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

uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

on the finite size scaling of clusters in compact directed percolation the exact finite size scaling properties of clusters in compact directed percolation on a square lattice are derived the results are implicit in previous work on the enumeration of staircase polygons but their explicit form has not been presented as such before the analysis provides important insights into the nature of this type of percolation transition

r norm bounds and metric properties for zero loci of real analytic functions abstract we consider the problem of deciding whether or not a zero locus x of multivariate real analytic functions crosses a given r norm ball in the real n dimensional affine space we perform a local study of the problem and we provide both necessary and sufficient conditions to answer the question our conditions derive from the analysis of differential geometric properties of x at the center of the ball an algorithm to evaluate r norms distances is proposed

a variant of jensen s inequality of mercer s type for operators with applications a variant of jensen s operator inequality for convex functions which is a generalization of mercer s result is proved obtained result is used to prove a monotonicity property for mercer s power means for operators and a comparison theorem for quasi arithmetic means for operators

combinatorial approach to feynman path integration combinatorial relations can be used to convert the non relativistic time sliced feynman path integral into perturbation expansions these methods reveal that when the time interval is sliced into n increments each order of perturbation theory sustains an error o 1 square root n in this way the author provides exact path integral results for the following potentials delta function comb finite well tunnelling barrier and a generalized exponential cusp for the tunnelling barrier it is seen how the celebrated 1 reflection factor arises in the limit of infinite barrier height the one dimensional coulomb problem is solved as a limiting case of the exponential cusp in addition for power potentials the author indicates how this path integral approach yields sometimes divergent nevertheless asymptotic perturbation expansions

zeta regularization for exact wkb resolution of a general 1d schrodinger equation we review an exact analytical resolution method for general one dimensional quantal anharmonic oscillators stationary schr dinger equations with polynomial potentials it is an exact form of wkb treatment involving spectral usual versus classical newer zeta regularizations in parallel the central results are a set of bohr sommerfeld like but exact quantization conditions directly drawn from wronskian identities and appearing to extend the bethe ansatz formulae of integrable systems such exact quantization conditions do not just select the eigenvalues some evaluate the spectral determinants and others the wavefunctions for the spectral parameter in general position this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

scaling analysis for the adsorption transition in a watermelon network of n directed non intersecting walks the partition function for the problem of n directed non intersecting walks interacting via contact potentials with a wall parallel to the direction of the walks has previously been calculated as an n n determinant here we describe how to analyse the scaling behaviour of this problem using alternative representations of the solution in doing so we derive the asymptotics of the partition function of a watermelon network of n such walks for all temperatures and so calculate the associated network exponents in the three regimes desorbed adsorbed and at the adsorption transition furthermore we derive the full scaling function around the adsorption transition for all n at the adsorption transition we also derive a simple product form for the partition function these results have in part been derived using recurrence relations satisfied by the original determinantal solution

on the hulls of directed percolation clusters the properties of the hulls of directed percolation clusters are studied the scaling and finite size scaling of many quantities around the percolation threshold pc are derived and a novel monte carlo algorithm which is more than twice as fast as the standard algorithm at pc has been formulated to study these properties simulations have been conducted that enable an estimation of all the exponents involved in particular the central exponent x relating the average hull length of clusters to their mass has been estimated to be 0 773 4 at the percolation threshold this same exponent is estimated to be 0 905 5 for p p c thus this second value should hold for directed animals

the proper formula for relative entropy and its asymptotics in quantum probability umegaki s relative entropys trd logd logd of states and with density operatorsd andd respectively is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint it is also proved that some other versions of the relative entropy give rise to the same asymptotics as umegaki s one as a byproduct the inequality tra logab tra loga logb is obtained for positive definite matricesa andb

