the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

further inequalities for the gamma function for x 0 and k 0 we present a method which permits us to obtain inequalities of the type k a x 1 r k x r k 1 k x 1 with the usual notation for the gamma function where a and are independent of k some examples are also given which improve well known inequalities finally we are also able to show in some cases that the values a and in the inequalities that we obtain cannot be improved

on gamma function inequalities on montre que certaines fonctions comportant les quotients de fonctions gamma sont completement monotones on en deduit des inegalites on etablit aussi la divisibilite infinie de plusieurs distributions de probabilite dont les transformees de laplace comportent des quotients de fonctions gamma

improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

comparametric equations with practical applications in quantigraphic image processing it is argued that hidden within the flow of signals from typical cameras through image processing to display media is a homomorphic filter while homomorphic filtering is often desirable there are some occasions where it is not thus cancellation of this implicit homomorphic filter is proposed through the introduction of an antihomomorphic filter this concept gives rise to the principle of quantigraphic image processing wherein it is argued that most cameras can be modeled as an array of idealized light meters each linearly responsive to a semi monotonic function of the quantity of light received integrated over a fixed spectral response profile this quantity depends only on the spectral response of the sensor elements in the camera a particular class of functional equations called comparametric equations is introduced as a basis for quantigraphic image processing these are fundamental to the analysis and processing of multiple images differing only in exposure the gamma correction of an image is presented as a simple example of a comparametric equation for which it is shown that the underlying quantigraphic function does not pass through the origin thus it is argued that exposure adjustment by gamma correction is inherently flawed and alternatives are provided these alternatives when applied to a plurality of images that differ only in exposure give rise to a new kind of processing in the amplitude domain the theoretical framework presented in this paper is applicable to the processing of images from nearly all types of modern cameras this paper is a much revised draft of a 1992 peer reviewed but unpublished report by the author entitled lightspace and the wyckoff principle

completely monotonic degrees for a difference between the logarithmic and psi functions abstract in the paper the authors firstly present a concise proof for complete monotonicity of a function involving a difference between the logarithmic and psi functions secondly compute completely monotonic degree of the above mentioned function and finally pose several conjectures on completely monotonic degrees of remainders and their derivatives for the asymptotic formula of the logarithm of the classical euler gamma function

complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions we present some complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions this extends some known results due to s l qiu and m vuorinen

some properties of the difference between the ramanujan constant and beta function abstract the authors present the power series expansions of the function r a b a at a 0 and at a 1 2 show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so called ramanujan constant r a and the beta function b a b a 1 a and obtain asymptotically sharp lower and upper bounds for r a in terms of b a and polynomials in addition some properties of the riemann zeta function n n n and its related sums are derived

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

completely monotonic and related functions abstract we study completely monotonic and related functions whose firstnderivatives are definite on an interval let l ndenote the class of functions defined by f in cal l n leftrightarrow 1 k f k t ge0 qquad forall t 0 forall k 0 le k le n forn we writef l such functions are called completely monotonic on 0 we prove in particular that the implication f in cal l n rightarrow bigl forall alpha 1 colon f alpha in cal l n bigr is true for 0 n 5 but false forn 6 in contrast l lorch and d j newman j london math soc 2 28 1983 31 45 theorem 8 claimed this implication to be false forn 5 further we prove that the implication f in cal l rightarrow bigl forall alpha 1 colon f alpha in cal l n bigr is true for 0 n 6 but false forn 20

achievability of asymptotic minimax regret by horizon dependent and horizon independent strategies the normalized maximum likelihood distribution achieves minimax coding log loss regret given a fixed sample size or horizon n it generally requires that n be known in advance furthermore extracting the sequential predictions from the normalized maximum likelihood distribution is computationally infeasible for most statistical models several computationally feasible alternative strategies have been devised we characterize the achievability of asymptotic minimaxity by horizon dependent and horizon independent strategies we prove that no horizon independent strategy can be asymptotically minimax in the multinomial case a weaker result is given in the general case subject to a condition on the horizon dependence of the normalized maximum likelihood motivated by these negative results we demonstrate that an easily implementable bayes mixture based on a conjugate dirichlet prior with a simple dependency on n achieves asymptotic minimaxity for all sequences simplifying earlier similar proposals our numerical experiments for the bernoulli model demonstrate improved finite sample performance by a number of novel horizon dependent and horizon independent algorithms

solving maximum entropy sampling problems using factored masks we present a practical approach to anstreicher and lee s masked spectral bound for maximum entropy sampling and we describe favorable results that we have obtained with a branch and bound algorithm based on our approach by representing masks in factored form we are able to easily satisfy a semidefiniteness constraint moreover this representation allows us to restrict the rank of the mask as a means for attempting to practically incorporate second order information

on some properties of digamma and polygamma functions in this note we present some new and structural inequalities for digamma polygamma and inverse polygamma functions we also extend generalize and refine some known inequalities for these important functions

supplements to a class of logarithmically completely monotonic functions associated with the gamma function abstract in this article the authors present a necessary condition a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic as a consequence of these results supplements to the recent investigation by chen and qi c p chen f qi logarithmically completely monotonic functions relating to the gamma function j math anal appl 321 2006 405 411 are provided and a new keckic vasic type inequality is concluded

a completely monotone function related to the gamma function abstract we show that the reciprocal of the function f z log z 1 z log z z c 0 is a stieltjes transform as a corollary we obtain that the derivative of f is completely monotone in the sense that 1 n 1f n x 0 for all n 1 and all x 0 this answers a question raised by dimitar dimitrov at the fifth international symposium on orthogonal polynomials special functions and applications held in patras in september 1999 to prove the result we examine the imaginary part of 1 f in the upper half plane in particular close to the negative real axis where stirling s formula is not valid

monotonicity rules for the ratio of two laplace transforms with applications abstract let f and g be both continuous functions on 0 with g t 0 for t 0 and let f x l f g x l g be respectively the laplace transforms of f and g converging for x 0 we prove that if there is a t 0 such that f g is strictly increasing on 0 t and strictly decreasing on t then the ratio f g is decreasing on 0 if and only if h f g 0 lim x 0 f x g x g x f x 0 with lim x 0 f x g x lim t f t g t and lim x f x g x lim t 0 f t g t provided the indicated limits exist while h f g 0 0 there is at least one x 0 such that f g is increasing on 0 x and decreasing on x as applications a unified treatment for certain bounds of psi function is presented and some properties of the modified bessel functions of the second are established these show that the monotonicity rules in this paper may contribute to study for certain special functions because many special functions can be expressed as corresponding laplace transforms

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

chaos games for wavelet analysis and wavelet synthesis abstract in this paper we consider chaos gamesto generate wavelet decompositions and wavelet synthesis of functions the primary technique involved is mixing several chaos games for the individual wavelets in order to get the overall wavelet expansion generalizations to dilation factors other than two and to multidimensional wavelets are given

some properties of the divided difference of psi and polygamma functions abstract let n 1 n 1 n for n 0 where n stands for the psi and polygamma functions for p q r and min p q let d x p x q n 1 n x be the divided difference of the functions n 1 for x in this paper we establish the necessary and sufficient conditions for the function n x n 1 x 2 n x n 2 x to be completely monotonic on in particular we find that the function n 1 2 n n 2 is strictly decreasing from 0 onto n n 1 n 1 n 2 these not only generalize and strengthen some known results but also yield many new and interesting ones

refinable functions functionals and iterated function systems invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage that although usually considered in different contexts are deeply linked we outline these links in particular with respect to existence and regularity of these objects

logarithmic convexity and inequalities for the gamma function abstract we propose a method based on logarithmic convexity for producing sharp bounds for the ratio x x as an application we present an inequality that sharpens and generalizes inequalities due to gautschi chu boyd lazarevic lupas and kershaw

remarks on some completely monotonic functions applying the euler maclaurin summation formula we obtain upper and lower polynomial bounds for the function xex 1 x 0 with coefficients the bernoulli numbers bk this enables us to give simpler proofs of some results of h alzer and f qi et al concerning complete monotonicity of certain functions involving the gamma function x the psi function x and the polygamma functions n x n 1 2

