the fully nonlinear development of gortler vortices in growing boundary layers the fully nonlinear development of small wavelength gortler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods the starting point for the analysis is the weakly nonlinear theory of hall j inst math applies 29 173 1982 who discussed the initial development of large wavenumber small amplitude vortices in a neighbourhood of the location where they first become linearly unstable that development is unusual in the context of nonlinear stability theory in that it is not described by the stuart watson approach in fact the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means it is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer in fact the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state thus the mean flow adjusts so as to make these large amplitude vortices locally neutral moreover in the region where the vortices exist the mean flow has a square root profile and the vortex velocity field can be written down in closed form the upper and lower boundaries of the region of vortex activity are determined by a free boundary problem involving the boundary layer equations in general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream in this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

extreme value distributions and renormalization group in the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent identically distributed random variables are studied the vast majority of the literature on the subject deals with affine normalization we argue that more general normalizations are natural from a mathematical and physical point of view and work them out the problem is approached using the language of renormalization group transformations in the space of probability densities the limit distributions are fixed points of the transformation and the study of its differential around them allows a local analysis of the domains of attraction and the computation of finite size corrections

von mises type conditions in second order regular variation we give a thorough treatment concerning sufficient conditions involving derivatives for extended regular variation of second order most of the results are new a summary of the analogous known results for first order extended regular variation is given first

on the existence of moments for the busy period in a single server queue we obtain sufficient conditions for the finiteness of the moments of the single server queue busy period in discrete and continuous time in the cases of power and exponential moments the conditions found coincide with the known necessary ones

fully polynomial approximation schemes for single item capacitated economic lot sizing problems np hard cases of the single item capacitated lot sizing problem have been the topic of extensive research and continue to receive considerable attention however surprisingly few theoretical results have been published on approximation methods for these problems to the best of our knowledge until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant in this paper we show that such methods do exist by presenting an even stronger result the existence of fully polynomial approximation schemes the approximation scheme is first developed for a quite general model which has concave backlogging and production cost functions and arbitrary monotone holding cost functions subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models

approximation formulations for the single product capacitated lot size problem we consider two approximation formulations for the single product capacitated lot size problem one formulation restricts the number of production policies and the other rounds demands up to multiples of a constant after briefly reviewing the literature within a new framework we discuss the relations between these approximation formulations next we provide relative error bounds and algorithms for solving the approximation problems we demonstrate that these approximation formulations require a significantly smaller number of calculations than the original formulation and that the relative error bounds are satisfactory for practical purposes

renormalization group theory for finite size scaling in extreme statistics we present a renormalization group rg approach to explain universal features of extreme statistics applied here to independent identically distributed variables the outlines of the theory have been described in a previous paper the main result being that finite size shape corrections to the limit distribution can be obtained from a linearization of the rg transformation near a fixed point leading to the computation of stable perturbations as eigenfunctions here we show details of the rg theory which exhibit remarkable similarities to the rg known in statistical physics besides the fixed points explaining universality and the least stable eigendirections accounting for convergence rates and shape corrections the similarities include marginally stable perturbations which turn out to be generic for the fisher tippett gumbel class distribution functions containing unstable perturbations are also considered we find that after a transitory divergence they return to the universal fixed line at the same or at a different point depending on the type of perturbation

an efficient algorithm for the capacitated single item dynamic lot size problem abstract a dynamic programming based algorithm is developed for the single item lot size problem with concave costs and arbitrary capacities by making use of the extreme point properties of the problem first the set of all feasible cumulative production levels that may occur in an optimal solution is generated in the second stage a dynamic programming procedure is carried out over this set the worst case computational effort is equal to that of the standard dynamic programming approach but extensive computational tests with the algorithm indicate that for t period problems the computational effort does not exceed o t4 the performance of the algorithm is compared with the performance of the existing procedures in the literature for the general the constant capacity and the constant unit cost problems the computational results demonstrate that our algorithm is at least three times faster than the other procedures for all problem types considered

