by any unbiased estimator of the hybrid fm pps parameters is investigated by means of the cramer rao lower bounds crlbs both exact and large sample approximate expressions of the bounds are derived and compared with the performance of the proposed methods based on monte carlo simulations

making classical ground state spin computing fault tolerant we examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation this model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions in its most primitive form systems constructed in this model cannot compute in an error free manner when working at nonzero temperature however by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free we achieve this by using techniques in fault tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature we further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class merlin arthur this provides an interesting connection between the physical behavior of certain many body spin systems and computational complexity

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

quasi ml period estimation from incomplete timing data given a noisy sequence of possibly shifted integer multiples of a certain period it is often of interest to accurately estimate the period with known integer regressors the problem is classical linear regression in many applications however the regressors are unknown integers and only loose bounds on the period are available examples include hop period and timing estimation wherein hops may be missed at the output of the frequency discriminator or the emitter may hop out of band pulse repetition interval pri analysis and passive rotating beam radio scanning we study several pertinent period estimators our emphasis is on a quasi maximum likelihood approach developed herein and an earlier method based on the fourier transform of a dirac delta train representation of the data surprisingly both are capable of attaining the clairvoyant crame spl acute r rao bound at moderate signal to noise ratios snrs even for short e g 10 samples we carefully address parameter identifiability issues and corroborate our findings with extensive simulations

near optimum low complexity smoothing loops for dynamical phase estimation application to bpsk modulated signals this correspondence provides and analyzes a low complexity near optimum fixed interval smoothing algorithm that approaches the performance of an optimal smoother for the price of two low complexity sequential estimators i e two phase locked loops plls based on a linear approximation of the problem a theoretical performance evaluation is given the theoretical results are compared to some simulation results and to the bayesian and hybrid cramer rao bounds they illustrate the good performance of the proposed smoothing pll s pll algorithm

on complexity of the quantum ising model we study complexity of several problems related to the transverse field ising model tim first we consider the problem of estimating the ground state energy known as the local hamiltonian problem lhp it is shown that the lhp for tim on degree 3 graphs is equivalent modulo polynomial reductions to the lhp for general k local stoquastic hamiltonians with any constant k ge 2 this result implies that estimating the ground state energy of tim on degree 3 graphs is a complete problem for the complexity class mathsf stoqma an extension of the classical class mathsf ma as a corollary we complete the complexity classification of 2 local hamiltonians with a fixed set of interactions proposed recently by cubitt and montanaro secondly we study quantum annealing algorithms for finding ground states of classical spin hamiltonians associated with hard optimization problems we prove that the quantum annealing with tim hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k local stoquastic hamiltonians this subclass includes all hamiltonians representable as a sum of a k local diagonal hamiltonian and a 2 local stoquastic hamiltonian

robust rls in the presence of correlated noise using outlier sparsity relative to batch alternatives the recursive least squares rls algorithm has well appreciated merits of reduced complexity and storage requirements for online processing of stationary signals and also for tracking slowly varying nonstationary signals however rls is challenged when in addition to noise outliers are also present in the data due to e g impulsive disturbances existing robust rls approaches are resilient to outliers but require the nominal noise to be white an assumption that may not hold in e g sensor networks where neighboring sensors are affected by correlated ambient noise prewhitening with the known noise covariance is not a viable option because it spreads the outliers to noncontaminated measurements which leads to inefficient utilization of the available data and unsatisfactory performance in this correspondence a robust rls algorithm is developed capable of handling outliers and correlated noise simultaneously in the proposed method outliers are treated as nuisance variables and estimated jointly with the wanted parameters identifiability is ensured by exploiting the sparsity of outliers which is effected via regularizing the least squares ls criterion with the l1 norm of the outlier vectors this leads to an optimization problem whose solution yields the robust rls estimates for low complexity real time operation a suboptimal online algorithm is proposed which entails closed form updates per time step in the spirit of rls simulations demonstrate the effectiveness and improved performance of the proposed method in comparison with the nonrobust rls and its state of the art robust renditions

carrier phase and frequency estimation for pilot symbol assisted transmission bounds and algorithms in this paper we consider the cramer rao lower bound crb for the joint estimation of the carrier phase and the frequency offset from a noisy linearly modulated burst signal containing random data symbols dss as well as known pilot symbols pss we point out that the crb depends on the location of the pss in the burst the number of pss the number of dss the signal to noise ratio snr and the data modulation scheme distributing the pss symmetrically about the center of the burst and estimating the carrier phase in the center of the burst interval decouples the frequency and phase estimation making the crb for phase estimation independent of the specific location of the pss at low and moderate snr the crbs for both phase and frequency estimation decrease as the fraction of the pss in the burst increases in addition the crb for frequency estimation decreases as the pss are separated with more dss numerical evaluation of the crb indicates that the carrier phase and frequency of a hybrid burst i e containing pss and dss can be estimated more accurately when exploiting both the presence of the dss and the a priori knowledge about the pss instead of using only the knowledge about the pss and ignoring the dss or considering all the received symbols pss and dss as unknown and ignoring the knowledge about the pss comparison of the crb with the performance of existing carrier synchronizers shows that the iterative soft decision directed sdd estimator with data aided da initialization performs very closely to the crb and provides a large improvement over the classical non data aided nda estimator at lower snr