gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

eikonal equations on the sierpinski gasket we study the eikonal equation on the sierpinski gasket in the spirit of the construction of the laplacian in kigami analysis on fractals cambridge tracts in mathematics 2001 we consider graph eikonal equations on the prefractals and we show that the solutions of these problems converge to a function defined on the fractal set we characterize this limit function as the unique metric viscosity solution to the eikonal equation on the sierpinski gasket according to the definition introduced in giga et al trans am math soc 367 49 66 2015

gradient estimates for heat kernels and harmonic functions abstract let x d be a doubling metric measure space endowed with a dirichlet form e deriving from a carre du champ assume that x d e supports a scale invariant l 2 poincare inequality in this article we study the following properties of harmonic functions heat kernels and riesz transforms for p 2 i g p l p estimate for the gradient of the associated heat semigroup ii r h p l p reverse holder inequality for the gradients of harmonic functions iii r p l p boundedness of the riesz transform p iv g b e a generalised bakry emery condition we show that for p 2 i ii iii are equivalent while for p i ii iv are equivalent moreover some of these equivalences still hold under weaker conditions than the l 2 poincare inequality our result gives a characterisation of li yau s gradient estimate of heat kernels for p while for p 2 it is a substantial improvement as well as a generalisation of earlier results by auscher coulhon duong hofmann 7 and auscher coulhon 6 applications to isoperimetric inequalities and sobolev inequalities are given our results apply to riemannian and sub riemannian manifolds as well as to non smooth spaces and to degenerate elliptic parabolic equations in these settings

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

weak curvature conditions and functional inequalities we give sufficient conditions for a measured length space x d to admit local and global poincare inequalities along with a sobolev inequality we first introduce a condition dm on x d d ef ined in terms of transport of measures we show that dm together with a doubling condition on implies a scaleinvariant local poincare inequality we show that if x d has nonnegative n ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies dm with constant 2 n t he condition dm is preserved by measured gromov hausdorff limits we then prove a sobolev inequality for measured length spaces with n ricci curvature bounded below by k 0 finally we derive a sharp global poincare inequality

l variational problems associated to measurable finsler structures abstract we study l variational problems associated to measurable finsler structures in euclidean spaces we obtain existence and uniqueness results for the absolute minimizers

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

spectral factorization via hermitian pencils abstract the spectral factorization problem is solved for state space systems via results on the canonical forms and inertia of hermitian matrix pencils these algebraic results then give a deflation method for spectral factorization

intrinsic geometry and analysis of diffusion processes and l variational problems the aim of this paper is twofold first we obtain a better understanding of the intrinsic distance of diffusion processes precisely a for all n 1 the diffusion matrix a is weak upper semicontinuous on if and only if the intrinsic differential and the local intrinsic distance structures coincide b if n 1 or if n 2 and a is weak upper semicontinuous on the intrinsic distance and differential structures always coincide c if n 2 and a fails to be weak upper semicontinuous on the non coincidence of the intrinsic distance and differential structures depend on the geometry of the non weak upper semicontinuity set of a second for an arbitrary diffusion matrix a we show that the intrinsic distance completely determines the absolute minimizer of the corresponding l variational problem and then obtain the existence and uniqueness for given boundary data we also give an example of a diffusion matrix a for which there is an absolute minimizer that is not of class c1 when a is continuous we also obtain the linear approximation property of the absolute minimizer

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

heat flow on finsler manifolds this paper studies the heat flow on finsler manifolds a finsler manifold is a smooth manifold m equipped with a minkowski norm f x txm on each tangent space mostly we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x the particular case of a hilbert norm on each tangent space leads to the important subclasses of riemannian manifolds where the heat flow is widely studied and well understood we present two approaches to the heat flow on a finsler manifold as gradient flow on l2 m m for the energy as gradient flow on the reverse l2 wasserstein space 2 m of probability measures on m for the relative entropy both approaches depend on the choice of a measure m on m and then lead to the same nonlinear evolution semigroup we prove 1 regularity for solutions to the nonlinear heat equation on the finsler space m f m typically solutions to the heat equation will not be 2 moreover we derive pointwise comparison results a la cheeger yau and integrated upper gaussian estimates a la davies 2008 wiley periodicals inc

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a fast algorithm for the simulation of arterial pulse waves one dimensional models have been widely used in studies of the propagation of blood pulse waves in large arterial trees under a periodic driving of the heartbeat traditional numerical methods such as the lax wendroff method are employed to obtain asymptotic periodic solutions at large times however these methods are severely constrained by the cfl condition due to large pulse wave speed in this work we develop a new numerical algorithm to overcome this constraint first we reformulate the model system of pulse wave propagation using a set of riemann variables and derive a new form of boundary conditions at the inlet the outlets and the bifurcation points of the arterial tree the new form of the boundary conditions enables us to design a convergent iterative method to enforce the boundary conditions then after exchanging the spatial and temporal coordinates of the model system we apply the lax wendroff method in the exchanged coordinate system which turns the large pulse wave speed from a liability to a benefit to solve the wave equation in each artery of the model arterial system our numerical studies show that our new algorithm is stable and can perform 15 times faster than the traditional implementation of the lax wendroff method under the requirement that the relative numerical error of blood pressure be smaller than one percent which is much smaller than the modeling error