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

a uniform approximation to the right normal tail integral this paper presents two simple formulas for approximation of the standard normal right tail probabilities and indicates the method of computing approximations of higher order one of the approximations relies on two simple numerical constants has absolute error of 0 00071 and relative error 0 023 the other uses four numerical constants has absolute error of 0 000019 the relative error is 0 005 and gives at least two significant digits for probabilities over the entire range 0

second order regular variation and rates of convergence in extreme value theory rates of convergence of the distribution of the extreme order statistic to its limit distribution are given in the uniform metric and the total variation metric a second order regular variation condition is imposed by supposing a von mises type condition which allows a unified treatment rates are constructed from the parameters of the second order regular variation condition some connections with poisson processes are discussed

a single item economic lot sizing problem with a non uniform resource approximation we study a generalization of the classical single item capacitated economic lot sizing problem to the case of a non uniform resource usage for production the general problem and several special cases are shown to be non approximable with any polynomially computable relative error in polynomial time an optimal dynamic programming algorithm and its approximate modification are presented for the general problem fully polynomial time approximation schemes are developed for two np hard special cases 1 cost functions of total production are separable and holding and backlogging cost functions are linear with polynomially related slopes and 2 all holding costs are equal to zero

an ot2 algorithm for the ni g ni nd capacitated lot size problem in this paper we study a class of the capacitated dynamic lot size problem where over time the setup costs are nonincreasing the unit holding costs have arbitrary pattern the unit production costs are nonincreasing and the capacities are nondecreasing we investigate the properties of the optimal solution for the problem and develop the concept of candidate subplan it is proven that only the candidate subplans need to be examined in searching for an optimal solution a dynamic programming algorithm incorporating the concept of candidate subplan is then devised which has run time complexity of ot2

stochastic lot sizing problem with inventory bounds and constant order capacities abstract in this paper we study the stochastic version of lot sizing problems with inventory bounds and order capacities customer demands inventory bounds and costs are subject to uncertainty and dependent with each other throughout the finite planning horizon two models in stochastic programming are developed the first one has inventory bound constraints and the second one has both inventory bound and order capacity constraints we explore structural properties of the two models and develop o n 2 and o n 2 t log n dynamic programming algorithms for them respectively our model also generalizes the deterministic lot sizing problem with inventory bounds for some cases when applied to the deterministic versions our algorithms outperform existing deterministic algorithms

turan type inequalities for kratzel functions abstract complete monotonicity laguerre and turan type inequalities are established for the so called kratzel function z defined by z u 0 t 1 e t u t d t where u 0 and r moreover we prove the complete monotonicity of a determinant function of which entries involve the kratzel function

geometrically concave univariate distributions in this paper our aim is to show that if a probability density function is geometrically concave convex then the corresponding cumulative distribution function and the survival function are geometrically concave convex too under some assumptions the proofs are based on the so called monotone form of l hospital s rule and permit us to extend our results to the case of the concavity convexity with respect to holder means to illustrate the applications of the main results we discuss in details the geometrical concavity of the probability density function cumulative distribution function and survival function of some common continuous univariate distributions moreover at the end of the paper we present a simple alternative proof to schweizer s problem related to the mulholland s generalization of minkowski s inequality

computational complexity of the capacitated lot size problem in this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings for the single item case new properties are introduced classes of problems solvable by polynomial time algorithms are identified and efficient solution procedures are given we show that special classes are np hard and that the problem with two items and independent setups is np hard under conditions similar to those where the single item problem is easy topics for further research are discussed in the last section

strongly nonlinear vortices in the stokes layer on an oscillating cylinder the stokes layer on a torsionally oscillating circular cylinder is susceptible to axially periodic vortices which are driven by a centrifugal mechanism which has some similarities to that which is responsible for gortler vortices in boundary layers on concave surfaces for small wavelength vortices an analytic description of linear weakly nonlinear and fully nonlinear perturbations is possible and as the growth rates of these modes are asymptotically greater than the timescale of the underlying stokes layer a quasi steady stability analysis is attempted in preference to a floquet type approach it is shown that highly nonlinear vortices can be expected to form on the surface of the cylinder it is noted that these vortices alter the basic mean flow at leading order and occupy a zone which has depth comparable to the stokes thickness depending on the taylor number of the flow three distinct types of vortex behaviour are found in the first of these corresponding to taylor numbers just above that at which the stokes layer becomes unstable over a part of its cycle the vortices are relatively weak and remain attached to the cylinder throughout their lifetime at larger taylor numbers the vortices initially grow whilst remaining on the cylinder but at some stage they break away and concentrate in a region within the stokes layer however their strength reduces with time and they soon decay away in the third regime identified before this vortex activity region dies the residual flow next to the cylinder becomes unstable once more in this case a second vortex zone begins to grow so that for a certain time interval two active zones exist the governing equations for the strongly nonlinear modes take the form of a free boundary value problem subject to ordinary differential constraints numerical solutions are presented and although periodic solutions are by no means guaranteed by use of the quasi steady approach the computations show that remarkably periodic vortex configurations tend to result

