geometric interpolation in p schatten class the aim of this work is to apply the complex interpolation method to norms of n tuples of operators in the p schatten classes of a hilbert space h the norms considered define finsler metrics in a certain manifold of positive operators and can be regarded as weighted p norms the weight being a positive invertible operator as a by product we obtain clarkson s type inequalities

subcritical approach to sharp hardy littlewood sobolev type inequalities on the upper half space abstract in this paper we establish the reversed sharp hardy littlewood sobolev hls for short inequality on the upper half space and obtain a new hls type integral inequality on the upper half space extending an inequality found by hang wang and yan in 6 by introducing a uniform approach the extremal functions are classified via the method of moving spheres and the best constants are computed the new approach can also be applied to obtain the classical hls inequality and other similar inequalities

short turbo codes over high order fields two classes of turbo codes constructed on high order finite fields are introduced the codes are derived from a particular protograph sub ensemble of the 2 3 regular low density parity check ldpc code ensemble the first construction results in a parallel concatenation of two non binary time variant accumulators the second construction consists of the serial concatenation of a non binary time variant differentiator with a non binary time variant accumulator and provides a highly structured flexible encoding scheme for 2 4 ldpc codes a cycle graph representation is also provided the proposed codes can be decoded efficiently either as ldpc codes via belief propagation decoding on their bipartite graphs or as turbo codes via the forward backward algorithm applied to the component code trellises by means of the fast fourier transform the proposed codes provide remarkable coding gains more than 1 db at a codeword error rate 10 4 over binary ldpc and turbo codes in the moderate short block length regime

riemannian metrics on an infinite dimensional symplectic group fil lopez galvan alberto manuel consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa saavedra 15 instituto argentino de matematica alberto calderon argentina

coded continuous phase modulation using ring convolutional codes rate 1 2 convolutional codes over the ring of integers modulo m are combined with m ary continuous phase modulation cpm schemes whose modulation indices are of the form h 1 m an m ary cpm scheme with h 1 m can be modeled by a continuous phase encoder cpe followed by a memoryless modulator mm where the cpe is linear over the ring of integers modulo m the fact that the convolutional code and the cpe are over the same algebra allows the state of the cpe to be fed back and used by the convolutional encoder a modified euclidean distance function that substantially simplifies the search for good codes has been derived and used to find new codes numerical results show that this approach consistently improves the performance as compared to coded schemes using binary convolutional codes with the same decoding complexity

contractive maps on operator ideals and norm inequalities ii abstract let i be a norm ideal of operators equipped with a unitarily invariant norm we exploit integral representations of certain functions to prove that certain ratios of linear operators acting on operators in i are contractive this leads to some new and old norm inequalities we also lift a variety of inequalities to the operator setting which were proved in the matrix setting earlier

some results on kwong functions and related inequalities abstract we investigate some relations between kwong functions and operator monotone functions as an application we present an arithmetic geometric mean type inequality by showing that for two continuous functions f g on 0 such that h t f t g t is a kwong function and f t g t t any positive matrices a b and any matrix x it holds that f a x g b g a x f b a x x b for each unitarily invariant norm

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

norm inequalities in operator ideals abstract in this paper we introduce a new technique for proving norm inequalities in operator ideals with a unitarily invariant norm among the well known inequalities which can be proved with this technique are the lowner heinz inequality inequalities relating various operator means and the corach porta recht inequality we prove two general inequalities and from them we derive several inequalities by specialization many of them new we also show how some inequalities known to be valid for matrices or bounded operators can be extended with this technique to normed ideals in c algebras in particular to the noncommutative l p spaces of a semi finite von neumann algebra

positive semidefinite solutions of the operator equation j 1nan jxaj 1 b abstract let a be a positive definite operator and b be a self adjoint operator we discuss the existence of positive semidefinite solutions of the operator equation j 1 n a n j xa j 1 b via an order preserving operator inequality by using the solutions we also give concrete examples of positive semidefinite matrices

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

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

a search for good convolutional codes to be used in the construction of turbo codes recursive systematic convolutional encoders have been shown to play a crucial role in the design of turbo codes we recall some properties of binary convolutional encoders and apply them to a search for good constituent convolutional codes of turbo codes tables of the best recursive systematic convolutional encoders found are presented for various rates together with the average bit error probability performances of some turbo codes using them

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

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

construction of linear ring codes for 6 psk the algebraic construction of linear codes over zm for m psk is considered for the case that m is the product of two distinct primes by means of the chinese remainder theorem crt such codes can be constructed from linear codes over the corresponding prime fields it is shown that for given component codes the crt composition yields psk codes that are at least as good and often much better than the codes obtained from the standard coset coding technique a lower bound on the minimum euclidean distance of 6 psk codes is derived that is based on the hamming weight distribution of the component codes

a cauchy schwarz inequality for operators with applications abstract for any unitarily invariant norm on hilbert space operators it is shown that for all operators a b x and positive real numbers r we have a xb r 2 aa x r xbb r some consequences are then discussed a simple proof is given for the fact that for positive operators a b the function spr a t b t 1 t is monotone in t on the positive half line

convexity of the inverse and moore penrose inverse abstract convexity properties of the inverse of positive definite matrices and the moore penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied for the positive definite case null space characterizations are derived and lead naturally to a concept of strong convexity of a matrix function extending the conventional concept of strict convexity the positive definite results are shown to allow for a unified analysis of problems in reproducing kernel hilbert space theory and inequalities involving matrix means the main results comprise a detailed study of the convexity properties of the moore penrose inverse providing extensions and generalizations of all the earlier work in this area

