fluid dynamic description of flocking via the povzner boltzmann equation we introduce and discuss the possible dynamics of groups of indistinguishable agents which are interacting according to their relative positions with the aim of deriving hydrodynamic equations these models are developed to mimic the collective motion of groups of species such as bird flocks fish schools herds of quadrupeds or bacteria colonies our starting model for these interactions is the povzner equation 21 which describes a dilute gas in which binary collisions of elastic spheres depend on their relative positions following the cucker and smale model 9 we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance under the assumption of weak dissipation it is shown that the povzner equation is modified through a correction in the form of a nonlinear friction type operator using this correction we formally obtain from the povzner equation in a direct way a fluid dynamic description of a system of agents with weak dissipative interactions with a coefficient of restitution that depends on their relative distance

interlacing of zeros of gegenbauer polynomials of non consecutive degree from different sequences a theorem due to stieltjes states that if p n n 0 infty is any orthogonal sequence then between any two consecutive zeros of p k there is at least one zero of p n whenever k frac 12 if 0 t k 1 and also to the zeros of c n 1 lambda and c n 2 lambda k if k in 1 2 3 more generally we prove that stieltjes interlacing holds between the zeros of the kth derivative of c n lambda and the zeros of c n 1 lambda k in 1 2 dots n 1 and we derive associated polynomials that play an analogous role to the de boor saff polynomials in completing the interlacing process of the zeros

flocking of the cucker smale model on general digraphs in the literature the study of the cucker smale c s flocking model is all restricted to connected undirected graphs or digraphs of some special types containing only a single root in this technical note we establish the flocking behavior of the c s model on general digraphs which contain all existing interaction topologies in relation to the c s model as special cases in particular the topology with multiple leaders is included the flocking results are guaranteed under some conditions on the initial state of the flock only the critical exponent below which unconditional convergence holds is given depending only on the interaction topology it is consistent with the ones obtained under some known special topologies in the literature

fast block toeplitz orthogonalization algorithms are presented which compute theqr factorization of a block toeplitz matrix ino n 2 block operations wheren is the block order of the matrix and a block operation is a multiplication inversion or a set of householder operations involving one or two blocks the algorithms are in general analogous to those presented in the scalar toeplitz case in a previous paper but the basic operation is the householder transform rather than the givens transform and the computation of the householder coefficients and other working matrices requires special treatment two algorithms are presented the first computes onlyr explicitly and the second computes bothq andr

avoiding collisions in flocks among the many models for flocking systems of interacting particles the one introduced by cucker and smale has attracted attention due to the fact that a convergence to flocking i e to a common velocity could be established depending on conditions on the initial state of the system in this note we extend this model by adding to it a repelling force between particles we show that for this modified model convergence to flocking is established along the same lines while in addition avoidance of collisions i e the respect of a minimal distance between particles is ensured

cucker smale flocking under hierarchical leadership a mathematical theory on flocking serves the foundation for several ubiquitous multiagent phenomena in biology ecology sensor networks and economics as well as social behavior like language emergence and evolution directly inspired by the recent fundamental works of cucker and smale on the construction and analysis of a generic flocking model we study the emergent behavior of cucker smale flocking under hierarchical leadership the rates of convergence towards asymptotically coherent group patterns in different scenarios are established the consistent convergence towards coherent patterns may well reveal the advantages and necessities of having leaders and leadership in a complex biological technological economic or social system with sufficient intelligence

fast toeplitz orthogonalization algorithms are presented which compute theqr factorization of an order n toeplitz matrix ino n 2 operations the first algorithm computes onlyr explicitly and the second computes bothq andr the algorithms are derived from a well known procedure for performing the rank 1 update ofqr factors using the shift invariance property of the toeplitz matrix the algorithms can be used to solve the toeplitz least squares problem and can be modified to solve toeplitz systems ino n space

fast orthogonalization algorithm and parallel architecture for ar spectral estimation based on forward backward linear prediction truncated qr methods have been shown to be comparable to svd based methods for the sinusoidal frequency estimation based on the forward backward linear prediction fblp model however without exploiting the special structure of the fblp matrix the qr decomposition qrd of the fblp matrix has a computational complexity on the order of 2 6m n n sup 2 3 o n sup 2 for a 2m n fblp matrix a fast algorithm to perform the qrd of the fblp matrix by exploiting its special toeplitz hankel form is proposed the computational complexity is reduced to 10n sup 2 4mn o n the fast algorithm can also be easily implemented by a linear systolic array reducing the number of time steps required to 2m 5n 4

