A Liapunov functional for a matrix neutral difference-differential equation with one delay

For the matrix neutral difference-differential equation $\text{x}\text{?}\text{(t) + A}\text{x}\text{?}\text{(t ? τ)  Bx(t) + Cx(t ? τ)}$ we construct a quadratic Liapunov functional which gives necessary and sufficient conditions for the asymptotic stability of the solutions of that equation. We consider a difference equation approximation of the difference-differential equation, and for this difference equation we construct a Liapunov function from which we obtain the desired Liapunov functional by an appropriate limiting process. The Liapunov functional thus obtained gives the best possible estimate for the rates of growth or decay of the solutions of the matrix neutral difference-differential equation. The results obtained are natural generalizations of previous results obtained for a matrix retarded difference-differential equation with one delay.     

Approximate and complete controllability of nonlinear systems to a convex target set

The research deals with complete and approximate controllability of the system $\text{(1)}\text{dx}\text{dy}\text{= f(t, x, u)}$, without control restraints to an arbitrary convex target set. First, some characterizations of complete controllability, to the target of (1) and a special case of (1) namely $\text{x}\text{?}\text{= A(t)x + k(t, u)}{}^7$ are given. As a consequence complete controllability is equivalent to null-controllability. Next certain equations are formulated. These are in the same spirit as J. P. Dauer's “A Controllability Technique for Nonlinear Systems” (J. Math. Anal. Appl. oo (1972), 442–451) and are utilized in the main contribution of the paper: Under certain convexity assumption, bounded perturbations of systems which are completely controllable to a fixed target G are completely controllable to G. Without the convexity assumption, but with perturbations satisfying a Lipschitz condition, approximate controllability to G of a perturbed system is equivalent to complete controllability to G of the unperturbed equation.     

Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups

The theory of symmetric local semigroups due to A. Klein and L. Landau (J. Funct. Anal.44 (1981), 121–136) is generalized to semigroups indexed by subsets of $\text{R}$ⁿ for n > 1. The result implies a similar result of A. E. Nussbaum (J. Funct. Anal.48 (1982), 213–223). It is further generalized to semigroups that are symmetric local in some directions and unitary in others. The results are used to give a simple proof of A. Devinatz's (Duke Math. J.22 (1955), 185–192) and N. I. Akhiezer's (“the Classical Moment Problem and Some Related Questions,” Hafner, New York, 1965) generalization of a theorem of Widder concerning the representation of functions as Laplace integrals. This result is extended to the representation as a Laplace integral of a function taking values in $\text{B}$($\text{R}$), the set of bounded linear operators on a Hilbert space $\text{R}$. Also, a theorem is proved encompassing both the result of Devinatz and Akhiezer, and Bochner's theorem on the representation of positive definite functions as Fourier integrals.     

On the C1 normal forms for hyperbolic vector fields

Given two germs of hyperbolic vector fields associated to autonomous ordinary differential equations $\text{x}\text{?}\text{=Ax+?}$ and $\text{y}\text{?}\text{=By+?}$, where $\text{x,y∈}\text{R}{}^{\mathrm{n}}$, and A and B are n×n matrices, we prove that under some algebraic conditions on the eigenvalues of the matrices and genericity condition on the nonlinear terms, they are at least C¹ conjugate if and only if A and B are similar. To cite this article: Z. Ren, J. Yang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

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).     

On pancyclic digraphs

We show that a strongly connected digraph with n vertices and minimum degree ? n is pancyclic unless it is one of the graphs K_p,p. This generalizes a result of A. Ghouila-Houri. We disprove a conjecture of J. A. Bondy by showing that there exist hamiltonian digraphs with n vertices and $\text{1}\text{2}\text{n(n + 1) – 3}$ edges which are not pancyclic. We show that any hamiltonian digraph with n vertices and at least $\text{1}\text{2}\text{n(n + 1) – 1}$ edges is pancyclic and we give some generalizations of this result. As applications of these results we determine the minimal number of edges required in a digraph to guarantee the existence of a cycle of length k, k ? 2, and we consider the corresponding problem where the digraphs under consideration are assumed to be strongly connected.     

Residue properties of certain quadratic units

Explicit formulas are given for the quadratic and quartic characters of units of certain quadratic fields in terms of representations by positive definite binary quadratic forms, as conjectured by Leonard and Williams (Pacific J. Math.71 (1977), Rocky Mountain J. Math.9 (1979)), and by Lehmer (J. Reine Angew. Math.268/69 (1974)). For example, if p and a are primes such that p≡1 (mod 8), q≡5 (mod 8) and the Legendre symbol $\text{(}\text{q}\text{p}\text{)=1}$, and if ε is the fundamental unit of $\text{Q}$(√q), then $\text{(}\text{ε}\text{p}\text{)}{}_4\text{=(?1)}{}^{\mathrm{b+d}}$, where p=a²+16b² and p^k=c²+16qd² with k odd.     

