qualitative behavior of non abelian toda like flows in this paper we analyze and compare the long term behavior of certain non abelian generalizations of the integrable toda lattice flow in particular we show that these flows exhibit generalized sorting behavior analogous to the sorting behavior of the original tridiagonal toda lattice we show that one of these generalizations the block double bracket equation is gradient with respect to a normal metric and exactly solvable this provides a new solvable case of the double bracket system

differential equations and matrix inequalities on isospectral families abstract motivated by the study of certain classes of hamiltonian and gradient flows we develop a set of inequalities that bound the size of a commutator x n m in terms of the eigenvalues of x and n these results apply directly to elements of matrix lie algebras such as skew hermitian matrices skew symmetric matrices etc and with a trivial modification can also be applied to hermitian matrices

an adjoint for likelihood maximization the process of likelihood maximization can be found in many different areas of computational modelling however the construction of such models via likelihood maximization requires the solution of

kriging hyperparameter tuning strategies response surfaces have been extensively used as a method of building effective surrogate models of high fidelity computational simulations of the numerous types of response surface models kriging is perhaps one of the most effective due to its ability to model complicated responses through interpolation or regression of known data while providing an estimate of the error in its prediction there is however little information indicating the extent to which the hyperparameters of a kriging model need to be tuned for the resulting surrogate model to be effective the following paper addresses this issue by investigating how often and how well it is necessary to tune the hyperparameters of a kriging model as it is updated during an optimization process to this end an optimization benchmarking procedure is introduced and used to assess the performance of five different tuning strategies over a range of problem sizes the results of this benchmark demonstrate the performance gains that can be associated with reducing the complexity of the hyperparameter tuning process for complicated design problems the strategy of tuning hyperparameters only once after the initial design of experiments is shown to perform poorly

exploiting hessian matrix and trust region algorithm in hyperparameters estimation of gaussian process gaussian process gp regression is a bayesian non parametric regression model showing good performance in various applications however it is quite rare to see research results on log likelihood maximization algorithms instead of the commonly used conjugate gradient method the hessian matrix is first derived simplified in this paper and the trust region optimization method is then presented to estimate gp hyperparameters numerical experiments verify the theoretical analysis showing the advantages of using hessian matrix and trust region algorithms in the gp context the trust region optimization method is a robust alternative to conjugate gradient method also in view of future researches on approximate and or parallel gp implementation

a cross dialectal acoustic comparison of vowels in northern and southern british english this study compares the duration and first two formants f1 and f2 of 11 nominal monophthongs and five nominal diphthongs in standard southern british english ssbe and a northern english dialect f1 and f2 trajectories were fitted with parametric curves using the discrete cosine transform dct and the zeroth dct coefficient represented formant trajectory means and the first dct coefficient represented the magnitude and direction of formant trajectory change to characterize vowel inherent spectral change visc cross dialectal comparisons involving these measures revealed significant differences for the phonologically back monophthongs u and also and the diphthongs e a most cross dialectal differences are in zeroth dct coefficients suggesting formant trajectory means tend to characterize such differences while first dct coefficient differences were more numerous for diphthongs with respect to visc the most striking differences are that u is considerably more diphthongized in the northern dialect and that the f2 trajectory of proceeds in opposite directions in the two dialects cross dialectal differences were found to be largely unaffected by the consonantal context in which the vowels were produced the implications of the results are discussed in relation to visc consonantal context effects and speech perception

iso spectral deformations of general matrix and their reductions on lie algebras we study an iso spectral deformation of the general matrix which is a natural generalization of the nonperiodic toda lattice equation this deformation is equivalent to the cholesky flow a continuous version of the cholesky algorithm introduced by watkins we prove the integrability of the deformation and give an explicit formula for the solution to the initial value problem the formula is obtained by generalizing the orthogonalization procedure of szego using the formula the solution to the lu matrix factorization can the constructed explicitly based on the root spaces for simple lie algebras we consider several reductions of the equation this leads to generalized toda equations related to other classical semi simple lie algebras which include the integrable systems studied by bogoyavlensky and kostant we show these systems can be solved explicitly in a unified way the behaviors of the solutions are also studied generically there are two types of solutions having either sorting property or blowing up to infinity in finite time