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

the statistical performance of some instantaneous frequency estimators the authors examine the class of smoothed central finite difference scfd instantaneous frequency if estimators which are based on finite differencing of the phase of the analytic signal these estimators are closely related to if estimation via the periodic first moment with respect to frequency of discrete time frequency representations tfrs in l cohen s 1966 class the authors determine the distribution of this class of estimators and establish a framework which allows the comparison of several other estimators such as the zero crossing estimator and one based on linear regression on the signal phase it is found that the regression if estimator is biased and exhibits a large threshold for much of the frequency range by replacing the linear convolution operation in the regression estimator with the appropriate convolution operation for circular data the authors obtain the parabolic scfd pscfd estimator which is unbiased and has a frequency independent variance yet retains the optimal performance and simplicity of the original estimator

special linear combinations of orthogonal polynomials abstract the paper lists a number of problems that motivate consideration of special linear combinations of polynomials orthogonal with the weight p x on the interval a b we study properties of the polynomials as well as the necessary and sufficient conditions for their orthogonality the special linear combinations of chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations

the relationship between instantaneous frequency and time frequency representations the relationship between instantaneous frequency estimation based on the derivative of the phase of the analytic signal and the first moment of the general time frequency representation from cohen s 1989 class is given in the continuous and discrete time domains application of the standard linear definition of first moment to discrete time frequency representations leads to biased instantaneous frequency estimators with high variance to correct this problem it is shown that periodic circular definitions of moments must be used to account for the periodization of the frequency variable due to sampling

a new model for self organized dynamics and its flocking behavior we introduce a model for self organized dynamics which we argue addresses several drawbacks of the celebrated cucker smale c s model the proposed model does not only take into account the distance between agents but instead the influence between agents is scaled in term of their relative distance consequently our model does not involve any explicit dependence on the number of agents only their geometry in phase space is taken into account the use of relative distances destroys the symmetry property of the original c s model which was the key for the various recent studies of c s flocking behavior to this end we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems which covers systems with non symmetric influence matrices in particular we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with leaders

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

estimation of multiple sinusoidal frequencies using truncated least squares methods to reduce the computational complexity of resolving closely spaced frequencies three truncated qr methods are proposed 1 truncated qr without column pivoting tqr 2 truncated qr with reordered columns tqrr and 3 truncated qr with column pivoting tqrp it is demonstrated that many of the benefits of the singular value decomposition based methods are achievable under the truncated qr methods with much lower computational cost based on the forward backward linear prediction model computer simulations and comparisons are provided for different truncation methods under various snrs comparisons of asymptotic performance for large data samples are also given

a fast but unstable orthogonal triangularization technique for toeplitz matrices abstract d sweet s clever qr decomposition algorithm for toeplitz matrices is considered it requires only o n 2 flops to factor an n n matrix we analyze the algorithm to point out its deficiencies and compare it against two other toeplitz solvers

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

computing the minimum phase filter using the ql factorization we investigate the ql factorization of a time invariant convolutive filtering matrix and show that this factorization not only provides the finite length equivalent to the minimum phase filter but also gives the associated all pass filter the convergence properties are analyzed and we derive the exact convergence rate and an upper bound for a simple single input single output system with filter length finally this upper bound is used to derive an approximation of the convergence rate for systems of arbitrary length implementation wise the method has the advantage of being numerically stable and straight forward to extend to the multiple input multiple output case furthermore due to the existence of fast ql factorization methods it is possible to compute the filters efficiently

a unified approach to the stft tfds and instantaneous frequency cohen s class of time frequency representations tfrs is reformulated into a discrete time discrete frequency computer implementable form it is shown how in this form many of the properties of the continuous time continuous frequency formulation are either lost or altered intuitions applicable in the continuous time case do not necessarily carry over to the discrete time case examined the properties of the discrete variable formulation examined are the presence and form of cross terms instantaneous frequency estimation and relationships between cohen s class of tfrs a parameterized class of distributions which is a blending between the short time fourier transform stft and the wigner ville distribution the two main conclusions are that all tfrs of cohen s class implementable in the given form which includes all commonly used tfrs possess cross terms and that instantaneous frequency estimation using periodic moments of these tfrs is purposeless since simpler methods obtain the same result

