Binomial coefficients and Jacobi sums
Richard H.
Hudson;
Kenneth S.
Williams
431-505
Abstract: Throughout this paper $e$ denotes an integer $ \geqslant 3$ and $ p$ a prime $\equiv \;1 \pmod e$. With $f$ defined by $ p = ef + 1$ and for integers $r$ and $s$ satisfying $1 \leqslant s < r \leqslant e - 1$ , certain binomial coefficients $ \left( {\begin{array}{*{20}{c}} {rf} {sf} \end{array} } \right)$ have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [13], Jacobi [19, 20], Stern [37-40], Lehmer [23] and Whiteman [42, 45, 46]. In $\S2$ we determine for each $e$ the exact number of binomial coefficients $\left( {\begin{array}{*{20}{c}} {rf} {sf} \end{array} } \right)$ not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefficient is said to be of order $e$ if and only if $(r,s) = 1$. In $\S\S3-4$, we show how the Davenport-Hasse relation [7], in a form given by Yamamoto [50], leads to determinations of $ {n^{(p - 1)/m}}$ in terms of binomial coefficients modulo $p = ef + 1 = mnf + 1$. These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which was first obtained in a slightly different form by Whiteman [45], we systematically investigate in $\S\S6-21$ all representative binomial coefficients of orders $e = 3,4,6,7,8,9,11,12,14,15,16,20$ and $24$, which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefficients of orders $ e = 5,10,13,15,16$ and $ 20$, which we are able to explicitly determine in terms of quaternary quadratic decompositions of $16p$ given by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these results have been obtained by previous authors and many new ones are included. For $ e = 7$ and $14$ we are unable to explicitly determine representative binomial coefficients in terms of the six variable quadratic decomposition of $ 72p$ given by Dickson [9] for reasons given in $\S10$, but we are able to express these binomial coefficients in terms of the parameter ${x_1}$ in this system in analogy to a recent result of Rajwade [34]. Finally, although a relatively rare occurrence for small $ e$, it is possible for representative binomial coefficients of order $ e$ to be congruent to one another $\pmod p$. Representative binomial coefficients which are congruent to $\pm 1$ times at least one other representative for all $p = ef + 1$ are called Cauchy-Whiteman type binomial coefficients for reasons given in [17] and $\S21$. All congruences between such binomial coefficients are carefully examined and proved (with the sign ambiguity removed in each case) for all values of $ e$ considered. When $ e = 24$ there are $ 48$ representative binomial coefficients, including those of lower order, and it is shown in $\S21$ that an astonishing $43$ of these are Cauchy-Whiteman type binomial coefficients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form ${n^{(p - 1)/m}}$ in contrast to the case for all $ e < 24$.
An algebraic classification of some even-dimensional spherical knots. I
M. Š.
Farber
507-527
Abstract: The main result of the paper is the classification of simple even-dimensional spherical knots in terms of their algebraic invariants.
An algebraic classification of some even-dimensional spherical knots. II
M. Š.
Farber
529-570
Abstract: The paper reduces the problem of classification of simple even-dimensional spherical knots of codimension two to an algebraic problem.
Homomorphisms of cocompact Fuchsian groups on ${\rm PSL}\sb{2}(Z\sb{p\sp{n}}[x]/(f(x)))$
Jeffrey
Cohen
571-585
Abstract: We obtain conditions under which $ {\text{PSL}}_2({Z_{{p^n}}}[x]/(f(x)))$ is a factor of $(l,m,n)$. Using this, certain results about factors of cocompact Fuchsian groups are obtained. For example, it is shown that: (i) $\Gamma$ has infinitely many simple nonabelian factors. (ii) $\Gamma$ has factors with nontrivial center. (iii) For each $n$, there exists $m$ such that $\Gamma$ has at least $n$ factors of order $m$. Further, all factored normal subgroups can be taken torsion-free. Also, new Hurwitz groups and noncongruence subgroups of the modular group are obtained.
The $\bar \partial $-Neumann solution to the inhomogeneous Cauchy-Riemann equation in the ball in ${\bf C}\sp{n}$
F. Reese
Harvey;
John C.
Polking
587-613
Abstract: Let $\vartheta$ denote the formal adjoint of the Cauchy-Riemann operator $\overline \partial$ on ${{\mathbf{C}}^n}$, and let $N$ denote the Kohn-Neumann operator on the unit ball in $ {{\mathbf{C}}^n}$. The operator $ \vartheta \; \circ \;N$ provides a natural fundamental solution for $\overline \partial f = g$ on the ball. It is our purpose to present the kernel $P$ for this operator $\vartheta \; \circ \;N$ explicitly--the coefficients are exhibited as rational functions.
