A geometric approach to regular perturbation theory with an application to hydrodynamics
Carmen
Chicone
4559-4598
Abstract: The Lyapunov-Schmidt reduction technique is used to prove a persistence theorem for fixed points of a parameterized family of maps. This theorem is specialized to give a method for detecting the existence of persistent periodic solutions of perturbed systems of differential equations. In turn, this specialization is applied to prove the existence of many hyperbolic periodic solutions of a steady state solution of Euler's hydrodynamic partial differential equations. Incidentally, using recent results of S. Friedlander and M. M. Vishik, the existence of hyperbolic periodic orbits implies the steady state solutions of the Eulerian partial differential equation are hydrodynamically unstable. In addition, a class of the steady state solutions of Euler's equations are shown to exhibit chaos.
A genealogy for finite kneading sequences of bimodal maps on the interval
John
Ringland;
Charles
Tresser
4599-4624
Abstract: We generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval, building each sequence uniquely from a pair of shorter ones. There is a single pair at generation 0, with members of length $ 1$. Concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits. (See Figure 0.) Figure 0. Loci of some finite kneading sequences for a two-parameter cubic family
A categorical approach to matrix Toda brackets
K. A.
Hardie;
K. H.
Kamps;
H. J.
Marcum
4625-4649
Abstract: In this paper we give a categorical treatment of matrix Toda brackets, both in the pre- and post-compositional versions. Explicitly the setting in which we work is, à la Gabriel-Zisman, a $2$-category with zeros. The development parallels that in the topological setting but with homotopy groups replaced by nullity groups of invertible $ 2$-morphisms. A central notion is that of conjugation of $2$-morphisms. Our treatment of matrix Toda brackets is carried forward to the point of establishing appropriate indeterminacies.
Topological entropy for finite invariant subsets of $Y$
Shi Hai
Li;
Xiang Dong
Ye
4651-4661
Abstract: Let $Y$ be the space $\{ z \in {\mathbf{C}}:{z^3} \in [0,1]\}$ with a metric defined by the arc length. Suppose that $f$ is a continuous map from $Y$ to itself and $P$ is a finite $f$-invariant subset. In this paper we construct a continuous map ${C_P}$ from $Y$ to itself satisfying ${C_P}{\vert _P} = f{\vert _P}$ which achieves the infimum topological entropies of continuous maps from $ Y$ to itself which agree with $f$ on $P$.
Periodic orbits of $n$-body type problems: the fixed period case
Hasna
Riahi
4663-4685
Abstract: This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form $\displaystyle ({\text{A}})\quad {\text{ }}\left\{ {\begin{array}{*{20}{c}} {{m_... ...q) = 0} {q(t + T) = q(t),\quad \forall t \in \Re .} \end{array} } \right.$ where $ {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n})$ and with $ {V_{ij}}(t,\xi )$ $ T$-periodic in $ t$ and singular in $ \xi$ at $\xi = 0$ Under additional hypotheses on $V$, when (A) is posed as a variational problem, the corresponding functional, $I$, is shown to have an unbounded sequence of critical values if the singularity of $V$ at 0 is strong enough. The critical points of $I$ are classical $T$-periodic solutions of (A). Then, assuming that $ I$ has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index $ k$ grows exponentially with $k$, at least when $k \equiv 0,1( \mod \ell - 2)$. The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the $3$-body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the $4$-body problem then generalized to the $ n$-body problem.
$\scr A$-generators for the Dickson algebra
\textviet{Nguyễn H. V.}
Hu’ng;
Franklin P.
Peterson
4687-4728
Abstract: Let ${D_k}$ denote the Dickson algebra in $ k$ variables over the field of two elements. We study the problem of determining a minimal set of generators for ${D_k}$ as a module over the Steenrod algebra $\mathcal{A}$. This is easy for $k = 1$ and $2$. In this paper we answer this question for $k = 3$ and $4$ and give techniques which may help solve the problem for general $k$.
Courbures scalaires des vari\'et\'es d'invariant conforme n\'egatif
Antoine
Rauzy
4729-4745
Abstract: In this paper, we are interested in the problem of prescribing the scalar curvature on a compact riemannian manifold of negative conformal invariant. We give a necessary and sufficient condition when the prescribed function $ f$ is nonpositive. When $\sup(f) > 0$, we merely find a sufficient condition. This is the subject of the first theorem. In the second one, we prove the multiplicity of the solutions of subcritical (for the Sobolev imbeddings) elliptic equations. In another article [8], we will prove the multiplicity of the solutions of the prescribing curvature problem, i.e. for a critical elliptic equation.
