On the duality operator of a convex cone

Let C be a convex set in Rⁿ. For each y?C, the cone of C at y, denoted by cone(y, C), is the cone {α(x ? y): α ? 0 and x?C}. If K is a cone in Rⁿ, we shall denote by $\text{K}{}^1$ its dual cone and by F(K) the lattice of faces of K. Then the duality operator of K is the mapping $\text{d}{}_{\mathrm{K}}\text{: F(K) → F(K}{}^1\text{)}$ given by $\text{d}{}_{\mathrm{K}}\text{(F) = (}\text{span}\text{F)}{}^{\mathrm{\perp }}\text{∩ K}{}^1$. Properties of the duality operator d_K of a closed, pointed, full cone K have been studied before. In this paper, we study d_K for a general cone K, especially in relation to d_{cone(y, K)}, where y?K. Our main result says that, for any closed cone K in Rⁿ, the duality operator d_K is injective (surjective) if and only if the duality operator d_{cone(y, K)} is injective (surjective) for each vector y?K ～ [K ∩ (? K)]. In the last part of the paper, we obtain some partial results on the problem of constructing a compact convex set C, which contains the zero vector, such that cone (0, C) is equal to a given cone.     

Positivity of Riesz functionals and solutions of quadratic and quartic moment problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K⊆Rn if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n,k)=(2,4) and K=R²; (ii) n?1, k=2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.     

Quasimonotonie und Ungleichungen in halbgeordneten Räumen

Let K be a cone in Rⁿ, K¹ the polar cone. A n × n-matrix A is called quasimonotone with respect to K, if x ? K ? ?? ? K¹, ? ≠ 0, (?,x) = 0, (?, Ax) ? 0. It is shown, that several properties are equivalent with this, especially the monotonicity of exp(tA) for all t? 0. This was already shown by Schneider-Vidyasagar, here we give a new approach by considering the differential equation x′(t) = Ax(t), x(0) ? K. Two open questions in [5] are settled. We then consider irreducible quasimonotone matrices and show that most of the usual properties of irreducible monotone matrices hold also for this class. In the last section it is shown, that the connections between A quasimonotone and exp(tA) monotone are special cases of differential inequalities for initial value problems in partially ordered topological vector spaces. The concept “quasimonotone,” introduced here in a similar manner as by Volkmann, is not only sufficient, but proves also to be essentially necessary for inequalities of this type.     

Conjugate cone characterization of positive definite and semidefinite matrices

Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rⁿ. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+A^T)^?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+A^T)(K) is closed, A is positive semidefinite on both K and the conjugate cone K^A={s∣x^T(A+ A^T)s?0?x∈K}, and (A+A^T)x=0 for all x in K such that x^TAx=0.     

Existence Criterion for Estimates of Derivatives of Rational Functions

Suppose that K is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov-Bernstein type for derivatives of a rational function R(z) at any fixed point z ₀ ∈ K. We prove that, for a fixed integer s, the estimate of the form |R ^(s) (z ₀)| ≤ C(K, z ₀, s)n‖R‖ _C(K), where R is an arbitrary rational function of degree n without poles on K and C is a bounded function depending on three arguments K, z ₀, and s, holds if and only if the supremum $$\omega (K,z_0 ,s) = \sup \left\{ {\frac{{\operatorname{dist} (z,K)}}{{\left| {z - z_0 } \right|^{s + 1} }}} \right\}$$ over z in the complement of K is finite. Under this assumption, C is less than or equal to const ·s!ω(K, z ₀, s).     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

The product of linear nonhomogeneous forms

We show that for an arbitrary unimodular lattice Λ of dimension n and an arbitrary point C =(c₁, c₂...c_n) ? Rⁿ a point Y = (y₁, y₂,..., y_n) ε Λ can be found and also a number h, satisfying the condition 1 ?h ? 2^?n/2 θ^?1 + 1 (0 < θ ? 2^?n/2), such that the inequality $$\prod\nolimits_{i = 1}^n {\left| {Y_i + hc_i } \right|}< \theta $$ will be satisfied.     