Constructions arising from N\'eron's high rank curves
M.
Fried
615-631
Abstract: Many papers quote Néron's geometric construction of elliptic curves of rank $11$ over $ \mathbb{Q}\;[{\mathbf{N}}]$--still, at the writing of this paper, the elliptic curves of highest demonstrated rank. The purported reason for the ordered display of "creeping rank" in [ $ {\mathbf{PP}},{\mathbf{GZ}},{\mathbf{Na}}$ and $ {\mathbf{BK}}$] is to make $ [{\mathbf{N}}]$ explicit. Excluding $ [{\mathbf{BK}}]$, however, these papers derive little from Néron's constructions. All show some lack of confidence in the details of $ [{\mathbf{N}}]$. The core of this paper ($\S3$), meets objections to $[{\mathbf{N}}]$ raised by correspondents. Our method adds a novelty as it magnifies the constructions of $ [{\mathbf{N}}]$--"generation of pencils of cubics from their singular fibers". This has two advantages: it displays (Remark 4.2) the free parameters whose specializations give high rank curves; and it demonstrates the existence of rank $11$ curves through one appeal only to Hilbert's irreducibility theorem. That is, we have eliminated the unusual analogue of Hilbert's result that takes up most of $ [{\mathbf{N}}]$. In particular $(\S4(c))$, the explicit form of the irreducibility theorem in $ [{\mathbf{Fr}}]$ applies to give explicit rank $11$ curves over $ \mathbb{Q}$: with Selmer's conjecture, rank $12$.
Nonstable reflexive sheaves on ${\bf P}\sp{3}$
Timothy
Sauer
633-655
Abstract: The spectrum is defined for nonstable rank two reflexive sheaves on ${{\mathbf{P}}^3}$ and is used to establish vanishing theorems for intermediate cohomology in terms of the Chern classes and the order of nonstability. These results are shown to be best possible and the extremal cases are classified. Some applications to Cohen-Macaulay generically local complete intersection curves in $ {{\mathbf{P}}^3}$ are given.
A minimal model for $\neg{\rm CH}$: iteration of Jensen's reals
Uri
Abraham
657-674
Abstract: A model of ${\text{ZFC}} + {2^{\aleph_0}} = {\aleph_2}$ is constructed which is minimal with respect to being a model of $ \neg {\text{CH}}$. Any strictly included submodel of $ {\text{ZF}}$ (which contains all the ordinals) satisfies ${\text{CH}}$. In this model the degrees of constructibility have order type $ {\omega_2}$. A novel method of using the diamond is applied here to construct a countable-support iteration of Jensen's reals: In defining the $ \alpha {\text{th}}$ stage of the iteration the diamond "guesses" possible $\beta > \alpha$ stages of the iteration.
Weak restricted and very restricted operators on $L\sp{2}$
J. Marshall
Ash
675-689
Abstract: A battlement is a real function with values in $\{ 0,1\}$ that looks like a castle battlement. A commuting with translation linear operator $ T$ mapping step functions on ${\mathbf{R}}$ into the set of all measurable functions on $ {\mathbf{R}}$ and satisfying $\parallel Tb{\parallel_2} \leqslant C\parallel b{\parallel_2}$ for all battlements $b$ is bounded on ${L^2}({\mathbf{R}})$. This remains true if the underlying space is the circle but is demonstrably false if the underlying space is the integers. Michael Cowling's theorem that linear commuting with translation operators are bounded on ${L^2}$ if they are weak restricted $(2,2)$ is reproved and an application of this result to sums of exponentials is given.
Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds
Hyeong In
Choi
691-716
Abstract: We define the asymptotic Dirichlet problem and give a sufficient condition for solving it. This proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete Riemannian manifolds.
Supercompactness of compactifications and hyperspaces
Murray G.
Bell
717-724
Abstract: We prove a theorem which implies that if $ \gamma \omega$ is a supercompact compactification of the countable discrete space $ \omega$ then $\gamma \omega - \omega$ is separable. This improves an earlier result of the author's that such a $\gamma \omega$ must have $\gamma \omega - \omega \;{\text{ccc}}$. We prove a theorem which implies that the hyperspace of closed subsets of $ {2^{\omega_2}}$ is not a continuous image of a supercompact space. This improves an earlier result of $ {\text{L}}$. Šapiro that the hyperspace of closed subsets of ${2^{\omega_2}}$ is not dyadic.
The level sets of the moduli of functions of bounded characteristic
Robert D.
Berman
725-744
Abstract: For $f$ a nonconstant meromorphic function on $ \Delta = \{ \vert z\vert < 1\}$ and $r \in (\inf \vert f\vert,\sup \vert f\vert)$, let $\mathcal{L}(f,r) = \{ z \in \Delta :\vert f(z)\vert = r\}$. In this paper, we study the components of $ \Delta \backslash \mathcal{L}(f,r)$ along with the level sets $\mathcal{L}(f,r)$. Our results include the following: If $f$ is an outer function and $\Omega$ a component of $ \Delta \backslash \mathcal{L}(f,r)$, then $\Omega$ is a simply-connected Jordan region for which $({\text{fr}}\;\Omega ) \cap \{ \vert z\vert = 1\}$ has positive measure. If $f$ and $g$ are inner functions with $ \mathcal{L}\,(f,r) = \mathcal{L}\,(g,s)$, then $g = \eta {f^\alpha }$, where $\vert\eta \vert = 1$ and $ \alpha > 0$. When $ g$ is an arbitrary meromorphic function, the equality of two pairs of level sets implies that $ g = c{f^\alpha }$, where $ c \ne 0$ and $\alpha \in ( - \infty ,\infty )$. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.
Endomorphisms of the cohomology of complex Grassmannians
Michael
Hoffman
745-760
Abstract: For any complex Grassmann manifold $G$, we classify all endomorphisms of the rational cohomology ring of $G$ which are nonzero on dimension $2$. Some applications of this result are given.
Approximation in the mean by solutions of elliptic equations
Thomas
Bagby
761-784
Abstract: A result analogous to the Vituškin approximation theorem is proved for mean approximation by solutions of certain elliptic equations.
On the paths of symmetric stable processes
Burgess
Davis
785-794
Abstract: It is shown that if $ X(t), t \geqslant 0$, is a symmetric stable process of index $\alpha, 0 < \alpha < 2$, then $\sup_t \lim \inf_{h \downarrow 0} (X(t + h) - X(t))h^{-1/\alpha} = \infty$ a.s. This settles a question of Fristedt about strictly stable subordinators.
Finitely generated extensions of partial difference fields
Peter
Evanovich
795-811
Abstract: A proof of the following theorem is given: If $ \mathcal{M}$ is a finitely generated extension of a partial difference field $\mathcal{K}$ then every subextension of $\mathcal{M}/\mathcal{K}$ is finitely generated. An integral measure of partial difference field extensions having properties similar to the dimension of field extensions and the limit degree of ordinary difference field extensions and a new method of computing transformal transcendence degree are developed.
Invariant densities for random maps of the interval
S.
Pelikan
813-825
Abstract: A random map is a discrete time process in which one of a number of functions is selected at random and applied. Here we study random maps of $[0,1]$ which represent dynamical systems on the square $[0,1] \times [0,1]$. Sufficient conditions for a random map to have an absolutely continuous invariant measure are given, and the number of ergodic components of a random map is discussed.
On Block's condition for simple periodic orbits of functions on an interval
Chung-Wu
Ho
827-832
Abstract: Recently, L. Block has shown that for any mapping $f$ of an interval, whether $ f$ has a periodic point whose period contains an odd factor greater than $ 1$ depends entirely on the periodic orbits of $f$ whose periods are powers of $2$. In this paper the author shows that Block's result is a special case of a more general phenomenon.
A property of complete minimal surfaces
Thomas
Hasanis;
Dimitri
Koutroufiotis
833-843
Abstract: If $M$ is a complete minimal surface in $ {R^n}$, we denote by $ W$ the set of points in $ {R^n}$ that do not lie on any tangent plane of $M$. By taking a point in $W$ as origin, the position vector of $ M$ determines a global unit normal vector field $e$ to $M$. We prove that if $e$ is a minimal section, then $M$ is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in ${R^3}$ covers all ${R^3}$. We also prove a similar result for a complete minimal surface $M$ in ${S^3}$, and deduce from it that if the spherical image of $M$ lies in a closed hemisphere, then $ M$ is a great $ {S^2}$.