Shadow forms of Brasselet-Goresky-MacPherson
Belkacem
Bendiffalah
4747-4763
Abstract: Brasselet, Goresky and MacPherson constructed an explicit morphism, providing a De Rham isomorphism between the intersection homology of a singular variety $X$ and the cohomology of some complex of differential forms, called "shadow forms" and generalizing Whitney forms, on the smooth part of $X$. The coefficients of shadow forms are integrals of Dirichlet type. We find an explicit formula for them; from that follows an alternative proof of Brasselet, Goresky and MacPherson's theorem. Next, we give a duality formula and a product formula for shadow forms and construct the correct algebra structure, for which shadow forms yield a morphism.
Equivalence relations induced by actions of Polish groups
Sławomir
Solecki
4765-4777
Abstract: We give an algebraic characterization of those sequences $({H_n})$ of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of ${H_0} \times {H_1} \times {H_2} \times \cdots$ are Borel. In particular, the equivalence relations induced by Borel actions of $ {H^\omega }$, $ H$ countable abelian, are Borel iff $H \simeq { \oplus _p}({F_p} \times \mathbb{Z}{({p^\infty })^{{n_p}}})$, where $ {F_p}$ is a finite $ p$-group, $\mathbb{Z}({p^\infty })$ is the quasicyclic $ p$-group, ${n_p} \in \omega$, and $p$ varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.
Periods for transversal maps via Lefschetz numbers for periodic points
A.
Guillamon;
X.
Jarque;
J.
Llibre;
J.
Ortega;
J.
Torregrosa
4779-4806
Abstract: Let $f:M \to M$ be a ${C^1}$ map on a ${C^1}$ differentiable manifold. The map $ f$ is called transversal if for all $m \in \mathbb{N}$ the graph of ${f^m}$ intersects transversally the diagonal of $M \times M$ at each point $(x,x)$ such that $x$ is a fixed point of ${f^m}$. We study the set of periods of $f$ by using the Lefschetz numbers for periodic points. We focus our study on transversal maps defined on compact manifolds such that their rational homology is $ {H_0} \approx \mathbb{Q}$, ${H_1} \approx \mathbb{Q} \oplus \mathbb{Q}$ and ${H_k} \approx \{ 0\}$ for $ k \ne 0,1$.
Radially symmetric internal layers in a semilinear elliptic system
Manuel A.
del Pino
4807-4837
Abstract: Let $B$ denote the unit ball in ${R^N},\quad N \geqslant 1$. We consider the problem of finding nonconstant solutions to a class of elliptic systems including the Gierer and Meinhardt model of biological pattern formation, $\displaystyle (1.1)\qquad {\varepsilon ^2}\Delta u - u + \frac{{{u^2}}} {{1 + k{u^2}}} + p = 0\quad {\text{in}}B,$ $\displaystyle (1.2)\quad D\Delta v - v + {u^2} = 0\quad {\text{in}}B,$ $\displaystyle (1.3)\quad \frac{{\partial u}} {{\partial n}} = 0 = \frac{{\partial v}} {{\partial n}}\quad {\text{on}}\partial B,$ where $ \varepsilon$, $ D$, $k$ and $\rho$ denote positive constants and $n$ the unit outer normal to $\partial B$. Assuming that the parameters $ \rho$, $k$ are small and $D$ large, we construct a family of radially symmetric solutions to (1.1)-(1.3) indexed by the parameter $ \varepsilon$, which exhibits an internal layer in $B$, as $ \varepsilon \to 0$.
Multiple solutions for a semilinear elliptic equation
Manuel A.
del Pino;
Patricio L.
