finite time entropy a probabilistic approach for measuring nonlinear stretching abstract transport and mixing processes in dynamical systems are often difficult to study analytically and therefore a variety of numerical methods have been developed finite time lyapunov exponents ftles or related stretching indicators are frequently used as a means to estimate transport barriers alternatively eigenvectors singular vectors or oseledets vectors of numerical transfer operators find almost invariant sets finite time coherent sets or time asymptotic coherent sets respectively which are minimally dispersed under the dynamics while these families of approaches geometric ftles and the probabilistic transfer operator often give compatible results a formal link is still missing here we present a small step towards providing a mathematical link we propose a new entropy based methodology for estimating finite time expansive behaviour along trajectories in autonomous and nonautonomous dynamical systems we introduce the finite time entropy fte field as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localised density evolved by the transfer operator the fte construction elucidates in a straightforward way the connection between the evolution of probability densities and the local stretching experienced we develop an extremely simple and numerically efficient method of constructing an estimate of the fte field the fte field is instantaneously calculable from a numerical transfer operator a transition matrix of conditional probabilities that describes a discretised version of the dynamical system once one has such a transition matrix the fte field may be computed for free we also show i how to avoid long time integrations in autonomous and time periodic systems ii how to perform backward time computations by a fast matrix manipulation rather than backward time integration and iii how to easily employ adaptive methods to focus on high value fte regions

ϕs orthogonal matrices abstract let s m n r be such that s 2 i or s 2 i for a m n c define s a s 1 a t s we study s orthogonal matrices those a m n c that satisfy s a a 1 let f r or f c we show that every s orthogonal a m n f has a polar decomposition a pu with p u m n f p is positive definite u is unitary and both factors are s orthogonal we show that if a is s orthogonal and normal and if 1 is not an eigenvalue of a then there exists a normal s skew symmetric n that is s n n such that a e in we also take a look at the particular cases s h k 0 i k i k 0 and s l k i k i n k

a variational theory of hyperbolic lagrangian coherent structures abstract we develop a mathematical theory that clarifies the relationship between observable lagrangian coherent structures lcss and invariants of the cauchy green strain tensor field motivated by physical observations of trajectory patterns we define hyperbolic lcss as material surfaces i e codimension one invariant manifolds in the extended phase space that extremize an appropriate finite time normal repulsion or attraction measure over all nearby material surfaces we also define weak lcss wlcss as stationary solutions of the above variational problem solving these variational problems we obtain computable sufficient and necessary criteria for wlcss and lcss that link them rigorously to the cauchy green strain tensor field we also prove a condition for the robustness of an lcs under perturbations such as numerical errors or data imperfection on several examples we show how these results resolve earlier inconsistencies in the theory of lcs finally we introduce the notion of a constrained lcs clcs that extremizes normal repulsion or attraction under constraints this construct allows for the extraction of a unique observed lcs from linear systems and for the identification of the most influential weak unstable manifold of an unstable node

the ψs polar decomposition when the cosquare of s is normal abstract let s m n c be nonsingular such that s t s is normal that is the cosquare of s is normal set s a s 1 a t s for a m n c we determine conditions on a so that a has a s polar decomposition we also find the possible jordan canonical forms of a s orthogonal matrix and of a s skew symmetric matrix in the cases a 1 s t s and b 1 s t s

detecting coherent structures using braids abstract the detection of coherent structures is an important problem in fluid dynamics particularly in geophysical applications for instance knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills existing methods detect lagrangian coherent structures which are barriers to transport by examining the stretching field as given by finite time lyapunov exponents these methods are very effective when the velocity field is well determined but in many applications only a small number of flow trajectories are known for example when dealing with oceanic float data we introduce a topological method for detecting invariant regions based on a small set of trajectories in this method we regard the two dimensional trajectory data as a braid in three dimensions with time being the third coordinate invariant regions then correspond to trajectories that travel together and do not entangle other trajectories we detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories and searching for loops that show negligible growth