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

wendel s and gautschi s inequalities refinements extensions and a class of logarithmically completely monotonic functions abstract in the article sufficient and necessary conditions that a class of functions involving ratio of euler s gamma functions and originating from wendel s and gautschi s inequalities are logarithmically completely monotonic are presented from this wendel s gautschi s kershaw s laforgia s bustoz ismail s merkle s and elezovic giordano pecaric s inequalities are refined extended and sharpened and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly

some gamma function inequalities a class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function as a consequence new upper and lower bounds for the ratio f x 1 f x s are obtained and compared with related bounds given in part by j d keckic and p m vasic our results are further applied to obtain functions which are laplace transforms of infinitely divisible probability measures

unified treatment of gautschi kershaw type inequalities for the gamma function abstract gautschi kershaw lorch laforgia and other authors gave several inequalities for the ratio x 1 x s where as usual denotes the gamma function in this paper we give a unified treatment of all their results and prove among other things new inequalities for the above ratio which involve the psi function inequalities for the ratio of two gamma functions are useful for example to deduce bernstein type inequalities for ultraspherical polynomials we give an example of this type

a new upper bound in the second kershaw s double inequality and its generalizations in the paper a new upper bound in the second kershaw s double inequality involving ratio of gamma functions is established and as generalizations of the second kershaw s double inequality the divided differences of the psi and polygamma functions are bounded

continued fraction estimates for the psi function we present continued fraction estimates for the psi function as a consequence we educe inequality for the euler mascheroni constant which improves a result of detemple

monotonicity of some functions involving the gamma and psi functions let l x x x t x s x 8 t 1 where x is euler s gamma function we determine conditions for the numbers s t so that the function x x s x t x t s 1 l x is strongly completely monotonic on 0 oo through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation we also give several other examples of strongly completely monotonic functions defined in terms of and functions some limiting and particular cases are also considered

sharp bounds for psi function in this paper we present the greatest value a 1 15 and the least value b 4 15 such that the double inequality l x a 0 where l x a 1 90 a 2 2 log x 2 x 3 a 1 3 45 a 2 90 a 2 2 log x 2 x 15 a 1 45 a and x is the classical psi function

asymptotic expansions for the gamma function in terms of hyperbolic functions abstract the smith s approximation formula for the gamma function is given by x 1 2 2 x e x 2 x tanh 1 2 x x 2 1 o 1 x 5 as x in this paper by means of a little known power series we develop this formula to an asymptotic expansions x 1 2 2 x e x 2 x tanh 1 2 x x 2 exp n 3 2 n 2 n 1 2 2 n 1 1 2 1 2 n 2 n 2 n 1 2 n b 2 n x 2 n 1 as x and give an estimate of remainder in the above asymptotic series where b 2 n is the bernoulli number moreover as relevant results ramanujan type asymptotic expansions for the gamma function x 1 2 are obtained an asymptotic expansion for wallis ratio and an estimate of the remainder are established

a comparison theorem for two divided differences and applications to special functions abstract in this paper we present a general comparison theorem for two divided differences of a three times differentiable function this gives a unified treatment for logarithmically complete monotonicity monotonicity and inequalities involving some special functions including gamma psi and polygamma functions as their consequences we not only refine and generalize some important results but also present simple and interesting alternative proofs of certain earlier results

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

an exact algorithm for maximum entropy sampling we study the experimental design problem of selecting a most informative subset having prespecified size from a set of correlated random variables the problem arises in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy in the gaussian case the problem is recast as that of maximizing the determinant of the covariance matrix of the chosen subset we demonstrate that this problem is np hard we establish an upper bound for the entropy based on the eigenvalue interlacing property and we incorporate this bound in a branch and bound algorithm for the exact solution of the problem we present computational results for estimated covariance matrices that correspond to sets of environmental monitoring stations in the united states

a linear integer programming bound for maximum entropy sampling we introduce a new upper bound for the maximum entropy sampling problem our bound is described as the solution of a linear integer program the bound depends on a partition of the underlying set of random variables for the case in which each block of the partition has bounded cardinality we describe an efficient dynamic programming algorithm to calculate the bound for the very special case in which the blocks have no more than two elements we describe an efficient algorithm for calculating the bound based on maximum weight matching this latter formulation has particular value for local search procedures that seek to find a good partition we relate our bound to recent bounds of hoffman lee and williams finally we report on the results of some computational experiments

