Complementarity forms of theorems of Lyapunov and Stein, and related results

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA^*+Q is positive semidefinite and X[AX+XA^*+Q]=0. By considering cone complementarity problems corresponding to linear transformations of the form I−S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA^*+Q is positive semidefinite and X[X−AXA^*+Q]=0. By specializing Q (to −I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X−AXA^* is positive definite.     

The mixed volume optimization problem

The mixed volume optimization problem is to determine the point of duality Q for a given convex set K that minimizes the “mixed volume” of the associated polar set (K^*;Q). In the plane, the mixed volumes translate as the area and length; in space, the mixed volumes include the volume, surface area, and mean width. In this paper, the geometric optimization problems associated with minimizing mixed volumes are examined from two perspectives: enumerative search and symbolic computation. The problem of minimizing the polar area through an enumerative search is first considered. The dual polygon (P_m^*;Q) is constructed for an arbitrary point of duality QP_m° by using an algebraic correspondence between the edges of P_m and the vertices of (P_m^*;Q), and the area of (P_m^*;Q), A(P^*_m;Q), is calculated and minimized using naive search techniques. A result due to Santaló is applied to verify the minimizing solution, and computational tests are described for various classes of randomly generated polygons. Statistical evidence indicates that a “good” approximation to the minimum area polar polygon occurs when the duality point is located at the center-of-gravity of P_m. The polar area problem is then investigated using symbolic procedures. Explicit symbolic expressions for the polar area and length functionals are computed and solved directly using the differential optimality conditions and Newton's iterative method of solution. The mixed volume and surface area functionals are formulated and solved using numerical products, and the mean width functional is described. Examples are used throughout to illustratethe methodology.     

On the adjoint of a matrix associated with trees

Let Q denote the vertex-edge incidence matrix of a tree T. we give an elementary derivation of an extension of a result of Merris on a graph-theoretic interpretation of the entries in the adjoint of the matrix Q¹Q.     

The deviation, density, and depth of partially ordered sets

Let P be a poset, and let γ be a linear order type with |γ| ≥ 3. The γ-deviation of P, denoted by γ-dev P, is defined inductively as follows: (1) γ-dev P=0, if P contains no chain of order type γ; (2) γ-dev P = , if γ-dev P and each chain C of type γ in P contains elements a and b such that a<b and [a, b] as an interval of P has γ-deviation <. There may be no ordinal such that γ-dev P = ; i.e., γ-dev P does not exist. A chain is γ-dense if each of its intervals contains a chain of order type γ. If P contains a γ-dense chain, then γ-dev P fails to exist. If either (1) P is linearly ordered or (2) a chain of order type γ does not contain a dense interval, then the converse holds. For an ordinal ξ, a special set S(ξ) is used to study ω_ξ-deviation. The depth of P, denoted by δ(P) is the least ordinal β that does not embed in P^*. Then the following statements are equivalent: (1) ω_ξ-dev P does not exist; (2) S(ξ) embeds in P; and (3) P has a subset Q of cardinality _ξ such that δ(Q^*) = ω_{ξ + 1}. Also ω_ξ-dev P = <ω_{ξ + 1} if and only if |δ(P^*)|_ξ; if these equivalent conditions hold, then ω^β_ξ < δ(P^*) ≤ ω ^{+ 1}_ξ for all β < . Applications are made to the study of chains of submodules of a module over an associative ring.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

The construction of irreducible normalized representations of the general linear group

A method is described for constructing in an explicit form an irreducible representation T of M_n(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A^*)=T^*(A) for all A∈M_n(F).     

Between local and global logarithmic averages

We obtain an approximation for the logarithmic averages of I{k^1/2a(k) S(k) k^1/2b(k)}, where a(k) → 0, b(k) → 0 (k → ∞) and S(k) is partial sum of independent, identically distributed random variables.     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

On the section of a convex polyhedron

Let P be a convex polyhedron in R³, and E be a plane cutting P. Then the section P_E=P∩E is a convex polygon. We show a sharp inequality , where L(P) denotes the sum of the edge-lengths of P.     