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

collective motion from consensus with cartesian coordinate coupling collective motions including rendezvous circular patterns and logarithmic spiral patterns can be achieved by introducing cartesian coordinate coupling to existing consensus algorithms we study the collective motions of a team of vehicles in 3d by introducing a rotation matrix to an existing consensus algorithm for double integrator dynamics it is shown that the network topology the damping gain and the value of the euler angle all affect the resulting collective motions we show that when the nonsymmetric laplacian matrix has certain properties the damping gain is above a certain bound and the euler angle is below equal or above a critical value the vehicles will eventually rendezvous move on circular orbits or follow logarithmic spiral curves lying on a plane perpendicular to the euler axis in particular when the vehicles eventually move on circular orbits the relative radii of the orbits respectively the relative phases of the vehicles on their orbits are equal to the relative magnitudes respectively the relative phases of the components of a right eigenvector associated with a critical eigenvalue of the nonsymmetric laplacian matrix simulation results are presented to demonstrate the theoretical results

a cucker smale model with noise and delay a generalization of the cucker smale model for collective animal behavior is investigated the model is formulated as a system of delayed stochastic differential equations it incorporates two additional processes which are present in animal decision making but are often neglected in modeling i stochasticity imperfections of individual behavior and ii delayed responses of individuals to signals in their environment sufficient conditions for flocking for the generalized cucker smale model are derived by using a suitable lyapunov functional as a by product a new result regarding the asymptotic behavior of delayed geometric brownian motion is obtained in the second part of the paper results of systematic numerical simulations are presented they not only illustrate the analytical results but hint at a somehow surprising behavior of the system namely that the introduction of an intermediate time delay may facilitate flocking

fast algorithms for computing qr and cholesky factors of toeplitz operators levinson lattice and schur recursions provide alternative ways of computing reflection coefficients for stationary time series the authors show that when a toeplitz correlation matrix is a product of two toeplitz data matrices as in the correlation method of linear prediction then the levinson recursions can be used to derive the lattice recursions and the lattice recursions can be used to derive the schur recursions they explore the close relation between qr and cholesky algorithms in the toeplitz case and present the underlying matrix factorization

schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

extended lagrange interpolation in weighted uniform norm the author studies the uniform convergence of extended lagrange interpolation processes based on the zeros of generalized laguerre polynomials

when do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials given p n n 0 a sequence of monic orthogonal polynomials we analyze their linear combinations with constant coefficients and fixed length i e q n x p n x a 1p n 1 x a kp n k a k 0 n k necessary and sufficient conditions are given for the orthogonality of the sequence q n n 0 an interesting interpretation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is shown moreover in the case k 2 we characterize the families p n n 0 such that the corresponding polynomials q n n 0 are also orthogonal

generalized associated polynomials and functions of second kind let pv v n0 pv v v 1 be a sequence of polynomials generated by a three term recurrence relation shifting the recurrence coefficients of the elements of pv v n0 we get a sequence of so called associated polynomials which play an important role in the theory of orthogonal polynomials we generalize this concept of associating for arbitrary polynomials vn n especially if vn is expanded in terms of pv v 0 n their associated polynomials are clenshaw polynomials which are used in numerical mathematics as a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of clenshaw polynomials analogously as for orthogonal polynomials we define functions of second kind for vn we prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials

characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials in the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients as an application we obtain quadrature rules for polynomial modifications of the bernstein measures on 1 1 having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients

zeros of linear combinations of laguerre polynomials from different sequences we study interlacing properties of the zeros of two types of linear combinations of laguerre polynomials with different parameters namely r n l n a al n a and s n l n a bl n 1 a proofs and numerical counterexamples are given in situations where the zeros of r n and s n respectively interlace or do not in general with the zeros of l k a l k a k n or n 1 the results we prove hold for continuous as well as integral shifts of the parameter a

