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Let X be a curve with an involution T which fixes
r points. We show that the
Weierstrass weight of a fixed point is at least
(r−2)(r−4)/8. Our proof is
independent of the recent result of Torres.
We consider the case where X=Fn,
the nth Fermat curve, and T is any of the
involutions of Fn. We find that our bound
equal to the actual weight in all known
cases (n[les ]7) and compute then n=8 case to demonstrate
the equality continues to hold.
The purpose of this paper is to prove irrationality results from
Jacobi's triple product identity, which can be written, for
x∈[Copf ]*, y∈[Copf ], [mid ]y[mid ]<1:
There are various proofs of this identity; the classical one rests on
theory of theta functions (, theorem 6, p. 69). An
alternative proof uses Heine's summation
formula (, p. 12). An elementary, self-contained
can be found in , p. 227.
In this paper, we will use the same elementary methods as in
 and , and prove
the following theorems.
We prove Du's positivity conjecture for the canonical basis
the q-Schur algebra, using elementary arguments and the positivity
result for Lusztig's canonical basis for
We also describe a family of subalgebras of the q-Schur algebra,
which is spanned by the canonical basis elements it contains.
Let [Sfr ]n and [Afr ]n denote
the symmetric and alternating groups of degree n∈ℕ
respectively. Let p be a prime number and let F be an
arbitrary field of characteristic
p. We say that a partition of n is p-regular
no p (non-zero) parts of it are equal;
otherwise we call it p-singular. Let
denote the Specht module corresponding to λ.
For λ a p-regular partition of n let
DλF denote the unique irreducible
top factor of SλF.
Denote by ΔλF
↓[Afr ]n its restriction to
[Afr ]n. Recall also that, over F, the ordinary
quiver of the modular group algebra FG is a finite directed
graph defined as follows:
the vertices are labelled by the set of all simple FG-modules,
L1, ..., Lr, and the
number of arrows from Li to
Lj). The quiver gives important
information about the block structure of G.
The theory of reductive monoids has been well developed by Renner
author, cf. [8, 13]. This paper concerns the
finite reductive monoids MF
where M is
a reductive monoid and F[ratio ]M→M
is a Frobenius map. We show that the
Deligne–Lusztig characters RT, θ
of finite reductive groups GF
have natural extensions to MF. We accomplish
by first showing that the action of a finite monoid (by
partial transformations) on an algebraic variety gives rise to a virtual
the monoid via étale cohomology. We then find the correct analogue
of the algebraic set L−1(U) (where
L is the Lang map) for M. The action of
MF on this algebraic
variety gives rise to virtual characters
RT¯, θ where T is an
F-stable maximal torus of
G, T¯ the Zariski closure of T in M
and θ an irreducible character of the finite
commutative monoid T¯F. We go on to show
that any irreducible character of MF is
a component of some RT¯, θ.
A classical fact is that Seifert manifolds with non-empty
boundary are covered by
surface bundles over the circle S1
and closed Seifert manifolds may or may not be
covered by surface bundles over S1. Some closed
graph manifolds are not covered by
surface bundles over S1
([LW] and [N]). Thurston asked if
complete hyperbolic 3-manifolds of finite volume are covered by surface
[T]. J. Luecke and Y. Wu
asked if graph manifolds with non-empty boundary are covered by surface
over S1 ([LW]). In this paper we
THEOREM 0·1. Each graph manifold with non-empty
is finitely covered by a surface bundle over the circle S1.
Questions concerning extensions of polynomials or analytic functions
Banach space E to its bidual E", and others about
reflexivity and related properties on spaces of homogeneous polynomials
Pk(E), have recently spurred interest
regarding the structure of the bidual of a space of polynomials
[4, 14, 22].
In this paper we study the relationship between the bidual of
Pk(E) and the space
of polynomials over E". We define a map through which
elements of the bidual of Pk(E)
may be viewed as polynomials over E"
and study this map to obtain information about
Pk(E)". We have tried as far
possible to avoid imposing restricting conditions on E.
We give a hyperkähler analogue of the classification of
Kähler manifolds. Namely we show that, for a compact semisimple
group G, the
complete G-invariant hyperkähler manifolds which are also
with respect to one of the complex structures are precisely the
of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits of
Gc. As a
consequence the only complete hyperkähler metrics which are of
with respect to a triholomorphic action of compact semisimple Lie group
the flat metric on ℍn
and the Calabi metric on T*[Copf ]Pn.
