Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.     

The Cauchy-Green formula and rational approximation on the sets with a finite perimeter in the complex plane

Integral representations of Lipschitz functions on the sets with a finite perimeter in $\text{C}$ are studied. These formulas can be viewed as generalizations of the classical Cauchy-Green theorem. Also, it is shown that those results lead to a convenient approach to certain problems in the theory of rational approximation.     

Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions

The stability of systems of ordinary differential equations of the form

$\text{dx}\text{dt}\text{= f(t, x, y, ?), ?}\text{dy}\text{dt}\text{= g(t, x, y, ?)}$

, where ? is a real parameter near zero, is studied. It is shown that if the reduced problem

$\text{dx}\text{dt}\text{= f(t, x, y, 0), 0 = g(t, x, y, 0)}$

, is stable, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied, then the full problem is asymptotically stable as t → ∞, and a domain of stability is determined which is independent of ?. Moreover, under certain additional conditions, it is shown that the solutions of the perturbed problem have limits as t → ∞. In this case, it is shown how these limits can be calculated directly from the equations

$\text{f(∞, x, y, ?) = 0 g(∞, x, y, ?) = 0}$

as expansions in powers of ?.     

The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” [J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms

$\text{th}{}_{\mathrm{r}}\text{: MU}{}_{\mathrm{s}}\text{(X) → H}{}_{\mathrm{s\to r}}\text{(X;}\text{Q}\text{)}$

.In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc.57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.     

Double unconditional bases associated with a quadratic characteristic parameter problem

A detailed discussion is given of the quadratic characteristic value problem, $\text{x = λPx + (}\text{1}\text{λ}\text{)Qx}$, where P and Q are compact selfadjoint operators in Hilbert space. Theorems are proven showing that unconditional bases of various types may be constructed from the root vectors of $\text{A(λ) = λP + (}\text{1}\text{λ}\text{)Q}$. These theorems are of interest in the stability theory of linear evolution equations, particularly for application to certain problems in the linearized theory of hydrodynamic stability.     

Second-order terms for the variances and covariances of the number of prime factors—Including the square free case

We obtain second-order terms for the variance and covariance of $\text{Ω(n)}$ and ω(n), the number of prime divisors counted with and without multiplicity, and connect these results to a formula of Renyi. We discuss the heuristic connection with the Landau-Sathe extension of the prime number theorem and develop new expansions for the mean and variance of ω(n) in the square free case.     

Uniqueness for algebraically regular solutions to pathological differential equations

A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices $\text{M}$ with standard norm. It is not assumed that $\text{A}$ ∈ $\text{M}$, but it is required that A have an inverse in $\text{M}$, a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary.The theorem may be easily generalized to any Banach algebra with identity.     

Vector structures and solutions of linear matrix equations

A variety of systems problems give rise to special cases of the linear matrix equation

$\text{∑}\text{i}\text{A}{}_{\mathrm{i}}\text{p×s}\text{X}\text{s×t}\text{B}{}_{\mathrm{i}}\text{t×q}\text{+}\text{∑}\text{j}\text{C}{}_{\mathrm{j}}\text{p×t}\text{X}{}^{\mathrm{T}}\text{t×s}\text{D}{}_{\mathrm{j}}\text{s×q}\text{=}\text{F}\text{p×q}\text{(Z)}$

. F(Z) may be a matrix-valued function of a matrix argument Z. It is the purpose here to summarize and extend some of the applicable solution procedures through a systematic use of operators which convert the matrix equation to vector and dimension-reduced vector forms. The format and details of the results are convenient for machine computation.     

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

Circulants,inversion of circulants,and some related matrix algebras

The spectral theory for circulants over $\text{R}$ or $\text{C}$ is discussed, followed by a discussion of inversion of circulants. The algebraic theory of one-generator matrix algebras and applications to circulants and some related algebras are presented. The theory of symmetric functions plays an important role. Thus the latter methods apply over arbitrary commutative rings.     

On product formulas for nonlinear semigroups

We study a number of sufficient conditions which guarantee the convergence of semigroup product formulas of the type

$\text{H}{}_{\mathrm{t}}\text{=}\text{lim}\text{n→∞}\text{(F}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{°G}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{)}{}^{\mathrm{n}}$

and its generalizations. Our hypotheses differ from those of other authors in that we do not assume in advance that the limit operator is a generator. Rather this is a consequence and hence the above formula yields an existence theorem (local in time) for nonlinear semigroups. A number of smoothness properties are studied as well. The results may be applied to and are motivated by the Navier-Stokes equations.     

Common solutions to the Lyapunov equations

Let A be an n×n matrix with complex entries. A necessary and sufficient condition is established for the existence of a Hermitian solution H to the equations

$\text{AH+HA}{}^1\text{=HA+A}{}^1\text{H=I}$

.     

On the rational spectra of graphs with abelian singer groups

Let G be a finite abelian group. We investigate those graphs $\text{G}$ admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra $\text{P}$(G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in $\text{P}$(G) for various G's, we relate subschemes of a related association scheme, subalgebras of $\text{P}$(G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors.     

Stability of a chain of linear differential equations

The behavior of an infinite sequence of ordinary differential equations of the form:

$\text{dXn}\text{dt}\text{=}\text{∑}\text{i=?M}\text{N}\text{L}{}_{\mathrm{i}}\text{X}{}_{\mathrm{i+n}}\text{, 0 ? n, 0 < N, M < ∞}$

,

$\text{X}{}_{\mathrm{n}}\text{(0) = C}{}_{\mathrm{n}}\text{, (1) X}{}_{\mathrm{n}}\text{≡ 0, n < 0}$

, where X_n(t) is a vector valued function of R⁺, is studied in spaces of infinite sequences of vectors. In particular, sufficient conditions for asymptotic stability of this sequence of linear equations are established and applied to the stability analysis of a string of vehicles with a simple form of automatic control.     

Approximation numbers of bounded operators

The theory of ideals of linear operators is well developed and has a lot of applications in theory and practise. The purpose of this paper is to give a first idea of a similar theory for bounded (nonlinear) operators. In view of applications we will not give an abstract (perhaps general nonsense) theory, but an example of a class λ_p of bounded operators with a structure similar to an $\text{L}$-module($\text{L}$ represents the class of all linear operators between Banach spaces), and applications to projection methods for solving equations with λ_p-type operators.     

A comparison result for a class of quasilinear elliptic partial differential equations

A comparison theorem and a uniqueness corollary for positive solutions to the equation

on the closure of a bounded open set are found. The important hypotheses on the nonlinear coefficients are that each p_i is positive and monotone increasing in u while q is monotone decreasing in u. An application is made to equations arising in the theory of chemical reactors.     

Self-adjoint extensions of the Laplace operator with respect to electric and magnetic boundary conditions

Systems of the type {$\text{A, KJ, 2iK1, I}$} are studied where $\text{J  J1}$ is unitary and $\text{KJK1 }\text{Im}\text{A}$. A complete realization theory as well as state space isomorphism theorem are given. Coupling problems are considered.     

On a neutral functional differential equation in a fading memory space

The linear autonomous, neutral system of functional differential equations $\text{d}\text{dt}\text{(μ 1 x(t) + ?(t)) = v 1 s(t) + g(t) (t ? o)}$, (1) x(t) = ?(t) (t ? 0), in a fading memory space is studied. Here μ and ν are matrix-valued measures supported on [0, ∞), finite with respect to a weight function, and $\text{?, g,}\text{and}\text{?}$ are Cⁿ-valued, continuous or locaily integrable functions, bounded with respect to a fading memory norm. Conditions which imply that solutions of (1) can be decomposed into a stable part and an unstable part are given. These conditions are of frequency domain type. The usual assumption that the singular part of μ vanishes is not needed. The results can be used to decompose the semigroup generated by (1) into a stable part and an unstable part.     

Some Extensions of Loewner's Theory of Monotone Operator Functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.