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

perturbation of polynomials and applications to the hough transform abstract let f and g be complex polynomials of the same degree we provide a new lower bound on the euclidean distance of points belonging to their zero loci in terms of bombieri s norm we also present a minimization of the bombieri s norm of the difference g f for c in the real case we apply the results above in the setting of the hough transform a standard technique to detect curves in images suggesting a bombieri s norm based recognition algorithm

plane partition vesicles we examine partitions and their natural three dimensional generalizations plane partitions as models of vesicles undergoing an inflation deflation transition the phase diagrams of these models include a critical point corresponding to an inflation deflation transition and exhibits multicritical scaling in the vicinity of a multicritical point located elsewhere on the critical curve we determine the locations of the multicritical points by analysing the generating functions using analytic and numerical means in addition we determine the numerical values of the multicritical scaling exponents associated with the multicritical scaling regimes in these models

complementary inequalities to inequalities of jensen and ando based on the mond pecaric method abstract applying the mond pecaric method to unital positive linear maps we shall show several complementary inequalities to jensen s inequalities on positive linear maps and consequently obtain complementary inequalities to ando s inequalities associated with operator means we shall apply them to obtain complementary estimates for the results by ando aujla vasudeva and fujii on hadamard product and operator means

around choi inequalities for positive linear maps abstract we show an inequality for unital positive linear maps interpolating choi s inequality p 0 with a recent result of bourin and ricard r 1 let a a 0 p q and q q p r 2 q q p then a p r a q r a p q r

on a question of bhatia and kittaneh abstract we settle in the affirmative a question of bhatia and kittaneh for p and q positive semidefinite n n matrices the inequality r pq 1 2 r p q holds for r 1 2 n

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

evaluation of a ln tan integral arising in quantum field theory we analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry we additionally obtain new representations of the clausen function cl2 and the catalan constant g cl2 2 as well as new relations between sine and clausen function values

a key inequality for functions of matrices an inequality is proved for convex functions applied to self adjoint matrices several known inequalities are shown to be consequences but properly weaker

moore penrose approach in the hough transform framework abstract let f x a be a real polynomial in two sets of variables x and a that is linear with respect to one of the variable sets say a in this paper we deal with two of the main steps of the hough transform framework for the pattern recognition technique to detect loci in images more precisely we present an algorithmic process based on the moore penrose pseudo inverse to provide a region of analysis in the parameter space in addition we state an upper bound for the sampling distance of the discretization of the parameter space region

inequalities for quantum relative entropy some logarithmic trace inequalities involving the notions of relative entropy are reobtained from a log majorization result the thermodynamic inequality is generalized and a chain of equivalent statements involving this inequality and the peierls bogoliubov inequality is obtained

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

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

two reverse inequalities associated with tsallis relative operator entropy via generalized kantorovich constant and their applications recently tsallis relative operator entropy tp a b and tsallis relative entropy dp a b are discussed by furuichi yanagi kuriyama we shall show two reverse inequalities involving tsallis relative operator entropy tp a b via generalized kantorovich constant k p as some applications of two reverse inequalities we shall show two trace reverse inequalities involving tr tp a b and dp a b and also a known reverse trace inequality involving the relative operator entropy s a b by fujii kamei and the umegaki relative entropy s a b is shown as a simple corollary we show the following result let a and b be strictly positive operators on a hilbert space h such that m1 i a m1 i 0 and m2 i b m2 i 0 put m m2m1 m m2m1 h mm m1m2m1m2 1 and p 0 1 let be normalized positive linear map on b h then the following inequalities hold i 1 k p p a p b tp a b tp a b tp a b and ii f p a tp a b tp a b tp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and k p 0 1 and f p mpphp hh 11 k p 1p 1 0 in addition let a and b be strictly positive definite matrices iii 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b and iv f p tr a dp a b tr tp a b dp a b in particular both iii and iv yield the following known result logs 1 tr a s a b tr s a b s a b where s 1 h1h 1elogh1h 1 is said to be the specht ratio and s 1 1

on a connection between the polylogarithm function and the bass diffusion model the purpose of this paper is to establish a connection between the polylogarithm function and a continuous probability distribution we provide closed form expressions in terms of the polylogarithm

the brownian excursion area a numerical analysis abstract using an explicit expression for the laplace transform of the brownian excursion area generating function a numerical analysis gives moments density and distribution function an asymptotic formula is given for small argument

