  We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical
structures that connect, through smooth paths parameterized by $\kappa$, a
finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions,
vectors, coefficients of a generating function...) to their ordinary sum at
$\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-embeddings can be used
to design nonlinear irreversible processes through this connection. A number of
examples of increasing complexity are worked out to illustrate the
possibilities uncovered by this concept. These include not only smooth
functions but also fractals on the real line and on the complex plane. As an
application, we use $\mathcal{B}_{\kappa}$-embeddings to formulate a robust
method for finding all roots of a univariate polynomial without factorizing or
deflating the polynomial. We illustrate this method by finding all roots of a
polynomial of 19th degree.