Felmer
4839-4853
Abstract: Let $\Omega$ be a bounded, smooth domain in ${\mathbb{R}^N}$, $N \geqslant 1$. We consider the problem of finding nontrivial solutions to the elliptic boundary value problem \begin{displaymath}\begin{array}{*{20}{c}} {\Delta u + \lambda u = h(x)\vert u{\... ...a } {u = 0\quad {\text{on}}\partial \Omega } \end{array} \end{displaymath} where $h \geqslant 0$, $h\not\equiv0$ is Hölder continuous on $\overline \Omega $ and $p > 1$, $\lambda$ are constants. Let ${\Omega _0}$ denote the interior of the set where $ h$ vanishes in $ \Omega$. We assume $ h > 0$ a.e. on $\Omega \backslash {\Omega _0}$ and consider the eigenvalues ${\lambda _i}(\Omega )$ and ${\lambda _i}({\Omega _0})$ of the Dirichlet problem in $\Omega$ and $ {\Omega _0}$ respectively. We prove that no nontrivial solution of the equation exists if $\lambda$ satisfies, for some $k \geqslant 1$, $\displaystyle {\lambda _k}({\Omega _0}) \leqslant \lambda \leqslant {\lambda _{k + 1}}(\Omega )$ On the other hand, if, for some nonnegative integers $ s$, $k$ with $ s \geqslant k + 1$, $ \lambda$ satisfies $\displaystyle {\lambda _s}(\Omega ) < \lambda < {\lambda _{k + 1}}({\Omega _0})$ then the equation above possesses at least $ s - k$ pairs of nontrivial solutions. For the proof of these results we use a variational approach. In particular, the existence result takes advantage of the even character of the associated functional.
On the ideal class group of real biquadratic fields
Patrick J.
Sime
4855-4876
Abstract: We discuss the structure of the ideal class group of real biquadratic fields $K$, concentrating on the case that the $4$-rank of the ideal class groups of the quadratic subfields of $K$ is 0. In this case, we give estimates for the $ 4$-rank of the ideal class group of $K$. As an example, let $K = \mathbb{Q}(\sqrt p ,\sqrt {627} )$, where $ p$ is a prime satisfying certain congruence conditions. The $ 2$-primary part of the ideal class group of $K$ is then isomorphic to $ {(\mathbb{Z}/4\mathbb{Z})^2},\mathbb{Z}/4\mathbb{Z} \times {(\mathbb{Z}/2\mathbb{Z})^2}$, or $ {(\mathbb{Z}/2\mathbb{Z})^4}$. Further, each of the above occurs infinitely often.
Linear Chevalley estimates
Ti
Wang
4877-4898
Abstract: A Chevalley estimate for a germ of an analytic mapping $ f$ is a function $ l:\mathbb{N} \to \mathbb{N}$ such that if the composite with $ f$ of a germ of an analytic function on the target vanishes to order at least $ l(k)$, then it vanishes on the image to order at least $k$. Work of Izumi revealed the equivalence between regularity of a mapping (in the sense of Gabrielov, see $\S1$) and the existence of a linear Chevalley estimate $l(k)$. Bierstone and Milman showed that uniformity of the Chevalley estimate is fundamental to several analytic and geometric problems on the images of mappings. The central topic of this article is uniformity of linear Chevalley estimates for regular mappings. We first establish the equivalence between uniformity of a linear Chevalley estimate and uniformity of a "linear product estimate" on the image: A linear product estimate on a local analytic ring (or, equivalently, on a germ of an analytic space) means a bound on the order of vanishing of a product of elements which is linear with respect to the sum of the orders of its factors. We study the linear product estimate in the central case of a hypersurface (i.e., the zero set of an analytic function). Our results show that a linear product estimate is equivalent to an explicit estimate concerning resultants. In the special case of hypersurfaces of multiplicity $2$, this allows us to prove uniformity of linear product estimates.
On the set of periods for $\sigma$ maps
M. Carme
Leseduarte;
Jaume
Llibre
4899-4942
Abstract: Let $\sigma$ be the topological graph shaped like the letter $\sigma$. We denote by 0 the unique branching point of $\sigma$, and by $ {\mathbf{O}}$ and ${\mathbf{I}}$ the closures of the components of $ \sigma \backslash \{ 0\}$ homeomorphics to the circle and the interval, respectively. A continuous map from $\sigma$ into itself satisfying that $ f$ has a fixed point in ${\mathbf{O}}$, or $f$ has a fixed point and $f(0) \in {\mathbf{I}}$ is called a $\sigma$ map. These are the continuous self-maps of $\sigma$ whose sets of periods can be studied without the notion of rotation interval. We characterize the sets of periods of all $\sigma$ maps.
Discontinuous robust mappings are approximatable
Shu Zhong
Shi;
Quan
Zheng;
Deming
Zhuang
4943-4957
Abstract: The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set $A$ of a topological space $X$ is said to be robust iff ${\text{cl}}A = {\text{cl}}$ int $ A$. A mapping $ f:X \to Y$ is said to be robust iff for each open set ${U_Y}$ of $Y$, $ {f^{ - 1}}({U_Y})$ is robust. We prove that if $X$ is a Baire space and $Y$ satisfies the second axiom of countability, then a mapping $f:X \to Y$ is robust iff it is approximatable in the sense that the set of points of continuity of $ f$ is dense in $ X$ and that for any other point $x \in X$, $(x,f(x))$ is the limit of $\{ ({x_\alpha },f({x_\alpha }))\}$, where for all $\alpha$, $ {x_\alpha }$ is a continuous point of $f$. This result justifies the notion of robustness.
Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula
Michael A.
Bean
4959-4983
Abstract: In a previous paper, it was shown that if $F$ is a binary form with complex coefficients having degree $n \geqslant 3$ and discriminant ${D_F} \ne 0$, and if ${A_F}$ is the area of the region $ \left\vert {F(x,y)} \right\vert \leqslant 1$ in the real affine plane, then $ {\left\vert {{D_F}} \right\vert^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac{1} {3},\frac{1} {3})$, where $B(\frac{1} {3},\frac{1} {3})$ denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence $\{ {M_n}\}$ defined by ${M_n} = \max \vert{D_F}{\vert^{1/n(n - 1)}}{A_F}$, where the maximum is taken over all forms of degree $n$ with ${D_F} \ne 0$, is decreasing, and then by showing that ${M_3} = 3B(\frac{1} {3},\frac{1} {3})$. The resulting estimate, ${A_F} \leqslant 3B(\frac{1} {3},\frac{1} {3})$ for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence $ \{ {M_n}\}$. By appealing to the theory of hypergeometric functions, it is shown that ${M_4} = {2^{7/6}}B(\frac{1} {4},\frac{1} {2})$ and that ${M_4}$ is attained for the form $XY({X^2} - {Y^2})$. It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that ${A_F}$ is the perimeter of the polygon corresponding to $F$. Based on this correspondence and a representation theorem for $\vert{D_F}{\vert^{1/n(n - 1)}}{A_F}$, it is conjectured that ${M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}}$, where $F_n^*(X,Y) = \prod_{k = 1}^n \left(X \sin\left(\frac{k\pi}{n}\right) - Y \cos\left(\frac{k\pi}{n}\right)\right)$, and that the limiting value of the sequence $\{ {M_n}\}$ is $2\pi$.
The variations of Hodge structure of maximal dimension with associated Hodge numbers $h\sp {2,0}>2$ and $h\sp {1,1}=2q+1$ do not arise from geometry
Azniv
Kasparian
4985-5007
Abstract: The specified variations are proved to be covered by a bounded contractible domain $\Omega$. After classifying the analytic boundary components of $\Omega$ with respect to a fixed realization, the group of the biholomorphic automorphisms ${\text{Aut}}\Omega$ and the ${\text{Aut}}\Omega $-orbit structure of $ \Omega$ are found explicitly. Then $\Omega$ is shown to admit no quasiprojective arithmetic quotients, whereas the lack of geometrically arising variations, covered by $\Omega$.
$e$-invariants and finite covers. II
Larry
Smith
5009-5021
Abstract: Let $\widetilde{M} \downarrow M$ be a finite covering of closed framed manifolds. By the Pontrijagin-Thom construction both $ \widetilde{M}$ and $ M$ define elements in the stable homotopy ring of spheres $\pi _*^s$. Associated to $\widetilde{M}$ and $M$ are their $e$invariants $ {e_L}(\widetilde{M})$, $ {e_L}(M) \in \mathbb{Q}/\mathbb{Z}$. If $\widetilde{N} \downarrow N$ is a finite covering of closed oriented manifolds, then there is a related invariant ${I_\Delta }(\widetilde{N} \downarrow N) \in \mathbb{Q}$ of the diffeomorphism class of the covering. In a previous paper we examined the relation between these invariants. We reduced the determination of $ {e_L}(\widetilde{M}) - p{e_L}(M)$, as well as ${I_\Delta }(\widetilde{N} \downarrow N)$, for a $ p$-fold cover, to the evaluation of certain sums of roots of unity. In this sequel we show how the invariant theory of the cyclic group $\mathbb{Z}/p$ may be used to evaluate these rums. For example we obtain $\displaystyle \sum\limits_{\mathop {{\zeta ^p} = 1}\limits_{\zeta \ne 1} } {\fr... ... - 1}})}} {{(1 - \zeta )(1 - {\zeta ^{ - 1}})}} = \frac{{(p - 1)(p - 2)}} {3}}$ which may be used to determine the value of $ {I_\Delta }$ in degrees congruent to $3$ $\mod 2(p - 1)$ for odd primes $ p$.