Problems in One-Dimension: General Considerations, Infinite Well Potential, Piecewise Constant Potentials, and Delta Function Potentials

2017 ◽  
pp. 115-162
Author(s):  
Paul R. Berman
Keyword(s):  
Delta Function ◽  
One Dimension ◽  
Piecewise Constant

scholarly journals A General Total Variation Minimization Theorem for Compressed Sensing Based Interior Tomography

2009 ◽  
Vol 2009 ◽  
pp. 1-3 ◽  
Author(s):  
Weimin Han ◽  
Hengyong Yu ◽  
Ge Wang
Keyword(s):  
Compressed Sensing ◽  
Total Variation ◽  
General Theorem ◽  
Region Of Interest ◽  
Delta Function ◽  
Piecewise Constant Function ◽  
Mathematical Tool ◽  
Total Variation Minimization ◽  
Piecewise Constant ◽  
Dirac Delta

Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant (Yu and Wang, 2009). Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009).

Thermodynamic properties of a finite one-dimensional system of bosons with repulsive delta-function interaction

1995 ◽  
Vol 73 (3-4) ◽  
pp. 245-247
Author(s):  
K. L. Poon ◽  
K. Young ◽  
D. Kiang
Keyword(s):  
Thermodynamic Properties ◽  
Thermodynamic Limit ◽  
General Model ◽  
Finite Size ◽  
Delta Function ◽  
One Dimension ◽  
Dimensional System ◽  
One Dimensional ◽  
Finite Size Corrections

The thermodynamics of N bosons in a length L in one dimension, with repulsive delta-function interaction, is studied numerically for finite N, L. The results show the nature of finite-size corrections and how the thermodynamic limit is approached, and hopefully will be of some guidance in seeking the solution of a more general model.

scholarly journals Shadow wave solutions for a scalar two-flux conservation law with Rankine–Hugoniot deficit

2021 ◽  
Vol 18 (03) ◽  
pp. 539-556
Author(s):  
Tanja Krunić ◽  
Marko Nedeljkov
Keyword(s):  
Shock Wave ◽  
Weak Solution ◽  
Hyperbolic Conservation Laws ◽  
Jump Condition ◽  
Delta Function ◽  
Piecewise Constant ◽  
Hyperbolic Conservation ◽  
Unbounded Solutions ◽  
Wave Solutions ◽  
Flux Conservation

This paper deals with hyperbolic conservation laws exhibiting a flux discontinuity at the origin and which does not admit a weak solution satisfying the Rankine–Hugoniot jump condition. We therefore seek unbounded solutions in the form of shadow waves supported by at the origin. The shadow waves are defined as nets of piecewise constant functions approximating a shock wave to which we add a delta function and possibly another unbounded part.

Exact and asymptotic solutions to a PDE that arises in time-dependent queues

2000 ◽  
Vol 32 (1) ◽  
pp. 256-283 ◽  
Author(s):  
Charles Knessl
Keyword(s):  
Limit Theorems ◽  
Asymptotic Properties ◽  
Mathematical Physics ◽  
Linear Functions ◽  
Time Dependent ◽  
One Dimension ◽  
Asymptotic Results ◽  
Piecewise Constant ◽  
Space And Time ◽  
Diffusion Problems

We consider a diffusing particle in one dimension that is subject to a time-dependent drift or potential field. A reflecting barrier constrains the particle's position to the half-line X ≥ 0. Such models arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We solve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the solution, for various ranges of space and time. We also relate our asymptotic results to those obtained by probabilistic approaches, such as central limit theorems and large deviations. We consider drifts that are either piecewise constant or linear functions of time.

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