scholarly journals On the Oval Shapes of Beach Stones

AppliedMath ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 16-38
Author(s):  
Theodore P. Hill
Keyword(s):  
Continuous Time ◽  
Basic Equation ◽  
Numerical Experiments ◽  
Physical Mechanism ◽  
Wave Process ◽  
Random Wave ◽  
Local Shape ◽  
Standard Curve ◽  
Non Local ◽  
Curve Shortening

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

scholarly journals A carbuncle cure for the Harten-Lax-van Leer contact (HLLC) scheme using a novel velocity-based sensor

2021 ◽  
Author(s):  
U. S. Vevek ◽  
B. Zang ◽  
T. H. New
Keyword(s):  
Data Structures ◽  
Numerical Experiments ◽  
Additional Data ◽  
Shear Layers ◽  
Hybrid Scheme ◽  
Local Pressure ◽  
Numerical Flux ◽  
Shear Sensor ◽  
Non Local

AbstractA hybrid numerical flux scheme is proposed by adapting the carbuncle-free modified Harten-Lax-van Leer contact (HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact (HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.

scholarly journals Multivalued Discrete Tomography Using Dynamical System That Describes Competition

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Takeshi Kojima ◽  
Tetsushi Ueta ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Theoretical Analysis ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Nonlinear Dynamical System ◽  
Reconstructed Image ◽  
Optimization Approach ◽  
Discrete Tomography ◽  
Nonlinear Dynamical ◽  
Time Optimization

Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

A Note on External Uniformization for Finite Markov Chains in Continuous Time

1992 ◽  
Vol 6 (1) ◽  
pp. 127-131 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Strong Law ◽  
Large Numbers ◽  
Finite Markov Chains

An external uniformization technique was developed by Ross [4] to obtain approximations of transition probabilities of finite Markov chains in continuous time. Yoon and Shanthikumar [7] then reported through extensive numerical experiments that this technique performs quite well compared to other existing methods. In this paper, we show that external uniformization results from the strong law of large numbers whose underlying distributions are exponential. Based on this observation, some remarks regarding properties of the approximation are given.

scholarly journals Coherent modification of entanglement: benefits due to extended Hilbert space

2016 ◽  
Vol 16 (11&12) ◽  
pp. 954-968
Author(s):  
Dmitry Solenov
Keyword(s):  
Hilbert Space ◽  
Continuous Time ◽  
Computing System ◽  
Local Network ◽  
Toffoli Gate ◽  
Quantum Random Walks ◽  
Physical Systems ◽  
Time Quantum ◽  
Speed Up ◽  
Non Local

A quantum computing system is typically represented by a set of non-interacting (local) two-state systems—qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.

scholarly journals Refutation of a Rumor by the Mass Media: Mathematical Model and Numerical Experiments

2021 ◽  
Vol 24 (2) ◽  
pp. 371-386
Author(s):  
Александр Петрович Михайлов ◽  
Александр Пхоун Чжо Петров
Keyword(s):  
Mathematical Model ◽  
Mass Media ◽  
Discrete Time ◽  
Continuous Time ◽  
Numerical Experiments ◽  
Mathematical Properties ◽  
The Media ◽  
The Mass Media

The process is considered, in which an unreliable rumor spreads in society, which is opposed by the broadcasting of the mass media. In this case, the unreliability of hearing is understood so that the information of the media contains a refutation and thereby inoculates individuals, that is, makes them immune to hearing. At the same time, individuals who have managed to accept the rumor cease to trust the media and thereby become unavailable for persuasion. For this process, a mathematical model is proposed in two versions. The variant with continuous time reveals some of the mathematical properties of the model. The discrete time option is more convenient for analyzing real processes since it allows one to estimate the parameters of the model. To assess these parameters, data on the ratings of the main socio-political programs of Russian TV channels were used. Several scenario calculations of the model with these parameters are presented. The main conclusion is that if the information disseminated by the media is not viral, that is, it is not retold by viewers to their neighbors in society, then the media are unable to resist rumors.

A New Approach for Simultaneous Shape and Topology Optimization Based on Implicit Topology Description Functions

2004 ◽  
Author(s):  
Xu Guo ◽  
Kang Zhao ◽  
Michael Yu Wang
Keyword(s):  
Topology Optimization ◽  
Numerical Experiments ◽  
Shape Derivative ◽  
Structural Topology ◽  
Numerical Instability ◽  
New Approach ◽  
Shape And Topology Optimization ◽  
Topology Description Function ◽  
And Topology ◽  
Non Local

In the present paper, a new approach for structural topology optimization based on implicit topology description function (TDF) is proposed. TDF is used to describe the shape/topology of a structure, which is approximated in terms of the nodal values. Then a relationship is established between the element stiffness and the values of the topology description function on its four nodes. In this way and with some non-local treatments of the design sensitivities, not only the shape derivative but also the topological derivative of the optimal design can be incorporated in the numerical algorithm in a unified way. Numerical experiments demonstrate that by employing this approach, the computational efforts associated with TDF (and level set) based algorithms can be saved. Clear optimal topologies and smooth structural boundaries free from any sign of numerical instability can be obtained simultaneously and efficiently.

A random wave process

1984 ◽  
Vol 12 (1) ◽  
pp. 97-114 ◽  
Author(s):  
D. A. Dawson ◽  
G. C. Papanicolaou
Keyword(s):  
Wave Process ◽  
Random Wave

scholarly journals On the numerical approximation of vectorial absolute minimisers in $$L^\infty $$

2020 ◽  
Vol 27 (6) ◽  
Author(s):  
Nikos Katzourakis ◽  
Tristan Pryer
Keyword(s):  
Calculus Of Variations ◽  
Boundary Data ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Lipschitz Mappings ◽  
Open Set ◽  
Locally Lipschitz ◽  
Non Local ◽  
A New Technique ◽  
Local Nature

AbstractLet $$\Omega $$ Ω be an open set. We consider the supremal functional $$\begin{aligned} \text {E}_\infty (u,{\mathcal {O}})\, {:}{=}\, \Vert \text {D}u \Vert _{L^\infty ( {\mathcal {O}} )}, \ \ \ {\mathcal {O}} \subseteq \Omega \text { open}, \end{aligned}$$ E ∞ ( u , O ) : = ‖ D u ‖ L ∞ ( O ) , O ⊆ Ω open , applied to locally Lipschitz mappings $$u : \mathbb {R}^n \supseteq \Omega \longrightarrow \mathbb {R}^N$$ u : R n ⊇ Ω ⟶ R N , where $$n,N\in \mathbb {N}$$ n , N ∈ N . This is the model functional of Calculus of Variations in $$L^\infty $$ L ∞ . The area is developing rapidly, but the vectorial case of $$N\ge 2$$ N ≥ 2 is still poorly understood. Due to the non-local nature of (1), usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $$n,N\ge 2$$ n , N ≥ 2 . We present numerical experiments aimed at understanding the behaviour of minimisers through a new technique involving p-concentration measures.

scholarly journals A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150629 ◽  
Author(s):  
Michael C. Dallaston ◽  
Scott W. McCue
Keyword(s):  
Flow Problem ◽  
Initial Conditions ◽  
Plane Curve ◽  
Rate Of Change ◽  
Plane Curves ◽  
Flow Rule ◽  
Closed Curves ◽  
Curve Shortening Flow ◽  
Standard Curve ◽  
Curve Shortening

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

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