Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients
John
M.
Holte
3837-3873
Abstract: This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime $p$, we consider the number of $(x, y)$ with $0 \leq x, y < p^n$ for which $\binom {x+y}{x}$ is divisible by $p^{zn}$ (but not $p^{zn+1}$) when $zn$ is an integer and $\alpha < z < \beta$, say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately $p^{n D((\alpha , \beta ))}$, where $D((\alpha , \beta )) := \sup \{ D(z) : \alpha < z < \beta \}$ and $D$ is given by an explicit formula. We also develop a ``$p$-adic multifractal'' theory and show how $D$ may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the $q$-binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.
Incompressible reacting flows
Joel
D.
Avrin
3875-3892
Abstract: We establish steady-state convergence results for a system of reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.
Spherical classes and the algebraic transfer
Nguyen
H. V.
Hu'ng
3893 - 3910
Degenerate parabolic equations with initial data measures
Daniele
Andreucci
3911-3923
Abstract: We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.
Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes
Manoj
K.
Chari
3925-3943
Abstract: This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on $h$-vectors of simplicial polytopes, we show that $h$-vectors of pure rank-$d$ simplicial complexes that have this property satisfy $h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]}$ and $h_{i} \leq h_{d-i}$ for $0 \leq i \leq [d/2]$. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the $h$-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi that the $h$-vector of a matroid complex satisfies the above two sets of inequalities.
Shellable nonpure complexes and posets. II
Anders
Björner;
Michelle
L.
Wachs
3945-3975
Abstract: This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327.
The Wills conjecture
Noah
Samuel
Brannen
3977-3987
Abstract: Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to $n$-dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.
Monge-Ampère equations relative to a Riemannian metric
A.
Atallah;
C.
Zuily
3989-4006
Abstract: We prove that in a bounded strictly convex open set $\Omega$ in $\mathbb {R}^n$, the problem \begin{displaymath}\begin {cases} \det \nabla ^2u=f(x), u|_{\partial \Omega }=\varphi , \end {cases}\end{displaymath} where $f>0,f\in C^\infty (\overline \Omega ), \varphi \in C^\infty (\partial \Omega )$, has a unique strictly convex solution $u\in C^\infty (\overline \Omega )$. This result extends to an arbitrary metric a theorem which has been proved by Caffarelli-Nirenberg-Spruck in the case of the Euclidean metric.
Four-Manifolds With Surface Fundamental Groups
Alberto
Cavicchioli;
Friedrich
Hegenbarth;
Dusan
Repovs
4007-4019
Abstract: We study the homotopy type of closed connected topological $4$-manifolds whose fundamental group is that of an aspherical surface $F$. Then we use surgery theory to show that these manifolds are $s$-cobordant to connected sums of simply-connected manifolds with an $\mathbb {S}^{2}$-bundle over $F$.
Graded Lie Algebras of Maximal Class
A.
Caranti;
S.
Mattarei;
M.
F.
Newman
4021-4051
Abstract: We study graded Lie algebras of maximal class over a field $\mathbf {F}$ of positive characteristic $p$. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct $| \mathbf {F}|^{\aleph _{0}}$ pairwise non-isomorphic such algebras, and $\max \{| \mathbf {F}|, \aleph _{0} \}$ soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.
A hypergeometric function approach to the persistence problem of single sine-Gordon breathers
Jochen
Denzler
4053-4083
Abstract: It is shown that for an interesting class of perturbation functions, at most one of the continuum of sine-Gordon breathers can persist for the perturbed equation. This question is much more subtle than the question of persistence of large portions of the family, because analytic continuation arguments in the amplitude parameter are no longer available. Instead, an asymptotic analysis of the obstructions to persistence for large Fourier orders is made, and it is connected to the asymptotic behaviour of the Taylor coefficients of the perturbation function by means of an inverse Laplace transform and an integral transform whose kernel involves hypergeometric functions in a way that is degenerate in that asymptotic analysis involves a splitting monkey saddle. Only first order perturbation theory enters into the argument. The reasoning can in principle be carried over to other perturbation functions than the ones considered here.
The Szego curve, zero distribution and weighted approximation
Igor
E.
Pritsker;
Richard
S.
