rotfel d type inequalities for norms abstract in this note we consider some norm inequalities related to the rotfel d trace inequality tr f a b tr f a f b for concave functions f 0 0 and arbitrary n by n matrices for instance we show that for a large class of non negative concave functions f t and for all symmetric norms we have f a b 2 f a f b and we conjecture that this holds for all non negative concave functions

convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

trace functions as laplace transforms we study trace functions on the form t trf a tb where f is a real function defined on the positive half line and a and b are matrices such that a is positive definite and b is positive semidefinite if f is non negative and operator monotone decreasing then such a trace function can be written as the laplace transform of a positive measure the question is related to the bessis moussa villani conjecture

λ aluthge transforms and schatten ideals abstract let t l h and let t u t t u be the polar decomposition of t then for every 0 1 the aluthge transform is defined by t t u t 1 we show that several properties which are known for the usual aluthge transform i e the case 1 2 also hold for aluthge transforms with 0 1 moreover we get several results which are new even for the usual aluthge transform

convexity and matrix means abstract in this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic geometric and harmonic means for numbers and matrices our first main inequality will be 1 f 0 f 1 f 1 f 0 f 1 f 1 1 for the convex function f when 1 and 0 1 moreover when 1 the inequality will be valid for operator convex functions then by selecting an appropriate convex function we obtain certain matrix inequalities in particular we obtain several mixed mean inequalities for operators using real and operator convexity our discussion will lead to new multiplicative refinements and reverses of the heinz and holder inequalities for matrices new and refined trace and determinant inequalities the significance of this work is its general treatment where convexity is the only needed property

adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

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

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

whitney extension theorems for convex functions of the classes c1 and c1 ω let c be a subset of n not necessarily convex f c r be a function and g c n be a uniformly continuous function with modulus of continuity we provide a necessary and sufficient condition on f g for the existence of a convex function f cc1 n such that f f on c and f g on c with a good control of the modulus of continuity of f in terms of that of g on the other hand assuming that c is compact we also solve a similar problem for the class of c1 convex functions on n with a good control of the lipschitz constants of the extensions namely lip f g finally we give a geometrical application concerning interpolation of compact subsets k of n by boundaries of c1 or c1 1 convex bodies with prescribed outer normals on k

on the global shape of continuous convex functions on banach spaces abstract we make some remarks on the global shape of continuous convex functions defined on a banach space z for instance we show that if z is separable then for every continuous convex function f z r there exists a unique closed linear subspace y f of z such that for the quotient space x f z y f and the natural projection z x f the function f can be written in the form f z z l z for all z z where l x and x f r is a convex function such that lim t x t v for every x v x f with v 0 this kind of result is generally false if z is nonseparable even in the hilbertian case z l 2 with an uncountable set

singular values of compressions restrictions and dilations for an operator a and a subspace e denote by a e the restriction of a to e and by ae the compression of a to e a pair s t of hermitian operators is said to be monotone if there exist two nondecreasing functions f g and a hermitian z such that s f z and t g z using this notion 1 we give a simple analytic characterisation of invertible operators x such that sing a x e sing a e for all subspaces e where sing stands for the sequence of singular values equivalently if a is hermitian we characterise invertible operators x satisfying eig ax e eig ae for all subspaces e eig standing for the sequence of eigenvalues 2 we prove many inequalities involving monotone pairs of positive operators for instance we show that detae detbe det ab e in some other circumstances the opposite inequality holds we also show that pairs of hermitians can be dilated into monotone pairs

operator log convex functions and operator means we study operator log convex functions on 0 and prove that a continuous nonnegative function on 0 is operator log convex if and only if it is operator monotone decreasing several equivalent conditions related to operator means are given for such functions operator log concave functions are also discussed

computation of differential operators in wavelet coordinates in found comput math 2 2002 pp 203 245 cohen dahmen and devore proposed an adaptive wavelet algorithm for solving general operator equations assuming that the operator defines a boundedly invertible mapping between a hilbert space and its dual and that a riesz basis of wavelet type for this hilbert space is available the operator equation is transformed into an equivalent well posed infinite matrix vector system this system is solved by an iterative method where each application of the infinite stiffness matrix is replaced by an adaptive approximation it was shown that if the errors of the best linear combinations from the wavelet basis with n terms are o n s for some s 0 which is determined by the besov regularity of the solution and the order of the wavelet basis then approximations yielded by the adaptive method with n terms also have errors of o n s moreover their computation takes only 0 n operations provided s s assuming that each entry of the stiffness matrix is exactly available at unit cost generally these entries have to be approximated using numerical quadrature in this paper restricting ourselves to differential operators we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error whereas in each column the average computational cost per entry is 0 1 as a consequence we can conclude that the adaptive wavelet algorithm has optimal computational complexity

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

contraction coefficients for noisy quantum channels generalized relative entropy monotone riemannian metrics geodesic distance and trace distance are all known to decrease under the action of quantum channels we give some new bounds on and relationships between the maximal contraction for these quantities

geometry of linear pairs for self dual gauge fields a linear pair for self dual gauge fields is constructed for the metric ds2 gzzdzdz gyydyd it is shown that for consistency gzz and gyy apart from a possible overall conformal factor are given in terms of two liouville fields of equal and opposite curvatures the null surface corresponding to the pair and the homogeneous solutions playing a fundamental role are constructed explicitly the five dimensional space of y y z z and the spectral parameter is studied the proper transformation of corresponding to holomorphic ones of y and z is found known monopole instanton and quasi periodic solutions are all shown to emerge systematically as particular cases of our formalism as examples of new possibilities the case of accelerated observers and that of cosmic string backgrounds are presented

uniform approximation of convex function in smooth banach spaces abstract this article is devoted to smooth approximation of convex functions on banach spaces with smooth norm we prove that if x is a smooth space and f is a w lower semicontinuous lipschitzian convex function on x then there exist two w lower semicontinuous g teaux differentiable convex function sequences f n n 1 and g n n 1 such that 1 f n f n 1 f and f g n 1 g n 2 f n f and g n f uniformly on x 3 cl x x d f n x x cl x x d g n x x x

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

bandlimited shearlet type frames with nice duals the present paper constructs a frame dual frame pair of shearlet type such that both frames possess the distinctive time frequency localization properties needed in establishing their desirable approximation properties our construction is based on a careful pasting together of two bandlimited shearlet parseval frames associated with two different frequency cones inspired by domain decomposition methods used primarily for the solution of pdes

more accurate operator means inequalities abstract our main target in this paper is to present new sharp bounds for inequalities that result when weighted operator means are filtered through positive linear maps and operator monotone functions as an application we prove a refined reverse of the celebrated golden thompson inequality furthermore we show how these inequalities can be squared

on the quantum renyi relative entropies and related capacity formulas following csiszar s approach in classical information theory it is shown that the quantum relative entropies with parameter 0 1 can be represented as generalized cutoff rates and hence a direct operational interpretation of the quantum relative entropies are provided it is also shown that various generalizations of the holevo capacity defined in terms of the relative entropies coincide for the parameter range 0 2 and an upper bound on the one shot e capacity of a classical quantum channel in terms of these capacities is given

trace inequalities on a generalized wigner yanase skew information abstract we introduce a generalized wigner yanase skew information and then derive the trace inequality related to the uncertainty relation this inequality is a non trivial generalization of the uncertainty relation derived by s luo for the quantum uncertainty quantity excluding the classical mixture in addition several trace inequalities on our generalized wigner yanase skew information are argued

global and fine approximation of convex functions let u subset of r d be open and convex we prove that every not necessarily lipschitz or strongly convex function f u r can be approximated by real analytic convex functions uniformly on all of u we also show that c 0 fine approximation of convex functions by smooth or real analytic convex functions on r d is possible in general if and only if d 1 nevertheless for d 2 we give a characterization of the class of convex functions on r d which can be approximated by real analytic or just smoother convex functions in the c 0 fine topology it turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function but by its global geometrical behaviour we also show that every c 1 convex and proper function on u can be approximated by c infinity convex functions in the c 1 fine topology and we provide some applications of these results concerning prescription of sub differential boundary data to convex real analytic functions and smooth surgery of convex bodies

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

trace inequalities for products of matrices in this short paper we study some trace inequalities of the products of the matrices and the power of matrices by the use of elementary calculations

on the laplacian energy like invariant abstract let g be a connected graph of order n with laplacian eigenvalues 1 2 n 1 n 0 the laplacian energy like invariant of the graph g is defined as lel lel g i 1 n 1 i lower and upper bounds for lel are obtained in terms of n number of edges maximum vertex degree and number of spanning trees

higher order extension of lowner s theory operator k tone functions the new notion of operator matrix k tone functions is introduced which is a higher order extension of operator matrix monotone and convex functions differential properties of matrix k tone functions are shown characterizations properties and examples of operator k tone functions are presented in particular integral representations of operator k tone functions are given generalizing familiar representations of operator monotone and convex functions

improved young and heinz inequalities for matrices we give refinements of the classical young inequality for positive real numbers and we use these refinements to establish improved young and heinz inequalities for matrices

families of completely positive maps associated with monotone metrics abstract an operator convex function on 0 which satisfies the symmetry condition k x 1 x k x can be used to define a type of non commutative multiplication by a positive definite matrix or its inverse using the primitive concepts of left and right multiplication and the functional calculus the operators for the inverse can be used to define quadratic forms associated with riemannian metrics which contract under the action of completely positive trace preserving maps we study the question of when these operators define maps which are also completely positive cp although a d 1 2 a d 1 2 is the only case for which both the map and its inverse are cp there are several well known one parameter families for which either the map or its inverse is cp we present a complete analysis of the behavior of these families as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points our primary tool is an order relation based on the concept of positive definite functions although some results can be obtained from known properties we also prove new results based on the positivity of the fourier transforms of certain functions concrete computations of certain fourier transforms not only yield new examples of positive definite functions but also examples in the much stronger class of infinitely divisible functions

space time adaptive wavelet methods for parabolic evolution problems with respect to space time tensor product wavelet bases parabolic initial boundary value problems are equivalently formulated as bi infinite matrix problems adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate in case the spatial domain is of product type the use of spatial tensor product wavelet bases is proved to overcome the so called curse of dimensionality i e the reduction of the convergence rate with increasing spatial dimension

regularization by sup inf convolutions on riemannian manifolds an extension of lasry lions theorem to manifolds of bounded curvature abstract we show how lasry lions s result on regularization of functions defined on r n or on hilbert spaces by sup inf convolutions with squares of distances can be extended to finite or infinite dimensional riemannian manifolds m of bounded sectional curvature more specifically among other things we show that if the sectional curvature k of m satisfies k 0 k k 0 on m for some k 0 0 and if the injectivity and convexity radii of m are strictly positive then every bounded uniformly continuous function f m r can be uniformly approximated by globally c 1 1 functions defined by f sup z m inf y m f y 1 2 d z y 2 1 2 d x z 2 as 0 with 0 2 our definition of global c 1 1 smoothness is intrinsic and natural and it reduces to the usual one in flat spaces but we warn the reader that in the noncompact case this definition differs from other notions of rather local c 1 1 smoothness that have been recently used by other authors based on charts the importance of this regularization method lies rather than on the degree of smoothness obtained on the fact that the correspondence f f is explicit and preserves many significant geometrical properties that the given functions f may have such as invariance by a set of isometries infima sets of minimizers ordering local or global lipschitzness and only when one additionally assumes that k 0 local or global convexity we also give two examples showing that this result completely fails even for nonflat cartan hadamard manifolds whenever f or k are not bounded