nonautonomous analysis of steady korteweg de vries waves under nonlocalised forcing abstract recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the korteweg de vries kdv equation arbitrary but small nonlocalised bottom topographies are amenable to this method two classes of free surface solutions hyperbolic and homoclinic solutions of the associated augmented dynamical system are characterised the first of these corresponds to near uniform free surface flows for which explicit formulae are developed for a range of topographies the second corresponds to solitary waves on the free surface and a method for determining their number is developed formulae for the shape of these solitary waves are also obtained theoretical free surface profiles are verified using numerical kdv solutions and excellent agreement is obtained

hyperbolic and elliptic transport barriers in three dimensional unsteady flows abstract we develop a general theory of transport barriers for three dimensional unsteady flows with arbitrary time dependence the barriers are obtained as two dimensional lagrangian coherent structures lcss that create locally maximal deformation along hyperbolic lcss this deformation is induced by locally maximal normal repulsion or attraction along shear lcss the deformation is created by locally maximal tangential shear hyperbolic lcss therefore play the role of generalized stable and unstable manifolds while closed shear lcss elliptic lcss act as generalized kam tori or kam type cylinders all these barriers can be computed from our theory as explicitly parametrized surfaces we illustrate our results by visualizing two dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the abc flow

on the ϕj polar decomposition of matrices abstract we present new results on the j polar decomposition of matrices we show that every symplectic matrix may be written as a product of symplectic operation matrices we present a simple form attained under symplectic equivalence which makes it easy to determine if a matrix does not have a j polar decomposition we also determine the rank 4 matrices with j polar decomposition

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

novel metrics for evaluating improvement in discrimination net reclassification and integrated discrimination improvement for normal variables and nested models net reclassification and integrated discrimination improvements have been proposed as alternatives to the increase in the auc for evaluating improvement in the performance of risk assessment algorithms introduced by the addition of new phenotypic or genetic markers in this paper we demonstrate that in the setting of linear discriminant analysis under the assumptions of multivariate normality all three measures can be presented as functions of the squared mahalanobis distance this relationship affords an interpretation of the magnitude of these measures in the familiar language of effect size for uncorrelated variables furthermore it allows us to conclude that net reclassification improvement can be viewed as a universal measure of effect size our theoretical developments are illustrated with an example based on the framingham heart study risk assessment model for high risk men in primary prevention of cardiovascular disease

nonautonomous control of stable and unstable manifolds in two dimensional flows abstract we outline a method for controlling the location of stable and unstable manifolds in the following sense from a known location of the stable and unstable manifolds in a steady two dimensional flow the primary segments of the manifolds are to be moved to a user specified time varying location which is near the steady location we determine the nonautonomous perturbation to the vector field required to achieve this control and give a theoretical bound for the error in the manifolds resulting from applying this control the efficacy of the control strategy is illustrated via a numerical example

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

the sum of two ϕs orthogonal matrices when s ts is normal and 1 σ s ts abstract let a nonsingular s m n c be given for a m n c set s a s 1 a t s we say that a is s symmetric if s a a we say that a is s orthogonal if a gl n and s a a 1 we say that a has a s polar decomposition if a u p for some s orthogonal u and s symmetric p suppose that s t s is normal and 1 s t s we determine conditions on a m n c so that a can be written as a sum of two s orthogonal matrices

detecting barriers to transport a review of different techniques abstract we review and discuss some different techniques for describing local dispersion properties in fluids a recent lagrangian diagnostics based on the finite scale lyapunov exponent fsle is presented and compared to the finite time lyapunov exponent ftle to the okubo weiss ow and hua klein hk criteria we show that the ow and hk are the limiting case of the ftle and that the fsle is the most efficient method for detecting the presence of cross stream barriers we illustrate our findings by considering two examples of geophysical interest a kinematic model of a meandering jet and lagrangian tracers advected by stratospheric circulation

the jordan canonical forms of complex orthogonal and skew symmetric matrices abstract we study the jordan canonical forms of complex orthogonal and skew symmetric matrices and consider some related results

