From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

   Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

An alternative approach for non-linear optimal control problems based on the method of moments

We propose an alternative method for computing effectively the solution of non-linear, fixed-terminal-time, optimal control problems when they are given in Lagrange, Bolza or Mayer forms. This method works well when the nonlinearities in the control variable can be expressed as polynomials. The essential of this proposal is the transformation of a non-linear, non-convex optimal control problem into an equivalent optimal control problem with linear and convex structure. The method is based on global optimization of polynomials by the method of moments. With this method we can determine either the existence or lacking of minimizers. In addition, we can calculate generalized solutions when the original problem lacks of minimizers. We also present the numerical schemes to solve several examples arising in science and technology.     

Moments of claims in a Markovian environment

This paper considers discounted aggregate claims when the claim rates and sizes fluctuate according to the state of the risk business. We provide a system of differential equations for the Laplace–Stieltjes transform of the distribution of discounted aggregate claims under this assumption. Using the differential equations, we present the first two moments of discounted aggregate claims in a Markovian environment. We also derive simple expressions for the moments of discounted aggregate claims when the Markovian environment has two states. Numerical examples are illustrated when the claim sizes are specified.     

On a resonance energy model based on expansion in terms of acyclic moments: Exact results

Summary The new concept of the resonance energy in conjugated hydrocarbons introduced by Jiang Y, Zhang H (1989) Theor Chim Acta 75:279 is further developed. This model is based on expansion of the -electron energy in terms of moments which are also equal to numbers of closed walks in a molecular graph. The reference system is established by counting only acyclic walks, i.e. those tracing only on acyclic subgraphs. Because acyclic walks could be counted only up to some finite length, the energy of the reference system has been evaluated by truncating higher terms in the expansion. In this paper a finite expression for the energy of the same reference system is derived, thus allowing its exact evaluation. The exact values differ significantly from the truncated ones. This difference, as well as the discrepancy between exact results and chemical experience, are discussed.     

Bootstrapping and Askey–Wilson polynomials

The mixed moments for the Askey–Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey–Wilson generating function. Modified generating functions of orthogonal polynomials are shown to generate polynomials satisfying recurrences of known degree greater than three. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.     

A study of the precision and accuracy of peak quantification in comprehensive two-dimensional liquid chromatography in time

Simulated chromatographic data were used to determine the precision and accuracy in the estimation of peak volumes (i.e., peak sizes) in comprehensive two-dimensional liquid chromatography in time (LC × LC). Peak volumes were determined both by summing the areas in the second dimension chromatograms and by fitting the second dimension areas to a Gaussian peak. The Gaussian method is better at predicting the peak volume than the moments method provided there are at least three second dimension injections above the limit of detection (LOD). However, when only two of the second dimension signals are substantially above baseline, the accuracy and precision of the Gaussian fit method become quite poor because the results from the fitting algorithm become indeterminate. Based on simulations in which the modulation ratio (M_R = 4¹σ/t_s) and sampling phase (?) were varied, we conclude for well-resolved peaks that the optimum precision in peak volumes in 2D separations will be obtained when the M_R is between two and five, such that there are typically four to ten second dimension peaks recorded over the eight σ width of the first dimension peak. This sampling rate is similar to that suggested by the Murphy–Schure–Foley criterion. This provides an RSD of approximately 2% for the signal-to-noise ratio used in the present simulations. The precision of the peak volume of experimental data was also assessed, and RSD values were in the range of 4–5%. We conclude that the poorer precision found in the LC × LC experimental data as compared to LC may be due to experimental imprecision in sampling the effluent from the first dimension column.     

Comparing Fleming-Viot and Dawson-Watanabe processes

Fleming-Viot processes and Dawson-Watanabe processes are two classes of “superprocesses” that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same “underlying particle motion”, and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.     

Moments of L-Functions Attached to the Twist of Modular Form by Dirichlet Characters

Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(Z). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for k =1/n with n ∈ N,and the result also holds for any real number 0 k 1 under the GRH for L(s, f ■χ).The authors also prove that under the GRH for L(s, f ■χ),for any real number k 0 and any large prime q.