the geometric nature of the flaschka transformation we show that the flaschka map originally introduced to analyze the dynamics of the integrable toda lattice system is the inverse of a momentum map we discuss the geometrical setting of the map and apply it to the generalized toda lattice systems on semisimple lie algebras the rigid body system on toda orbits and to coadjoint orbits of semidirect products groups in addition we develop an infinite dimensional generalization for the group of area preserving diffeomorphisms of the annulus and apply it to the analysis of the dispersionless toda lattice pde and the solvable rigid body pde

design and analysis of noisy computer experiments of functions calculated by long running computer codes the literature in this area commonly assumes that the objective function is a smooth deterministic function of the inputs yet it is well known that many computer simulations especiallythoseofcomputational fluidandstructuraldynamicscodes oftendisplaywhatonemightcall numerical noise rather than lying on a smooth curve results appear to contain a random scatter about a smooth trend this paper extends previous optimization methods based on the interpolating method ofkriging to the case of such noisy computer experiments firstly we review how the kriging interpolation can be modified to filter out numerical noise we then show how to adjust the estimate of the error in a kriging prediction so that previous approaches to optimization such as the method of maximizing the expected improvement continue to work effectively we introduce the problems associated with noise and demonstrate our approach using computational fluid dynamics based problems

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

nonholonomic double bracket equations and the gauss thermostat in this rapid communication we consider certain equations that arise from imposing a constant kinetic energy constraint on a one dimensional set of oscillators this is a nonlinear nonholonomic constraint on these oscillators and the dynamics are consistent with gauss s law of least constraint dynamics of this sort are of interest in nonequilibrium molecular dynamics we show that under certain choices of external potential these equations give rise to a generalization of the so called double bracket equations which are of interest in studying gradient flows and integrable systems such as the toda lattice in the case of harmonic potentials the flow is described by a symmetric bracket and periodic solutions are obtained

lie algebra representations nilpotent matrices and the c numerical range abstract in this paper we survey some applications of the representation theory of lie algebras to linear algebra this includes the derivation of the jordan form for tensor products of invertible matrices the study of normal form problems for nilpotent matrices as well as the derivation of explicit formulas for the c numerical radius of certain nilpotent block shift matrices arising in quantum mechanics and nuclear magnetic resonance nmr spectroscopy in this latter case a conjecture concerning an explicit formula for the c numerical radius is stated we show the existence of unitary transformations that realize the prospective maxima our approach depends on the clebsch gordan decomposition for unitary representations of su 2 c

a differential equation for diagonalizing complex semisimple lie algebra elements abstract in this paper we consider a generalization of ebenbauer s differential equation for non symmetric matrix diagonalization to a flow on arbitrary complex semisimple lie algebras the flow is designed in such a way that the desired diagonalizations are precisely the equilibrium points in a given cartan subalgebra we characterize the set of all equilibria and establish a morse bott type property of the flow global convergence to single equilibrium points is shown starting from any semisimple lie algebra element for strongly regular initial conditions we prove that the flow converges to an element of the cartan subalgebra and thus achieves asymptotic diagonalization

spectral analysis of fokker planck and related operators arising from linear stochastic differential equations we study spectral properties of certain families of linear second order differential operators arising from linear stochastic differential equations we construct a basis in the hilbert space of square integrable functions using modified hermite polynomials and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed in particular we completely describe the spectrum of the fokker planck operator on an appropriate invariant subspace of rapidly decaying functions the eigenvalues of the fokker planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state which is important for stochastic estimation and control applications we show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator lie algebra which leads to a simple expression for the flow of their eigenfunctions

use of kriging models to approximate deterministic computer models the use of kriging models for approximation and metamodel based design and optimization has been steadily on the rise in the past decade the widespread usage of kriging models appears to be hampered by 1 the lack of guidance in selecting the appropriate form of the kriging model 2 computationally efficient algorithms for estimating the model s parameters and 3 an effective method to assess the resulting model s quality in this paper we compare 1 maximum likelihood estimation mle and cross validation cv parameter estimation methods for selecting a kriging model s parameters given its form and 2 and an r 2 of prediction and the corrected akaike information criterion for assessing the quality of the created kriging model permitting the comparison of different forms of a kriging model these methods are demonstrated with six test problems finally different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

structural optimization using kriging approximation an optimization method using kriging approximation is applied to a structural optimization problem the method involves two main processes the first is a space estimation process that uses the kriging method and the second is an optimization process the use of the kriging method makes it easier to perform the approximation optimization as an example of the estimation performed as part of structural optimization a response surface for layout optimization of beam reinforcement is estimated to evaluate the applicability of the kriging method kriging estimation is compared with neural network approximation as a numerical example the optimization of a stiffened cylinder for an eigenfrequency problem is illustrated the obtained results clearly show the applicability of the method