On orthogonal groups over the integers of a non-dyadic local field

Let L be a lattice in a quadratic space over a non-dyadic local field. We shall answer the question: What are the lattices whose unit groups coincide with that of L? If the residue class field has more than three elements the question is easy. In this case such a lattice must be $\text{a}$L or $\text{a}$L^# with a fractional ideal $\text{a}$ and the dual lattice L^# by Satz 2 of A. Kallmann, M. Kneser, and U. Stuhler (J. Reine Angew. Math.258 (1978), 51–54) or Theorem 5.2 of C. R. Riehm (Amer. J. Math.89 (1967), 549–577). But it is not easy in the case of the residue class field of three elements.     

Contractive Korovkin approximations

Our results are related to $\text{L}$¹-shadows in L_p-spaces. For p = 1 we will complete the characterization of $\text{L}$¹-shadows and $\text{L}$^1,1-shadows. For 1 < p < ∞ S. J. Bernau has shown that the $\text{L}$¹-shadow of a set in L_p is the range of a contractive projection. We will show that the corresponding theorem is not true for all reflexive spaces, but is true for locally uniformly convex reflexive spaces.     

Coloring n-sets red and blue

A short proof is given of the following result of J. Beck. Given any $\text{m = 2}{}^{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext><mtext>+o(1)</mtext>}}$ sets of cardinality n, there exists a 2-coloring of the underlying points so that no set is monochromatic.     

A theorem of the transversal theory for matroids of finite character

Let M = (S, I) be a matroid of finite character on the infinite set S. Let $\text{A}\text{= 〈A}{}_1\text{:i ∈ I〉}$ be any system of subsets of S each having finite rank and let $\text{B}\text{= 〈B}{}_1\text{: j ∈ J〉}$ be a finite system of sets of arbitrary rank. Necessary and sufficient conditions are given for the system $\text{A}$ ? $\text{B}$ to have an independent system of distinct representatives.     

Decay of solutions of nonlinear wave equations in three space dimensions

Given a solution of the Cauchy problem for nonlinear wave equations of the type $\text{?}{}^2\text{u}\text{?t}{}^2\text{? Δu + f(u) = 0}$ in three space dimensions the asymptotic behaviour in time is considered. It is shown that for nonlinearities which behave like powers $\text{¦u¦}{}^{\mathrm{\sigma \; ?\; 1}}\text{u}$ uniform decay holds with a certain rate depending on σ if $\text{5 > σ >}\text{1}\text{2}\text{+}\text{1}\text{2}\text{√13}$, and moreover scattering states exist if σ is not too small. This improves former results of W. A. Strauss (J. Funct. Anal. 2 (1968), 409–457).     

A new algorithm for summing divergent series Part 1. Basic theory and illustrations

If $\text{S}\text{(}\text{1}\text{n}\text{)}$ is a descending series in n for a Stieltjes continued fractions, polynomials A(n), B(n) are chosen so that $\text{A}\text{(}\text{n}\text{) +}\text{B}\text{(}\text{n}\text{)}\text{S}\text{(}\text{1}\text{n}\text{)}$ has a set of coefficients of powers of n (and n^?1) equal to those in a given divergent series $\text{T}\text{(}\text{1}\text{n}\text{)}$. The polynomials A(n), B(n) are related to the continued fraction convergents. A number of choices of S are discussed and asymptotic formulae developed.     

Instability of certain traveling wave solutions to the Fitz Hugh-Nagumo Nerve Axon equations

A study is made of the existence and stability of certain traveling wave solutions to the Fitz Hugh-Nagumo system $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(u ? a)(1 ? u) ? v,}\text{?v}\text{?t}\text{= bu}$. In particular it is shown that for the wave speed c sufficiently small such waves are unstable in the temporal linearized sense. The study complements recent work of J. M. Greenberg and D. A. Larson.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.     

The free fermion field as a Markov field

A definition of a Markov field is given which allows for noncommuting fields. In the commutative case, we recover Nelson's definition (E. Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis12 (1973), 97–112). Conditional expectations are shown to exist in a regular probability gage space, and, using an independence property of these in the free fermion gage space, it is shown that the free fermion field over $\text{H}$^?1($\text{R}$^d) is a Markov field.     

Convex sets of some doubly stochastic matrices

Let Ω denote the set of all n by n doubly stochastic matrices. Let t be a real number such that $\text{1}\text{t}\text{?}\text{1}\text{n}$ and let m be a real number such that $\text{1}\text{m}\text{? 1 ?}\text{1}\text{t}$. The set $\text{Ω}{}_{\mathrm{s}}\text{= {A ? Ω :}\text{1}\text{m}\text{? a}{}_{\mathrm{ij}}\text{?}\text{1}\text{t}\text{, 1 ? i, j ? n}}$ is the convex hull of the matrices in $\text{Ω}{}_{\mathrm{s}}$ having as many largest entries, namely, $\text{1}\text{t}$, as possible in each row and column while filling out the remaining entries with the value $\text{1}\text{m}$ and if necessary at most one entry in each row and column which has a value between $\text{1}\text{m}$ and $\text{1}\text{t}$.     

Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.