Varga
4085-4105
Abstract: In 1924, Szeg\H{o} showed that the zeros of the normalized partial sums, $s_{n}(nz)$, of $e^{z}$ tended to what is now called the Szeg\H{o} curve $S$, where \begin{displaymath}S:= \left \{ z \in {\mathbb {C}}:|ze^{1-z}|=1 \text { and } |z| \leq 1 \right \}. \end{displaymath} Using modern methods of weighted potential theory, these zero distribution results of Szeg\H{o} can be essentially recovered, along with an asymptotic formula for the weighted partial sums $\{e^{-nz}s_{n} (nz)\}^{\infty }_{n=0}$. We show that $G:= {\operatorname {Int}} S$ is the largest universal domain such that the weighted polynomials $e^{-nz} P_{n}(z)$ are dense in the set of functions analytic in $G$. As an example of such results, it is shown that if $f(z)$ is analytic in $G$ and continuous on $\overline {G}$ with $f(1)=0$, then there is a sequence of polynomials $\left \{P_{n}(z)\right \}^{\infty }_{n=0}$, with $\deg P_{n} \leq n$, such that \begin{displaymath}\displaystyle \lim_{n \rightarrow \infty } \|e^{-nz} P_{n}(z)-f(z)\|_{\overline {G}} =0, \end{displaymath} where $\| \cdot \|_{\overline {G}}$ denotes the supremum norm on $\overline {G}$. Similar results are also derived for disks.
Asymptotic analysis for linear difference equations
Katsunori
Iwasaki
4107-4142
Abstract: We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.
On the convergence of $\sum c_nf(nx)$ and the Lip 1/2 class
István
Berkes
4143-4158
Abstract: We investigate the almost everywhere convergence of $\sum c_{n} f(nx)$, where $f$ is a measurable function satisfying \begin{equation*}f(x+1) = f(x), \qquad \int _{0}^{1} f(x) \, dx =0.\end{equation*} By a known criterion, if $f$ satisfies the above conditions and belongs to the Lip $\alpha$ class for some $\alpha > 1/2$, then $\sum c_{n} f(nx)$ is a.e. convergent provided $\sum c_{n}^{2} < +\infty$. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions $f$ and almost exponentially growing sequences $(n_{k})$ such that $\sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $\sum c_{k}^{2} < +\infty$. For functions $f$ with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.
Equilibria of set-valued maps on nonconvex domains
H.
Ben-El-Mechaiekh;
W.
Kryszewski
4159-4179
Abstract: We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if $K$ is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space $E$ , and $\Phi :K\longrightarrow 2^E$ is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition \begin{equation*}\Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*} then there exists $x_0\in K$ such that $0\in \Phi (x_0).$ Here, $T_K^r(x)$ denotes a new concept of retraction tangent cone to $K$ at $x$ suited for compact neighborhood retracts. When $K$ is locally convex at $x,T_K^r(x)$ coincides with the usual tangent cone of convex analysis. Special attention is given to neighborhood retracts having ``lipschitzian behavior'', called $L-$retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.
On the conjecture of Birch and Swinnerton-Dyer
Cristian
D.
Gonzalez-Avilés
4181-4200
Abstract: In this paper we complete Rubin's partial verification of the conjecture for a large class of elliptic curves with complex multiplication by ${\mathbb {Q}}(\sqrt {-7})$.
Hamiltonian torus actions on symplectic orbifolds and toric varieties
Eugene
Lerman;
Susan
Tolman
4201-4230
Abstract: In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.
Existence of positive solutions for some problems with nonlinear diffusion
A.
Cañada;
P.
Drábek;
J.
L.
Gámez
4231-4249
Abstract: In this paper we study the existence of positive solutions for problems of the type \begin{equation*}\begin {array}{cl} -\Delta _pu(x)=u(x)^{q-1}h(x,u(x)), & x\in \Omega , u(x)=0, & x\in \partial \Omega , \end {array} \end{equation*} where $\Delta _p$ is the $p$-Laplace operator and $p,q>1$. If $p=2$, such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases $p=q$, $p<q$ and $p>q$, respectively. Also, some systems of equations are considered.
Tame Combings of Groups
Michael
L.
Mihalik;
Steven
T.
Tschantz
4251-4264
Abstract: In this paper, we introduce the idea of tame combings for finitely presented groups. If $M$ is a closed irreducible 3-manifold and $\pi _{1}(M)$ is tame combable, then the universal cover of $M$ is homeomorphic to ${\mathbb {R}}^{3}$. We show that all asynchronously automatic and all semihyperbolic groups are tame combable.