a simple fptas for a single item capacitated economic lot sizing problem with a monotone cost structure abstract the single item capacitated economic lot sizing cels problem is a fundamental problem of production and inventory management the first fully polynomial approximation scheme fptas for this problem with concave cost functions was developed by van hoesel and wagelmans c p m van hoesel a p m wagelmans fully polynomial approximation schemes for single item capacitated economic lot sizing problems mathematics of operations research 26 2001 339 357 chubanov et al s chubanov m y kovalyov e pesch an fptas for a single item capacitated economic lot sizing problem mathematical programming series a 106 2006 453 466 later presented a sophisticated fptas for the general case of the cels problem with a monotone cost structure in this paper we present a better fptas for this case the ideas and presentation of our fptas are simple and straightforward its running time is about n 4 e 2 times faster than that of chubanov et al 5 where n is the number of production periods and e is the anticipated relative error of the approximate solution

jensen cotes upper and lower bounds on the gaussian q function and related functions we present new families of lower and upper bounds on q functions first we consider the craig form of the gaussian q function q x and shown that its integrand phi thetas x can be partitioned into a pair of complementary convex and concave segments this property is then exploited in conjunction with the jensen inequality and the newton cotes quadrature rule to produce a complete family of upper and lower bounds on q x which can be made arbitrarily tight by finer segmentation the basic idea is then utilized to derive families of upper and lower bounds also for the squared gaussian q function q2 x the 2d joint gaussian q function q x y p and the generalized marcum q function qm x y the bounds are shown to be tighter than alternatives found in the literature and in some cases the lower bounds provided find no equivalent in current literature the generality of the principle is the elegant point of the method and the resulting jensen cotes bounds are easy to implement and evaluate since only elementary transcendental functions are involved as an example of application to the analysis of communication systems we consider the bit error rates ber s of decode and forward df cooperative relaying schemes with coherent and differential phase shift keying psk modulations which have been shown to have an intricate dependence on the gaussian q function complicated by crossproducts irrational functional arguments and multiple numerical integrations in that example the bounds substantially reduce the complexity required to evaluate the expressions retaining tightness despite multiple numerical integrations with infinite limits

large wavenumber vortices in time periodic flows the linear stability properties of short wavelength vortices within time periodic flows are examined using a wkb based asymptotic analysis such vortices typically have an instantaneous growth rate considerably larger than the time scale of the underlying flow and thus have often been studied using quasi steady techniques a major drawback of these quasi steady approaches is that although they are able to determine the instantaneous flow structure they are of no use in describing the evolution of the modes it is shown how rational asymptotic analysis based on wkb ideas overcomes this deficiency and enables the form of the neutral stability curve to be obtained for short wavelengths the technique is applied to three problems first the key features of the analysis are exemplified by careful consideration of a model problem which is motivated by the physical situation of the sinusoidal heating of a horizontal flat plate bounding a semi infinite layer of fluid the governing linearized stability equations take the form of a system which is sixth order in the spatial variable and second order in time however the important features of this system are captured by our model which is only second order in the spatial coordinate and first order in time the second physically important flow examined is the motion induced in a viscous fluid surrounding a long torsionally oscillating cylinder a curved stokes flow detailed asymptotic and numerical studies of both the model and the two physically motivated flows are undertaken and it is shown how the analytical results provide good qualitative agreement with the numerical findings at quite modest wavenumbers the methodology adopted here may be used as a basis for the rational study of the stability properties of other time periodic flows a key result forthcoming from this study is the demonstration that the first few terms in the relevant high wavenumber form of the neutral curve for modes in an oscillatory flow may be derived relatively quickly with minimal computational effort

on the norming constants for normal maxima abstract given n independent standard normal random variables it is well known that their maxima m n can be normalized such that their distribution converges to the gumbel law in a remarkable study hall proved that the kolmogorov distance d n between the normalized m n and its associated limit distribution is less than 3 log n in the present study we propose a different set of norming constants that allow this upper bound to be decreased with d n c m log n for n m 5 furthermore the function c m is computed explicitly which satisfies c m 1 and lim m c m 1 3 as a consequence some new and effective norming constants are provided using the asymptotic expansion of a lambert w type function