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

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

subcritical approach to conformally invariant extension operators on the upper half space abstract in this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators this family includes among others the classical poisson extension operator and the extension operator with riesz kernel we show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions we also compute the limiting behavior at the boundary of the extensions of the extremal functions

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

on infinite divisibility of positive definite functions arising from operator means abstract for a point wisely non negative positive definite function a certain criterion for its infinite divisibility i e all its fractional powers are also positive definite is obtained this criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means in particular we determine when the function cosh x s cosh x s 0 1 s s 1 1 is infinitely divisible

an arithmetic geometric harmonic mean inequality involving hadamard products abstract given matrices of the same size a a ij and b b ij we define their hadamard product to be a b a ij b ij we show that if x i 0 and q p 0 then the n n matrices x i x j x i x j x i 1 x j 1 x i x j and x i p x j p x i q x j q are positive definite and we relate these facts to some matrix valued arithmetic geometric harmonic mean inequalities some of which involve hadamard products and others unitarily invariant norms it is known that if a is positive semidefinite then max a b b 1 max a ij where denotes the spectral norm we show that the converse of this statement is false and give a useful partial converse

some results on matrix monotone functions abstract let a and b be hermitian matrices we say that a b if a b is nonnegative definite a function 0 r is said to be matrix monotone m m if a b 0 implies that a b matrix convex and concave functions can be similarly defined in this paper several known results some in sharper forms are given alternative proofs these include the following a a positive function t is m m iff t t is m m b if t is m m then both t t and t t are matrix convex and are almost always strictly convex on the other hand t is always concave and almost always strictly concave the concavity of m m functions and an inequality of hansen are consequences of a more general inequality on m m functions some complementary inequalities are also established we then prove an extension of lyapunov s theorem let a and p be positive definite matrices and either or 1 be a m m function then the unique matrix solution x of the matrix equation ax xa a p p a is positive definite a similar result holds for a related equation most of the results are obtained by exploiting the integral representation of m m functions

on the definiteness of the solutions of certain matrix equations abstract it is shown that for each dimension n 2 there is a real number t n such that for all t 2 t n the unique n n solution x of the matrix equation a 2 x xa 2 taxa b must be positive semidefinite for any positive definite a and positive semidefinite b this generalizes the classical lyapunov result it is established that t n 8 4 for n 2 3 4 respectively and that t n 2 for all n upper bounds are given for n up to 6 related equations are also discussed

positive definite solutions of some matrix equations in this paper we investigate some existence questions of positive semi definite solutions for certain classes of matrix equations known as the generalized lyapunov equations we present sufficient and necessary conditions for certain equations and only sufficient for others

blowup analysis for integral equations on bounded domains abstract consider the integral equation f q 1 x f y x y n d y f x 0 x where r n is a smooth bounded domain for 1 n the existence of energy maximizing positive solution in the subcritical case 2 q 2 n n and nonexistence of energy maximizing positive solution in the critical case q 2 n n are proved in 6 for n the existence of energy minimizing positive solution in the subcritical case 0 q 2 n n and nonexistence of energy minimizing positive solution in the critical case q 2 n n are also proved in 4 based on these in this paper the blowup behaviour of energy maximizing positive solution as q 2 n n in the case of 1 n and the blowup behaviour of energy minimizing positive solution as q 2 n n in the case of n are analyzed we see that for 1 n the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical sobolev exponent but for n different phenomena appear

strong monotonicity for various means abstract point wise monotonicity in parameters for various one parameter families of scalar means such as power difference means binomial means and stolarsky means is well known but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness among other things we obtain monotonicity in the sense of infinite divisibility which is much stronger than that in the sense of positive definiteness these strong monotonicity results are proved based on explicit computations for measures in relevant levy khintchine or actually kolmogorov formulas

iterative decoding of binary block and convolutional codes iterative decoding of two dimensional systematic convolutional codes has been termed turbo de coding using log likelihood algebra we show that any decoder can be used which accepts soft inputs including a priori values and delivers soft outputs that can be split into three terms the soft channel and a priori inputs and the extrinsic value the extrinsic value is used as an a priori value for the next iteration decoding algorithms in the log likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code the iteration is controlled by a stop criterion derived from cross entropy which results in a minimal number of iterations optimal and suboptimal decoders with reduced complexity are presented simulation results show that very simple component codes are sufficient block codes are appropriate for high rates and convolutional codes for lower rates less than 2 3 any combination of block and convolutional component codes is possible several interleaving techniques are described at a bit error rate ber of 10 sup 4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block convolutional component codes interleaver sizes less than 1000 and for three to six iterations

