Improved Bounds for Taylor Coefficients of Analytic Functions

The practical computation of verified bounds for Taylor coefficients of analytic functions is considered. Using interval arithmetic, the bounds are constructed from Cauchy's estimate and from some of its modifications. By employing the mean value form for intermediate function evaluations, the accuracy of the bounds is improved by several powers of ten, compared to earlier results.     

Volume estimate for a cone with a submanifold as vertex

We give some estimates for the volume of a cone with vertex a submanifold P of a Riemannian or Kaehler manifold M. The estimates are functions of bounds of the mean curvature of P and the sectional curvature of M. They are sharp on cones having a basis which is contained in a tubular hypersurface about P in a space form or in a complex space form.Work partially supported by DGICYT Grant No. PB90-0014-C03-01.     

A Complex Mean Value Form for Curves

A new mean value form for analytic functions defined on curves in the complex plane is discussed.     

Chernoff bounds for mean overflow rates

A number of independent traffic streams arrive at a queueing node which provides a finite buffer and a non-idling service at constant rate. Customers which arrive when the buffer is full are dropped and counted as overflows. We present Chernoff type bounds for mean overflow rates in the form of finite-dimensional minimization problems. The results are based on bounds for moment generating functions of buffer and bandwidth usage of the individual streams in an infinite buffer with constant service rate. We calculate these functions for regulated, Poisson and certain on/off sources. The achievable statistical multiplexing gain and the tightness of the bounds are demonstrated by several numerical examples.     

Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution to the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte-Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yaari utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.     

THE CAUCHY PROBLEMS FOR DISSIPATIVE HYPERBOLIC MEAN CURVATURE FLOW

In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Amp`ere equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.     

Scale-independent complementary bivariational principles in a complex Hilbert space,and their application to a scattering problem

Scale-independent complementary bivariational principles in a complex Hilbert space are derived from the stationary principle. These principles consist of two scale-independent functionals which yield upper bounds and lower bounds, respectively, to both the real and the imaginary part of a particular quantity associated with an inhomogeneous linear equation. They have the advantage that one need only guess the form of solutions of the equation and its auxiliary equation, not their size. Moreover, for a given pair of trial functions, they yield better bounds than the scale-dependent complementary bivariational principles obtained by Barnsley and Baker. Their application to a scattering problem yields scale-independent complementary bivariational principles for the scattering amplitude as well as those for the total scattering cross section.     

D.C. programming approach for multicommodity network optimization problems with step increasing cost functions

We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x ^* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x^*)-LB)/{f(x^*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained.     

Functional inequalities for modified Bessel functions

In this paper, our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kind. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Turán-type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind, we prove that the cumulative distribution function of the gamma–gamma distribution is log-concave. At the end of this paper, several open problems are posed, which may be of interest for further research.     

Green’s Function for Second Order Elliptic Equations in Non-divergence Form

We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C^1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.

     

On the optimum choice of weight functions in a class of variational calculations

Summary We consider the variational solution of a particular class of second order differential equations and show that expansions in terms of Chebychev and a range of ultraspherical polynomials lead to operator matrices that are asymptotically diagonal, and that hence their convergence properties can be completely characterised using a previously developed analysis. For a given class of weight functions bounds are given on the convergence of the coefficients and of the weighted mean square error, in terms of the analyticity properties of the coefficients in the differential equation. These bounds are used to discuss the optimum choice of weight function for such a calculation.     

On the Proportional Odds Model in Survival Analysis

The proportional odds (PO) model with its property of convergent hazard functions is of considerable value in modeling survival data with non-proportional hazards. This paper explores the structure, implications, and properties of the PO model. Results proved include connections with geometric minima and maxima, ageing characteristics, and bounds on mean and variance of survival times.     

下半连续函数的Proximal—次微分与广义中值定理

该文对定义在Hilbert空间E上的一般下半连续函数证明了如［9］中形式的逼近中值定理在Proximal-次微分意义下也成立．若E＝［a，b］R，则得到了不等式形式的中值定理．作为应用给出了函数凸性、Lipschitz性质及常数性质的Proximal-次微分刻划．     

New inequalities from classical Sturm theorems

Inequalities satisfied by the zeros of the solutions of second-order hypergeometric equations are derived through a systematic use of Liouville transformations together with the application of classical Sturm theorems. This systematic study allows us to improve previously known inequalities and to extend their range of validity as well as to discover inequalities which appear to be new. Among other properties obtained, Szegő's bounds on the zeros of Jacobi polynomials for , are completed with results for the rest of parameter values, Grosjean's inequality (J. Approx. Theory 50 (1987) 84) on the zeros of Legendre polynomials is shown to be valid for Jacobi polynomials with |β|1, bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions are derived as well as an inequality involving the geometric mean of zeros of Bessel functions.     

Invariant Measures and Asymptotic Gaussian Bounds for Normal Forms of Stochastic Climate Model

The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.     

Bounded knapsack sharing

A bounded knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more linear inequality constraints, an objective function composed of single variable continuous functions called tradeoff functions, and lower and upper bounds on the variables. A single constraint problem which can have negative or positive constraint coefficients and any type of continuous tradeoff functions (including multi-modal, multiple-valued and staircase functions) is considered first. Limiting conditions where the optimal value of a variable may be plus or minus infinity are explicitly considered. A preprocessor procedure to transform any single constraint problem to a finite form problem (an optimal feasible solution exists with finite variable values) is developed. Optimality conditions and three algorithms are then developed for the finite form problem. For piecewise linear tradeoff functions, the preprocessor and algorithms are polynomially bounded. The preprocessor is then modified to handle bounded knapsack sharing problems with multiple constraints. An optimality condition and algorithm is developed for the multiple constraint finite form problem. For multiple constraints, the time needed for the multiple constraint finite form algorithm is the time needed to solve a single constraint finite form problem multiplied by the number of constraints. Some multiple constraint problems cannot be transformed to multiple constraint finite form problems.     

Semiparametric bounds of mean and variance for exotic options

Finding semiparametric bounds for option prices is a widely studied pricing technique. We obtain closed-form semiparametric bounds of the mean and variance for the pay-off of two exotic (Collar and Gap) call options given mean and variance information on the underlying asset price. Mathematically, we extended domination technique by quadratic functions to bound mean and variances. This work was supported by National Science Foundation of the United States (Grant Nos. DMS-0720977 and DMS-0805929)     

Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns

This paper is concerned with determination of optimal run time for an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. In this study, a portion of the defective items is considered to be scrap, while the other is assumed to be repairable. Total production-inventory cost functions are derived respectively for both EPQ models with breakdown (no-resumption policy is adopted) and without breakdown taking place. These cost functions are integrated and the renewal reward theorem is used to cope with the variable cycle length. Theorems on conditional convexity of the integrated overall costs and bounds of the production run time are proposed and proved. We conclude that the optimal run time falls within the range of bounds and it can be pinpointed by the use of the bisection method based on the intermediate value theorem. Numerical example is provided to demonstrate its practical usages.     

Bounds for the expectation of the kth root of a random variable

Exact bounds for the mean value of a fractional moment, such as the sample standard deviation, are considered. These bounds are compared with those found by summation techniques applied to computer extended series. An expansion involving an arbitrary parameter is introduced, and now best bounds are found by optimization. There are applications to bounds for a certain class of quadrature problem.