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

zeros of orthogonal polynomials generated by canonical perturbations of measures abstract in this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the uvarov perturbation of a positive borel measure d x when the measure is semiclassical then its electrostatic interpretation is given

positive quadrature formulas iii asymptotics of weights first we discuss briefly our former characterization theorem for positive interpolation quadrature formulas abbreviated qf provide an equivalent characterization in terms of jacobi matrices and give links and applications to other qf in particular to gauss kronrod quadratures and recent rediscoveries then for any polynomial t n which generates a positive qf a weight function depending on n is given with respect to which t n is orthogonal to p n 1 with the help of this result an asymptotic representation of the quadrature weights is derived in general the asymptotic behaviour is different from that of the gaussian weights only under additional conditions do the quadrature weights satisfy the so called circle law corresponding results are obtained for positive qf of radau and lobatto type

positivity of the weights of extended clenshaw curtis quadrature rules we prove that some extended clenshaw curtis quadrature rules have all weights positive we also present extended filippi rules of open type having all weights positive conjectures on the possibility of other positive quadrature rules embedded in clenshaw curtis or filippi rule are suggested

cucker smale flocking under rooted leadership with fixed and switching topologies in recent years a number of articles have focused on mathematical models for emergent phenomena for instance the flocking of birds or the schooling of fish in 2007 cucker and smale proposed an ingenious model which captures many of the observed features of moving animals subsequently shen extended the result to hierarchically structured flocks motivated by these works in this paper we study the discrete cucker smale flocking under rooted leadership which means that there exists an overall leader such that any other agent is led directly or indirectly by the leader the feature of our proposal departing from the existing models is that both the assumption of symmetry and the partial ordering of a hierarchy are dropped the rooted leadership topology is a necessary condition for the group to converge towards a single leader s fixed constant velocity the rates of convergence are established for flocks with fixed and switching topologies the results may reveal the applicability and advantage o

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

a kinetic flocking model with diffusion we study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the cucker smale flocking in presence of diffusion whose strength depends on the density this kinetic equation describes the collective behavior of an ensemble of organisms animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration our analysis takes advantage both from the fact that the global equilibrium is a maxwellian distribution function and on the contrary to what happens in the cucker smale model ieee trans autom control 52 852 862 2007 the interaction potential is an integrable function precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

stochastic flocking dynamics of the cucker smale model with multiplicative white noises we present a strong asymptotic stochastic flocking estimate for the stochastically perturbed cucker smale model we characterize a form of multiplicative white noises and present sufficient conditions on the control parameters to guarantee the almost sure exponential convergence toward constant equilibrium states when the strength of multiplicative noises is sufficiently large we show that the strong stochastic flocking occurs even for negative communication weights

a fast least square deconvolution algorithm for vocal tract cross section estimation an efficient computation algorithm based on unitary jacobi type rotations is developed for fast least square deconvolution specifically this algorithm is able to solve a positive definite covariance matrix with a special toeplitz structure in o n sup 2 operations instead of o n sup 3 operations this algorithm which takes advantage of the inherent structure of the underlying deconvolution problem is guaranteed to be numerically stable since only unitary transformation is used it is implemented on an experimental system for the estimation of human vocal tract cross section a significant gain in processing is observed

qr factorization of toeplitz matrices this paper presents a new algorithm for computing theqr factorization of anm n toeplitz matrix ino mn operations the algorithm exploits the procedure for the rank 1 modification and the fact that both principal m 1 n 1 submatrices of the toeplitz matrix are identical an efficient parallel implementation of the algorithm is possible

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

some properties of zeros of sobolev type orthogonal polynomials abstract for polynomials orthogonal with respect to a discrete sobolev product we prove that for each n q n has at least n m zeros on the convex hull of the support of the measure where m denotes the number of terms in the discrete part interlacing properties of zeros are also described

the fast adaptive rotor s rls algorithm a fast algorithm for implementation of the qr factorization based recursive least squares rls adaptive filter is discussed this fast adaptive rotors far algorithm can be implemented with a pipelined array of processors called rotors and cisors the rotors compute 2 2 orthogonal givens rotations and the cisors compute the cosines and sines of the angles used in the rotors the algorithm requires 4n rotors and 2n cisors at each iteration to compute the solution to the rls problem the algorithm is numerically stable the far algorithm is derived using a single generic updating formula for orthogonal matrices which is introduced and derived whereas the generic updating formula is reminiscent of previous fast transversal filters and fast lattice algorithms the set of internally propogated adaptive filter quantities is entirely different and constitutes yet another complete characterization of the rls covariance and the forward backward and pinning estimation problems

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

hybrid algorithm for fast toeplitz orthogonalization new techniques for fast toeplitz qr decomposition are presented the methods are based on the shift invariance property of a toeplitz matrix the numerical properties of the algorithms are discussed and some comparisons are made with two other fast toeplitz orthogonalization methods