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

an arithmetic geometric mean inequality for products of three matrices abstract consider the following noncommutative arithmetic geometric mean inequality given positive semidefinite matrices a 1 a n the following holds for each integer m n 1 n m j 1 j 2 j m 1 n a j 1 a j 2 a j m n m n j 1 j 2 j m 1 all distinct n a j 1 a j 2 a j m where denotes a unitarily invariant norm including the operator norm and schatten p norms as special cases while this inequality in full generality remains a conjecture we prove that the inequality holds for products of up to three matrices m 3 the proofs for m 1 2 are straightforward to derive the proof for m 3 we appeal to a variant of the classic araki lieb thirring inequality for permutations of matrix products

symmetric function means of positive operators abstract many authors have studied means involving ratios of the elementary symmetric functions this work involves generalizing these means to an operator theoretic setting and extending many of the classical scalar inequalities to operators it is shown that these operator means are related to electrical networks

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

some convex and monotone matrix functions we prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections as a consequence we prove that a certain product of matrix monotone functions is a matrix monotone function similar results are proved for matrix convex functions these results include some known results for matrix monotone and matrix convex functions

comparison of various means for operators for hilbert space operatorsh k xwithh k 0 the norm inequality h1 2xk1 2 12 hx xk is known where is an arbitrary unitarily invariant norm a refinement of this arithmetic geometric mean inequality is studied similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean

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

rate 1 2 component codes for nonbinary turbo codes the iterative decoding structure and component maximum a posteriori decoders used for decoding binary concatenated codes can be extended to the nonbinary domain this paper considers turbo codes over nonbinary rings specifically ternary quaternary penternary hexernary and octernary codes the best rate 1 2 component codes are determined using a practical search algorithm the performance of the resulting rate 1 3 turbo codes on an additive white gaussian noise channel using q ary phase shift keying modulation is given

negative power nonlinear integral equations on bounded domains abstract this is the continuation of our previous work 10 where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp hardy littlewood sobolev inequality in this paper we introduce some nonlinear integral equations on bounded domains that are related to the sharp reversed hardy littlewood sobolev inequality these are integral equations with nonlinear term involving negative exponents existence results as well as nonexistence results are obtained

cauchy schwarz norm inequalities for weak integrals of operator valued functions for a finite measures on and weakly measurable families at t and bt t of hilbert space operators we have the non commutative cauchy schwarz inequalities in schatten p ideals axbd p a aa d q 1ad 2qx b b bd r 1b d 2rp for all x p h and for all p q r 1 such that 1q 1r 2p if both at t and bt t consists of commuting normal operators then axbd a ad x b bd for all unitarily invariant norms and all x h if additionally a ad i and b bd i then i a ad xi b bd h and i a ad xi b bd x axbd applications include young s and arithmetic geometric logarithmic means inequalities for operators and the mean value theorem for operator monotone functions

geometric interpolation in symmetrically normed ideals abstract the aim of this work is to apply the complex interpolation method to norms of n tuples of operators in a symmetrically normed ideal j b h defined by a symmetric norming function s n f the norms considered define finsler metrics in a certain manifold of positive operators and can be regarded as weighted norms the weight being a positive invertible operator

the hasegawa petz mean properties and inequalities abstract we study a family of means introduced by h hasegawa and d petz 1996 13 properties with respect to the parameter such as monotonicity and logarithmic concavity further monotonicity and concavity in the mean variables are shown besides the comparison between the hasegawa petz mean and the geometric mean is completely solved the connection to earlier results on operator monotonicity and some applications are also discussed

trellis coded continuous phase frequency shift keying with ring convolutional codes trellis coding techniques are applied to continuous phase frequency shift keying cpfsk a new coding scheme based on convolutional codes on the ring of integers modulo p is shown to be a natural way to apply trellis coding to cpfsk previous work has decomposed cpfsk into two parts a linear encoder with memory called the continuous phase encoder cpe and a memoryless modulator mm where the cpe often has a code structure defined over the ring of integers modulo p the combination of a modulo p convolutional channel encoder ce and the cpe is a linear modulo p encoder design examples are given for rate 1 2 coded quaternary cpfsk with modulation indexes 1 2 and 1 4 and rate 2 3 coded octal cpfsk with modulation index 1 8 combinations are optimized in the normalized minimum euclidean distance sense for a given total number of states in the overall mlse receiver numerical results show that this new coding scheme consistently obtains better performance than previous schemes

completely positive mappings and mean matrices abstract some functions f r r induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices moreover such a function f can define a linear mapping j d f 1 m n m n on matrices which is basic in the constructions of monotone metrics the present subject is to check the complete positivity of j d f 1 in the case of a few concrete functions f this problem has been motivated by applications in quantum information

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