Let μ and ν be Radon probability measures on ℝ with
compact supports. We will
consider natural intersections of μ and the image of ν under an isometry
Our main result gives a lower bound for the packing dimensions of these
measures, if μ and ν satisfy some regularity assumptions. Our method,
originally from , is the same as used in
 where similarities were considered instead of isometries.
Let f[ratio ]X→[Copf ]ℙp
be a finite complex analytic map into complex projective space,
with dimX<p. We obtain a result on the equivalence
low homotopy groups between the image of f and
[Copf ]ℙp, the level of comparison is a function
of p, the
maximal number of preimages of f and how bad the singularities
X are. This global
result is deduced from a generalisation of a theorem of H. Hamm on the
structure of singularities, see .
Let X be a smooth projective manifold over the complex
field [Copf ] and L a
Cartier divisor on X. Then (X, L) is called
a polarized (resp. quasi-polarized) manifold
if L is ample (resp. nef and big). For a polarized manifold
(X, L), Takao Fujita
introduced the notion of the Kodaira energy
κε(X, L). The Kodaira energy of
(X, L) is thought to have some interesting phenomena
κ(X)=−∞ (for example,
Spectrum Conjecture which was proposed by T. Fujita). In order to study
Kodaira energy of (X, L), we consider the case in which
X has a fibre space, that is,
there exist a smooth projective manifold Y with
dimX>dimY[ges ]1 and a surjective
morphism f[ratio ]X→Y with connected fibres.
In this case we introduce the notion of
relative Kodaira energy and study some property of it. By using some results
relative Kodaira energy, we study some properties of the Kodaira energy.
In a recent paper, K. B. Lee introduced the notion of an
infra-solvmanifold of type
(R). These manifolds are completely determined by their fundamental group
a Π is a finite extension of a lattice Γ of a solvable
Lie group of type (R) and this
lattice Γ is called the translational part of Π.
Having fixed an abstract group Π occurring as the fundamental group
infra-solvmanifold of type (R), it seems to be hard
to describe, in a formal algebraic
language, which subgroup of Π is the translational part. In
his paper Lee formulated
a conjecture which would solve this problem, however, we show that this
fails. Nevertheless, by defining a concept of eigenvalues for
automorphisms of certain
solvable groups (both Lie groups and discrete groups), we are able to prove
theorem, characterizing completely the translational part of the fundamental
of an infra-solvmanifold of type (R).
The Fourier binest algebra is defined as the intersection of the
Volterra nest algebra on L2(ℝ) with its conjugate
by the Fourier transform. Despite the absence of
nonzero finite rank operators this algebra is equal to the closure in the
weak operator topology of the Hilbert–Schmidt bianalytic
pseudo-differential operators. The
(non-distributive) invariant subspace lattice is determined as an
augmentation of the
Volterra and analytic nests (the Fourier binest) by a continuum of nests
with the unimodular functions
exp(−isx2/2) for s>0. This multinest
is the reflexive closure of the Fourier binest and, as a topological space
weak operator topology, it is shown to be homeomorphic to the unit disc.
identification the unitary automorphism group of the algebra is determined
semi-direct product ℝ2×κℝ
action κt(λ, μ)
A locally non-trivial holomorphic family of Riemann surfaces over
hyperbolic Riemann surface B determines a homomorphism (monodromy)
group of B to the modular group. We give a necessary condition
to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate
number of holomorphic families in terms of topological information for
families over Riemann surfaces of genus 0.
We describe a method of constructing explicitly eigenfunctions
Laplacian with a prescribed boundary behaviour on a class of manifolds
hyperbolic type. These are manifolds of the form
where X is an n-dimensional
Riemannian manifold and the metric of M is a perturbation of the
Let X(t) (t ∈ RN)
be a fractional Brownian motion in Rd of index
α. Let GrX([0, 1]N)
be the graph of X and let
It is proved that if N<αd, then almost surely
and if N>αd, then almost surely
where ϕ-m is the ϕ-Hausdorff measure and K1,
K2 are positive finite constants. The
exact Hausdorff measure of the image and graph of certain Gaussian random
with independent fractional Brownian motion components are also obtained.