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

integrals and potentials of differential 1 forms on the sierpinski gasket abstract we provide a definition of the integral along paths in the sierpinski gasket k for differential smooth 1 forms associated to the standard dirichlet form e on k we show how this tool can be used to study the potential theory on k in particular we prove i a de rham reconstruction of a 1 form from its periods around lacunas in k ii a hodge decomposition of 1 forms with respect to the hilbertian energy norm iii the existence of potentials of smooth 1 forms on a suitable covering space of k we finally show that this framework provides versions of the de rham duality theorem for the fractal k

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

a rademacher type theorem on l2 wasserstein spaces over closed riemannian manifolds abstract let p be any borel probability measure on the l 2 wasserstein space p 2 m w 2 over a closed riemannian manifold m we consider the dirichlet form e induced by p and by the wasserstein gradient on p 2 m under natural assumptions on p we show that w 2 lipschitz functions on p 2 m are contained in the dirichlet space d e and that w 2 is dominated by the intrinsic metric induced by e we illustrate our results by giving several detailed examples

efficient computation of the matrix exponential by generalized polar decompositions in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the method of generalized polar decompositions which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation we develop techniques valid for a range of lie groups the orthogonal group the symplectic group lorentz isotropy and scaling groups however the gpd approach is equally promising in a more general context even when lie group structure is not at issue our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential

short time asymptotics in dirichlet spaces this work is involved with the short time asymptotics of the heat semigroup in a general setting a generalization of fang s version of varadhan s formula is proved a spectral gap or the possibility of obtaining one by an appropriate change of measure is required 2001 john wiley sons inc

the parameterized sr algorithm for symplectic butterfly matrices the sr algorithm is a structure preserving algorithm for computing the spectrum of symplectic matrices any symplectic matrix can be reduced to symplectic butterfly form a symplectic matrix b in butterfly form is uniquely determined by 4n 1 parameters using these 4n 1 parameters we show how one step of the symplectic sr algorithm for b can be carried out in o n arithmetic operations compared to o n 3 arithmetic operations when working on the actual symplectic matrix moreover the symplectic structure which will be destroyed in the numerical process due to roundoff errors when working with a symplectic butterfly matrix will be forced by working just with the parameters

pulse wave propagation in a model human arterial network assessment of 1 d visco elastic simulations against in vitro measurements a numerical model based on the nonlinear one dimensional 1 d equations of pressure and flow wave propagation in conduit arteries is tested against a well defined experimental 1 1 replica of the human arterial tree the tree consists of 37 silicone branches representing the largest central systemic arteries in the human including the aorta carotid arteries and arteries that perfuse the upper and lower limbs and the main abdominal organs the set up is mounted horizontally and connected to a pulsatile pump delivering a periodic output similar to the aortic flow terminal branches end in simple resistance models consisting of stiff capillary tubes leading to an overflow reservoir that reflects a constant venous pressure the parameters required by the numerical algorithm are directly measured in the in vitro set up and no data fitting is involved comparison of experimental and numerical pressure and flow waveforms shows the ability of the 1 d time domain formulation to capture the main features of pulse wave propagation measured throughout the system test as a consequence of the simple resistive boundary conditions used to reduce the uncertainty of the parameters involved in the simulation the experimental set up generates waveforms at terminal branches with additional non physiological oscillations the frequencies of these oscillations are well captured by the 1 d model even though amplitudes are overestimated adding energy losses in bifurcations and including fluid inertia and compliance to the purely resistive terminal models does not reduce the underdamped effect suggesting that wall visco elasticity might play an important role in the experimental results nevertheless average relative root mean square errors between simulations and experimental waveforms are smaller than 4 for pressure and 19 for the flow at all 70 locations studied

convex analysis and global optimization of joint actuator location and control problems it is shown that the optimal value of the continuous time linear quadratic problem regarded as a function of the system model and index parameters exhibits properties convexity concavity and monoticity specially suitable for optimization purposes based on this fact a procedure for the global solution determination of eventually nonconvex problems involving the above mentioned function is proposed such problems embody some known designs such as filtering under noise uncertainty or precision constraints and optimal actuator sensor location the last problem is deeply analyzed and two practical applications namely satellite attitude control and large flexible system control are included

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

sr and sz algorithms for the symplectic butterfly eigenproblem abstract sr and sz algorithms for the symplectic generalized eigenproblem that are based on the reduction of a symplectic matrix to symplectic butterfly form are discussed a 2 n 2 n symplectic butterfly matrix has 8 n 4 generically nonzero entries which are determined by 4 n 1 parameters while the sr algorithm operates directly on the matrix entries the sz algorithm works with the 4 n 1 parameters the algorithms are made more compact and efficient by using laurent polynomials instead of standard polynomials to drive the iterations