production planning with time dependent capacity bounds abstract in this paper we examine the problem of finding minimum cost production schedules that satisfy known demands over a finite planning horizon a dynamic programming algorithm is developed to find these schedules for cases in which production in each period is constrained by a time dependent capacity bound the costs considered are production and inventory holding costs and all cost functions are assumed to be nondecreasing and concave the algorithm is an extension of florian and klein s method developed for problems in which capacity bounds are the same in all periods although the problem with time dependent bounds is np complete the algorithm is shown to be efficient when the capacity bounds are integer multiples of a common divisor and the largest multiplier is small hence it is useful in applications in which production capacity is periodically increased by adding facilities of the same size

meta heuristics for dynamic lot sizing a review and comparison of solution approaches proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve because these problems are so difficult various solution techniques have been proposed to solve them in the past decade meta heuristics such as tabu search genetic algorithms and simulated annealing have become popular and efficient tools for solving hard combinatorial optimization problems we review the various meta heuristics that have been specifically developed to solve lot sizing problems discussing their main components such as representation evaluation neighborhood definition and genetic operators further we briefly review other solution approaches such as dynamic programming cutting planes dantzig wolfe decomposition lagrange relaxation and dedicated heuristics this allows us to compare these techniques understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms finally we discuss general guidelines for computational experiments and illustrate these with several examples

generalized convexity and inequalities abstract let r 0 and let m be the family of all mean values of two numbers in r some examples are the arithmetic geometric and harmonic means given m 1 m 2 m we say that a function f r r is m 1 m 2 convex if f m 1 x y m 2 f x f y for all x y r the usual convexity is the special case when both mean values are arithmetic means we study the dependence of m 1 m 2 convexity on m 1 and m 2 and give sufficient conditions for m 1 m 2 convexity of functions defined by maclaurin series the criteria involve the maclaurin coefficients our results yield a class of new inequalities for several special functions such as the gaussian hypergeometric function and a generalized bessel function

approximating mills ratio abstract consider the mills ratio f x 1 x x x 0 where is the density function of the standard gaussian law and its cumulative distribution we introduce a general procedure to approximate f on the whole 0 which allows to prove interesting properties where f is involved as applications we present a new proof that 1 f is strictly convex and we give new sharp bounds of f involving rational functions functions with square roots or exponential terms also chernoff type bounds for the gaussian q function are studied

convergence rates of limit distribution of maxima of lognormal samples abstract let m n denote the partial maximum of an independent and identically distributed lognormal random sequence in this paper we derive the exact uniform convergence rate of the distribution of the normalized maximum m n b n a n to its extreme value distribution where the constants a n and b n are chosen by an optimal way

chernoff type bounds for the gaussian error function we study single term exponential type bounds also known as chernoff type bounds on the gaussian error function this type of bound is analytically the simplest such that the performance metrics in most fading channel models can be expressed in a concise closed form we derive the conditions for a general single term exponential function to be an upper or lower bound on the gaussian error function we prove that there exists no tighter single term exponential upper bound beyond the chernoff bound employing a factor of one half regarding the lower bound we prove that the single term exponential lower bound of this letter outperforms previous work numerical results show that the tightness of our lower bound is comparable to that of previous work employing eight exponential terms

some new upper and lower bounds for the mills ratio abstract in this paper we present new upper and lower bounds for the mills ratio of the standard gaussian law several different methods are used to derive these new bounds one of the methods reproduces the bounds of several different authors in previous works as special cases and is a very general method that produces many new bounds one of the bounds can be written in terms of hyperbolic sine and inverse hyperbolic sine functions some of the bounds involve exponential functions and are improved versions of previously proposed bounds or are improved versions of the new bounds introduced earlier in this paper some results from reliability theory and jensen s inequality are used to improve determinantal inequalities some open problems are discussed and conjectures are made

