Remarks on CCR and CAR flows over closed convex cones

For a closed convex cone [Formula: see text] in [Formula: see text] which is spanning and pointed, i.e. [Formula: see text] and [Formula: see text] we consider a family of [Formula: see text]-semigroups over [Formula: see text] consisting of a certain family of CCR flows and CAR flows over [Formula: see text] and classify them up to the cocycle conjugacy.

Generation of Chow Parameters and Reduced Variables Through Nearest Neighbor Relations in Threshold Networks

Generation of useful variables and features is an important issue throughout the machine learning, artificial intelligence, and applied fields for their efficient computations. In this paper, the nearest neighbor relations are proposed for the minimal generation and the reduced variables of the functions in the threshold networks. First, the nearest neighbor relations are shown to be minimal and inherited for threshold functions and they play an important role in the iterative generation of the Chow parameters. Further, they give a solution for the Chow parameters problem. Second, convex cones are made of the nearest neighbor relations for the generation of the reduced variables. Then the edges of convex cones are compared for the discrimination of variables. Finally, the reduced variables based on the nearest neighbor relations are shown to be useful for documents classification.

De Branges Type Lemma and Approximation in Weighted Spaces

The purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

On Notations for Conic Hulls and Related Considerations on Tangent Cones

We propose two di erent notations for cones generated by a set and for convex cones generated by a set, usually denoted by a same notation. We make some remarks on the Bouligand tangent cone and on the Clarke tangent cone for star-shaped sets and for locally convex sets. We give some applications of these remarks to a di erentiable optimization problem with an abstract constraint.

Some applications of the open mapping theorem in locally convex cones

UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set: $$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$ Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$ . As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.