More examples on ordered set reconstruction

We show that there are nonisomorphic ordered sets P and Q that have the same maximal and minimal decks and a rank k such that there is a map B from the elements of rank k in P to the elements of rank k in Q such that P{x} is isomorphic to Q{B(x)} for all x of rank k in P. The examples are preceded by a criterion as to when two nonisomorphic ordered sets will have a rank k and a map B as above.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Pseudoarcs, pseudocircles, Lakes of Wada and generic maps on

We prove a Bruckner–Garg type theorem for the fiber structure of a generic map from a continuum X into the unit interval I. We also study the specific case of X=S². We show that each nondegenerate component of each fiber of a generic map in C(S²,I) is figure-eight-like. This together with a result by Krasinkiewicz and Levin gives that each nondegenerate component of each fiber of a generic map in C(S²,I) is hereditarily indecomposable and figure-eight-like. We also show that pseudoarcs, pseudocircles and Lakes of Wada appear in abundance in fibers of a generic map in C(S²,I). We also exhibit a general method for proving when a P-like hereditarily indecomposable continuum is Q-like when Q is a certain graph containing P.     

Zero-term rank preservers

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.     

Hypernormal matrices

An nxn matrix A is hypernormal if APA^*=A^*PA for all permutation matrices P. We shall explain how to construct hypernormal matrices.     

Linear time algorithms for the weighted tailored 2-partition problem and the weighted 2-center problem under l∞-distance

In this paper, the weighted tailored 2-partition problem and the weighted 2-center problem under l_∞-distance are considered. An O(2^d−1·d·n) algorithm to solve the weighted tailored 2-partition problem and an O(d²·n + d²·log*d) time algorithm to solve the weighted 2-center problem in the d-dimensional case are presented.     

Classical and vector sturm—liouville problems: recent advances in singular-point analysis and shooting-type algorithms

Significant advances have been made in the last year or two in algorithms and theory for Sturm—Liouville problems (SLPs). For the classical regular or singular SLP −(p(x)u′)′ + q(x)u = λw(x)u, a < x < b, we outline the algorithmic approaches of the recent library codes and what they can now routinely achieve.

For a library code, automatic treatment of singular problems is a must. New results are presented which clarify the effect of various numerical methods of handling a singular endpoint.

For the vector generalization −(P(x)u′)′+Q(x)u = λW(x)u where now u is a vector function of x, and P, Q, W are matrices, and for the corresponding higher-order vector self-adjoint problem, we outline the equally impressive advances in algorithms and theory.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.     

On the minimal reducible bound for outerplanar and planar graphs

Let L be the set of all additive and hereditary properties of graphs. For P₁, P₂ L we define the reducible property R = P₁ P₂ as follows: G P₁P₂ if there is a bipartition (V₁, V₂) of V(G) such that V₁ P₁ and V₂ P₂. For a property P L, a reducible property R is called a minimal reducible bound for P if P R and for each reducible property R′, R′ R → P R′. It is proved that the class of all outerplanar graphs has exactly two minimal reducible bounds in L. Some related problems for planar graphs are discussed.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

New constructions of divisible designs

A construction is given for a (p^2a(p+1),p²,p^2a+1(p+1),p^2a+1,p^2a(p+1)) (p a prime) divisible difference set in the group H×Z²_pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ₁≠0, and those are fairly rare. We also give a construction for a (p^a−1+p^a−2+…+p+2,p^a+2, p^a(p^a+p^a−1+…+p+1), p^a(p^a−1+…+p+1), p^a−1(p^a+…+p²+2)) divisible difference set in the group H×Z_p2×Z^a_p. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ₁=λ₂, and this corresponds to the parameters for the ordinary Menon difference sets.