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

matrix versions of some classical inequalities abstract some natural inequalities related to rearrangement in matrix products can also be regarded as extensions of classical inequalities for sequences or integrals in particular we show matrix versions of chebyshev and kantorovich type inequalities the matrix approach may also provide simplified proofs and new results for classical inequalities for instance we show a link between cassel s inequality and the basic rearrangement inequality for sequences of hardy littlewood polya and we state a reverse inequality to the hardy littlewood polya inequality in which matrix technics are essential

eigenvalue inequalities for differences of means of hilbert space operators abstract we prove several eigenvalue inequalities for the differences of various means of two positive invertible operators a and b on a separable hilbert space under the assumption that a b is compact equality conditions of these inequalities are also obtained

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

eigenvalue inequalities for convex and log convex functions abstract we give a matrix version of the scalar inequality f a b f a f b for positive concave functions f on 0 we show that choi s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences some inequalities for log convex functions are presented and a new arithmetic geometric mean inequality for positive matrices is given we also point out a simple proof of the bhatia kittaneh arithmetic geometric mean inequality

an optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems in this paper we modify the adaptive wavelet algorithm from gantumur et al an optimal adaptive wavelet method without coarsening of the iterands technical report 1325 department of mathematics utrecht university march 2005 math comp to appear so that it applies directly i e without forming the normal equation not only to self adjoint elliptic operators but also to operators of the form l a b where a is self adjoint elliptic and b is compact assuming that the resulting operator equation is well posed we show that the algorithm has optimal computational complexity

numerical radius inequalities via convexity abstract the main goal of this article is to present numerical radius inequalities for matrices based on convexity of certain numerical radius functions this simple approach extends some unitarily invariant norms inequalities such as heinz and young inequalities to the context of numerical radius however due to weak unitary invariance of the numerical radius these extended inequalities will be weaker than the corresponding unitarily invariant norms versions

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

young s inequality in trace class operators if a and b are trace class operators and if u is a partial isometry then where 1 denotes the norm in the trace class the present paper characterises the cases of equality in this young inequality and the characterisation is examined in the context of both the operator and the hilbert schmidt forms of young s inequality

extensions of lieb s concavity theorem the operator function a b trf a b k k defined in pairs of bounded self adjoint operators in the domain of a function f of two real variables is convex for every hilbert schmidt operator k if and only if f is operator convex we obtain as a special case a new proof of lieb s concavity theorem for the function a b tra p k b q k where p and q are non negative numbers with sum p q 1 in addition we prove concavity of the operator function a b to tr left frac a a mu 1 k frac x1b b mu 2 k right in its natural domain d 2 1 2 cf definition 3

construction of hyperbolic monopoles methods are presented for explicit construction of hyperbolic monopoles as defined by atiyah m f atiyah commun math phys 93 437 1984 and magnetic monopoles in hyperbolic space unpublished lecture notes 1984 the techniques permit the extraction of flat space self dual monopoles trivially through a simple rescaling relationships to previous works of the present author are indicated

remarks on two recent results of audenaert abstract audenaert recently obtained some new matrix norm inequalities in audenaert 2015 2 3 in this note we provide alternative proofs for these results

a generalized skew information and uncertainty relation a generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by fujii in addition we prove the trace inequality conjectured by luo and zhang finally we point out that theorem 1 in s luo and q zhang ieee trans inf theory vol 50 pp 1778 1782 no 8 aug 2004 is incorrect in general by giving a simple counter example

new inequalities for the generalized karcher mean abstract recently palfia introduced a generalized karcher mean as a solution of an operator equation in this article we present several relations for this new mean in particular we investigate the behavior of this generalized mean when filtered through operator monotone functions and positive linear maps revealing its information monotonicity