One-sided maximal functions and Hp

Let N be the nontangential maximal function of a function u harmonic in the Euclidean half-space Rⁿ × (0, ∞) and let N^? be the nontangential maximal function of its negative part. If u(0, y) = o(y^?n) as y → ∞, then ∥N∥_p ? c_p ∥N^?∥_p, 0 < p < 1, and more. The basic inequality of the paper (Theor. 2.1) can be used not only to derive such global results but also may be used to study the behavior of u near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of u. For example, if u belongs to H^p for some 0 < p < 1, then every thick cone in the half-space must contain a zero of u.     

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

On indecomposable subcones

Let K₁,K₂ be cones. We say that K₁ is a subcone of K₂ if ExtK₁?ExtK₂. Furthermore, if K₁≠K₂, K₁ is called a proper subcone; if dimK₁=dimK₂, K₁ is called a non-degenerate subcone. We first prove that every n-dimensional indecomposable cone, n?3, contains a non-degenerate indecomposable subcone which has no more than 2n-2 extremals. Then we construct for each n?3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable.     

Maximal exponents of polyhedral cones (II)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rⁿ. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that the maximum value of γ(K) as K runs through all n-dimensional minimal cones (i.e., cones having n+1 extreme rays) is n²-n+1 if n is odd, and is n²-n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ(A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

Constant mean curvature hypersurfaces with single valued projections on planar domains

A classical problem in constant mean curvature hypersurface theory is, for given H?0, to determine whether a compact submanifold Γn−1 of codimension two in Euclidean space , having a single valued orthogonal projection on Rn, is the boundary of a graph with constant mean curvature H over a domain in Rn. A well known result of Serrin gives a sufficient condition, namely, Γ is contained in a right cylinder C orthogonal to Rn with inner mean curvature HC?H. In this paper, we prove existence and uniqueness if the orthogonal projection Ln−1 of Γ on Rn has mean curvature and Γ is contained in a cone K with basis in Rn enclosing a domain in Rn containing Ln−1 such that the mean curvature of K satisfies HK?H. Our condition reduces to Serrin's when the vertex of the cone is infinite.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I

Let A be an n × n complex matrix, and write A = H + iK, where i² = ?1 and H and K are Hermitian matrices. The characteristic polynomial of the pencil xH + yK is f(x, y, z) = det(zI ? xH ? yK). Suppose f(x, y, z) is factored into a product of irreducible polynomials. Kippenhahn [5, p. 212] conjectured that if there is a repeated factor, then there is a unitary matrix U such that U^?1AU is block diagonal. We prove that if f(x, y, z) has a linear factor of multiplicity greater than n?3, then H and K have a common eigenvector. This may be viewed as a special case of Kippenhahn’s conjecture.     

Rings of matrix invariants in positive characteristic

Denote by R_n,m the ring of invariants of m-tuples of n×n matrices (m,n?2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R_n,m. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))?n, then R_n,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R_2,m is determined for the case char(K)=2: it consists of 2^m+m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)≠2. We prove that the characterization of the R_n,m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R_2,4, and analyze the difference between the cases char(K)=2 and char(K)≠2.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Existence of triple solutions of discrete (n,p) boundary value problems

We consider the following boundary value problem: −Δⁿy = F(k,y, Δy,…,Δⁿ⁻¹y), k ϵ Z[n − 1, N], Δⁱy(0) = 0, 0 ≤ i ≤ n − 2, Δ^py(N + n - p) = 0, where n ≥ 2 and p is a fixed integer satisfying 0 ≤ p ≤ n − 1. Using a fixed-point theorem for operators on a cone, we shall yield the existence of at least three positive solutions.     

Approximation by rational modules in Sobolev and Lipschitz norms

Let X?C be compact, 0>n∈Z, and g a continuous function on X. Let R(n,g,X) be the rational module consisting of the functions on X of the type r₀ + r₁g + ··· + r_ngⁿ, where r_j is a rational function with poles off X, 0 ? j ? n. It is shown that if X is nowhere dense, g is sufficiently smooth, and $\text{}\text{?}\text{t6g(z) ≠ 0, z ∈ X}$, then the restriction to X of each function in C∈(C) is approximable in the Lip(n ? 1, X)-norm, n ? 2, by functions in R(n, g, X). Also dealt with are approximation problems in Sobolev norms by more general types of rational modules.