extreme statistics for time series distribution of the maximum relative to the initial value the extreme statistics of time signals is studied when the maximum is measured from the initial value in the case of independent identically distributed iid variables we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn then we turn to correlated periodic gaussian signals with a 1 f ensuremath alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height mathrm mrh mathrm i the exact mathrm mrh mathrm i distribution is derived for ensuremath alpha 0 iid variables ensuremath alpha 2 random walk ensuremath alpha 4 random acceleration and ensuremath alpha ensuremath infty single sinusoidal mode for other intermediate values of ensuremath alpha the distribution is determined from simulations we find that the mathrm mrh mathrm i distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all ensuremath alpha the two main distinguishing features of the mathrm mrh mathrm i distribution are the much larger weight for small relative heights and the divergence at zero height for ensuremath alpha g3 we also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions finally we show that for signals arising from time translationally invariant distributions the density of near extreme states is the same as the mathrm mrh mathrm i distribution this is used in developing a scaling theory for the threshold singularities of the two distributions

heavy tail modeling and teletraffic data special invited paper huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence various estimation methods for heavy tailed time series with positive innovations are reviewed these include parameter estimation and model identification methods for autoregressions and moving averages parameter estimation methods include those of yule walker and the linear programming estimators of feigin and resnick as well estimators for tail heaviness such as the hill estimator and the qq estimator examples are given using call holding data and interarrivals between packet transmissions on a computer network the limit theory makes heavy use of point process techniques and random set theory

a fully polynomial approximation scheme for single product scheduling in a finite capacity facility this paper considers a version of the economic lot sizing problem for a single product produced in a facility of finite capacity over a finite time horizon with specifiable start and end conditions a set of algorithms is presented that will approximate the optimal production schedule to a given allowable error e algorithms with computation time bounds of o1 e2 are presented which allow for setups of finite length setups with or without direct cash flow quite general cost and demand functions and a wide variety of production policy constraints the procedures make no a priori assumptions about the form of the optimal solution numerical results are included

asymptotic expansions of the moments of extremes from general error distribution abstract in this paper we derive the asymptotic expansions of the moments of normalized partial maxima for general error distribution a byproduct is to deduce the convergence rates of the moments of normalized maxima to the moments of the corresponding extreme value distribution

a new heuristic approach for the multi item dynamic lot sizing problem abstract in this paper a framework for a new heuristic approach for solving the single level multi item capacitated dynamic lot sizing problem is presented the approach uses an iterative item by item strategy for generating solutions to the problem in each iteration a set of items are scheduled over the planning horizon and the procedure terminates when all items are scheduled an algorithm that implements this approach is developed in which in each iteration a single item is selected and scheduled over the planning horizon each item is scheduled by the solution of a bounded single item lot sizing problem where bounds on inventory and production levels are used to ensure feasibility of the overall problem the performance of this algorithm is compared to some well known heuristics over a set of test problems the computational results demonstrated that on the average our algorithm outperforms other algorithms the suggested algorithm especially appears to outperform other algorithm for problems with many periods and few items in the literature these problems are considered as hard

maximal height statistics for 1 f alpha signals numerical and analytical results are presented for the maximal relative height distribution of stationary periodic gaussian signals one dimensional interfaces displaying a 1 f alpha power spectrum for 0 1 regime of strong correlations and a highly accurate picture gallery of distribution functions can be constructed numerically analytical results can be obtained in the limit alpha infinity and for large alpha by perturbation expansion furthermore using path integral techniques we derive a trace formula for the distribution function valid for alpha 2n even integer from the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha 1 is found to be in agreement with simulations comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions

solving multi item capacitated lot sizing problems using variable redefinition mixed integer programming models are typically not used to solve realistic sized production scheduling problems because they require exorbitant solution times we impose a useful taxonomy on production scheduling problems and develop alternative formulations for a wide variety of problems within the taxonomy the linear programming relaxation of the new models is very effective in generating bounds we show that these bounds are equal to those that could be generated using lagrangian relaxation or column generation the linear programming bounds increase in effectiveness as the problems become larger perhaps of greatest significance is that practitioners can obtain our results using only standard off the shelf codes such as lindo or mpsx 370 we report computational experience in several computing environments hardware and software on problems with up to 200 products and 10 time periods 2000 0 1 variables

mills ratio monotonicity patterns and functional inequalities in this paper we study the monotonicity properties of some functions involving the mills ratio of the standard normal law from these we deduce some new functional inequalities involving the mills ratio and we show that the mills ratio is strictly completely monotonic at the end of this paper we present some turan type inequalities for mills ratio