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

on the lie algebra associated to s unitary matrices abstract a natural generalization of unitary groups arising from sesquilinear forms which are assumed neither hermitian nor skew hermitian is considered let s m n c an s unitary matrix a is a matrix a gl n c such that a s a s the set u s of all s unitary matrices is a matrix lie group a formula for the real dimension of the associated lie algebra u s when s is nonsingular and normal is derived when s is invertible and unitary it is shown that u s is the direct sum of some lie algebras associated to the indefinite unitary groups finally the dimension formula is applied to a class of permutation matrices

measures for evaluation of prognostic improvement under multivariate normality for nested and nonnested models when comparing performances of two risk prediction models several metrics exist to quantify prognostic improvement including the change in the area under the receiver operating characteristic curve the integrated discrimination improvement the net reclassification index at event rate the change in standardized net benefit the change in brier score and the change in scaled brier score we explore the behavior and interrelationships between these metrics under multivariate normality in nested and nonnested model comparisons we demonstrate that within the framework of linear discriminant analysis all six statistics are functions of squared mahalanobis distance a robust metric that properly measures discrimination by quantifying the separation between the risk scores of events and nonevents these relationships are important for overall interpretability and clinical usefulness through simulation we demonstrate that the performance of the theoretical estimators under normality is comparable or superior to empirical estimation methods typically used by investigators in particular the theoretical estimators for the net reclassification index and the change in standardized net benefit exhibit less variability in their estimates as compared to their empirically estimated counterparts finally we explore how these metrics behave with potentially nonnormal data by applying these methods in a practical example based on the sex specific cardiovascular disease risk models from the framingham heart study our findings aim to give greater insight into the behavior of these measures and the connections existing among them and to provide additional estimation methods with less variability for the net reclassification index and the change in standardized net benefit

asymptotic distribution of auc nris and idi based on theory of u statistics the change in area under the curve auc the integrated discrimination improvement idi and net reclassification index nri are commonly used measures of risk prediction model performance some authors have reported good validity of associated methods of estimating their standard errors se and construction of confidence intervals whereas others have questioned their performance to address these issues we unite the auc idi and three versions of the nri under the umbrella of the u statistics family we rigorously show that the asymptotic behavior of auc nris and idi fits the asymptotic distribution theory developed for u statistics we prove that the auc nris and idi are asymptotically normal unless they compare nested models under the null hypothesis in the latter case asymptotic normality and existing se estimates cannot be applied to auc nris or idi in the former case se formulas proposed in the literature are equivalent to se formulas obtained from u statistics theory if we ignore adjustment for estimated parameters we use sukhatme randles dewet condition to determine when adjustment for estimated parameters is necessary we show that adjustment is not necessary for ses of the auc and two versions of the nri when added predictor variables are significant and normally distributed the ses of the idi and three category nri should always be adjusted for estimated parameters these results allow us to define when existing formulas for se estimates can be used and when resampling methods such as the bootstrap should be used instead when comparing nested models we also use the u statistic theory to develop a new se estimate of auc copyright 2017 john wiley sons ltd

comparing roc curves derived from regression models in constructing predictive models investigators frequently assess the incremental value of a predictive marker by comparing the roc curve generated from the predictive model including the new marker with the roc curve from the model excluding the new marker many commentators have noticed empirically that a test of the two roc areas often produces a non significant result when a corresponding wald test from the underlying regression model is significant a recent article showed using simulations that the widely used roc area test produces exceptionally conservative test size and extremely low power in this article we demonstrate that both the test statistic and its estimated variance are seriously biased when predictions from nested regression models are used as data inputs for the test and we examine in detail the reasons for these problems although it is possible to create a test reference distribution by resampling that removes these biases wald or likelihood ratio tests remain the preferred approach for testing the incremental contribution of a new marker