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

a matrix reverse cauchy schwarz inequality abstract the matrix geometric mean is concave we complete this important fact with a reverse result this follows from an interesting non commutative extension of a classical reverse cauchy schwarz inequality cassel 1951 our investigation also leads to state a reverse result to von neuman s trace inequality

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

a difference counterpart to a matrix holder inequality abstract we point out a sharp reverse cauchy schwarz holder matrix inequality the cauchy schwarz version involves the usual matrix geometric mean let a i and b i be positive definite matrices such that 0 ma i b i ma i for some scalars 0 m m and i 1 2 n then i 1 n a i i 1 n b i i 1 n a i b i m m 2 4 m m i 1 n a i where the matrix geometric mean of positive definite a and b is defined by a b a 1 2 a 1 2 ba 1 2 1 2 a 1 2

zeta function of the bessel operator on the negative real axis we investigate the pole structure of the zeta function zeta anu s sigma k lambda k s built out of the eigenvalues lambda k of a bessel operator anu subject to dirichlet boundary conditions at one end of the domain this leads us to the study of zeta anu on the negative real axis where most of the singularities occur

on an operator kantorovich inequality for positive linear maps abstract we improve the operator kantorovich inequality as follows let a be a positive operator on a hilbert space with 0 m a m then for every unital positive linear map a 1 2 m m 2 4 m m 2 a 2 as a consequence a 1 a a a 1 m m 2 2 m m

the distribution of the area under a bessel excursion and its moments a bessel excursion is a bessel process that begins at the origin and first returns there at some given time t we study the distribution of the area under such an excursion which recently found application in the context of laser cooling the area a scales with the time as a sim t 3 2 independent of the dimension d but the functional form of the distribution does depend on d we demonstrate that for d 1 the distribution reduces as expected to the distribution for the area under a brownian excursion known as the airy distribution deriving a new expression for the airy distribution in the process we show that the distribution is symmetric in d 2 with nonanalytic behavior at d 2 we calculate the first and second moments of the distribution as well as a particular fractional moment we also analyze the analytic continuation from d 2 in the limit where d rightarrow 4 from below this analytically continued distribution is described by a one sided levy alpha stable distribution with index 2 3 and a scale factor proportional to 4 d t 3 2

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

squaring operator polya szego and diaz metcalf type inequalities abstract we square operator polya szego and diaz metcalf type inequalities as follows if operator inequalities 0 m 1 2 a m 1 2 and 0 m 2 2 b m 2 2 hold for some positive real numbers m 1 m 1 and m 2 m 2 then for every unital positive linear map the following inequalities hold a b 2 m 1 m 2 m 1 m 2 2 m 1 m 2 m 1 m 2 4 a b 2 and m 2 m 2 m 1 m 1 a b 2 m 1 m 1 m 2 2 m 2 2 m 2 m 2 m 1 2 m 1 2 2 8 m 2 m 1 m 1 m 2 m 1 2 m 1 2 m 2 m 2 2 a b 2

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

matrix power means and the information monotonicity abstract lim and palfia established the notion of the matrix power means for k positive definite matrices k 3 and showed that the matrix power means have the information monotonicity for a unital positive linear mapping in this note by virtue of the generalized kantorovich constant we show counterparts to the information monotonicity of the matrix power means

finite length scaling of collapsing directed walks we study the finite length scaling of self interacting partially directed self avoiding walks utilising enumeration data up to a total length of 6000 steps this facilitates the evaluation of the numerical techniques available for calculating exponents at the point and in the collapsed phase of walk type models another consequence is the conjecture of a new scaling theory for the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of undirected problems including interacting saw we provide a phenomenological picture of the phase transition in terms of the condensation of droplets that allows us to understand the various length scales involved in the problem pacs numbers 05 50 q 05 70 fh 61 41 e email prel aleks brak tonyg mundoe maths mu oz au

