the geometry of a collector is one of the important factors that can increase the incident radiation on the collector surface. in the present study, the incident radiation for a stationary collector with cone geometry, i.e. a conical collector, is theoretically and experimentally investigated. this type of collector is always stable and does not need a fixture to install. moreover, it has a sym...

In this paper, we are concerned with the existence and multiplicity of symmetric positive solutions for the following second-order three-point boundary value problem u′′(t)+ a(t)f (t, u(t))= 0, 0 < t < 1, u(t)= u(1 − t), u′(0)− u′(1)= u(1/2), where a : (0,1) → [0,∞) is symmetric on (0,1) and maybe singular at t = 0 and t = 1, f : [0,1] × [0,∞) → [0,∞) is continuous and f (·, u) is symmetric on ...

We consider the problem of finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices S + . Assume that Γ is defined by a separating oracle, which, for any given m×m symmetric matrix Ŷ , either confirms that Ŷ ∈ Γ or returns several selected cuts, i.e., a number of symmetric matrices Ai, i = 1, ..., p, p ≤ pmax, such that Γ is in the polyhedron {...

1. The purpose of this note is to describe the extension of the almost everywhere convergence of Poisson integrals to the case of the bounded symmetric domains of Cartan. It is useful to realize such a bounded symmetric domain D as a generalized upper half-plane (i.e., a Siegel domain of type II), and this is done as follows. Let V1 and V2 be two finite-dimensional vector spaces over C. Assume ...

We study the second-order m-point boundary value problem u′′(t) + a(t) f (t,u(t)) = 0, 0 < t < 1, u(0) = u(1) = ∑m−2 i=1 αiu(ηi), where 0 < η1 < η2 < ··· < ηm−2 ≤ 1/2, αi > 0 for i= 1,2, . . . ,m− 2 withm−2 i=1 αi < 1,m≥ 3. a : (0,1)→ [0,∞) is continuous, symmetric on the interval (0,1), and maybe singular at t = 0 and t = 1, f : [0,1]× [0,∞)→ [0,∞) is continuous, and f (·,x) is symmetric on th...

In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift” of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equival...

We introduce a first-order method for solving very large convex cone programs. Themethod uses an operator splittingmethod, the alternating directionsmethod of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to inter...

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the...

This paper explores the geometric structure of spectrahedral cone, called symmetry adapted positive semidefinite (PSD) and Gram spectrahedron a symmetric polynomial. In particular, we determine dimension PSD describe its extremal rays, discuss matrix representations. We also consider spectrahedra for specific families polynomials including binary polynomials, quadratics, ternary quartics sextic...