Related papers: On Separation of Variables
Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives…
The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
A discrete function of $n$ variables is a mapping $g : X_1 \times \ldots \times X_n \rightarrow A$, where $X_1, \ldots, X_n$, and $A$ are arbitrary finite sets. Function $g$ is called {\em separable} if there exist $n$ functions $g_i : X_i…
A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
Let G be a block matrix function with one diagonal block A being positive definite and the off diagonal blocks complex conjugates of each other. Conditions are obtained for G to be factorable (in particular, with zero partial indices) in…
Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these…
We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which…
For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For…
When dealing with certain mathematical problems, it is sometimes necessary to show that some function induces a metric on a certain space. When this function is not a well renowned example of a distance, one has to develop very particular…
Mathematical models are sometime given as functions of independent input variables and equations or inequations connecting the input variables. A probabilistic characterization of such models results in treating them as functions with…
For two independent, almost surely finite random variables, independence of their minimum (time) and the event that one of them is either greater, equal or less than the other (cause) is completely characterized. It is shown that, other…
In this paper we present the necessary and sufficient conditions of separability for multipartite pure states. These conditions are very simple, and they don't require Schmidt decomposition or tracing out operations. We also give a…
It is shown that, for the block matrices belonging to $M(nd,\mathbb{C})$ with commuting and normal block entries of dimension $d$, the separability of such a block matrices is equivalent to its semi-positive definity. The separability…
Two subsets of a given set are path-disconnected if they lie in different connected components of the larger set. Verification of path-disconnectedness is essential in proving the infeasibility of motion planning and trajectory optimization…
The separability problem is formulated in terms of a characterization of a single entanglement witness. More specifically, we show that any (in general multipartite) state \varrho is separable if and only if a specially constructed…
Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of…
It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.
We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an…