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

modulus and the poincare inequality on metric measure spaces the purpose of this paper is to develop the understanding of modulus and the poincare inequality as defined on metric measure spaces various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces consequently modulus is shown to be upper semi continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces moreover several competing definitions for the poincare inequality are shown to coincide if the underlying measure is doubling one such characterization considers only continuous functions and their continuous upper gradients and extends work of heinonen and koskela applications include showing that the p poincare inequality with a doubling measure for p 1 persists through to the limit of a sequence of converging pointed metric measure spaces this extends results of cheeger a further application is the construction of new doubling measures in euclidean space which admit a 1 poincare inequality

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

an anatomically based model of transient coronary blood flow in the heart an efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network by constraining the form of the velocity profile across the vessel radius the three dimensional navier stokes equations are reduced to one dimensional equations governing conservation of mass and momentum these equations are coupled to a pressure radius relationship characterizing the elasticity of the vessel wall to describe the transient blood flow through a vessel segment the two step lax wendroff finite difference method is used to numerically solve these equations the flow through bifurcations where three vessel segments join is governed by the equations of conservation of mass and momentum the solution to these simultaneous equations is calculated using the multidimensional newton raphson method simulations of blood flow through a geometric model of the coronary network are presented demonstrating phy

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

measurable riemannian geometry on the sierpinski gasket the kusuoka measure and the gaussian heat kernel estimate we study the standard dirichlet form and its energy measure called the kusuoka measure on the sierpinski gasket as aprototype of measurable riemannian geometry the shortest pathmetric on the harmonic sierpinski gasket is shown to be thegeodesic distance associated with the measurable riemannianstructure the kusuoka measure is shown to have the volumedoubling property with respect to the euclidean distance and alsoto the geodesic distance li yau type gaussian off diagonal heatkernel estimate is established for the heat kernel associated withthe kusuoka measure

on local poincare via transportation it is shown that curvature dimension bounds cd n k for a metric measure space x d m in the sense of sturm imply a weak l1 poincare inequality provided x d has m almost surely no branching points

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

cheeger s energy on the harmonic sierpinski gasket abstract koskela and zhou have proven that on the harmonic sierpinski gasket with kusuoka s measure the natural dirichlet form coincides with cheeger s energy we give a different proof of this result which uses the properties of the lyapunov exponent of the gasket

energy measures and indices of dirichlet forms with applications to derivatives on some fractals we introduce the concept of index for regular dirichlet forms by means of energy measures and discuss its properties in particular it is proved that the index of strong local regular dirichlet forms is identical with the martingale dimension of the associated diffusion processes as an application a class of self similar fractals is taken up as an underlying space we prove that first order derivatives can be defined for functions in the domain of the dirichlet forms and their total energies are represented as the square integrals of the derivatives

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

optimal transport cheeger energies and contractivity of dynamic transport distances in extended spaces abstract we introduce the setting of extended metric topological measure spaces as a general wiener like framework for optimal transport problems and nonsmooth metric analysis in infinite dimension after a brief review of optimal transport tools for general radon measures we discuss the notions of the cheeger energy of the radon measures concentrated on absolutely continuous curves and of the induced dynamic transport distances we study their main properties and their links with the theory of dirichlet forms and the bakry emery curvature condition in particular concerning the contractivity properties and the evi formulation of the induced heat semigroup

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

a numerical method for computing the hamiltonian schur form we derive a new numerical method for computing the hamiltonian schur form of a hamiltonian matrix mathcal m in mathbb r 2n times 2n that has no purely imaginary eigenvalues we demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous time algebraic riccati equations despite the fact that no complete error analysis for the method is yet available the numerical results indicate that if no eigenvalues of mathcal m are close to the imaginary axis then the method computes the exact hamiltonian schur form of a nearby hamiltonian matrix and thus is numerically strongly backward stable the new method is of complexity mathbf o n 3 and hence it solves a long standing open problem in numerical analysis

parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network reduced models of human arterial networks are an efficient approach to analyze quantitative macroscopic features of human arterial flows the justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks although these types of models have been employed extensively and many issues associated with their implementations have been widely researched the issue of data uncertainty has received comparatively little attention similar to many biological systems a large amount of uncertainty exists in the value of the parameters associated with the models clearly reliable assessment of the system behaviour cannot be made unless the effect of such data uncertainty is quantified in this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system the uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic this type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions we demonstrate how the application of a high order stochastic collocation method based on the generalized polynomial chaos expansion combined with a discontinuous galerkin spectral hp element discretization in physical space can successfully simulate this type of hyperbolic system subject to uncertain inputs with bounds building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved