A quadratic approximation to the Sendov radius near the unit circle
Michael
J.
Miller
851-873
Abstract: Define $S(n,\beta)$ to be the set of complex polynomials of degree $n\ge2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $\vert P\vert _\beta$ to be the distance between $\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\beta)=\sup \{ \vert P\vert _\beta : P \in S(n,\beta) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta)\le1$. In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\beta)\sim1+C_1(1-\vert\beta\vert)+C_2(1-\vert\beta\vert)^2$ for $\vert\beta\vert$near $1$. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.
On the degenerate Beltrami equation
V.
Gutlyanskii;
O.
Martio;
T.
Sugawa;
M.
Vuorinen
875-900
Abstract: We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient $\mu(z)$ has the norm $\Vert\mu\Vert _\infty=1.$Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of $\mu.$A uniqueness theorem is also proved when the singular set $\operatorname{Sing} (\mu)$ of $\mu$is contained in a totally disconnected compact set with an additional thinness condition on $\operatorname{Sing}(\mu).$
Geometry of Fermat adeles
Alexandru
Buium
901-964
Abstract: If $L(a,s):=\sum_n c(n,a)n^{-s}$ is a family of ``geometric'' $L-$functions depending on a parameter $a$, then the function $(p,a)\mapsto c(p,a)$, where $p$ runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by ``adjoining a Fermat quotient operation'', then the functions $c(p,a)$ become functions in the enlarged geometry at least for $L-$functions of curves and Abelian varieties.
Resultants and discriminants of Chebyshev and related polynomials
Karl
Dilcher;
Kenneth
B.
Stolarsky
965-981
Abstract: We show that the resultants with respect to $x$ of certain linear forms in Chebyshev polynomials with argument $x$ are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.
On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions
Hui
Li
983-998
Abstract: Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$. In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types''. In this paper, we construct such manifolds with these ``types''. As a consequence, we have a precise list of the values of these invariants.
The number of certain integral polynomials and nonrecursive sets of integers, Part 1
Tamás
Erdélyi;
Harvey
Friedman
999-1011
Abstract: Given $r > 2$, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the ``cube'' with real coordinates from $[-r,r]$ into $[-t,t]$. This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable.
The number of certain integral polynomials and nonrecursive sets of integers, Part 2
Harvey
M.
Friedman
1013-1023
Abstract: We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, from recursion theory, and from the negative solution to Hilbert's 10th Problem.
Elliptic equations with BMO coefficients in Lipschitz domains
Sun-Sig
Byun
1025-1046
Abstract: In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the $W^{1,p} (1<p<\infty)$ estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the $W^{1,p}$ theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.
On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$
Andreas
Schweizer
1047-1059
Abstract: We study the extension generated by the $x$-coordinates of the $p^e$-torsion points of an elliptic curve over a function field of characteristic $p$. If $S\to C$ is a non-isotrivial elliptic surface in characteristic $p$ with a $p^e$-torsion section, then for $p^e>11$ our results imply restrictions on the genus, the gonality, and the $p$-rank of the base curve $C$, whereas for $p^e\le 11$ such a surface can be constructed over any base curve $C$. We also describe explicitly all occurring $p^e$ in the cases where the surface $S$ is rational or $K3$ or the base curve $C$ is rational, elliptic or hyperelliptic.
Parametric Bäcklund transformations I: Phenomenology
Jeanne
N.
Clelland;
Thomas
A.
Ivey
1061-1093
Abstract: We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.
Gröbner bases of associative algebras and the Hochschild cohomology
Yuji
Kobayashi
1095-1124
Abstract: We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let $A$ be a finitely generated algebra over a commutative ring $K$ with a (possibly infinite) Gröbner base $G$ on a free algebra $F$, that is, $A$ is the quotient $F/I(G)$ with the ideal $I(G)$ of $F$ generated by $G$. Given a Gröbner base $H$ for an $A$-subbimodule $L$ of the free $A$-bimodule $A \cdot X \cdot A = A_K \otimes K \cdot X \otimes_KA$ generated by a set $X$, we have a morphism $\partial$ of $A$-bimodules from the free $A$-bimodule $A \cdot H \cdot A$ generated by $H$ to $A \cdot X \cdot A$ sending the generator $[h]$to the element $h \in H$. We construct a Gröbner base $C$ on $F \cdot H \cdot F$ for the $A$-subbimodule Ker($\partial$) of $A \cdot H \cdot A$, and with this $C$ we have the free $A$-bimodule $A \cdot C \cdot A$ generated by $C$ and an exact sequence $A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A$. Applying this construction inductively to the $A$-bimodule $A$ itself, we have a free $A$-bimodule resolution of $A$.
Weakly compact approximation in Banach spaces
Edward
Odell;
Hans-Olav
Tylli
1125-1159
Abstract: The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$so that for any weakly compact set $D \subset E$ and $\varepsilon > 0$there is a weakly compact operator $V: E \to E$ satisfying $\sup _{x\in D} \Vert x - Vx \Vert < \varepsilon$ and $\Vert V\Vert \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^{*}$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^{*})$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.
Long-time behavior for a nonlinear fourth-order parabolic equation
María
J.
Cáceres;
J.
A.
Carrillo;
G.
Toscani
1161-1175
Abstract: We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.
Harmonic maps $\mathbf{M^3 \rightarrow S^1}$ and 2-cycles, realizing the Thurston norm
Gabriel
Katz
1177-1224
Abstract: Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a harmonic map $f: M^3 \rightarrow S^1$ with Morse-type singularities delivers the Thurston norm $\chi_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \mathbb{Z} )$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\Sigma \subset M$ in the homology class of a fiber, we show that the Thurston number $\chi_-(\Sigma)$ satisfies an inequality \begin{displaymath}\chi_-(\Sigma) \geq \chi_-(F_{best}) - \rho^\circ(\Sigma, f)\cdot Var_{\chi_-}(f).\end{displaymath} Here the variation $Var_{\chi_-}(f)$ is can be expressed in terms of the $\chi_-$-invariants of the fiber components, and the twist $\rho^\circ(\Sigma, f)$ measures the complexity of the intersection of $\Sigma$ with a particular set $F_R$ of ``bad" fiber components. This complexity is tightly linked with the optimal ``$\tilde f$-height" of $\Sigma$, being lifted to the $f$-induced cyclic cover $\tilde M^3 \rightarrow M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its twist $\rho_{\chi_-}(f)$. We prove that, for a harmonic $f$, $\chi_-([F_{best}]) = \, \chi_-(F_{best})$ if and only if $\rho_{\chi_-}(f) = 0$.
Stable and finite Morse index solutions on $\mathbf{R}^n$ or on bounded domains with small diffusion
E.
N.
Dancer
1225-1243
Abstract: In this paper, we study bounded solutions of $- \Delta u = f (u)$ on $\mathbf{R}^n$ (where $n = 2$ and sometimes $n = 3$) and show that, for most $f$'s, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of $- \epsilon^2 \Delta u = f (u)$ on $\Omega$ with Dirichlet or Neumann boundary conditions for small $\epsilon$.
Brownian motion in twisted domains
Dante
DeBlassie;
Robert
Smits
1245-1274
Abstract: The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius'' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out $\pm g(r)$ units along the unit normal to the curve when the traveler is $r$ units away from the origin. The function $g$ is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case $g(r) = \gamma r^p$, $0<p\le 1$. When $p=1$, a twisted domain can reasonably be interpreted as a ``twisted cone.''