961 research outputs found
Dynamic programming approach to structural optimization problem – numerical algorithm
In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed
Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit
The 1D discrete fractional Laplacian operator on a cyclically closed
(periodic) linear chain with finitenumber of identical particles is
introduced. We suggest a "fractional elastic harmonic potential", and obtain
the -periodic fractionalLaplacian operator in the form of a power law matrix
function for the finite chain ( arbitrary not necessarily large) in explicit
form.In the limiting case this fractional Laplacian
matrix recovers the fractional Laplacian matrix ofthe infinite chain.The
lattice model contains two free material constants, the particle mass and
a frequency.The "periodic string continuum limit" of the
fractional lattice model is analyzed where lattice constant and
length of the chain ("string") is kept finite: Assuming finiteness of
the total mass and totalelastic energy of the chain in the continuum limit
leads to asymptotic scaling behavior for of thetwo material
constants,namely and . In
this way we obtain the -periodic fractional Laplacian (Riesz fractional
derivative) kernel in explicit form.This -periodic fractional Laplacian
kernel recovers for the well known 1D infinite space
fractional Laplacian (Riesz fractional derivative) kernel. When the scaling
exponentof the Laplacian takesintegers, the fractional Laplacian kernel
recovers, respectively, -periodic and infinite space (localized)
distributionalrepresentations of integer-order Laplacians.The results of this
paper appear to beuseful for the analysis of fractional finite domain problems
for instance in anomalous diffusion (L\'evy flights), fractional Quantum
Mechanics,and the development of fractional discrete calculus on finite
lattices
Wave equation in higher dimensions - periodic solutions
We discuss the solvability of the periodic-Dirichlet problem for the wave equation with forced vibrations xtt(t, y) − ∆x(t, y) + l(t, y, x(t, y)) = 0 in higher dimensions with sides length being irrational numbers and superlinear nonlinearity. To this effect we derive a new dual variational method
O niektórych trudnościach pojęciowych związanych z próbą wykazania fałszywości twierdzenia, że nic nie istnieje
Czy coś istnieje? Tak, oczywiście; samo zadanie tego pytania powinno być sprawą wstydliwą dla pytającego – zalicza go przecież w poczet głupców. Nikt na serio go nie zadaje ani nie zadawał, ani w życiu codziennym, ani nawet w najbardziej ekstrawaganckich rozważaniach filozoficznych. Jeden Gorgiasz twierdził (chociaż nie pytał, czy), że nic nie istnieje, lecz to był zapewne żart wielkiego retora, cyrkowa sztuczka ku uciesze publiczności
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