arXiv subjects
Publications and source records attributed to Alexandra Teodor.
We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.
We prove new formulas for $\operatorname{DD}_k(n)$, the number of plane partition diamonds of length $k$ of $n$, and, also, for its polynomial part.
Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number of strict plane partitions of $n$ and $\operatorname{pp}^{so}(n)$, the number of symmetric plane partitions of $n$. Also, we give new formulas for $P_r(n)$, the number of $r$-component multipartitions of $n$.
Let $\mathbf a=(a_1,\ldots,a_r)$ be a sequence of positive integers and $k\geq 2$ an integer. We study $p_{k,\mathbf a}(n)$, the restricted $k$-multipartition function associated to $\mathbf a$ and $k$. We prove new formulas for $p_{k,\mathbf a}(n)$, its waves $W_j(n,k,\mathbf a)$'s and its polynomial part $P_{k,\mathbf a}(n)$. Also, we give a lower bound for the density of the set $\{n\geq 0\;:\;p_{k,\mathbf a}(n)\not\equiv 0(\bmod\;m)\}$, where $m\geq 2$ is an integer.
We give a probabilistic representation of the solution to a semilinear elliptic Dirichlet problem with general (discontinuous) boundary data. The boundary behaviour of the solution is in the sense of the controlled convergence initiated by A. Cornea. Uniqueness results for the solution are also provided.