some properties of functions related to the gamma psi and tetragamma functions in this paper some properties associated with the complete monotonicity and the logarithmically complete monotonicity of functions related to the gamma psi and tetragamma functions are obtained relevant connections of the results presented here with those derived in earlier works are also pointed out

on the method of logarithmic cumulants for parametric probability density function estimation parameter estimation of probability density functions is one of the major steps in the area of statistical image and signal processing in this paper we explore several properties and limitations of the recently proposed method of logarithmic cumulants molc parameter estimation approach which is an alternative to the classical maximum likelihood ml and method of moments mom approaches we derive the general sufficient condition for a strong consistency of the molc estimates which represents an important asymptotic property of any statistical estimator this result enables the demonstration of the strong consistency of molc estimates for a selection of widely used distribution families originating from but not restricted to synthetic aperture radar image processing we then derive the analytical conditions of applicability of molc to samples for the distribution families in our selection finally we conduct various synthetic and real data experiments to assess the comparative properties applicability and small sample performance of molc notably for the generalized gamma and k families of distributions supervised image classification experiments are considered for medical ultrasound and remote sensing sar imagery the obtained results suggest that molc is a feasible and computationally fast yet not universally applicable alternative to mom molc becomes especially useful when the direct ml approach turns out to be unfeasible

logarithmically completely monotonic functions relating to the gamma function abstract in this paper the logarithmically complete monotonicity of the function e x x x x in 0 for r and 0 is considered and the corresponding result by g d anderson r w barnard k c richards m k vamanamurthy and m vuorinen is generalized as applications of these results some inequalities between identric mean and ratio of two gamma functions by j d keckic and p m vasic are extended

random affine iterated function systems curve generation and wavelets it is shown how refinement methods for smooth curve generation can be carried out efficiently through iterated function systems ifs affine transformations are constructed so that when composed randomly they generate the desired smooth curve underlying this random algorithm is the tree traversal property of ifs under a refinement method the points on the curve correspond to leaves on some n tree ifs theory enables one to generate all of these leaves through a single orbit of an appropriate markov chain applications include bezier curves splines wavelets and various interpolants

series associated to some expressions involving the volume of the unit ball and applications the aim of this paper is to construct asymptotic series associated to some expressions involving the volume of the n dimensional unit ball new refinements and sharpenings of some old and recent inequalities on the volume of the n dimensional unit ball are presented

a complete monotonicity property of the gamma function abstract a logarithmically completely monotonic function is completely monotonic the function 1 ln x 1 x ln x 1 is strictly completely monotonic on 0 the function x 1 x x is strictly logarithmically completely monotonic on 0

the data correcting algorithm for the minimization of supermodular functions the data correcting dc algorithm is a recursive branch and bound type algorithm in which the data of a given problem instance are heuristically corrected at each branching in such a way that the new instance will be as close as possible to polynomially solvable and the result satisfies a prescribed accuracy the difference between optimal and current solution in this paper the dc algorithm is applied to determining exact or approximate global minima of supermodular functions the working of the algorithm is illustrated by an instance of the simple plant location spl problem computational results obtained for the quadratic cost partition problem qcp show that the dc algorithm outperforms a branch and cut algorithm not only for sparse graphs but also for nonsparse graphs with density more than 40 often with speeds 100 times faster

nonsmooth optimization via quasi newton methods we investigate the behavior of quasi newton algorithms applied to minimize a nonsmooth function f not necessarily convex we introduce an inex act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the armijo and wolfe conditions if f is absolutely continuous along the line furthermore the line search is guaranteed to terminate if f is semi algebraic it seems quite difficult to establish a convergence theorem for quasi newton methods applied to such general classes of functions so we give a care ful analysis of a special but illuminating case the euclidean norm in one variable using the inexact line search and in two variables assuming that the line search is exact in practice we find that when f is locally lipschitz and semi algebraic with bounded sublevel sets the bfgs broyden fletcher goldfarb shanno method with the inexact line search almost always generates sequences whose cluster points are clarke stationary and with function values converging r linearly to a clarke station ary value we give references documenting the successful use of bfgs in a variety of nonsmooth applications particularly the design of low order controllers for linear dynamical systems we conclude with a challenging open question

