Class Ck Research Articles

Complete set of steady states for the general stoichiometric dynamical system

The complete set of steady states for rate constants in the range 0?ki<∞ and concentrations in the range 0?Xi<∞ is given explicitly in parametric form for the general chemical reaction system. The only assumptions are that the stoichiometries are real numbers, and the reaction rates are proportional to functions of class Ck, k≳0; the functions are assumed to be positive in the interior of the domain. Hence, these results apply far more generally than just to chemical systems and should be valid for many ecological and economic models as well. The set of steady states in the interior of the domain is in general a simply connected differentiable manifold M of dimension n+r−d, where n = number of species, r = number of reactions, and d = rank ν (ν = stoichiometric matrix). The full set of steady states M* consists of those in the interior (M) plus a frequently very complicated set of steady states lying in the boundary. M* is the union of a set of differentiable manifolds but is not itself a differentiable manifold in general. If the reaction rate laws are all monomials (i.e., products of concentrations raised to some ’’order of kinetics’’), a working APL algorithm is given which provides enough information to construct explicitly the set of manifolds comprising M*. This algorithm is used to treat an early version of the Belousov–Zhabotinski reaction mechanism. The algorithm may be used for more general rate laws with minor modifications.

Famiglie di dischi analitici con bordo su sottovarietàCR non generiche

In this work the authors prove that all sufficiently small analytic discs in the tangent space of a non generic embedded CR manifold M′⊂CN (M′ at least of class C4) can be lifted uniquely to analytic discs in CN with boundaries on M′ Moreover if M′ is real analytic or C∞ then real analytic or C∞ discs lift without loss of derivatives. If M′ is of class CK then there is a 1+ε loss of derivatives in the lifting. A stability result is also proven.

Calculation of the Molien generating function for invariants of space groups

The new algorithms for computing the Molien generating function for a representation of a finite group obtained in the preceding paper are applied to obtain an expression which can be used for irreducible representation (*kn) of any crystallographic space group G. It proves convenient to express M (Γ,G;z) as a sum: M (Γ,G;z) =1/‖P‖Σkckm̄ (Γ,gk;z), where the partial Molien function m̄ (Γ;gk;z) is labelled by a coset representative gk carrying the class index k of the point group P=G/T, T being the translation group, the sum is over classes k, and ck is the order of class Ck(P). The resulting form was used to compute M (Γ,G;z) for irreducible representations *Γn, *Xn, *Rn of nonsymmorphic space group A-15 or O3h-Pm3n in which many high temperature superconducting crystals occur. Certain of these representations (matrix groups) are identified as generalized Coxeter groups, i.e., unitary groups generated by reflections. The Molien function for these groups as the required form given by Shephard and Todd: M (Γ,G;z) =[Πi(1−zdi)]−1. The di satisfy dimensionality theorems.

Erratum: Calculation of the Molien generating function for invariants of space groups

The new algorithms for computing the Molien generating function for a representation of a finite group obtained in the preceding paper are applied to obtain an expression which can be used for irreducible representation (*kn) of any crystallographic space group G. It proves convenient to express M (Γ,G;z) as a sum: M (Γ,G;z) =1/‖P‖Σkckm (Γ,gk;z), where the partial Molien function m (Γ;gk;z) is labelled by a coset representative gk carrying the class index k of the point group P=G/T, T being the translation group, the sum is over classes k, and ck is the order of class Ck(P). The resulting form was used to compute M (Γ,G;z) for irreducible representations *Γn, *Xn, *Rn of nonsymmorphic space group A‐15 or O3h‐Pm3n in which many high temperature superconducting crystals occur. Certain of these representations (matrix groups) are identified as generalized Coxeter groups, i.e., unitary groups generated by reflections. The Molien function for these groups as the required form given by Shephard and Todd: M (Γ...

