scholarly journals HALF-SPACE MACDONALD PROCESSES

2020 ◽  
Vol 8 ◽  
Author(s):  
GUILLAUME BARRAQUAND ◽  
ALEXEI BORODIN ◽  
IVAN CORWIN
Keyword(s):  
Half Space ◽  
Symmetric Functions ◽  
Universality Class ◽  
Integer Partitions ◽  
Probabilistic Systems ◽  
Integral Formulas ◽  
Macdonald Processes ◽  
The Laplace Transform ◽  
Macdonald Symmetric Functions ◽  
Average Size

Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.

scholarly journals Thermoelastic Diffusion Multicomponent Half-Space under the Effect of Surface and Bulk Unsteady Perturbations

2019 ◽  
Vol 24 (1) ◽  
pp. 26 ◽  
Author(s):  
Sergey Davydov ◽  
Andrei Zemskov ◽  
Elena Akhmetova
Keyword(s):  
Half Space ◽  
Green's Functions ◽  
Green’S Functions ◽  
Local Equilibrium ◽  
Linear Equations ◽  
Integral Form ◽  
External Effects ◽  
Thermoelastic Diffusion ◽  
One Dimensional ◽  
The Laplace Transform

This article presents an algorithm for solving the unsteady problem of one-dimensional coupled thermoelastic diffusion perturbations propagation in a multicomponent isotropic half-space, as a result of surface and bulk external effects. One-dimensional physico-mechanical processes, in a continuum, have been described by a local-equilibrium model, which included the coupled linear equations of an elastic medium motion, heat transfer, and mass transfer. The unknown functions of displacement, temperature, and concentration increments were sought in the integral form, which was a convolution of the surface and bulk Green’s functions and external effects functions. The Laplace transform on time and the Fourier sine and cosine transforms on the coordinate were used to find the Green’s functions. The obtained Green’s functions was analyzed. Test calculations were performed on the examples of some technological processes.

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.

Viscous-Elastic Half-Space Vibration Stimulated by High-Speed Moving Loads of Different Speeds

2012 ◽  
Vol 518-523 ◽  
pp. 3874-3877
Author(s):  
Tao Qian ◽  
Xiao Ping Shui ◽  
Yong Fa Zhang ◽  
Yong Gang Guo ◽  
Meng Ma
Keyword(s):  
Half Space ◽  
Coordinate Transformation ◽  
High Speed ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Moving Loads ◽  
Cylindrical Coordinates ◽  
Relative Coordinate ◽  
The Laplace Transform ◽  
Practical Security

A rule of response of an infinite viscous-elastic half-space stimulated by the moving loads of different speeds is outlined in this paper. In order to obtain a three-dimensional analytical solution of the Viscous-elastic half-space with the moving loads of different speeds, the Laplace transform and relative coordinate transformation in cylindrical coordinates are used. Then, the IFFT and relative coordinate transformation are used to solve two-dimensional infinite integration which can greatly improve the operational efficiency. The rules of responses of different velocities from the results by using the principle of dynamics and energy dissipation are also analyzed and induced in this paper, and obtain the incentives of displacement distortion by the super-Rayleigh wave velocity at surface. The results could be referred in improving the practical security in the project.

scholarly journals On the Matchings-Jack Conjecture for Jack Connection Coefficients Indexed by Two Single Part Partitions

10.37236/5085 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Andrei L. Kanunnikov ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Arbitrary Number ◽  
Symmetric Functions ◽  
Recurrence Formula ◽  
Integer Partitions ◽  
Integer Coefficients ◽  
Power Sum ◽  
Connection Coefficients ◽  
Algebraic Framework ◽  
Commutative Subalgebras ◽  
Single Part

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $\alpha$ are defined as the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions. Goulden and Jackson conjectured that they are polynomials in $\beta = \alpha-1$ with non negative integer coefficients of combinatorial significance, the Matchings-Jack conjecture.In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$. We use an algebraic framework of Lasalle (2008) for Jack symmetric functions and a bijective construction in order to show that the coefficients satisfy a simple recurrence formula and prove the Matchings-Jack conjecture in this case. Furthermore we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.

scholarly journals NUMERICAL MODELLING OF DYNAMIC RESPONSE OF A PARTIALLY SATURATED POROELASTIC HALF-SPACE IN CASE OF A LOAD ACTING INSIDE A CUBIC CAVITY

2020 ◽  
Vol 82 (4) ◽  
pp. 507-523
Author(s):  
A.N. Petrov ◽  
M.V. Grigoryev
Keyword(s):  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Boundary Integral ◽  
Poroelastic Medium ◽  
Partially Saturated ◽  
Pore Pressures ◽  
The Laplace Transform ◽  
Poroelastic Material

