developed matrix inequalities via positive multilinear mappings abstract utilizing the notion of positive multilinear mappings we present some matrix inequalities in particular the choi davis jensen inequality f a b f a f b does not hold in general for a matrix convex function f and a positive multilinear mapping we prove this inequality under certain condition on f moreover some kantorovich and convexity type inequalities including positive multilinear mappings are presented

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

operator means of probability measures abstract let p be the complete metric space consisting of positive invertible operators on an infinite dimensional hilbert space with the thompson metric we introduce the notion of operator means of probability measures on p in parallel with kubo and ando s definition of two variable operator means and show that every operator mean is contractive for the wasserstein distance by means of a fixed point method we consider deformation of such operator means and show that the deformation of any operator mean becomes again an operator mean in our sense based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

geometric mean flows and the cartan barycenter on the wasserstein space over positive definite matrices abstract we introduce a class of flows on the wasserstein space of probability measures with finite first moment on the cartan hadamard riemannian manifold of positive definite matrices and consider the problem of differentiability of the corresponding cartan barycentric trajectory as a consequence we have a version of lie trotter formula and a related unitarily invariant norm inequality furthermore a fixed point theorem related to the karcher equation and the cartan barycentric trajectory is also presented as an application

diagonality measures of hermitian positive definite matrices with application to the approximate joint diagonalization problem in this paper we introduce properly invariant diagonality measures of hermitian positive definite matrices these diagonality measures are defined as distances or divergences between a given positive definite matrix and its diagonal part we then give closed form expressions of these diagonality measures and discuss their invariance properties the diagonality measure based on the log determinant divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case these diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of hermitian positive definite matrices numerical computations based on a modified newton method are presented and commented

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

geometry of midpoint sets for thompson s metric abstract it is well known that the convex cone p m of m m positive definite matrices is a cartan hadamard riemannian manifold with respect to the riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them in this paper we consider the thompson metric on p m inherited from the spectral norm and study some geometric structures of the thompson midpoints we prove that there is a unique midpoint minimal geodesic between a and b if and only if the spectrum of a 1 b is contained in a a 1 for some a 0 and the set of thompson midpoints between a and b is compact and is convex in both riemannian and euclidean sense it is further shown that the set of all weighted midpoints between a and b is compact and convex in riemannian sense

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

bounds for the wasserstein mean with applications to the lie trotter mean abstract since barycenters in the wasserstein space of probability distributions have been introduced the wasserstein metric and the wasserstein mean of positive definite hermitian matrices have been recently developed in this paper we explore some properties of wasserstein mean of positive definite matrices and find bounds for the wasserstein mean with respect to operator norm and loewner order using bounds for the wasserstein mean we verify that the wasserstein mean is the multivariate lie trotter mean

an order inequality characterizing invariant barycenters on symmetric cones abstract this paper is concerned with invariant and contractive barycenters on the wasserstein space of probability measures on metric spaces of non positive curvature where the center of gravity also called the cartan barycenter is the canonical barycenter on hadamard spaces we establish an order inequality of probability measures on partially ordered symmetric spaces of non compact type namely symmetric cones self dual homogeneous cones characterizing the cartan barycenter among other invariant and contractive barycenters the derived inequality and partially ordered structures on the probability measure space lead also to significant results on norm inequalities including the ando hiai inequality for probability measures on symmetric cones

regular operator mappings and multivariate geometric means we introduce the notion of regular operator mappings of several variables generalising the notion of spectral function this setting is convenient for studying maps more general than what can be obtained from the functional calculus and it allows for jensen type inequalities and multivariate non commutative perspectives as a main application of the theory we consider geometric means of k operator variables extending the geometric mean of k commuting operators and the geometric mean of two arbitrary positive definite matrices we propose dierent types of updating conditions that seems natural in many applications and prove that each of these conditions together with a few other natural axioms uniquely defines the geometric mean for any number of operator variables the means defined in this way are given by explicit formulas and are computationally tractable 1 msc2010 classification 26b25 47a64

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

riemannian metrics on positive definite matrices related to means abstract the riemannian metric on the manifold of positive definite matrices is defined by a kernel function in the form k d h k i j i j 1 tr p i hp j k when i i p i is the spectral decomposition of the foot point d and the hermitian matrices h k are tangent vectors for such kernel metrics the tangent space has an orthogonal decomposition the pull back of a kernel metric under a mapping d g d is a kernel metric as well several riemannian geometries of the literature are particular cases for example the statistical metric for multivariate gaussian distributions and the quantum fisher information in the paper the case x y m x y is mostly studied when m x y is a mean of the positive numbers x and y there are results about the geodesic curves and geodesic distances the geometric mean the logarithmic mean and the root mean are important cases

on the exponential metric increasing property a short and simple proof is given for the inequality that shows that positive definite matrices constitute a riemannian manifold of negative curvature the idea of the proof leads to generalisations to non riemannian metrics and to connections with some well known inequalities of mathematical physics

noether symmetries and conserved quantities for fractional forced birkhoffian systems abstract in this paper noether symmetries and conserved quantities for fractional forced birkhoffian systems are studied firstly a new fractional pfaff birkhoff variational principle with riemann liouville derivatives and generalized force is established and from which the fractional forced birkhoff equations are derived secondly the definitions and the criteria of the invariance of a functional under an e parameter group of infinitesimal transformations are given finally the fractional noether theorems for forced birkhoffian systems are obtained and an illustrative example is presented

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

operator power means due to lawson lim palfia for 1 t 2 abstract for 1 t 1 lim palfia defined a new family of operator power means of positive definite matrices and subsequently by lawson lim their notion and most of their results extend to the setting of positive invertible operators on a hilbert space each of these means except t 0 arises as a unique positive invertible solution of a non linear operator equation and satisfies all desirable properties of power arithmetic means of positive real numbers the purpose of this paper is to extend the range in which operator power means due to lawson lim palfia are defined we investigate some properties of operator power means for t 2 2 1 1

variational integrators for forced birkhoffian systems in this letter we first extend the pfaff birkhoff principle to include forces and obtain the pfaff birkhoff d alembert principle consequently well then motions of forced birkhoffian systems coincide with extremals of the modified principle subsequently compared with the continuous case the discrete forced birkhoffian equations are constructed by discretizing the pfaff birkhoff d alembert principle considered as algorithms the discrete equations have good numerical behavior in terms of getting the correct amounts by which the energy changes over the integration process demonstrated by the given example

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

the effect of test signal type and bandwidth on the categorical scaling of loudness recently several methods for obtaining clinical measures of loudness growth through the use of categorical scaling cs have been proposed for the selection of hearing aids these methods use differing test signals or suggest frequency specific level corrections in an attempt to reflect the loudness perception of hearing aid processed speech while some decisions regarding the stimuli utilized for loudness perception procedures are based on measured relationships to speech signals the effect on loudness perception of changing signal type and bandwidth as measured by cs remains unclear the relationships between the cs loudness growth of signals of differing type pure tones noise bands filtered temporally inverted passband speech and bandwidth were examined for subjects with both normal and impaired hearing results suggest that when the bandwidth is similar e g pure tone and 1 3 oct signal type does not have significant bearing on loudness perception as expected increasing the bandwidth beyond the critical band affected loudness growth as wideband speech stimuli were judged to be significantly louder than narrow band speech at equivalent overall sound levels in this investigation similar loudness growth patterns were noted across category ratings for all test signal bandwidths in contrast loudness growth data obtained using an intelligible speech signal cox et al the american academy of audiology 1994a revealed loudness growth patterns for speech were fundamentally different than those obtained for tones implications for hearing aid fitting strategies are discussed recently several methods for obtaining clinical measures of loudness growth through the use of categorical scaling cs have been proposed for the selection of hearing aids these methods use differing test signals or suggest frequency specific level corrections in an attempt to reflect the loudness perception of hearing aid processed speech while some decisions regarding the stimuli utilized for loudness perception procedures are based on measured relationships to speech signals the effect on loudness perception of changing signal type and bandwidth as measured by cs remains unclear the relationships between the cs loudness growth of signals of differing type pure tones noise bands filtered temporally inverted passband speech and bandwidth were examined for subjects with both normal and impaired hearing results suggest that when the bandwidth is similar e g pure tone and 1 3 oct signal type does not have significant bearing on loudness perception as expected increasing the bandwidth bey

a fixed point mean approximation to the cartan barycenter of positive definite matrices abstract in this paper we define a new weighted inductive geometric mean h n on the riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the cartan barycenter karcher mean we show that for t 0 1 the weighted geometric mean equation x h n 1 t a 1 a n x where t t 1 t w 1 w n 1 t has a unique positive definite solution that approaches as t 0 to the weighted cartan mean of a 1 a n numerical computations for the stochastic convergence in terms of h n to derive a new contractive barycenter other than the cartan barycenter are presented

monotonic properties of the least squares mean we settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual loewner order indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity moreover we introduce a weighted least squares mean and derive our results in this more general setting

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

structure of hiai petz parametrized geometry for positive definite matrices abstract recently hiai petz 2009 10 discussed a parametrized geometry for positive definite matrices with a pull back metric for a diffeomorphism to the euclidean space though they also showed that the geodesic is a path of operator means their interest lies mainly in metrics of the geometry in this paper we reconstruct their geometry without metrics and then we show their metric for each unitarily invariant norm defines a finsler one also we discuss another type of geometry in hiai and petz 2009 10 which is a generalization of corach porta recht s one 3

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

the power mean and the least squares mean of probability measures on the space of positive definite matrices abstract in this paper we derive properties of the least squares or karcher mean of probability measures on the open cone of positive definite matrices of some fixed dimension endowed with the trace metric that generalize known properties of the weighted least squares mean of finitely many positive definite matrices our approach is based on first defining the t power mean of a probability measure as the unique fixed point of the contractive map x x t z d z with respect to the thompson metric establishing its properties analogous to those of the power mean for a finite number of positive definite matrices and showing the t power means converge to the karcher mean as t 0 we carry out this program first of all for the compactly supported probability measures and show that theory including the monotonicity extends to the general case

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

monotonicity of the matrix geometric mean an attractive candidate for the geometric mean of m positive definite matrices a1 am is their riemannian barycentre g one of its important operator theoretic properties monotonicity in the m arguments has been established recently by lawson and lim we give an elementary proof of this property using standard matrix analysis and some counting arguments we derive some new inequalities for g one of these says that for any unitarily invariant norm g is not bigger than the geometric mean of a1 am

multi marginal optimal transport on riemannian manifolds we study a multi marginal optimal transportation problem on a riemannian manifold with cost function given by the average distance squared from multiple points to their barycenter under a standard regularity condition on the first marginal we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable thus inducing a monge solution this result generalizes mccann s polar factorization theorem on manifolds from two to several marginals in the same sense that a well known result of gangbo and swiech generalizes brenier s polar factorization theorem on bbb r n

log majorization and lie trotter formula for the cartan barycenter on probability measure spaces abstract we extend ando hiai s log majorization for the weighted geometric mean of positive definite matrices into that for the cartan barycenter in the general setting of probability measures on the riemannian manifold of positive definite matrices equipped with trace metric the main key is the settlement of the monotonicity problem of the cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone we also derive a version of lie trotter formula and related unitarily invariant norm inequalities for the cartan barycenter as the main application of log majorization

the least squares mean of positive hilbert schmidt operators abstract we show that the least squares mean on the riemannian manifold of positive operators in the extended hilbert schmidt algebra of linear operators on a hilbert space equipped with the canonical trace metric is the unique solution of the corresponding karcher equation this allows us to conclude that the least squares mean is the restriction of the karcher mean on the open cone of all bounded positive definite operators and hence inherits the basic properties of that mean conversely the karcher mean on the positive definite operators is shown to be the unique monotonically strongly continuous extension of the least squares mean on