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

wavelets with patchwise cancellation properties we construct wavelets on general n dimensional domains or manifolds via a domain decomposition technique resulting in so called composite wavelets with this construction wavelets with supports that extend to more than one patch are only continuous over the patch interfaces normally this limited smoothness restricts the possibility for matrix compression and with that the application of these wavelets in adaptive methods for solving operator equations by modifying the scaling functions on the interval and with that on the n cube that serves as parameter domain we obtain composite wavelets that have patchwise cancellation properties of any required order meaning that the restriction of any wavelet to each patch is again a wavelet this is also true when the wavelets are required to satisfy zeroth order homogeneous dirichlet boundary conditions on part of the boundary as a result compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose also taking stability into account our composite wavelets have all the properties for the application to the adaptive solution of well posed operator equations of orders 2t for t 3 2

singular value estimates of oblique projections abstract let w and m be two finite dimensional subspaces of a hilbert space h such that h w m and let p w m denote the oblique projection with range w and nullspace m in this article we get the following formula for the singular values of p w m 2 s k p w m 1 min f h x w m 2 where the minimum is taken over the set of all operator pairs f h on h such that r f w r h m and fh p w m and k 1 dim w we also characterize all the pairs where the minimum is attained

connectedness properties of the space of complete nonnegatively curved planes we study the space of complete riemannian metrics of nonnegative curvature on the plane equipped with the c k topology if k is infinite we show that the space is homeomorphic to the separable hilbert space for any k we prove that the space cannot be made disconnected by removing a finite dimensional subset a similar result holds for the associated moduli space the proof combines properties of subharmonic functions with results of infinite dimensional topology and dimension theory a key step is a characterization of the conformal factors that make the standard euclidean metric on the plane into a complete metric of nonnegative sectional curvature

operator inequalities involving improved young and its reverse inequalities abstract this note aims to present some operator inequalities and matrices inequalities for the hilbert schmidt norm first the refined young and its reverse inequalities for scalars are given then based on these inequalities operator inequalities and matrices inequalities on the hilbert schmidt norm are established

commutator inequalities for the hilbert schmidt norm abstract two types of commutator inequalities for the hilbert schmidt norm are established the first type of these inequalities is related to a classical inequality of clarkson and the second type is related to the unitary approximation of positive and invertible operators

coding theorems for compound problems via quantum renyi divergences recently a new notion of quantum renyi divergences has been introduced by muller lennert dupuis szehr fehr and tomamichel and wilde winter and yang which found a number of applications in strong converse theorems here we show that these new renyi divergences are also useful tools to obtain coding theorems in the direct domain of various problems we demonstrate this by giving new and considerably simplified proofs for the achievability parts of stein s lemma with composite null hypothesis universal state compression and the classical capacity of compound classical quantum channels based on single shot error bounds already available in the literature and simple properties of the quantum renyi divergences the novelty of our proofs is that the composite compound coding theorems can be almost directly obtained from the single shot error bounds essentially with the same effort as for the case of simple null hypothesis single source single channel

mean matrices and infinite divisibility abstract we consider matrices m with entries mij m i j where 1 n are positive numbers and m is a binary mean dominated by the geometric mean and matrices w with entries wij 1 m i j where m is a binary mean that dominates the geometric mean we show that these matrices are infinitely divisible for several much studied classes of means

interpolated inequalities for unitarily invariant norms abstract in this article we interpolate the well known young and heinz inequalities for unitarily invariant norms and some of their known refinements then we prove new interpolated refinements in the end we use this interpolation idea to prove a hidden monotonicity behavior these inequalities obey

some inequalities for norms on matrices and operators we study several inequalities for norms on matrices in particular for the hilbert schmidt and operator norms these inequalities occur when comparing norms of the products xy and yx for matrices x and y with suitable assumptions we also point out some trace inequalities

trace inequalities in nonextensive statistical mechanics abstract in this short paper we establish a variational expression of the tsallis relative entropy in addition we derive a generalized thermodynamic inequality and a generalized peierls bogoliubov inequality finally we give a generalized golden thompson inequality