some properties of the gamma and psi functions with applications in this paper some monotoneity and concavity properties of the gamma beta and psi functions are obtained from which several asymptotically sharp inequalities follow applying these properties the authors improve some well known results for the volume n of the unit ball b n r n the surface area n 1 of the unit sphere s n 1 and some related constants

some completely monotonic functions of positive order we completely determine the set of 2 for which the function e x e x e x 1 is convex on 0 and use this result to give some special classes of completely monotonic functions of positive order related to gamma and psi functions

monotonicity properties of functions related to the psi function abstract in this paper the monotonicity properties of functions related to the psi function are obtained a double inequality for the euler mascheroni constant is established relevant connections of the results presented here with those derived in earlier works are also pointed out

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

inequalities for the volume of the unit ball in rn abstract let n n 2 1 n 2 be the volume of the unit ball in rn we determine the best possible constants a b a b and such that the inequalities a n n 1 n 1 n b n n 1 n 1 n a 2 n 1 n n b 2 and 1 1 n 2 n n 1 n 1 1 1 n are valid for all integers n 1 our results refine and complement inequalities proved by g d anderson et al k h borgwardt and d a klain and g c rota

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

a continuous analogue of sperner s theorem one of the best known results of extremal combinatorics is sperner s theorem which asserts that the maximum size of an antichain of subsets of an element set equals the binomial coefficient that is the maximum of the binomial coefficients in the last twenty years sperner s theorem has been generalized to wide classes of partially ordered sets it is the purpose of the present paper to propose yet another generalization that strikes in a different direction we consider the lattice of linear subspaces through the origin of the vector space because this lattice is infinite the usual methods of extremal set theory do not apply to it it turns out however that the set of elements of rank of the lattice that is the set of all subspaces of dimension of or grassmannian possesses an invariant measure that is unique up to a multiplicative constant can this multiplicative constant be chosen in such a way that an analogue of sperner s theorem holds for with measures on grassmannians replacing binomial coefficients we show that there is a way of choosing such constants for each level of the lattice that is natural and unique in the sense defined below and for which an analogue of sperner s theorem can be proven the methods of the present note indicate that other results of extremal set theory may be generalized to the lattice by similar reasoning c 1997 john wiley sons inc

maximization of submodular functions theory and enumeration algorithms abstract submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs after reviewing some long standing theoretical results about the structure of local and global maxima of submodular functions cherenin s selection rules and his dichotomy algorithm we revise the above mentioned theory and show that our revision is useful for creating new non binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

using continuous nonlinear relaxations to solve constrained maximum entropy sampling problems we consider a new nonlinear relaxation for the constrained maximum entropy sampling problem the problem of choosing the s x s principal submatrix with maximal determinant from a given n x n positive definite matrix subject to linear constraints we implement a branch and bound algorithm for the problem using the new relaxation the performance on test problems is far superior to a previous implementation using an eigenvalue based relaxation a parallel implementation of the algorithm exhibits approximately linear speed up for up to 8 processors and has successfully solved problem instances that were heretofore intractable

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

on the asymptotic representation of the euler gamma function by ramanujan abstract the problem of approximation to the euler gamma function on the basis of some ramanujan s formulas is considered the function h x g x 6 8x3 4x2 x where g x e x x 1 x is studied it is proved that on the interval 1 the function h x is increasing monotonically from h 1 0 0111976 to h 1 30 0 0333

monotonicity results for dirichlet l functions abstract we present some monotonicity results for dirichlet l functions associated to real primitive characters we show in particular that these dirichlet l functions are far from being logarithmically completely monotonic also we show that unlike in the case of the riemann zeta function the problem of comparing the signs of d k d s k log l s at any two points s 1 s 2 1 is subtler

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

applications of parametric programming and eigenvalue maximization to the quadratic assignment problem we investigate new bounding strategies based on different relaxations of the quadratic assignment problem in particular we improve the lower bound found by using an eigenvalue decomposition of the quadratic part and by solving a linear program for the linear part the improvement is accomplished by applying a steepest ascent algorithm to the sum of the two bounds

a subadditive property of the gamma function abstract let minx 0 2x x and log 2 log 0 946850 we prove that the function x x is subadditive on 0 if and only if 0