Computer modeling based on the boundary element method is performed for the problem of loading in terms of the Heaviside step function inside a cubic cavity located in a partially saturated poroelastic half-space. A poroelastic medium is represented by a heterogeneous material-based model consisting of an elastic matrix phase and two phases of fillers – liquid and gas filling the pore system. The material model corresponds to a three-component medium. The constitutive relations of poroelastic medium written in terms skeleton displacements and pore pressures of fillers are considered. The original initial-boundary value problem is reduced to a boundary value problem by using the formal application of the Laplace transform. The research technique is based on the direct approach boundary integral equations of 3D isotropic linear theory of poroelasticity. Boundary integral equations corresponding to the boundary value problem are solved by the boundary element method in combination with the collocation method. In this study 8-noded elements have been adopted to discretize the boundary of poroelastic half-space. It is assumed that the element is linear with respect to displacements and pore pressures, while only one central node is used to represent tractions and fluxes. Algorithms for eliminating singularities, decreasing the order and subdividing elements are employed to compute the integral coefficients of a discrete analogue of the boundary integral equation. Regular integrals are calculated using the Gauss quadrature formula. The solution in time is obtained by numerical inversion of the Laplace transform. The numerical inversion method relies on quadrature formulas for computing the convolution integral. The time dependences of unknown displacement functions and pore pressures at points on the surface of the half-space and the cavity are plotted. The corresponding graphs are given. The influence of the cavity depth and degree of saturation on dynamic responses is investigated. The solution obtained by using the model of a fully saturated poroelastic material is compared to that of partially saturated poroelastic material. It is noted that the model used for solving this problem leads to an underestimation of displacement and overestimation of pore pressure estimates.

scholarly journals Exact Solution of Thermoelastic Problem for a One-Dimensional Bar without Energy Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
A. M. Abd El-Latief ◽  
S. E. Khader
Keyword(s):  
Exact Solution ◽  
Relaxation Time ◽  
Thermal Stress ◽  
Energy Dissipation ◽  
Heat Source ◽  
Half Space ◽  
Time Dependent ◽  
One Dimensional ◽  
The Laplace Transform ◽  
Without Energy Dissipation

We consider a homogeneous isotropic thermoelastic half-space in the context of the theory of thermoelasticity without energy dissipation. There are no body forces or heat source acting on the half-space. The surface of the half-space is affected by a time dependent thermal shock and is traction free. The Laplace transform with respect to time is used. The inverse transforms are obtained in an exact manner for the temperature, thermal stress, and displacement distributions. These solutions are represented graphically and discussed for several cases of the applied heating. Comparison is made between the predictions here and those of the theory of thermoelasticity with one relaxation time.

scholarly journals Interactions due to Moving Heat Sources in Generalized Thermoelastic Half-Space using L-S Model

2013 ◽  
Vol 18 (3) ◽  
pp. 815-831 ◽  
Author(s):  
N. Sarkar ◽  
A. Lahiri
Keyword(s):  
Half Space ◽  
Heat Sources ◽  
Dimensional Problem ◽  
Eigenvalue Approach ◽  
One Dimensional ◽  
Coupled Equations ◽  
The Laplace Transform ◽  
Moving Plane ◽  
Generalized Thermoelastic ◽  
S Model

Abstract A one-dimensional problem for a homogeneous, isotropic and thermoelastic half-space subjected to a moving plane of heat source on the boundary of the space, which is traction free, is considered in the context of Lord- Shulaman model (L-S model) of thermoelasticity. The Laplace transform and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled equations. Numerical results for the temperature, thermal stress, and displacement distributions are represented graphically and discussed

Exact Transient Full-Field Analysis of an Antiplane Subsurface Crack Subjected to Dynamic Impact Loading

1994 ◽  
Vol 61 (3) ◽  
pp. 649-655 ◽  
Author(s):  
Chien-Ching Ma ◽  
Szu-Kuzi Chen
Keyword(s):  
Fundamental Solution ◽  
Half Space ◽  
Observation Point ◽  
Field Analysis ◽  
Full Field ◽  
Transient Solutions ◽  
Diffracted Waves ◽  
The Laplace Transform ◽  
Long Time ◽  
Close Form

The transient problem of a half-space containing a subsurface inclined semi-infinite crack subjected to dynamic antiplane loading on the boundary of the half-space has been investigated to gain insight into the phenomenon of the interaction of stress waves with material defects. The solutions are determined by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution is the exponentially distributed traction on crack faces. The exact close-form transient solutions of stresses and displacement are obtained in this study. These solutions are valid for an infinitely long time and have accounted for the contributions of incident, reflected, and diffracted waves. Numerical results of the transient stresses are obtained and compared with the corresponding static values. The transient solution has been shown to approach the static value after the first few diffracted waves generated from the crack tip have passed the observation point.

scholarly journals Lax Operator for Macdonald Symmetric Functions

2015 ◽  
Vol 105 (7) ◽  
pp. 901-916 ◽  
Author(s):  
Maxim Nazarov ◽  
Evgeny Sklyanin
Keyword(s):  
Symmetric Functions ◽  
Lax Operator ◽  
Macdonald Symmetric Functions
Sign in / Sign up

Export Citation Format

Share Document