Expand all Collapse all | Results 101 - 114 of 114 |
101. CMB 2000 (vol 43 pp. 239)
On the Number of Divisors of the Quadratic Form $m^2+n^2$ For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 |
102. CMB 2000 (vol 43 pp. 174)
Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces We show that the use of orbifold bundles enables some questions to
be reduced to the case of flat bundles. The identification of
moduli spaces of certain parabolic bundles over elliptic surfaces
is achieved using this method.
Categories:14J27, 32L07, 14H60, 14D20 |
103. CMB 2000 (vol 43 pp. 129)
Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves Let $E$ be a stable rank 2 vector bundle on a smooth projective
curve $X$ and $V(E)$ be the set of all rank~1 subbundles of $E$
with maximal degree. Here we study the varieties (non-emptyness,
irreducibility and dimension) of all rank~2 stable vector bundles,
$E$, on $X$ with fixed $\deg(E)$ and $\deg(L)$, $L \in V(E)$ and
such that $\card \bigl( V(E) \bigr) \geq 2$ (resp. $\card \bigl(
V(E) \bigr) = 2$).
Category:14H60 |
104. CMB 1999 (vol 42 pp. 499)
Characterizations of Simple Isolated Line Singularities A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$
with a smooth $1$-dimensional critical set $\Sigma=\{(x,y)\in \CC\times
\CC^n \mid y=0\}$. An isolated line singularity is defined by the
condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is
equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line
singularities were classified by Dirk Siersma and are analogous
of the famous $A-D-E$ singularities. We give two new
characterizations of simple isolated line singularities.
Categories:32S25, 14B05 |
105. CMB 1999 (vol 42 pp. 445)
Smooth Maps and Real Algebraic Morphisms Let $X$ be a compact nonsingular real algebraic variety and let $Y$
be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear
subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R})
\times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved
that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be
approximated by regular maps if and only if $f^* \bigl( H^1(Y,
\mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where
$H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X,
\mathbb{Z}/2)$ generated by the cohomology classes of algebraic
hypersurfaces in $X$. This follows from another result on maps
into generalized flag varieties.
Categories:14P05, 14P25 |
106. CMB 1999 (vol 42 pp. 307)
On the Moduli Space of a Spherical Polygonal Linkage We give a ``wall-crossing'' formula for computing the topology of
the moduli space of a closed $n$-gon linkage on $\mathbb{S}^2$.
We do this by determining the Morse theory of the function
$\rho_n$ on the moduli space of $n$-gon linkages which is given by
the length of the last side---the length of the last side is
allowed to vary, the first $(n - 1)$ side-lengths are fixed. We
obtain a Morse function on the $(n - 2)$-torus with level sets
moduli spaces of $n$-gon linkages. The critical points of $\rho_n$
are the linkages which are contained in a great circle. We give a
formula for the signature of the Hessian of $\rho_n$ at such a
linkage in terms of the number of back-tracks and the winding
number. We use our formula to determine the moduli spaces of all
regular pentagonal spherical linkages.
Categories:14D20, 14P05 |
107. CMB 1999 (vol 42 pp. 354)
A Real Holomorphy Ring without the SchmÃ¼dgen Property A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $-N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, N-f \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgen-W\"ormann result that for any preordering $T$ in a
finitely generated $\R$-algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.
Categories:12D15, 14P10, 44A60 |
108. CMB 1999 (vol 42 pp. 263)
Mellin Transforms of Mixed Cusp Forms We define generalized Mellin transforms of mixed cusp forms, show
their convergence, and prove that the function obtained by such a
Mellin transform of a mixed cusp form satisfies a certain
functional equation. We also prove that a mixed cusp form can be
identified with a holomorphic form of the highest degree on an
elliptic variety.
Categories:11F12, 11F66, 11M06, 14K05 |
109. CMB 1999 (vol 42 pp. 209)
Ample Vector Bundles of Curve Genus One We investigate the pairs $(X,\cE)$ consisting of a smooth complex
projective variety $X$ of dimension $n$ and an ample vector bundle
$\cE$ of rank $n-1$ on $X$ such that $\cE$ has a section whose
zero locus is a smooth elliptic curve.
Categories:14J60, 14F05, 14J40 |
110. CMB 1999 (vol 42 pp. 78)
Fermat Jacobians of Prime Degree over Finite Fields We study the splitting of Fermat Jacobians of prime
degree $\ell$ over an algebraic closure of a finite field of
characteristic $p$ not equal to $\ell$. We prove that their
decomposition is determined by the residue degree of $p$ in the
cyclotomic field of the $\ell$-th roots of unity. We provide a
numerical criterion that allows to compute the absolutely simple
subvarieties and their multiplicity in the Fermat Jacobian.
Categories:11G20, 14H40 |
111. CMB 1998 (vol 41 pp. 442)
A Mountain Pass to the Jacobian Conjecture. This paper presents an approach to injectivity theorems via the
Mountain Pass Lemma and raises an open question. The main result
of this paper (Theorem~1.1) is proved by means of the Mountain Pass
Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$
are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii
R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow
\hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is
injective. This was discovered in a joint attempt by the authors
to prove a stronger result conjectured by the first author: Namely,
that a sufficient condition for injectivity of class $\cC^{1}$ maps
$F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues
of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii
R}^n$. This is stated as Conjecture~2.1. If true, it would imply
(via {\it Reduction-of-Degree}) {\it injectivity of polynomial
maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$
{\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the
celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The
paper ends with several examples to illustrate a variety of cases
and known counterexamples to some natural questions.
Keywords:Injectivity, ${\cal C}^1$-maps, polynomial maps, Jacobian Conjecture, Mountain Pass Categories:14A25, 14E09 |
112. CMB 1998 (vol 41 pp. 267)
On the nonemptiness of the adjoint linear system of polarized manifold Let $(X,L)$ be a polarized manifold over the complex number field
with $\dim X=n$. In this paper, we consider a conjecture of
M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this
conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs
|L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the
result of Sommese.
Keywords:Polarized manifold, adjoint bundle Categories:14C20, 14J99 |
113. CMB 1997 (vol 40 pp. 456)
Approximation of smooth maps by real algebraic morphisms Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all
$q$-dimensional $\Bbb F$-vector subspaces of $\Bbb F^{p}$, where $\Bbb F$
stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here
$\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper
investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic
variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the
${\cal C}^\infty$ compact-open topology, by real algebraic morphisms.
Categories:14P05, 14P25 |
114. CMB 1997 (vol 40 pp. 352)
A New Proof of a Theorem of Magnus Using naive algebraic geometric methods a new proof of the
following celebrated theorem of Magnus is given:
Let $G$ be a group with a presentation having $n$ generators and $m$
relations. If $G$ also has a presentation on $n-m$ generators, then
$G$ is free of rank $n-m$.
Categories:20E05, 20C99, 14Q99 |