Monotone penalty approximation of extremal solutions for quasilinear noncoercive variational inequalities

This paper is about a monotone approximation scheme for extremal (least or greatest) solutions of the following variational inequality:$\text{u∈K:〈Au+F(u),v−u〉⩾0,}\text{∀v∈K,}$in the interval between some appropriately defined sub- and supersolutions. The variational inequality is approximated by a sequence of penalty equations. The extremal solutions of the penalty equations, constructed iteratively and forming a monotone sequence, are proved to converge to the corresponding solutions of the original inequality. We note that no monotoneity assumption on the lower-order term F is imposed.     

Geometric approach to a dilation theorem

Problem: Given operators A_j ? O on Hilbert space $\text{H}$, with ΣA_j = 1, to find commuting projectors E_j on a Hilbert space $\text{H}$ ? $\text{H}$ such that (for all _j) x¹A_jy = x¹E_jy for, x, y ∈ $\text{H}$. This paper gives an explicit construction, quite different from the familiar solution.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Matrices diagonally similar to a symmetric matrix

Let $\text{F}$ be field, and let A and B be n × n matrices with elements in $\text{F}$. Suppose that A is completely reducible and that B is symmetric. If the principal minors of A determined by the 1- and 2-circuits of the graph of B and by the chordless circuits of the graph of A are equal to the corresponding principal minors of B, then A is diagonally similar to B; and conversely.     

A note on matrix subalgebras containing a full Jordan block

Let A be a subalgebra of the full matrix algebra M_n(F), and suppose J∈A, where J is the Jordan block corresponding to xⁿ. Let $\text{S}$ be a set of generators of A. It is shown that the graph of $\text{S}$ determines whether A is the full matrix algebra M_n(F).     

Toeplitz matrices and determinants with Fisher-Hartwig symbols

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.     

Complexity of representation of graphs by set systems

Let $\text{F}$ be a family of subsets of S and let G be a graph with vertex set $\text{V={x}{}_{\mathrm{A}}\text{|A ∈}\text{F}\text{}}$ such that: (x_A, x_B) is an edge iff $\text{A?B≠}\text{0}\text{/}$. The family $\text{F}$ is called a set representation of the graph G.It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets.The set representations of random graphs are also considered.     

Structure of H-classes in the semigroup of nonnegative matrices

This paper investigates the structure of the $\text{H}$-classes in the semigroup N_n of nonnegative matrices. We obtain two sets of equivalent conditions for any two matrices A,B to satisfy $\text{A}\text{H}\text{B in N}{}_{\mathrm{n}}$. We establish a one-to-one and onto correspondence between the $\text{H}$-class $\text{H}$_A and the group W_A0 of the greatest cone independent submatrix A₀ of A. We find W_A0 can be made up from the groups of the connective submatrices of A₀.     

On ranges of Lyapunov transformations

A Lyapunov transformation is a linear transformation on the set $\text{H}$_n of hermitian matrices H ? $\text{C}$^n,n of the form $\text{L}$_A(H) = A¹H + HA, where A ?$\text{C}$^n,n. Given a positive stable A ?$\text{C}$^n,n, the Stein-Pfeffer Theorem characterizes those K ? $\text{H}$_n for which K = $\text{L}$_B(H), where B is similar to A and H is positive definite. We give a new proof of this result, and extend it in several directions. The proofs involve the idea of a controllability subspace, employed previously in this context by Snyders and Zakai.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.     

A presentation for the special automorphism group of a free group

A presentation is given for SA_n, the group of automorphisms of determinant 1 of a free group F_n of rank n. The canonical isomorphisms $\text{H}{}_2\text{(A}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{H}{}_2\text{(SA}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{K}{}_2\text{(}\text{Z}\text{)}$ are established for n ≥ 5, where A_n is the full group of automorphisms of F_n.     

Some special vapnik-chervonenkis classes

For a class $\text{C}$ of subsets of a set X, let V($\text{C}$) be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ $\text{C}$. The condition V($\text{C}$) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈$\text{C}$, then $\text{V(}\text{C}\text{)=2}$ if and only if $\text{C}$ is linearly ordered by inclusion. If $\text{C}$ is of the form $\text{C}\text{={∩}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{C}{}_{\mathrm{i}}\text{:C}{}_{\mathrm{i}}\text{∈}\text{C}{}_{\mathrm{i}}$, i=1,2,…,n}, where each $\text{C}{}_{\mathrm{i}}$ is linearly ordered by inclusion, then $\text{V(}\text{C}\text{)?n+1}$. If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and $\text{C}$ is the collection of all sets {x: f(x)>0} for f in H, then $\text{V(}\text{C}\text{)=n}$.     

Projective repesentations of the Hilbert Lie group U (H)2 via quasifree statres on the CAR algebra

The group $\text{U}$ (H)₂ of unitary operators (on a Hilbert space H) which differ from the identity by a Hilbert-Schmidt operator may be imbedded in the group of Bogoliubov automorphisms of the CAR algebra over H in such a way as to be weakly inner in any gauge-invariant quasifree representation. Consequently each such quasifree representation determines a projective representation of $\text{U}$ (H)₂. If 0 ? A ? I is the operator on H determining the quasifree representation π_A and ?_A denotes the cyclic projective representation of $\text{U}$ (H)₂ generated from the G.N.S. cyclic vector $\text{Ω}{}_{\mathrm{A}}\text{for}\text{π}{}_{\mathrm{A}}$, then the 2-cocycle in $\text{U}$ (H)₂ determined by ?_A can be given explicitly. We prove that this 2-cocycle is a coboundary if any only if A or 1 ? A is Hilbert-Schmidt. The representations ?_A, on restriction to the group $\text{U}$ (H)₁ consisting of unitaries which differ from the identity by a trace class operator, always determine 2-cocycles which are coboundaries. These representations of $\text{U}$ (H)₁ have already been investigated by 21., 22., 87–110). Thus the Stratila-Voiculescu representations of $\text{U}$ (H)₁ always extend to projective representations of $\text{U}$ (H)₂ and to ordinary representations when A or 1 ? A is Hilbert-Schmidt. This fact enables exploitation of the type analysis of Stratila and Voiculescu to determine the type of the von Neumann algebra ρ_A($\text{U}$(H)₂)″. In the special case where 0 and 1 are not eigenvalues of $\text{A, Ω}{}_{\mathrm{A}}$ is cyclic and separating for ρ_A($\text{U}$(H)₂)″ and hence determines a K.M.S. state on this algebra. It is shown that for special choices of A, type III_λ (0 < λ ? 1) factors ρ_A($\text{U}$(H)₂)″ may be constructed.     

C1-algebras with finite duals

A forest is a finite partially ordered set F such that for x, y, z?F with x ? z, y ? z one has x ? y or y ? x. In this paper we give a complete characterization of all separable C¹-algebras $\text{A}$ with a finite dual $\text{A}\text{?}$, for which Prim $\text{A}$ is a forest with inclusion as partial order. These results are extended to certain separable C¹-algebras $\text{A}$ with a countable dual$\text{A}\text{?}$. As an example these results are used to characterize completely all separable C¹-algebras $\text{A}$ with a three point dual.     

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The permanent function is used to determine geometrical properties of the set $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then $\text{F}$ corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of $\text{F}$. If A is fully indecomposable, then the dimension of $\text{F}$ equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of $\text{Ω}{}_{\mathrm{n}}$ are triangles and rectangles. For n ? 6, $\text{Ω}{}_{\mathrm{n}}$ has four types of three-dimensional faces. The facets of the faces of $\text{Ω}{}_{\mathrm{n}}$ are characterized. Faces of $\text{Ω}{}_{\mathrm{n}}$ which are simplices are determined. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is two-neighborly but not a simplex, then $\text{F}$ has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2^k. All k-dimensional faces with at least 2^k?1 + 1 vertices are determined.     

Existence and regularity of extrema

Consider the minimization of a possibly noncoercive Gâteaux differentiable functional $\text{F:X→}\text{R}$. A modified notion of coercivity is introduced which may be usable to show existence of a minimum. Alternatively, $\text{F}\text{?}\text{:D→}\text{R}$ has a minimum at $\text{y}\text{ε}$$\text{D}$ ($\text{F}\text{?}$ not differentiable but the restriction F of $\text{F}\text{?}$ to $\text{X}$?$\text{D}$ differentiable), one may be able to show $\text{y}\text{?}$ is actually in $\text{X}$. The latter case is related to justification of formally calculated “necessary conditions” for optimal controls. The arguments are applications of Ekeland's “approximate variational principle” (J. Math. Anal. Appl.47 (1974), 324–353).     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.