the ϕs polar decomposition let sn s mnr s2 i sn s mnr s2 i and let sn sn sn for s sn let s mnc mnc be given by sa s 1ats an a mnc is called s symmetric if sa a a is called s skew symmetric if sa a and a is called s orthogonal if a is nonsingular and sa a 1 we say that a has a s polar decomposition if a qt for some s orthogonal q mnc and some s symmetric t mnc let vs a mnc s sa a we show that every nonsingular a vs has a s polar decomposition we determine conditions for which an a mnc has a s polar decomposition we also determine the possible jordan canonical forms of a s orthogonal matrix and of a s skew symmetric matrix

hamiltonian square roots of skew hamiltonian matrices revisited abstract recently h fassbender et al linear algebra appl 287 1999 125 proved the following theorem every real skew hamiltonian matrix w has a real hamiltonian square root h i e h 2 w we prove an analog of this theorem for complex matrices our approach may be of independent interest namely we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product

a generalized polar decomposition abstract let g gl n c be a matrix group and g g an antiautomorphic involution of g then naturally defines a subgroup g a g a a i for many classical groups every member of g belongs to a left right coset of gf represented by a generalized symmetric matrix b b the paper remarks on the generalized existence of a polar decomposition and its numerical computation the relationship between the map x x 1 x and the subgroups conjugate to g is discussed

two guidelines for bootstrap hypothesis testing two guidelines for nonparametric bootstrap hypothesis testing are highlighted the first recommends that resampling be done in a way that reflects the null hypothesis even when the true hypothesis is distant from the null the second guideline argues that bootstrap hypothesis tests should employ methods that are already recognized as having good features in the closely related problem of confidence interval construction violation of the first guideline can seriously reduce the power of a test sometimes this reduction is spectacular since it is most serious when the null hypothesis is grossly in error the second guideline is of some importance when the conclusion of a test is equivocal it has no direct bearing on power but improves the level accuracy of a test

controlling the unsteady analogue of saddle stagnation points it is well known that saddle stagnation points are crucial flow organizers in steady autonomous flows due to their accompanying stable and unstable manifolds these have been extensively investigated experimentally numerically and theoretically in situations related to macro and micromixers in order to either restrict or enhance mixing saddle points are also important players in the dynamics of mechanical oscillators in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior the entity analogous to a saddle point in an unsteady nonautonomous flow is a time varying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time within the context of nearly steady flows the unsteady velocity perturbation required to ensure that such a hyperbolic saddle trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

the ϕj polar decomposition of matrices with rank 2 abstract it is known that if a matrix has a j polar decomposition then it is of even rank we provide necessary and sufficient conditions for a 2 n by 2 n matrix of rank 2 to have a j polar decomposition

definition and properties of lagrangian coherent structures from finite time lyapunov exponents in two dimensional aperiodic flows this paper develops the theory and computation of lagrangian coherent structures lcs which are defined as ridges of finite time lyapunov exponent ftle fields these ridges can be seen as finite time mixing templates such a framework is common in dynamical systems theory for autonomous and time periodic systems in which examples of lcs are stable and unstable manifolds of fixed points and periodic orbits the concepts defined in this paper remain applicable to flows with arbitrary time dependence and in particular to flows that are only defined computed or measured over a finite interval of time previous work has demonstrated the usefulness of ftle fields and the associated lcss for revealing the lagrangian behavior of systems with general time dependence however ridges of the ftle field need not be exactly advected with the flow the main result of this paper is an estimate for the flux across an lcs which shows that the flux is small and in most cases negligible for well defined lcss or those that rotate at a speed comparable to the local eulerian velocity field and are computed from ftle fields with a sufficiently long integration time under these hypotheses the structures represent nearly invariant manifolds even in systems with arbitrary time dependence the results are illustrated on three examples the first is a simplified dynamical model of a double gyre flow the second is surface current data collected by high frequency radar stations along the coast of florida and the third is unsteady separation over an airfoil in all cases the existence of lcss governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible

a real coninvolutory analog of the polar decomposition abstract we study properties of coninvolutory matrices e i and derive a canonical form under similarity as well as a canonical form under unitary consimilarity for them we show that any complex matrix has a coninvolutory dilation and we characterize the minimum size of a coninvolutory dilation of a square matrix we characterize the m by n complex matrices a that can be factored as a re with r real and e coninvolutory and we discuss the uniqueness of this factorization when a is square and nonsingular

skew coninvolutory matrices we study the properties of skew coninvolutory ee i matrices and derive canonical forms and a singular value decomposition we study the matrix function s a sa 1s 1 defined on nonsingular matrices and with s satisfying ss i or ss i we show that every square nonsingular a may be written as a xy with s x x and s y y 1 we also give necessary and sufficient conditions on when a nonsingular matrix may be written as a product of a coninvolutory matrix and a skew coninvolutory matrix or a product of two skew coninvolutory matrices moreover when a is similar to a 1 or when a is similar to a 1 or when a is similar to a or when a is similar to a we determine the possible jordan canonical forms of a for which the similarity matrix may be taken to be skew coninvolutory

s orthogonal matrices and s symmetries abstract let s m n c be nonsingular a q m n c is called s orthogonal if q t s q s an s orthogonal h is called an s symmetry if rank h i 1 we give conditions on s so that every s orthogonal can be written as a product of s symmetries we give conditions on s so that some s orthogonal cannot be written as a product of s symmetries moreover we determine conditions on s so that there are no s symmetries

the ψs polar decomposition abstract let s m n be nonsingular we set s a s 1 a 1 s for all nonsingular a m n a matrix a is called s symmetric if s a a it is called s orthogonal if s a a 1 and it is called s antiorthogonal if s a a 1 we show that the following are equivalent 1 a is s symmetric 2 there exists a s antiorthogonal z m n such that a e z 3 there exists a s orthogonal x m n such that a e ix and 4 there exists a s symmetric b m n such that a b 2 when s is coninvolutory s s i or skew coninvolutory s s i we show that every nonsingular matrix has a s polar decomposition that is every nonsingular matrix may be written as a re where r is s orthogonal and e is s symmetric if a is possibly singular we define a to be s orthogonal if s 1 a s a and determine which singular matrices have a s polar decomposition

polar decompositions in finite dimensional indefinite scalar product spaces general theory abstract several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied matrices that admit such polar decompositions are described in various ways in particular a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given

an analytic framework for identifying finite time coherent sets in time dependent dynamical systems abstract the study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems barriers to transport which mitigate mixing are currently the subject of intense study in the autonomous setting the use of transfer operators perron frobenius operators to identify invariant and almost invariant sets has been particularly successful in the nonautonomous time dependent setting coherent sets a time parameterised family of minimally dispersive sets are a natural extension of almost invariant sets the present work introduces a new analytic transfer operator construction that enables the calculation of finite time coherent sets sets are that minimally dispersive over a finite time interval this new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude the construction can also be applied to general markov processes on continuous state space

equivalence of improvement in area under roc curve and linear discriminant analysis coefficient under assumption of normality in this paper we investigate the addition of new variables to an existing risk prediction model and the subsequent impact on discrimination quantified by the area under the receiver operating characteristics curve auc of roc based on practical experience concerns have emerged that the significance of association of the variable under study with the outcome in the risk model does not correspond to the significance of the change in auc that is often the variable is significant but the change in auc is not this paper demonstrates that under the assumption of multivariate normality and employing linear discriminant analysis lda to construct the risk prediction tool statistical significance of the new predictor s is equivalent to the statistical significance of the increase in auc under these assumptions the result extends asymptotically to logistic regression we further show that equality of variance covariance matrices of predictors within cases and non cases is not necessary when lda is used however our practical example from the framingham heart study data suggests that the finding might be sensitive to the assumption of normality copyright 2011 john wiley sons ltd

skew ϕ polar decompositions abstract we present necessary and sufficient conditions so that a square complex matrix a can be written as a q s with orthogonal q and skew symmetric s we also present necessary and sufficient conditions so that a 2n by 2n complex matrix b can be written as b w r with symplectic w and hamiltonian r