Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.     

One property of best quadrature formulas

It is established that for classW _p ^r (r=i, 2, ...; 1?p?∞) the best quadrature formulas of the form $$\int_0^1 {f\left( x \right)dx = } \Sigma _{k = 0}^\rho \mathop {\Sigma _{i = 1}^n }\limits_{\left( {0 \leqslant \rho \leqslant r - 1} \right)} a_{ik} f^{\left( k \right)} \left( {x_i } \right) + R\left( f \right)$$ , when ρ = 2m and ρ = 2m + 1, coincide with one another. This same fact also supervenes for the class (r=1, 2, ...; 1?p?∞) of periodic functions.     

A substitute for the Sard-Smale theorem in theC 1 case

IfF is a Fredholm mapping of indexΝ ∃ ℤ and classC ^max(Ν,0)+1 between separable Banach spaces, the Sard—Smale theorem yields the existence of arbitrarily small perturbations ofF having 0 as a regular value. The smoothness requirement cannot be weakened in the Sard—Smale theorem itself, at least whenΝ ≥ 0, but we prove that the approximation result remains valid irrespective of the indexΝ whenF is only of classC ¹ and satisfies appropriate properness-like conditions. The separability of the spaces is not needed either. Everything carries over to the setting of Banach manifolds modeled on spaces with a norm of classC ¹ away from the origin. We also show that in Banach spaces, theC ¹ norm assumption can be dropped without major prejudice. The application to degree theory forC ¹ Fredholm mappings of index 0 is developed in a separate paper.     

Calcul fonctionnel a plusieurs variables pour des operateurs pseudodifferentiels dansR n

We study aC ^∞ functional calculus with several variables forv pseudodifferential operatorsP ₁, …,P _v inR ⁿ . Whenf is a function belonging to the classS _1.0 ^r (R ^v ) of Hörmander, we prove that, under some conditions,f(P ₁,…,P _v) is a pseudodifferential operator, and we give an asymptotic formula for its symbol.     

The lower order of functions of the classB

The class of functions Φ(z, t) defined for z∈ Cⁿ and t ≥0 such that the functions Φ(z, ¦w¦), w∈C, are plurisubharmonic in Cⁿ⁺¹ is called the classD. A typical example of functions of the classB are functions of the form \(\ln M_g (z,t) = \mathop {\ln \sup |}\limits_{|w| = t} g(z,w)|\) where g(z, w), z∈Cⁿ, w∈C, is an entire function in Cⁿ⁺¹. In this note it is proved under certain restrictions on the function Φ(z, t)εB that its lower order relative to the variable t is the same for all z∈Cⁿ except, possibly, for the points z of a set of zero Γ capacity. See [5].     

Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

We prove a strengthenedC ^r -closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC ^r -closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ ^k on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997. Translated by S. K. Lando     

Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Pic^d(C) such that h⁰(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in P^r as a projectively normal curve) if and only if (r+1)h¹≤g≤r(r−1)/2+2h¹, where h¹?g+r−d=h¹(C,L).     

Rigidity result on conjugacies of families of diffeomorphisms

Embedding flows are used to obtain a rigidity result on strongly topological conjugacy of families of diffeomorphisms, i.e. families of C^r(2?r?∞) diffeomorphisms, the strongly topologically conjugating homeomorphisms near degenerate saddle-nodes will be differentiable on center manifolds of the saddle-nodes.     

Topological stability for conservative systems

We prove that the C¹ interior of the set of all topologically stable C¹ incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.     

On spline function approximations to the solution of volterra integral equations of the first kind

A procedure, using spline functions of degreem, for the solution of linear Volterra integral equations of the first kind is presented. The method produces an approximate solution of classC ^m-1, is order (m+1) and is shown to be numerically stable form≦4.     

C 0 elements for generalized indefinite Maxwell equations

In this paper we develop the C ⁰ finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H ^r regularity for some r?<?1. The ingredients of our method are that two ??mass-lumping?? L ² projectors are applied to curl and div operators in the problem and that C ⁰ linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C ⁰ Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H ^r regularity where r may vary in the interval [0, 1), we obtain the error bound ${{\mathcal O}(h^r)}$ in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.     

On Riesz type kernels over local fields

Based on a representation lemma, Riesz type kernels on the local fieldK and on the integer ringO inK are constructed. Furthermore, we discuss approximation theorems for the Lipschitz classLip(L^r;a) and the L^p boundedness of such operators motivated by the open problem: Does $\sigma _n f\xrightarrow{{a,e}}f$ for f ∈ L¹(0) (see M. H. Taibleson [6] and [5])?     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Expectation of random polytopes

A random polytopeP _n in a convex bodyC is the convex hull ofn identically and independently distributed points inC. Its expectation is a convex body in the interior ofC. We study the deviation of the expectation ofP _n fromC asn→∞: while forC of classC ^k+1,k≥1, precise asymptotic expansions for the deviation exist, the behaviour of the deviation is extremely irregular for most convex bodiesC of classC ¹. Dedicated to my teacher and friend Professor Edmund Hlawka on the occasion of his 80^th birthday     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of classC ⁴ has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC ²-hypersurfaces.     

Covering numbers for Chevalley groups

LetG be a quasisimple Chevalley group. We give an upper bound for the covering number cn(G) which is linear in the rank ofG, i.e. we give a constantd such that for every noncentral conjugacy classC ofG we haveC ^rd =G, wherer=rankG. Research supported in part by NSERC Canada Grant A7251. Research supported in part by the Hermann Minkowski-Minerva Center for Geometry at Tel Aviv University.     

Regularity for some scalar variational problems under general growth conditions

We consider integrals of the calculus of variations over a set Ω of ? ⁿ , and the related regularity result: are the minimizers smooth functions, say for example of classC ^∞(Ω)? Classically, the so-called natural growth conditions on the integrand have been the main sufficient assumptions for regularity. In recent years, motivated also by application, the interest in the study of this problem has increased under more general growth assumptions. In this paper, we propose some general growth conditions that guarantee regularity for a class of scalar variational problems.     

Haefliger’s codimension-one singular foliations, open books and twisted open books in dimension 3

We consider singular foliations of codimension one on 3-manifolds, in the sense defined by André Haefliger as being ??₁-structures. We prove that under the obvious linear embedding condition, they are ??₁-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C ^r , r ?? 1. In particular, in dimension 3, this gives a very simple proof of Thurston??s 1976 regularization theorem without using Mather??s homology equivalence.     

An extension of the theory of Fredholm determinants

Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form ε μ(dω) ?_ω, where ?_ω φ(x) = ?_ω(x). φ(ψ_ω x), , φ belongs to a space of Hölder or C ^r functions, ?_ω is Hölder or C ^r , and ψ_ω is a contraction or a C ^r contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps.