In this paper, we consider the acoustic cloaking based on singular transformations and its approximation in the context of scattering problems in R. The cloaking based on singular transformations can not avoid a singularity of the resulting Laplace–Beltrami operator because singular transformations blow up one point to a sphere. In order to treat the singularity properly, we use the variational...

In a recent paper we constructed an integrable generalization of the Toda law on the square lattice. We construct other examples of integrable dynamics of Toda type on the square lattice, as well as on the triangular lattice, as nonlinear symmetries of the discrete Laplace equations on square and triangular lattices. We also construct the tau -function formulations and Darboux-Bäcklund transfor...

We propose a method to obtain the equilibrium distribution for positions and velocities of a one-dimensional particle via time-averaging and Laplace transformations. We apply it to the case of a damped harmonic oscillator in contact with a thermal bath. The present method allows us to treat, among other cases, a Gaussian noise function exponentially correlated in time, e.g., Gaussian colored no...

In this paper, we are interested in solving infinite linear systems of differential equations of the form x′ (t) = Tx (t) + b with x(0) = x0, where T is either the generalized Cesàro operator C (λ) or the weighted mean matrix Nq, x0 and b are two given infinite column matrices and λ is a sequence with non-zero entries. We use a new method based on Laplace transformations to solve these systems.

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized by the planarity of its elementary quadrilaterals. Therefore the linear property of planarity seems to be a basic geometric property underlying integrability....

In this paper we deal with the solvability of the infinite system of differential equations x′(t) = ∆(λ)x(t) + b with x(0) = a, where ∆(λ) is the triangle defined by the infinite matrix whose the nonzero entries are [∆(λ)]nn = λn and [∆(λ)]n,n−1 = λn−1 for all n ∈ N, for a given sequence λ and a, b are two given infinite column matrices. We use a new method based on Laplace transformations to s...

We motivate and summarise some recent results in the application of formally exact boundary conditions in nuclear time-dependent Hartree-Fock calculations, making use of Laplace transformations to calculate the values of the wave functions at the boundaries. We have realised the method in the case of giant monopole resonances of spherically-symmetric nuclei, and present strength functions of 16...

We propose a method to obtain the equilibrium distribution for positions and velocities of a one-dimensional particle via time-averaging and Laplace transformations. We apply it to the case of a damped harmonic oscillator in contact with a thermal bath. The present method allows us to treat, among other cases, a Gaussian noise function exponentially correlated in time, e.g., Gaussian colored no...

Engineers and students in communications and mathematics are confronted with transformations such as the z-Transform, the Fourier transform, or the Laplace transform. Often it is quite hard to quickly find the appropriate transform in a book or the Internet, much less to have a comprehensive overview of transformation pairs and corresponding properties. In this document I compiled a handy colle...