scholarly journals New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences

2002 ◽  
Vol 15 (4) ◽  
pp. 309-325 ◽  
Author(s):  
M. L. Kleptsyna ◽  
A. Le Breton ◽  
M. Viot
Keyword(s):  
Quadratic Forms ◽  
General Setting ◽  
Laplace Transforms ◽  
Recursive Equation ◽  
Explicit Formulas ◽  
Volterra Type ◽  
Alternative Approach ◽  
The Laplace Transform ◽  
Recursive Equations ◽  
Filtering Problem

Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Volterra-type recursions describing characteristics of the corresponding optimal filter. In the case of Gauss-Markov sequences, where the previous equations reduce to ordinary forward recursive equations, an alternative approach prices another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transforms, various relationships between the corresponding entries are identified. In particular, relationships between the solutions of matched forward and backward Riccati equations are thus proved probabilistically; they are proved again directly. In various specific cases, a further analysis of the concerned equations lead to completely explicit formulas for the Laplace transform.

scholarly journals Optimal linear filtering of general multidimensional Gaussian processes and its application to Laplace transforms of quadratic functionals

2001 ◽  
Vol 14 (3) ◽  
pp. 215-226 ◽  
Author(s):  
M. L. Kleptsyna ◽  
A. Le Breton
Keyword(s):  
Gaussian Processes ◽  
Laplace Transforms ◽  
Quadratic Functional ◽  
Linear Filtering ◽  
Volterra Type ◽  
Conditional Mean ◽  
Optimal Linear ◽  
Filtering Problem ◽  
Optimal Linear Filtering ◽  
Optimal Filtering Problem

The optimal filtering problem for multidimensional continuous possibly non-Markovian, Gaussian processes, observed through a linear channel driven by a Brownian motion, is revisited. Explicit Volterra type filtering equations involving the covariance function of the filtered process are derived both for the conditional mean and for the covariance of the filtering error. The solution of the filtering problem is applied to obtain a Cameron-Martin type formula for Laplace transforms of a quadratic functional of the process. Particular cases for which the results can be further elaborated are investigated.

On the Number of Representations of Integers by Quadratic Forms in Twelve Variables

1998 ◽  
Vol 5 (6) ◽  
pp. 545-564
Author(s):  
G. Lomadze
Keyword(s):  
Quadratic Forms ◽  
Explicit Formulas ◽  
Representations Of Integers ◽  
Positive Integers

Abstract A way of finding exact explicit formulas for the number of representations of positive integers by quadratic forms in 12 variables with integral coefficients is suggested.

A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
A. Caffagni ◽  
D. Angeli ◽  
G. S. Barozzi ◽  
S. Polidoro
Keyword(s):  
Heat Conduction ◽  
Convergence Rate ◽  
Boundary Conditions ◽  
One Dimensional ◽  
Integral Formulas ◽  
Alternative Approach ◽  
The Laplace Transform ◽  
Piecewise Continuous ◽  
Continuous Boundary ◽  
Duhamel’S Integral

Classical Green’s and Duhamel’s integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (1) ◽  
pp. 180-199 ◽  
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

A result on the Laplace transform associated with the sticky Brownian motion on an interval

2020 ◽  
pp. 2150031
Author(s):  
Shiyu Song
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Perturbation Approach ◽  
Integrated Process ◽  
Modified Bessel Function ◽  
Airy Functions ◽  
Occupation Times ◽  
The Laplace Transform

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab. 45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially equivalent to solving the associated differential equations with boundary conditions. We obtain the explicit expression for the joint Laplace transform in terms of the modified Bessel function and Airy functions.

Analysis of the fluid weighted fair queueing system

2003 ◽  
Vol 40 (01) ◽  
pp. 180-199
Author(s):  
Fabrice Guillemin ◽  
Ravi Mazumdar ◽  
Alain Dupuis ◽  
Jacqueline Boyer
Keyword(s):  
Laplace Transform ◽  
Performance Measures ◽  
Joint Distribution ◽  
Queueing System ◽  
Laplace Transforms ◽  
Fair Queueing ◽  
Mean Waiting Time ◽  
The Laplace Transform ◽  
The Mean

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.

scholarly journals Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 589
Author(s):  
Paolo Emilio Ricci ◽  
Diego Caratelli ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Laplace Transforms ◽  
Series Expansions ◽  
Connection Coefficients ◽  
General Series ◽  
The Laplace Transform

Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coefficients for the series of orthogonal polynomials, we find the possibility to extend the Tricomi method to more general series expansions. Some examples showing the effectiveness of the considered procedure are shown.

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

2007 ◽  
Vol 03 (03) ◽  
pp. 475-501 ◽  
Author(s):  
THOMAS A. SCHMIDT ◽  
MARK SHEINGORN
Keyword(s):  
Quadratic Forms ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Explicit Formulas ◽  
Indefinite Quadratic ◽  
Positive Integers ◽  
Simple Closed Geodesics ◽  
Hall Ray

The Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. In our previous work, we classified the low height-achieving non-simple geodesics of [Formula: see text] into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized.

scholarly journals On the optimal design of biased contests

2020 ◽  
Vol 15 (4) ◽  
pp. 1435-1470 ◽  
Author(s):  
Qiang Fu ◽  
Zenan Wu
Keyword(s):  
Optimal Design ◽  
General Setting ◽  
Programming Approach ◽  
Objective Functions ◽  
Equilibrium Solutions ◽  
Design Problems ◽  
Playing Field ◽  
Alternative Approach ◽  
Broad Array ◽  
Contest Design

This paper explores the optimal design of biased contests. A designer imposes an identity‐dependent treatment on contestants that varies the balance of the playing field. A generalized lottery contest typically yields no closed‐form equilibrium solutions, which nullifies the usual implicit programming approach to optimal contest design and limits analysis to restricted settings. We propose an alternative approach that allows us to circumvent this difficulty and characterize the optimum in a general setting under a wide array of objective functions without solving for the equilibrium explicitly. Our technique applies to a broad array of contest design problems, and the analysis it enables generates novel insights into incentive provisions in contests and their optimal design. For instance, we demonstrate that the conventional wisdom of leveling the playing field, which is obtained in limited settings in previous studies, does not generally hold.

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