Cone In Rn Research Articles

Generalized Vector-Valued Hardy Functions

We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone in Rn, an open convex cone in Rn, and a regular cone in Rn, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in Lp norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space Hp in Cn. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued Lp functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions Hp. We obtain boundary value results on the distinguished boundary Rn+i{0¯} and on the topological boundary Rn+i∂B of the tube for the analytic functions in the Lp and vector valued tempered distribution topologies. Suggestions for associated future research are given.

Convergence of two-stage iterative scheme for [formula omitted]-weak regular splittings of type II

Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, K-monotonicity is sufficient for the convergence of K-regular and K-weak regular splittings, where K is a proper cone in Rn. However, the convergence theory of a two-stage iteration scheme in a general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., K=R+n), is open. To this end, we propose convergence theory of two-stage iterative scheme for K-weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a K-regular splitting and then establish some comparison theorems. We also study K-monotone convergence theory of the stationary two-stage iterative method in case of a K-weak regular splitting of type II. The most interesting and important part of this work is on M-matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model. The computation of large PageRank matrices shows that the two-stage Gauss-Seidel method performs better than the Gauss-Seidel methods.

Uniqueness property for 2-dimensional minimal cones in $\mathbb{R}^3$

In this article we treat two closely related problems: 1) the upper semicontinuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the 2-dimensional minimal cones in R3 . Given an open set Ω ⊂ Rn, a closed set E ⊂ Ω is said to be Almgren minimal of dimension d in Ω if it minimizes the d-Hausdorff measure among all its Lipschitz deformations in Ω. We say that a d-dimensional minimal set E in an open set Ω admits upper semi-continuity if, whenever {fn(E)}n is a sequence of deformations of E in Ω that converges to a set F, then we have Hd(F) ≥ lim supn Hd(fn(E)). This guarantees in particular that E minimizes the d-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. As proved in [19], when several 2-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known 2-dimensional minimal cones in Rn to generate new families of minimal cones by taking their almost orthogonal unions. The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

A Brunn–Minkowski theory for coconvex sets of finite volume

Let C be a closed convex cone in Rn, pointed and with interior points. We consider sets of the form A=C∖K, where K⊂C is a closed convex set. If A has finite volume (Lebesgue measure), then A is called a C-coconvex set, and K is called C-close. The family of C-coconvex sets is closed under the addition ⊕ defined by C∖(A1⊕A2)=(C∖A1)+(C∖A2). We develop first steps of a Brunn–Minkowski theory for C-coconvex sets, which relates this addition to the notion of volume. In particular, we establish the equality condition for a Brunn–Minkowski type inequality (with reversed inequality sign) and introduce mixed volumes and their integral representations. For C-close sets, surface area measures and cone-volume measures can be defined, in analogy to similar notions for convex bodies. They are Borel measures on the intersection of the unit sphere with the interior of the polar cone of C. We prove a Minkowski-type uniqueness theorem for C-close sets with equal surface area measures. Concerning Minkowski-type existence problems, we give conditions for a Borel measure to be either the surface area measure or the cone-volume measure of a C-close set. These conditions are sufficient in the first case, and necessary and sufficient in the second case.

Permutation invariant proper polyhedral cones and their Lyapunov rank

The Lyapunov rank of a proper cone K in a finite dimensional real Hilbert space is defined as the dimension of the space of all Lyapunov-like transformations on K, or equivalently, the dimension of the Lie algebra of the automorphism group of K. This (rank) measures the number of linearly independent bilinear relations needed to express a complementarity system on K (that arises, for example, from a linear program or a complementarity problem on the cone). Motivated by the problem of describing spectral/proper cones where the complementarity system can be expressed as a square system (that is, where the Lyapunov rank is greater than equal to the dimension of the ambient space), we consider proper polyhedral cones in Rn that are permutation invariant. For such cones we show that the Lyapunov rank is either 1 (in which case, the cone is irreducible) or n (in which case, the cone is isomorphic to R+n). In the latter case, we show that the corresponding spectral cone is isomorphic to a symmetric cone.

The p-cones in dimension n ≥ 3 are not homogeneous when p ≠ 2

Using the T-algebra machinery we show that, up to linear isomorphism, the only strictly convex homogeneous cones in Rn (n≥3) are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when p≠2, 1<p<∞ and n≥3, thus answering a problem proposed by Gowda and Trott.

Optimal Hardy inequalities in cones

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operatoris subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequalityholds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) &gt; 0 is given explicitly.

Spectral cones in Euclidean Jordan algebras

A spectral cone in a Euclidean Jordan algebra V of rank n is of the form K=λ−1(Q), where Q is a permutation invariant convex cone in Rn and λ:V→Rn is the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). In this paper, we describe some properties of spectral cones. We show, for example, that spectral cones are invariant under automorphisms of V, that the dual of a spectral cone is a spectral cone when V is simple or carries the canonical inner product, and characterize the pointedness/solidness of a spectral cone. We also show that for any spectral cone K in V, dim⁡(K)∈{0,1,m−1,m}, where dim⁡(K) denotes the dimension of K and m is the dimension of V.

Regularity and Bernstein-type results for nonlocal minimal surfaces

Abstract We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi [6] stating that the validity of Bernstein’s theorem in dimension n + 1 {n+1} is a consequence of the nonexistence of n-dimensional singular minimal cones in ℝ n {\mathbb{R}^{n}} .

On the irreducibility, Lyapunov rank, and automorphisms of special Bishop–Phelps cones

Motivated by optimization considerations, we consider cones in Rn – to be called special Bishop–Phelps cones – of the form {(t,x):t≥||x||}, where ||⋅|| is a norm on Rn−1. We show that when n≥3, such cones are always irreducible. Defining the Lyapunov rank of a proper cone K as the dimension of the Lie algebra of the automorphism group of K, we show that the Lyapunov rank of any special Bishop–Phelps polyhedral cone is one. Extending an earlier known result for the l1-cone (which is a special Bishop–Phelps cone with 1-norm), we show that any lp-cone, for 1≤p≤∞, p≠2, has Lyapunov rank one. We also study automorphisms of special Bishop–Phelps cones, in particular giving a complete description of the automorphisms of the l1-cone.

The Vector-Valued Functions Associated with Circular Cones

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

Smooth and nonsmooth analyses of vector-valued functions associated with circular cones

Let Lθ be the circular cone in Rn which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function fc(x) on Rn by applying f to the spectral values of the spectral decomposition of x∈Rn with respect to Lθ. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

INVARIANT DIFFERENTIAL OPERATORS ON THE MINKOWSKI-EUCLID SPACE

For two positive integers m and n, let Pn be the open con- vex cone in Rn(n+1)/2 consisting of positive definite n×n real symmetric matrices and let R(m,n) be the set of all m × n real matrices. In this paper, we investigate differential operators on the non-reductive homo- geneous space Pn × R(m,n) that are invariant under the natural action of the semidirect product group GL(n,R) n R (m,n) on the Minkowski- Euclid space Pn × R (m,n) . These invariant differential operators play an important role in the theory of automorphic forms on GL(n,R)n R (m,n) generalizing that of automorphic forms on GL(n,R).

Minimal external representations of tropical polyhedra

Tropical polyhedra are known to be representable externally, as intersections of finitely many tropical half-spaces. However, unlike in the classical case, the extreme rays of their polar cones provide external representations containing in general superfluous half-spaces. In this paper, we prove that any tropical polyhedral cone in Rn (also known as “tropical polytope” in the literature) admits an essentially unique minimal external representation. The result is obtained by establishing a (partial) anti-exchange property of half-spaces. Moreover, we show that the apices of the half-spaces appearing in such non-redundant external representations are vertices of the cell complex associated with the polyhedral cone. We also establish a necessary condition for a vertex of this cell complex to be the apex of a non-redundant half-space. It is shown that this condition is sufficient for a dense class of polyhedral cones having “generic extremities”.

Upper and lower bounds for the iterates of order-preserving homogeneous maps on cones

We define upper bound and lower bounds for order-preserving homogeneous of degree one maps on a proper closed cone in Rn in terms of the cone spectral radius. We also define weak upper bounds for these maps. For a proper closed cone C⊂Rn, we prove that any order-preserving homogeneous of degree one map f:intC→intC has a lower bound. If C is polyhedral, we prove that the map f has a weak upper bound. We give examples of weak upper bounds for certain order-preserving homogeneous of degree one maps defined on the interior of R+n.

A duality between the metric projection onto a convex cone and the metric projection onto its dual

If K and L are mutually dual pointed convex cones in Rn with the metric projections onto them denoted by PK and PL respectively, then the following two assertions are equivalent: (i) PK is isotone with respect to the order induced by K (i.e. v−u∈K implies PKv−PKu∈K); (ii) PL is subadditive with respect to the order induced by L (i.e. PLu+PLv−PL(u+v)∈L for any u,v∈Rn).

Nonsymmetric conical upper density and 𝑘-porosity

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in R n \mathbb {R}^n . As an application, we find an upper bound close to n − k n-k for the Hausdorff dimension of sets with large k k -porosity. With k k -porous sets we mean sets which have holes in k k different directions on every small scale.

Transfinite diameter notions in ℂN and integrals of Vandermonde determinants

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂN. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann.337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂN. Our results extend to certain weights and measures defined on cones in ℝN.

Slicing surfaces and the Fourier restriction conjecture

AbstractWe deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid and the elliptic hyperboloid in ℝn implies that for the cone in ℝn+1. We also prove a new restriction estimate for any surface in ℝ3 locally isometric to the plane and of finite type.

Hardy-Sobolev Inequalities in a Cone

The attainability of the exact constant in the Hardy-Sobolev inequality is established in an arbitrary cone in ℝn. Bibliography: 17 titles.