Division Theorem Research Articles

Théorème de division et stabilité en géométrie analytique locale

Using a division theorem of convergent power series with estimate of the norms on a fundamental system of poly-discs, we give a theorem allowing to “pass from formal to convergent”. Then, we study stable and flat morphisms between singular analytic space-germs.

The Signature of Symplectic Manifolds

The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.

The signature of symplectic manifolds

The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.

Isomorphic factorisations. I. Complete graphs

An isomorphic factorisation of the complete graphKp{K_p}is a partition of the lines ofKp{K_p}intotisomorphic spanning subgraphsG; we then writeG|KpG|{K_p}, andG∈Kp/tG \in {K_p}/t. If the set of graphsKp/t{K_p}/tis not empty, then of courset|p(p−1)/2t|p(p - 1)/2. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever(t,p)=1(t,p) = 1or(t,p−1)=1(t,p - 1) = 1. We give a new and shorter proof of her result which involves permuting the points and lines ofKp{K_p}. The construction developed in our proof happens to give all the graphs inK6/3{K_6}/3andK7/3{K_7}/3. The Divisibility Theorem asserts that there is a factorisation ofKp{K_p}intotisomorphic parts whenevertdividesp(p−1)/2p(p - 1)/2. The proof to be given is based on our proof of Guidotti’s Theorem, with embellishments to handle the additional difficulties presented by the cases whentis not relatively prime toporp−1p - 1.

The divisibility theorem for isomorphic factorizations of complete graphs

The divisibility theorem for isomorphic factorizations of complete graphs

Weierstrass Division in Quasianalytic Local Rings

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Convolution equations for vector-valued entire functions of nuclear bounded type

Given two complex Banach spaces E and F, convolution operators “with scalar coefficients” are characterized among all convolution operators on the space H N b ( E ′ ; F ) {H_{Nb}}(E’;F) of entire mappings of bounded nuclear type of E’ into F. The transposes of such operators are characterized as multiplication operators in the space E x p ( E ; F ′ ) Exp(E;F’) of entire mappings of exponential type of E into F’. The division theorem for entire functions of exponential type of Malgrange and Gupta is then extended to the case when one factor is vector-valued. With this tool the following “vector-valued” existence and approximation theorems for convolution equations are proved: THEOREM 1. Nonzero convolution operators “of scalar type” are surjective on H N b ( E ′ ; F ) {H_{Nb}}(E’;F) . THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in H N b ( E ′ ; F ) {H_{Nb}}(E’;F) by exponential-polynomial solutions.

Generalization of Hermite’s Divisibility Theorems and the Mann - Shanks Primality Criterion for s-Fibonomial Arrays

Generalization of Hermite’s Divisibility Theorems and the Mann - Shanks Primality Criterion for s-Fibonomial Arrays

Principal Local Ideals in Weighted Spaces of Entire Functions

This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let $\rho > 1$ and $q > 1$, and define ${E_I}[\rho ,q]$ (respectively, ${E_P}[\rho ,q]$) to be the space of all entire functions f of one variable which satisfy $|f(x + iy)| = O(\exp (A|x{|^\rho } + A|y{|^q}))$ for some (respectively, all) $A > 0$. It is shown that in each of the spaces ${E_I}[\rho ,q]$ and ${E_P}[\rho ,q]$, the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when $\rho \ne q$ a division theorem fails to hold for either space ${E_I}[\rho ,q]$ or ${E_P}[\rho ,q]$.

Principal local ideals in weighted spaces of entire functions

This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let p > 1 and q > 1, and define E,[p, q] (respectively, Ef[p, q]) to be the space of all entire functions f of one variable which satisfy lf(x + iy)j = O(exp (A lxlx + Al y lq )) for some (respectively, all) A > 0. It is shown that in each of the spaces E,[p, q] and E,[p, q], the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when p # q a division theorem fails to hold for either space E,[p, q] or E,[p, q].

Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability

The purpose of this paper is to prove that a proper, infinitesimally stable mapping of one finite dimensional Co manifold into another is stable. This result is stated as Theorem 1 of ? 3. The method we use to show this result proves somewhat more; two things which it proves are stated as Theorems 2 and 3 in ? 3. The interest of this is the following. Stable mappings play an important role in the theory of singularities of differentiable mappings (cf. [8] and [9]). Therefore it is desirable to have methods for proving that particular mappings are stable. The result of this paper is a step towards such methods, since it is generally easier to prove that a mapping f is infinitesimally stable than it is to prove that f is stable. This paper is organized as follows. In ? 1, we set down our notation and terminology. All of this is quite standard except that manifolds have corners. However it seemed desirable to be explicit, since the precise meaning of the terminology varies from author to author. In ? 2, we introduce the topology W = WO, on the space of Co mappings of one manifold into another. The main results (? 3) are formulated in terms of this topology. In general, composition is not continuous with respect to the topology W. A major part of ? 2 is devoted to proving that various restrictions of the composition mapping are continuous. In ? 3, we state the main results. Note that Theorem 3 trivially implies Theorems 1 and 2. We will refer to the previous paper [5] in this series as I. In ? 4, we apply the results of I. Proposition 1 is essentially a restatement of the division theorem in terms of topologies introduced in ? 2. Proposition 2, on the other hand, introduces something new, in that it enlarges the set of functions that we can divide by. In ? 5, we interpret a result of Seeley in terms of the topologies that we

Stability of C ∞ Mappings: I. The Division Theorem

Let f: Ni Pi (i = 1, 2) be Co mappings of Co manifolds. We will say fi and f2 are isomorphic if there exist Co diffeomorphisms h: N1 N2 and h': P1 P2 such that h' o f1 = fo h. The main notion studied in this series of papers will be that of (oo -structurally) stable mapping. Roughly speaking, a Co mapping f: N P is said to be (ao -structurally) stable if every Co mapping g: N o P which is sufficiently close to f in a suitable topology (the direct limit of the fine Ck topologies, as k Ao) is isomorphic to f (cf. Thom and Levine [7]). There is an infinitesimal notion corresponding to stability. Namely, we say a Co mapping f: N o P is infinitesimally stable if for every Co vector field w along f there exist Co vector fields u on N and v on P such that w = Tf a u + v o f. (Our terminology and notation is the following: TN denotes the tangent bundle of N. N, denotes the tangent space to N at n, for each n e N. A vector field along f is a mapping w: N TP such that w(n) e Pf(,) for all n e N. The mapping of tangent bundles induced by f is denoted Tf: TN-+ TP.) In Stability of Mappings: II, we will show that a proper, infinitesimally stable, Co mapping is stable. In a later paper we will show that a proper, stable, Co mapping is infinitesimally stable. Thus, a proper, Co mapping is stable if and only if it is infinitesimally stable. A very simple example will be given in Stability of Mappings: II to show that infinitesimal stability of f does not imply stability of f if the hypothesis that f is proper is dropped. Also, if this hypothesis is dropped, stability of f does not imply infinitesimal stability. B. Morin has constructed an example which shows this. The purpose of this paper is to state and prove an analytic result which will be used in our proof that a proper, infinitesimally stable, Co mapping is stable. The result is that there exist mappings Q and H satisfying a certain identity (equation (1) of ? 2) and having certain differentiability properties.

A Divisibility Theorem

A Divisibility Theorem

Approximation of certain functions by exponentials on a half line

Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)