factorizations and geometric means of positive definite matrices abstract in this paper we provide a new class of metric geometric means of positive definite matrices varying over hermitian unitary matrices we show that each hermitian unitary matrix induces a factorization of the cone p m of m m positive definite hermitian matrices into geodesically convex subsets and a hadamard metric structure on p m an explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two variable mean is different to the standard geometric mean some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied

inequalities for the wasserstein mean of positive definite matrices abstract we prove majorization inequalities for different means of positive definite matrices these include the cartan mean the karcher mean the log euclidean mean the wasserstein mean and the power mean

operator means of probability measures and generalized karcher equations abstract in this article we consider means of positive bounded linear operators on a hilbert space we present a complete theory that provides a framework which extends the theory of the karcher mean its approximating matrix power means and a large part of kubo ando theory to arbitrary many variables in fact to the case of probability measures with bounded support on the cone of positive definite operators this framework characterizes each operator mean extrinsically as unique solutions of generalized karcher equations which are obtained by exchanging the matrix logarithm function in the karcher equation to arbitrary operator monotone functions over the positive real half line if the underlying hilbert space is finite dimensional then these generalized karcher equations are riemannian gradients of convex combinations of strictly geodesically convex log determinant divergence functions hence these new means are their global minimizers in analogy to the case of the karcher mean as pointed out our framework is based on fundamental contraction results with respect to the thompson metric which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above

cross modality matching and the loudness growth function for click stimuli loudness intensity functions for click stimuli were obtained from 30 adult listeners having normal n 10 flat n 10 or sloping n 10 high frequency cochlear hearing loss configurations the procedure of cross modality matching cmm between loudness and perceived line length r p hellman and c h meiselman j speech hear res 31 605 615 1988 j acoust soc am 88 2596 2606 1990 j acoust soc am 93 966 975 1993 was used to validate their loudness growth functions mean group loudness exponents were similar to those reported in recent investigations that utilized pure tone stimuli further supporting the validity and reliability of the cmm task as an estimate of the loudness growth function the results also suggest that the mean loudness function for clicks is similar to the function obtained with tonal stimuli at least for listeners with moderately impaired hearing or better moreover cmm produced less variability than the more conventional psychophysical methods of magnitude estimation and production for the groups with cochlear hearing loss toward direct application of the cmm technique in lieu of absolute exponential slope values the individually determined loudness growth function over a range of intensities should be compared to the normal mean functions for calculations of deviations

categorical loudness scaling and equal loudness contours in listeners with normal hearing and hearing loss this study describes procedures for constructing equal loudness contours elcs in units of phons from categorical loudness scaling cls data and characterizes the impact of hearing loss on these estimates of loudness additionally this study developed a metric level dependent loudness loss which uses cls data to specify the deviation from normal loudness perception at various loudness levels and as function of frequency for an individual listener with hearing loss cls measurements were made in 87 participants with hearing loss and 61 participants with normal hearing an assessment of the reliability of cls measurements was conducted on a subset of the data cls measurements were reliable there was a systematic increase in the slope of the low level segment of the cls functions with increase in the degree of hearing loss elcs derived from cls measurements were similar to standardized elcs international organization for standardization iso 226 2003 the presence of hearing loss decreased the vertical spacing of the elcs reflecting loudness recruitment and reduced cochlear compression representing cls data in phons may lead to wider acceptance of cls measurements like the audiogram that specifies hearing loss at threshold level dependent loudness loss describes deficit for suprathreshold sounds such information may have implications for the fitting of hearing aids

γ commuting triples of positive definite matrices and midpoint equations on unitary matrices triples of positive definite matrices that can be transformed to a commuting triple by a congruence transformation are called commuting in this paper we give various descriptions and characterizations of commuting triples and particularly focus on those involving the geometric mean and midpoint equations on a generating set of the special unitary group