isospectral flows on a class of finite dimensional jacobi matrices abstract we present a new matrix valued isospectral ordinary differential equation that asymptotically block diagonalizes n n zero diagonal jacobi matrices employed as its initial condition this o d e features a right hand side with a nested commutator of matrices and structurally resembles the double bracket o d e studied by r w brockett in 1991 we prove that its solutions converge asymptotically that the limit is block diagonal and above all that the limit matrix is defined uniquely as follows for n even a block diagonal matrix containing 2 2 blocks such that the super diagonal entries are sorted by strictly increasing absolute value furthermore the off diagonal entries in these 2 2 blocks have the same sign as the respective entries in the matrix employed as the initial condition for n odd there is one additional 1 1 block containing a zero that is the top left entry of the limit matrix the results presented here extend some early work of kac and van moerbeke

an automated hybrid genetic conjugate gradient algorithm for multimodal optimization problems the genetic algorithm ga have good global search characteristics and local optimizing algorithm loa have good local search characteristics in the present work best characteristics of ga and loa are combined to develop a hybrid genetic algorithm hga a bank of ga s are used to get a good starting solution for a conjugate gradient algorithm the number of ga banks is selected using an automated procedure based on fibonacci numbers this automated hybrid genetic algorithm ahga is used for solving general multimodal optimization problems while assuring global optimality to a significant degree the designed algorithm is also tested against a variety of standard test functions besides assuring global optimality to a significant extent ahga is also found to be an efficient algorithm requiring only one tuning error parameter saving considerable time on the part of the user the method also addresses the problem of selecting a good starting design for gradient based algorithm further in the few cases where the algorithm does not converge to a global minima a local minima is assured because of the use of the gradient based local search in the final stage of the algorithm further the algorithm assures one final solution to the optimization problem and addresses the problem of providing a deterministic output which inhibits the use of ga in engineering optimization software and engineering applications

double bracket equations and geodesic flows on symmetric spaces in this paper we consider the geometry of hamiltonian flows on the cotangent bundle of coadjoint orbits of compact lie groups and on symmetric spaces a key idea here is the use of the normal metric to define the kinetic energy this leads to hamiltonian flows of the double bracket type we analyze the integrability of geodesic flows according to the method of thimm we obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity we demonstrate for example integrability of the geodesic flow on the real and complex grassmannians we also consider right invariant systems and the generalized rigid body equations in this setting

lie algebraic aspects of the finite nonperiodic toda flows in this paper we review various aspects of nonperiodic full and tridiagonal toda flow theory in particular we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic toda equations we describe the structure of these systems on general semisimple lie algebras in the case of the full kostant asymmetric toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior we describe the relationship between the asymmetric toda flows and the symmetric and indefinite toda flows and show how one may conjugate from the full kostant toda flows to the full symmetric toda flows via a poisson map we describe briefly related toda systems such as nonabelian toda and the peakon flows

acoustic evidence for dynamic formant trajectories in australian english vowels the extent to which it is necessary to model the dynamic behavior of vowel formants to enable vowel separation has been the subject of debate in recent years to investigate this issue a study has been made on the vowels of 132 australian english speakers male and female the degree of vowel separation from the formant values at the target was contrasted to that from modeling the formant contour with discrete cosine transform coefficients the findings are that although it is necessary to model the formant contour to separate out the diphthongs the formant values at the target plus vowel duration are sufficient to separate out the monophthongs however further analysis revealed that there are formant contour differences which benefit the within class separation of the tense lax monophthong pairs

adjoint an approach for the rapid development of discrete adjoint solvers an automatic differentiation tool is used to develop the adjoint code for a three dimensional computational fluid dynamics solver rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver we have applied it selectively to produce code that computes the flux jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method the resulting linear discrete adjoint system is then solved using the portable extensible toolkit for scientific computation this selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint adjoint approach this approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver furthermore the approach is largely automatic thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations these significant advantages come at the cost of increased memory requirements for the adjoint solver derivatives of drag and lift coefficients are validated and the low computational cost and ease of implementation of the method are shown

evaluation of a strategy for automatic formant tracking variations on an automatic formant tracking strategy developed at alberta will be compared to manual formant measurements from two databases of vowels spoken by men women and children in texas or michigan correct vowel formant candidates for f1 f2 and f3 may be found roughly 85 90 percent of the time for adult male speakers using autocorrelation lpc with the following settings f3 maximum at 3000 hz lpc order of 14 sampling rate of 10 khz j markel and a gray linear prediction of speech springer new york 1975 experience shows good results are also often found with females and children s speech provided the sampling rate and f3 maximum are scaled appropriately for each speaker our new basic strategy involves analyzing each utterance at several distinct sampling rates and coordinated f3 cutoff frequencies with a fixed lpc order each scaling choice provides an independent set of candidates that is post processed by a simple tracking algorithm a correlation measure between a spectrogram of the original signal and one resynthesized from each estimated track set is defined this measure is combined with other heuristic figures of merit based on e g continuity formant ranges and bandwidths to choose the best analysis

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

estimation principal components and hamiltonian systems in this paper we show how symplectic geometry can be used to analyse a fundamental problem in estimation theory the fitting of lines and planes to a data set in n dimensions symplectic methods yield information not only on the critical point structure of the maximum likelihood estimation mle function see byrnes and willems 7 but also on the asymmetries in the distribution of the data points in n space

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

asymptotic hamiltonian dynamics the toda lattice the three wave interaction and the non holonomic chaplygin sleigh abstract in this paper we discuss asymptotic stability in energy preserving systems which have an almost poisson structure in particular we consider a class of poisson systems which includes the toda lattice in standard hamiltonian systems one of course does not expect asymptotic stability the key here is the structure of the phase space of the poisson or almost poisson systems and the nature of the equilibria our systems are in some sense generalizations of the integrable toda lattice system but are not integrable in general as a particular example we point out an interesting connection between three mechanical two degree of freedom systems that exhibit asymptotic stability two of them are classical hamiltonian systems while the third is a non holonomic system non holonomic systems are generally energy preserving but not hamiltonian but the system analyzed here turns out to have a phase space which is the union of hamiltonian ones we also discuss various higher dimensional examples

likelihood ratio forensic voice comparison using parametric representations of the formant trajectories of diphthongs non contemporaneous speech samples from 27 male speakers of australian english were compared in a forensic likelihood ratio framework parametric curves polynomials and discrete cosine transforms were fitted to the formant trajectories of the diphthongs a e o a and the estimated coefficient values from the parametric curves were used as input to a generative multivariate kernel density formula for calculating likelihood ratios expressing the probability of obtaining the observed difference between two speech samples under the hypothesis that the samples were produced by the same speaker versus under the hypothesis that they were produced by different speakers cross validated likelihood ratio results from systems based on different parametric curves were calibrated and evaluated using the log likelihood ratio cost function cllr the cross validated likelihood ratios from the best performing system for each vowel phoneme were fused using logistic regression the resulting fused system had a very low error rate thus meeting one of the requirements for admissibility in court

on explicitly solvable gradient systems of moser karmarkar type abstract a generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied it is shown that nonlinear dynamical systems of moser karmarkar type generically take both a lax representation and a gradient form a related optimization problem on a simplex is discussed an integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of moser karmarkar type are also investigated

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

spectral shape features versus formants as acoustic correlates for vowels the first three formants i e the first three spectral prominences of the short time magnitude spectra have been the most commonly used acoustic cues for vowels ever since the work of peterson and barney j acoust soc am 24 175 184 1952 however spectral shape features which encode the global smoothed spectrum provide a more complete spectral description and therefore might be even better acoustic correlates for vowels in this study automatic vowel classification experiments were used to compare formants and spectral shape features for monopthongal vowels spoken in the context of isolated cvc words under a variety of conditions the roles of static and time varying information for vowel discrimination were also compared spectral shape was encoded using the coefficients in a cosine expansion of the nonlinearly scaled magnitude spectrum under almost all conditions investigated in the absence of fundamental frequency f0 information automatic vowel classification based on spectral shape features was superior to that based on formants if f0 was used as an additional feature vowel classification based on spectral shape features was still superior to that based on formants but the differences between the two feature sets were reduced it was also found that the error pattern of perceptual confusions was more closely correlated with errors in automatic classification obtained from spectral shape features than with classification errors from formants therefore it is concluded that spectral shape features are a more complete set of acoustic correlates for vowel identity than are formants in comparing static and time varying features static features were the most important for vowel discrimination but feature trajectories were valuable secondary sources of information

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

spectral gradient flow and equilibrium configurations of point vortices we formulate the problem of finding equilibrium configurations of n point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew symmetric matrix m whose nullspa