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

scaling near the theta point for isolated polymers in solution recently questions have been raised as to the conclusions that can be drawn from currently proposed scaling theory for a single polymer in various types of solution in two and three dimensions here we summarize the crossover theory predicted for low dimensions and clarify the scaling arguments that relate thermal exponents for quantities on approaching the theta point from low temperatures to those associated with the asymptotics in polymer length at the theta point itself

simple connections between generalized hypergeometric series and dilogarithms connections between generalized hypergeometric series and dilogarithms are investigated some simple relations of an appell s function and dilogarithms are found

uniform asymptotics of area weighted dyck paths using the generalized method of steepest descents for the case of two coalescing saddle points we derive an asymptotic expression for the bivariate generating function of dyck paths weighted according to their length and their area in the limit of the area generating variable tending towards 1 the result is valid uniformly for a range of the length generating variable including the tricritical point of the model

area width scaling in generalised motzkin paths abstract we consider a generalised version of motzkin paths where horizontal steps have length l with l being a fixed positive integer we first give the general functional equation for the area width generating function of this model using a heuristic ansatz we then derive the area width scaling behaviour in terms of a scaling function in one variable for the special cases of dyck standard motzkin and schroder paths before generalising our approach to arbitrary l we then rigorously derive the tricritical scaling of schroder paths by applying the generalised method of steepest descents to the known exact solution for their area width generating function our results show that for dyck and schroder paths the heuristic scaling ansatz reproduces the rigorous results

the tricritical behavior of self interacting partially directed walks we present the thermodynamics of two variations of the interacting partially directed self avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values while the discrete model has been considered previously to varying degrees of success the continuous model we now define has not the examination of the continuous model leads to theexact derivation of several exponents for the discrete model some of these exponents can be calculated using a continued fraction representation for both models the crossover exponent is found to be 2 3 moreover we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function additionally the similarities noticed previously to other models but left unexplored are explained with the aid of necklacing arguments

recognition of feature curves on 3d shapes using an algebraic approach to hough transforms abstract feature curves are largely adopted to highlight shape features such as sharp lines or to divide surfaces into meaningful segments like convex or concave regions extracting these curves is not sufficient to convey prominent and meaningful information about a shape we have first to separate the curves belonging to features from those caused by noise and then to select the lines which describe non trivial portions of a surface the automatic detection of such features is crucial for the identification and or annotation of relevant parts of a given shape to do this the hough transform ht is a feature extraction technique widely used in image analysis computer vision and digital image processing while for 3d shapes the extraction of salient feature curves is still an open problem thanks to algebraic geometry concepts the ht technique has been recently extended to include a vast class of algebraic curves thus proving to be a competitive tool for yielding an explicit representation of the diverse feature lines equations in the paper for the first time we apply this novel extension of the ht technique to the realm of 3d shapes in order to identify and localize semantic features like patterns decorations or anatomical details on 3d objects both complete and fragments even in the case of features partially damaged or incomplete the method recognizes various features possibly compound and it selects the most suitable feature profiles among families of algebraic curves

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

a biologically inspired compound eye detector array part ii statistical performance analysis this is the second part of our paper in this paper we propose model and analyze the performance of a detector array for localizing far field particle emitting sources which is inspired by but generalizes the compound eye of insects the array consists of multiple eyelets each having a conical module with a lens on its top and an inner subarray containing multiple particle detectors using a parametric measurement model introduced for the array in part i in this part we analytically and numerically analyze the statistical performance of the array first we compute the statistical cramer rao bounds crbs on the errors in estimating the direction of arrival of the incident particles then we derive a lower bound on the mean square angular error msae of source localization for any specific array configuration thirdly we consider two source direction estimators the maximum likelihood estimator mle and the weighted direction estimator wde and analyze their msae performance in the numerical examples we quantitatively compare the performance of the proposed array with the biological compound eye show the array performance as a function of the array configuration variables optimally design the array configuration illustrate that the mle asymptotically attains the performance bound whereas the wde is nearly optimal for sufficiently large snr and analyze the hardware efficiency by comparing the two msae bounds potential applications of this work include artificial vision in medicine or robotics astronomy assisted security and particle communications

generalized matrix version of reverse holder inequality abstract a generalized matrix version of reverse cauchy schwarz holder inequality is proved this includes the recent results proved by bourin fujii lee niezgoda and seo

operator inequalities associated with aloga via specht ratio abstract an operator means a bounded linear operator on a hilbert space h we obtained the basic property between specht ratio s 1 and generalized kantorovich constant k p in math inequal appl in press that is specht ratio s 1 can be expressed by generalized kantorovich constant k p s 1 e k 1 we shall investigate several product type and difference type inequalities associated with a log a by applying this basic property to several kantorovich type inequalities

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

generalized shannon inequalities based on tsallis relative operator entropy abstract tsallis relative operator entropy is defined and then its properties are given shannon inequality and its reverse one in hilbert space operators derived by furuta linear algebra apll 381 2004 219 are extended in terms of the parameter of the tsallis relative operator entropy moreover the generalized tsallis relative operator entropy is introduced and then several operator inequalities are derived

higher order airy scaling in deformed dyck paths we introduce a deformed version of dyck paths ddp where additional to the steps allowed for dyck paths jumps orthogonal to the preferred direction of the path are permitted we consider the generating function of ddp weighted with respect to their half length area and number of jumps this represents the first example of an exactly solvable two dimensional lattice vesicle model showing a higher order multicritical point applying the generalized method of steepest descents we see that the associated two variable scaling function is given by the logarithmic derivative of a generalized higher order airy integral

the dilogarithm function for complex argument this paper summarizes the basic properties of the euler dilogarithm function often referred to as the spence function these include integral representations series expansions linear and quadrat

reverse inequalities involving two relative operator entropies and two relative entropies we shall discuss relation among tsallis relative operator entropy tp a b the relative operator entropy s a b by j i fujii kamei the tsallis relative entropy dp a b by furuichi yanagi kuriyama and the umegaki relative entropy s a b we show the following result let a and b be strictly positive definite matrices such that m1i a m1i 0 and m2i b m2i 0 put h m1m2m1m2 1 and p 0 1 then the following inequalities hold 1 k p p tr a 1 p tr b p dp a b tr tp a b dp a b where k p is the generalized kantorovich constant defined by k p hp h p 1 h 1 p 1 p hp 1 hp h p and the first inequality is the reverse one of the second known inequalty in particular logs 1 tr a s a b tr s a b s a b where s 1 is the specht ratio defined by s 1 h1h 1elogh1h 1 and the first inequality is the reverse one of the second known inequalty it is known that k p 0 1 for p 0 1 and s 1 1

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

zeta function regularization is uniquely defined and well hawking s zeta function regularization procedure 1977 is shown to be rigorously and uniquely defined thus putting an end to the spreading lore about different difficulties associated with it basic misconceptions misunderstandings and errors which keep appearing in important scientific journals when dealing with this beautiful regularization method and other analytical procedures are clarified and corrected

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

polylogarithms and riemann s ζ function riemann s z function perhaps first appeared in statistical mechanics in 1900 in planck s theory of the blackbody radiation and then in 1912 in debye s theory of the specific heats of solids 1 subsequently this function has played an important role in the statistical theory of the ideal bose gas especially for the understanding of bose einstein condensation bec 2 more recently this function together with the mellin transform has become a powerful tool for the analysis of the thermodynamic potentials 3 4 it would be no surprise to find fruitful applications of riemann s z function in other areas of today s theoretical physics 5 recently it was found that the statistical thermodynamics of ideal gases can be given a unified picture through polylogs defined in terms of the fugacity z and dimensions d 6 there is richness that this unified picture reveals such as the anomalous physics in null dimension 7 the fermi bose reflection in d 3 8 and the fermi bose equivalence in d 52 9 these physical results are consequences of some special properties of polylogs it has been long known that a polylog of integral order becomes riemann s z function of the same order when its argument attains unity 10 thus riemann s z function can enter into the unified theory of the statistical thermodynamics via the polylogs quite naturally interestingly we find that this formulation shows another way of evaluating riemann s z function which is presented in this work the classical theory of polylogs begins with euler s dilog and landen s trilog and extends to higher order polylogs such as the quadrilog in physical applications the order of a polylog is related to physical dimensions d thus polylogs of integral order lower than the dilog have been conceived such as the nil log and the monolog for the physics in d 50 and 2 6 there have been suggestions that negative dimensions can be of theoretical interest 11 they require the polylogs of still lower orders than the nil log departing from the direction of the classical theory of polylogs what we find is that in these circumstances there exists even a simpler relationship between the polylogs and riemann s z function we can use this relationship to evaluate riemann s z function very simply perhaps more simply than by most standard methods it also lends an interesting insight into the nature of the values of riemann s z function ii polylogs and their properties

sum rules for zeros of bessel functions and an application to spherical aharonov bohm quantum bags the authors study the sum zeta h s sigma jej s over the eigenvalues ej of the schrodinger equation in a spherical domain with dirichlet walls threaded by a line of magnetic flux rather than using green function techniques they tackle the mathematically non trivial problem of finding exact sum rules for the zeros of bessel functions jv which are extremely helpful when seeking numerical approximations to ground state energies these results are particularly valuable if v is neither an integer nor half an odd integer

cluster structure of collapsing polymers in order to better understand the geometry of the polymer collapse transition we study the distribution of geometric clusters made up of the nearest neighbor interactions of an interacting self avoiding walk we argue for this new correlated percolation problem that in two dimensions and possibly also in three dimensions a percolation transition takes place at a temperature lower than the collapse transition hence this novel transition should be governed by exponents unrelated to the point exponents this also implies that there is a temperature range in which the polymer has collapsed but has no long range cluster structure we use monte carlo to study the distribution of clusters on the simple cubic and manhattan lattices on the manhattan lattice where the data are most convincing we find that the percolation transition occurs at p 1 461 3 while the collapse transition is known to occur exactly at 1 414 we propose a finite size scaling form for the cluster distribution and estimate several of the critical exponents regardless of the value of p this percolation problem sheds new light on polymer collapse

scaling in the collapsed polymer phase exact results the author derives an exact scaling form for the low temperature partition function in a model of polymer collapse this confirms series work and so gives the exponents sigma 1 2 gamma 1 4 and chi 3 4 exactly the model considered is a variant of the self interacting partially directed self avoiding walk in two dimensions

on matrix inequalities between the power means counterexamples abstract we prove that the known sufficient conditions on the real parameters p q for which the matrix power mean inequality a p b p 2 1 p a q b q 2 1 q holds for every pair of matrices a b 0 are indeed best possible the proof proceeds by constructing 2 2 counterexamples the best possible conditions on p q for which a p 1 p a q 1 q holds for every unital positive linear map and a 0 are also clarified

sum rules for quantum billiards the authors write explicit integral expressions for sums of inverse powers of the eigenvalues of the laplacian with dirichlet boundary conditions in a simply connected bounded two dimensional domain

an algebraic approach to hough transforms abstract the main purpose of this paper is to lay the foundations of a general theory which encompasses the features of the classical hough transform and extend them to general algebraic objects such as affine schemes the main motivation comes from problems of detection of special shapes in medical and astronomical images the classical hough transform has been used mainly to detect simple curves such as lines and circles we generalize this notion using reduced grobner bases of flat families of affine schemes to this end we introduce and develop the theory of hough regularity the theory is highly effective and we give some examples computed with cocoa see 1

matrix inequalities for convex functions abstract many converses of jensen s inequality for convex functions can be found in the literature here we give matrix versions with matrix weights of these inequalities some applications to the hadamard product of matrices are also given

matrix trace inequalities on tsallis relative entropy of negative order abstract in this paper we show matrix trace inequalities on the quantum tsallis relative entropy of negative order which includes the quasi geometric mean of positive definite matrices moreover we show an estimate of the difference between two tsallis relative entropies of negative order

an asymmetric kadison s inequality abstract some inequalities for positive linear maps on matrix algebras are given especially asymmetric extensions of kadison s inequality and several operator versions of chebyshev s inequality we also discuss well known results around the matrix geometric mean and connect it with complex interpolation

rectangular vesicles in three dimensions the phase diagram of a model of rectangular vesicles in three dimensions is examined the scaling of the generating function is determined in the area perimeter volume area and volume perimeter ensembles the results are interpreted within the framework of tricritical scaling and the crossover exponents associated with the transitions are determined we identify three phases in the phase diagram a needle phase a disc phase and a cubical phase these phases are separated by three curves of transitions that meet in a multicritical point

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