On the Weyl tensor of a self-dual complex 4-manifold
Florin
Alexandru
Belgun
853-880
Abstract: We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the $\beta$-surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of $\mathbb{CP} ^2$.
Hilbert spaces of Dirichlet series and their multipliers
John
E.
McCarthy
881-893
Abstract: We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted $\ell^2$ norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in the multiplier algebra may or may not extend to be analytic on a larger half-plane than the functions in the Hilbert space.
Symplectic semifield planes and ${\mathbb Z}_4$--linear codes
William
M.
Kantor;
Michael
E.
Williams
895-938
Abstract: There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and $\mathbb{Z}_4$-linear Kerdock and Preparata codes on the other. These inter-relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non-isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of $\mathbb{Z}_4$-linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of ``boring'' affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes ``boring'' in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.
Low-degree points on Hurwitz-Klein curves
Pavlos
Tzermias
939-951
Abstract: We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.
Analytic order of singular and critical points
Eugenii
Shustin
953-985
Abstract: We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a germ of a holomorphic function in two variables with an isolated critical point, what is the minimal degree of a polynomial, equivalent to the given function up to a local holomorphic coordinate change? Classically known estimates for such a degree $d$ in these questions are $\sqrt{\mu}+1\le d\le \mu+1$, where $\mu$ is the Milnor number. Our result in both the problems is $d\le a\sqrt{\mu}$ with an absolute constant $a$. As a corollary, we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces.
Smooth solutions to a class of free boundary parabolic problems
Olivier
Baconneau;
Alessandra
Lunardi
987-1005
Abstract: We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem'' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive $t$, if the data are smooth.
The 2-twist-spun trefoil has the triple point number four
Shin
Satoh;
Akiko
Shima
1007-1024
Abstract: The triple point number of an embedded surface in 4-space is the minimal number of the triple points on all the projection images into 3-space. We show that the 2-twist-spun trefoil has the triple point number four.
Extensions contained in ideals
Dan
Kucerovsky
1025-1043
Abstract: We prove a Weyl-von Neumann type absorption theorem for extensions which are not full, and give a condition for constructing infinite repeats contained in an ideal. We also clear up some questions associated with the purely large criterion for full extensions to be absorbing.
Semilinear parabolic equations involving measures and low regularity data
H.
Amann;
P.
Quittner
1045-1119
Abstract: A detailed study of abstract semilinear evolution equations of the form $\dot u+Au=\mu(u)$ is undertaken, where $-A$ generates an analytic semigroup and $\mu(u)$ is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.
A Capelli Harish-Chandra homomorphism
Tomasz
Przebinda
1121-1154
Abstract: For a real reductive dual pair the Capelli identities define a homomorphism $\mathcal{C}$ from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a $\rho$-shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism $\mathcal{C}$ based solely on the Harish-Chandra's radial component maps. Thus we provide a geometric interpretation of the $\rho$-shift.
Character degree graphs and normal subgroups
I.
M.
Isaacs
1155-1183
Abstract: We consider the degrees of those irreducible characters of a group $G$whose kernels do not contain a given normal subgroup $N$. We show that if
Complete hyperelliptic integrals of the first kind and their non-oscillation
Lubomir
Gavrilov;
Iliya
D.
Iliev
1185-1207
Abstract: Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form \begin{displaymath}I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1 x+\ldots + \alpha_{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma, \end{displaymath} where $\alpha_i$ are real and $\Sigma\subset \mathbb{R}$ is a maximal open interval on which a continuous family of ovals $\{\delta(h)\}$ exists. We show that the $g$-dimensional real vector space of these integrals is not Chebyshev in general: for any $g>1$, there are hyperelliptic Hamiltonians $H$ and continuous families of ovals $\delta(h)\subset\{H=h\}$, $h\in\Sigma$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $\Sigma$. Our main result is Theorem 1 in which we show that when $g=2$, exceptional families of ovals $\{\delta(h)\}$ exist, such that the corresponding vector space is still Chebyshev.
How to do a $p$-descent on an elliptic curve
Edward
F.
Schaefer;
Michael
Stoll
1209-1231
Abstract: In this paper, we describe an algorithm that reduces the computation of the (full) $p$-Selmer group of an elliptic curve $E$ over a number field to standard number field computations such as determining the ($p$-torsion of) the $S$-class group and a basis of the $S$-units modulo $p$th powers for a suitable set $S$ of primes. In particular, we give a result reducing this set $S$ of `bad primes' to a very small set, which in many cases only contains the primes above $p$. As of today, this provides a feasible algorithm for performing a full $3$-descent on an elliptic curve over $\mathbb Q$, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of $E[p]$ is favorable, simplifications are possible and $p$-descents for larger $p$ are accessible even today. To demonstrate how the method works, several worked examples are included.
Universal covers for Hausdorff limits of noncompact spaces
Christina
Sormani;
Guofang
Wei
1233-1270
Abstract: We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature, then $Y$ has a universal cover.