On the multiplication of tensor fields

Let M be a paracompact n-dimensional manifold of class Ck+1, T(x) the tangent space of the point xeM and T(x) * the dual space. A real valued (p+r)-linear function 4(x) with p arguments in T(x) and r arguments in T(x) * is called a tensor of type (p, r) at the point x. The tensors of type (p, r) at x form a linear space Tp(x). The products of two tensors DE Tp(x) and 'E Ta(x) is defined by (Vk) (X; 61 .. * +q) t*1 . . . *8

On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures

THEOREM 1.1. Assume that (i) Md is a d-dimensional (d_ 2), complete Riemann manifold of class Ck, k> 2, with nonnegative sectional curvatures; (ii)f: Md -> Ed +e is a Ck isometric immersion of Md into a Euclidean space Ed + e, e >0; and (iii) the relative nullity v(p) of the immersion f at the point p C Md satisfies v(p) > Ofor all p. Let m=min v(p) for p C Md. Then f(Md) is Ckl nm-cylindrical; that is, there exists a complete Riemann manifold Md-m of class Ck-l such that Md, Ed+e, f have factorizations Md=Emx Md-m Ed+ e=EmxEd+ em, f=ixg, where i: Em ->Em is the identity map and g: Md-m Ed+ ermis a Ck-1 isometric immersion. Also, f(Md) is not (m + 1)-cylindrical.

On the differentiability of isometries

Let 9TC be an re-dimensional differentiable manifold of class Ck, k = l, • • • , oo, w (as usual O means analytic, and we use the conventions * ±m= and w + w=w for integers m). An automorphism of 9TC is a one-to-one map of 911 onto itself which is of class C* and is nonsingular. Now let 9TC be given a Riemannian structure, i.e. a positive definite covariant tensor field of rank two, and suppose that in some family of admissible coordinate systems that cover 9TC the components of this tensor field are of class Ck+1. This hypothesis will be satisfied, for example, if 911 is the manifold of class Ck associated with a manifold 9TC of class Ck+2 and the metric tensor field is a tensor field of highest possible class (namely Ck+1) on 911. Moreover this hypothesis implies that in the neighborhood of each point of 9TC we can define a Riemannian normal coordinate system and that such a coordinate system is admissible, i.e. of class C*. The Riemannian structure on 9H gives rise in a well-known way to a metric p on the underlying point set of 9ft, and we denote by M the metric space that results. An automorphism of M is of course a oneto-one map of M onto itself which preserves distance, i.e. an isometry. A remarkable result of Myers and Steenrod [l] states that an automorphism of M is always an automorphism of 9TC. It is natural to conjecture that the following stronger result is true.

On Singular Homology in Differentiable Spaces

In a recent paper [4], Eilenberg proved that in a manifold of class Uk the study of singular simplexes and the study of singular simplexes of class k in fact, we shall prove the theorem for a more general category of spaces, defined as follows. DEFINITION 1. A subset M of a euclidean space E' is said to be a differentiable space of class Ck if there exist an open set U of En containing M and a differentiable mapping 0: U -* M of class Ck such that 0(x) = x for each x e M. 0 is called a retraction of class Ck. It follows from a lemma of Whitney [5, p. 667] that every differentiable manifold M of class Ck in En is a differentiable space of class Ck. In the sequel, we shall assume M to be a given differentiable space of class Ck in the euclidean space En and 0 to be a given retraction of class Ck of some open set U onto M. LEMMA 1. Let X and Xo be closed bounded subsets of some euclidean space Em such that Xo C X. Let f: X -* M be a mapping which is of class Ck on Xo. Then there exists a homotopy ft: X M (O 0 such that the e-neighborhood N, of f(X) is contained in the open set U of En. According to Lemma 3 of Eilenberg [4, p. 673], there exists a mapping g: X -N. of class &k such that g(x) = f(x) for each x e Xo and that p(f(x), g(x)) < e for each x e X, where p denotes the distance function of En. Define a homotopy 4e: X -+ N. (O ? t < 1) by taking 4t(x) the point which divides the line-segment joining f(x) to g(x) in the ratio t:(1 t). Then the required homotopy ft: X -E M (O < t < 1) is given by ft = 0t . This completes the proof. Let X be a closed bounded subset of Em and 1 the closed interval (0, 1) of real numbers. Then the topological product X X I is a closed bounded subset of Em+l. DEFINITION 2. Two mappings f, g: X -* M of class Ck are said to be Ck -homotopic if there exists a mapping h: X X I -+ M of class Ck such that h(x, 0) = f(x) and h(x, 1) = g(x) for each x e X. LEMMA 2. Two mappings f, g: X -+ M of class Ck are Ck-homotopic if and only if they are homotopic. PROOF. The necessity is trivial. To prove the sufficiency, let us suppose