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

matrix extremal problems and shift invariant means abstract we consider an m m real symmetric matrix m a x with a 1 a m on the main diagonal and x in all off diagonal positions where m 2 and a a 1 a m is a given m tuple of positive real numbers we study the extremal problem of finding the minimum and maximum of x where m a x is positive semidefinite we show that the polynomial det m a x in variable x has only real roots with a unique negative root and that m a x is positive semidefinite resp definite if and only if x lies in the closed resp open interval determined by the negative and smallest positive roots it is further shown that the negative and smallest positive root maps over m tuples of positive real numbers contract the thompson metric and induce new multivariate means of positive real numbers satisfying the monotonicity homogeneity joint concavity and super multiplicativity in particular the smallest positive root map extends to such a mean of infinite variable of positive real numbers that realizes the limits for decreasing sequences and it eventually gives rise to a shift invariant mean of bounded sequences

computing the karcher mean of symmetric positive definite matrices we present and analyze an iterative method for approximating the karcher mean of a set ofn n positive denite

means in metric spaces and the center of mass in this article k convex metric spaces are considered where a several variable mapping is provided as a limit point of an iteration scheme based on the midpoint map in the metric space itself this mapping considered as a mean of its variables has some properties which relates it to the center of mass of these variables in the metric space sufficient conditions are given here for the two points to be identical as well as upper bounds on their distances from one another the asymptotic rate of convergence of the iterative process defining the mean is also determined here the case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied here as examples of k convex metric spaces

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

generalized polya szego type inequalities for some non commutative geometric means in this paper we shall show generalized polya szego type inequalities of n positive invertible operators on a hilbert space for any integer n 3 in terms of the following two typical non commutative geometric means that is one is the higher order weighted geometric mean due to lawson lim which is an extension of the ando li mathias geometric mean and the other is the weighted chaotic geometric mean among others the specht ratio plays an important role in our discussion which is the upper bound of a ratio type reverse of the weighted arithmetic geometric mean inequality

on the semidefinite representation of real functions applied to symmetric matrices abstract we present a new semidefinite representation for the trace of a real function f applied to symmetric matrices when a semidefinite representation of the convex or concave function f is known our construction is intuitive and yields a representation that is more compact than the previously known one we also show with the help of matrix geometric means and a riemannian metric over the set of positive definite matrices that for a rational exponent p in the interval 0 1 the matrix x raised to p is the largest element of a set represented by linear matrix inequalities this result further generalizes to the case of the matrix a p b which is the point of coordinate p on the geodesic from a to b we give numerical results for a problem inspired from the theory of experimental designs which show that the new semidefinite programming formulation can yield an important speed up factor

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

finsler metrics on symmetric cones let v be a simple euclidean jordan algebra with an associative inner product langle cdot cdot rangle and let omega be the corresponding symmetric cone let mathcal j v be the compact symmetric space of all primitive idempotents of v we show that the function s a b defined by

a converse of loewner heinz inequality and applications to operator means abstract let f t be an operator monotone function then a b implies f a f b but the converse implication is not true let a b be the geometric mean of a b 0 if a b then b 1 a i the converse implication is not true either we will show that if f b i 1 f a i i for all sufficiently small 0 then f a i f b i and a b moreover we extend it to multi variable matrices means

fractional variational problems from extended exponentially fractional integral in this letter we study the fractional variational problems from extended exponentially fractional integral both the lagrangian and hamiltonian formulations are explored and discussed in some details some interesting consequences are revealed

the relative operator entropy and the karcher mean abstract the karcher mean for positive invertible operators on a hilbert space which is the unique solution of the karcher operator equation was established by lawson lim based on the geometric considerations for positive invertible operators in this note we extend both this mean and the equation to the non invertible case the key concept is the relative operator entropy for operators whose ranges are closed we show that the karcher mean for the non invertible case is the unique solution of the extended karcher equation as a byproduct we can show the uniqueness of the karcher solution for the invertible case based only on that of the power means