Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this
paper is the following theorem.

{\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$.
There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that $G(k_j)=0$,
$k_j<k_{j+1}$, $\lim_{j\to \infty}k_j=\infty$, $\lim_{k\to \infty}|G(k)|$ does
not exist, $\limsup_{k\to +\infty}|G(k)|=\infty$. This $g$ oscillates
infinitely often in any interval $[1-\delta, 1]$, however small $\delta>0$ is.}

In this paper a new method for inverting the Laplace transform from the real
axis is formulated. This method is based on a quadrature formula. We assume
that the unknown function $f(t)$ is continuous with (known) compact support. An
adaptive iterative method and an adaptive stopping rule, which yield the
convergence of the approximate solution to $f(t)$, are proposed in this paper.

A method is given for creating material with a desired refraction
coefficient. The method consists of embedding into a material with known
refraction coefficient many small particles of size $a$. The number of
particles per unit volume around any point is prescribed, the distance between
neighboring particles is $O(a^{\frac{2-\kappa}{3}})$ as $a\to 0$, $0<\kappa<1$
is a fixed parameter. The total number of the embedded particle is
$O(a^{\kappa-2})$.

An evolution equation, arising in the study of the Dynamical Systems Method
(DSM) for solving equations with monotone operators, is studied in this paper.
The evolution equation is a continuous analog of the regularized Newton method
for solving ill-posed problems with monotone nonlinear operators $F$. Local and
global existence of the unique solution to this evolution equation are proved,
apparently for the firs time, under the only assumption that $F'(u)$ exists and
is continuous with respect to $u$. The earlier published results required more
smoothness of $F$.

Boundary integral equation is derived for the problem of scattering of
electromagnetic waves by 3D homogeneous body of arbitrary shape.