In this paper, we write down the separable Werner state in a two-qubit system as a convex combination of product states explicitly. The Werner state in a two-qubit system has a single real parameter and varies from inseparable state to separable state according to the value of its parameter. The convex combination of product states gives a hidden variable model to the separable Werner state.

We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a decomposition. Separable states correspond to mixing from one to four pure product states. Inseparable states can be described as pseudomixtures of four or five pure pr...

Support Vector Machines (SVMs) are known to be consistent and robust for classification and regression if they are based on a Lipschitz continuous loss function and on a bounded kernel with a dense and separable reproducing kernel Hilbert space. These facts are even true in the regression context for unbounded output spaces, if the target function f is integrable with respect to the marginal di...

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕H1. In particular, we prove the existence of a oneparameter family of dense no...

In this paper we present a probabilistic algorithm for multiview reconstruction from calibrated images. The algorithm is based on multi-resolution volumetric range image integration and is highly separable as it only employs local optimization. Dense depth maps are transformed in an octree data structure with variable voxel sizes. This allows for an efficient modeling of point clouds with very ...

For a bounded non-negative self-adjoint operator acting in a complex, infinitedimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M ⊂ H such that M ∩R = M ∩R = {0}. We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint opera...

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations ba...

Abstract The current bottleneck of globally solving mixed-integer (nonconvex) quadratically constrained problems (MIQCPs) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting-surface method based on multiple diagonal perturbations to derive convex quadratic relaxations for nonconvex quadratic probl...

Let A be an unbounded from above self-adjoint operator in a separable Hilbert space H and EA(·) its spectral measure. We discuss the inverse spectral problem for singular perturbations à of A (à and A coincide on a dense set in H). We show that for any a ∈ R there exists a singular perturbation à of A such that à and A coincide in the subspace EA((−∞, a))H and simultaneously à has an additional...

Few results have been obtained on the packing spheres constant or exact formula for separable Orlicz function spaces (Yang, 2002, P.895-899, Ye, 1987, P.487-493). In this paper, by using the continuity of ideal space norm, we firstly proved that simple function class is dense in L∗ Φ function space. This is a necessary condition of interpolation theorem. Hence, the exact value of packing sphere...