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

matrix form of the inverse young inequalities abstract we use operator monotone and operator convex functions to prove an inverse to the young inequality for eigenvalues of positive definite matrices and then apply it to obtain a matrix inverse young inequality which can be considered as a complement of a result of t ando also we give a necessary and sufficient condition for the equality

a trace inequality arising from quantum information theory abstract we introduce a skew information of lieb s type s f g a x tr f a xg a x tr f a g a x 2 for selfadjoint matrices a x we give conditions for f and g so that sf g is positive or negative as another important application we settle the problem posed by yanagi furuichi and kuriyama from quantum information theory

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

trace inequalities for completely monotone functions and bernstein functions abstract we prove a matrix trace inequality for completely monotone functions and for bernstein functions as special cases we obtain non trivial trace inequalities for the power function x x q which for certain values of q complement mccarthy s trace inequality and for others strengthen it

discontinuous legendre wavelet element method for elliptic partial differential equations abstract by incorporating the legendre multiwavelet into the discontinuous galerkin dg method this paper presents a novel approach for solving poisson s equation with dirichlet boundary which is known as the discontinuous legendre multiwavelet element dlwe method derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes one striking advantage of our method is that the differential operator boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost numerical experiments demonstrate the validity of this method furthermore this paper generalizes the dlwe method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm the proposed method and its generalizations are also applicable to some other kinds of partial differential equations

a complete refinement of young s inequality abstract in this article we present multiple term refinements of young s inequality for real numbers and operators in particular given a natural number n we find n positive terms refining young s inequality detailed properties of this refinement are studied leading to a new interesting mean

operator means and positivity of block operators the aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators it is shown that for positive operators a b c and for every non negative operator monotone function f on 0 infty the block matrix begin aligned left begin array cc f a f c f c f b end array right end aligned is positive if and only if c le a b in particular if c le a b then begin aligned left begin array cc a c c b end array right end aligned is positive

uncertainty relation on wigner yanase dyson skew information abstract we give a trace inequality related to the uncertainty relation of wigner yanase dyson skew information this inequality corresponds to a generalization of the uncertainty relation derived by s luo 2005 8 for the quantum uncertainty quantity excluding the classical mixture

uncertainty relation on generalized wigner yanase dyson skew information abstract we give a trace inequality related to the uncertainty relation of generalized wigner yanase dyson skew information which is two parameter s extension of our result in 12

implementation of an iterative map in the construction of quasi periodic instantons chaotic aspects and discontinuous rotation numbers an iterative map of the unit disk in the complex plane is used to explore certain aspects of self dual four dimensional gauge fields quasi periodic in the euclidean time these fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits the iterations do not correspond directly to a discretized time evolution of the gauge fields they are implemented in an indirect fashion first t r being the standard coordinates the r t half plane is mapped on the unit disk in an appropriate way this provides an r t parametrization of z0 the starting point of the iterations and makes the iterates increasingly complex functions of r and t these are then incorporated as building blocks in the generating function of the fields we explain in what sense and to what extent some remarkable features of our map indicated in the title are thus carried over into the continuous time development of the fields special features for quasiperiodicity are stu

concavity of the auxiliary function appearing in quantum reliability function reliability functions characterize the asymptotic behavior of the error probability for transmission of data on a channel holevo introduced the quantum channel and gave an expression for a random coding lower bound involving an auxiliary function holevo ogawa and nagaoka conjectured that this auxiliary function is concave here we give a proof of this conjecture

inequalities for generalized euclidean operator radius via young s inequality using a refinement of the classical young inequality we refine some inequalities of operators including the function p p where p p is defined for p 1p 1 and operators t1 tn b h t1 tn b h by p t1 tn sup x 1 i 1n tix x p 1p among other things we show that if t1 tn b h t1 tn b h and p q 1p q 1 with 1p 1q 1 then 1n i 1nti 2 p t1 tn q t1 tn 1n i 1nti 2 1p i 1n ti p 1q i 1n ti q inf x y 1 x y where x y 1p i 1n ti x x p i 1n ti y y q 2

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed