Expand all Collapse all | Results 26 - 50 of 70 |
26. CJM 2005 (vol 57 pp. 1314)
Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$-form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the Darboux-Givental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 |
27. CJM 2005 (vol 57 pp. 961)
Cone-Monotone Functions: Differentiability and Continuity We provide a porosity-based approach to the differentiability and
continuity of real-valued functions on separable Banach spaces,
when the function is monotone with respect to an ordering induced
by a convex cone $K$ with non-empty interior. We also show that
the set of nowhere $K$-monotone functions has a $\sigma$-porous
complement in the space of continuous functions endowed with the
uniform metric.
Keywords:Cone-monotone functions, Aronszajn null set, directionally porous, sets, GÃ¢teaux differentiability, separable space Categories:26B05, 58C20 |
28. CJM 2005 (vol 57 pp. 771)
The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of
an elliptic differential operator $A$ on a manifold with conical
singularities, acting as an unbounded operator on a weighted $L_p$-space.
Under suitable conditions we show that the resolvent
$(\lambda-\underline A)^{-1}$ exists
in a sector of the complex plane and decays like $1/|\lambda|$ as
$|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent
with enough precision to guarantee existence and boundedness of imaginary
powers of $\underline A$.
As an application we treat the Laplace--Beltrami operator for a metric with
straight conical degeneracy and describe domains yielding
maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.
Keywords:Manifolds with conical singularities, resolvent, maximal regularity Categories:35J70, 47A10, 58J40 |
29. CJM 2005 (vol 57 pp. 871)
Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the
Dirichlet problem for one type of vortex equation, which
generalizes the well-known Hermitian Einstein equation. Secondly,
we will give existence results for solutions of these vortex
equations over various complete noncompact K\"ahler manifolds.
Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 |
30. CJM 2005 (vol 57 pp. 506)
Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups
are now well understood when the generator, $A$, is a Dirichlet form
operator.
It has been shown that in some holomorphic function spaces the
semigroup operators, $e^{-tA}$, can be bounded {\it below} from
$L^p$ to $L^q$ when $p,q$ and $t$ are suitably related.
We will show that such lower boundedness occurs also in spaces
of subharmonic functions.
Keywords:Reverse hypercontractivity, subharmonic Categories:58J35, 47D03, 47D07, 32Q99, 60J35 |
31. CJM 2005 (vol 57 pp. 225)
Unbounded Fredholm Operators and Spectral Flow We study the gap (= ``projection norm'' = ``graph distance'') topology
of the space of all (not necessarily bounded) self-adjoint Fredholm
operators in a separable Hilbert space by the Cayley transform and
direct methods. In particular, we show the surprising result that
this space is connected in contrast to the bounded case. Moreover, we
present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions)
and prove the homotopy invariance. As an example, we discuss operator
curves on manifolds with boundary.
Categories:58J30, 47A53, 19K56, 58J32 |
32. CJM 2005 (vol 57 pp. 251)
Some New Results on $L^2$ Cohomology of Negatively Curved Riemannian Manifolds The present paper is concerned with the study of the $L^2$ cohomology
spaces of negatively curved manifolds. The first half presents a
finiteness and vanishing result obtained under some curvature
assumptions, while the second half identifies a class of metrics
having non-trivial $L^2$ cohomology for degree equal to the half
dimension of the space. For the second part we rely on the existence
and regularity properties of the solution for the heat equation for
forms.
Category:58J50 |
33. CJM 2005 (vol 57 pp. 17)
On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations We introduce and study several notions of amenability for unitary
corepresentations and $*$-representations of algebraic quantum groups,
which may be used to characterize amenability and co-amenability for
such quantum groups. As a background for this study, we investigate
the associated tensor C$^{*}$-categories.
Keywords:quantum group, amenability Categories:46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 |
34. CJM 2005 (vol 57 pp. 99)
Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic
differential operators on a homogenous bundle over a compact Lie
group. Subject to a technical condition, these operators are
homotopic to the Laplacian. The technical condition is further
investigated, with examples given where it holds and others where
it does not. Since many spectral invariants are also homotopy
invariants, these results provide information about the invariants
of these operators.
Categories:58J70, 43A77 |
35. CJM 2004 (vol 56 pp. 963)
A Berry-Esseen Type Theorem on Nilpotent Covering Graphs We prove an estimate for the speed of convergence of the
transition probability for a symmetric random walk
on a nilpotent covering graph.
To obtain this estimate, we give a complete proof of
the Gaussian bound for the gradient of the Markov kernel.
Categories:22E25, 60J15, 58G32 |
36. CJM 2004 (vol 56 pp. 926)
K-Homology of the Rotation Algebras $A_{\theta}$ We study the K-homology of the rotation algebras
$A_{\theta}$ using the six-term cyclic sequence
for the K-homology of a crossed product by
${\bf Z}$. In the case that $\theta$ is irrational,
we use Pimsner and Voiculescu's work on AF-embeddings
of the $A_{\theta}$ to search for the missing
generator of the even K-homology.
Categories:58B34, 19K33, 46L |
37. CJM 2004 (vol 56 pp. 590)
The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian
manifold with Heisenberg group $H_1$ as boundary. We calculate the heat
kernel and Green's function for $L$, and give global and small time
estimates of the heat kernel. A class of hypersurfaces in this
manifold can be regarded as approximations of $H_1$. We also restrict
$L$ to each hypersurface and calculate the corresponding heat kernel
and Green's function. We will see that the heat kernel and Green's
function converge to the heat kernel and Green's function on the
boundary.
Categories:35H20, 58J99, 53C17 |
38. CJM 2004 (vol 56 pp. 638)
Multisymplectic Reduction for Proper Actions We consider symmetries of the Dedonder equation arising from
variational problems with partial derivatives. Assuming a proper
action of the symmetry group, we identify a set of reduced equations
on an open dense subset of the domain of definition of the fields
under consideration. By continuity, the Dedonder equation is
satisfied whenever the reduced equations are satisfied.
Keywords:Dedonder equation, multisymplectic structure, reduction,, symmetries, variational problems Categories:58J70, 35A30 |
39. CJM 2003 (vol 55 pp. 1100)
Polar Homology For complex projective manifolds we introduce polar homology
groups, which are holomorphic analogues of the homology groups in
topology. The polar $k$-chains are subvarieties of complex
dimension $k$ with meromorphic forms on them, while the boundary
operator is defined by taking the polar divisor and the Poincar\'e
residue on it. One can also define the corresponding analogues for the
intersection and linking numbers of complex submanifolds, which have the
properties similar to those of the corresponding topological notions.
Keywords:Poincar\' e residue, holomorphic linking Categories:14C10, 14F10, 58A14 |
40. CJM 2003 (vol 55 pp. 266)
Two Algorithms for a Moving Frame Construction The method of moving frames, introduced by Elie Cartan, is a
powerful tool for the solution of various equivalence problems.
The practical implementation of Cartan's method, however, remains
challenging, despite its later significant development and
generalization. This paper presents two new variations on the Fels and
Olver algorithm, which under some conditions on the group action,
simplify a moving frame construction. In addition, the first
algorithm leads to a better understanding of invariant differential
forms on the jet bundles, while the second expresses the differential
invariants for the entire group in terms of the differential invariants
of its subgroup.
Categories:53A55, 58D19, 68U10 |
41. CJM 2003 (vol 55 pp. 247)
Differential Structure of Orbit Spaces: Erratum This note signals an error in the above paper by giving a counter-example.
Categories:37J15, 58A40, 58D19, 70H33 |
42. CJM 2002 (vol 54 pp. 1187)
On the Injectivity of $C^1$ Maps of the Real Plane Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Categories:34D05, 54H20, 58F10, 58F21 |
43. CJM 2002 (vol 54 pp. 1086)
Combinatorics of the Heat Trace on Spheres We present a concise explicit expression for the heat trace
coefficients of spheres. Our formulas yield certain combinatorial
identities which are proved following ideas of D.~Zeilberger. In
particular, these identities allow to recover in a surprising way
some known formulas for the heat trace asymptotics. Our approach is
based on a method for computation of heat invariants developed in [P].
Categories:05A19, 58J35 |
44. CJM 2001 (vol 53 pp. 715)
Differential Structure of Orbit Spaces We present a new approach to singular reduction of Hamiltonian systems
with symmetries. The tools we use are the category of differential
spaces of Sikorski and the Stefan-Sussmann theorem. The former is
applied to analyze the differential structure of the spaces involved
and the latter is used to prove that some of these spaces are smooth
manifolds.
Our main result is the identification of accessible sets of the
generalized distribution spanned by the Hamiltonian vector fields of
invariant functions with singular reduced spaces. We are also able
to describe the differential structure of a singular reduced space
corresponding to a coadjoint orbit which need not be locally closed.
Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds Categories:37J15, 58A40, 58D19, 70H33 |
45. CJM 2001 (vol 53 pp. 780)
Seiberg-Witten Invariants of Lens Spaces We show that the Seiberg-Witten invariants of a lens space determine
and are determined by its Casson-Walker invariant and its
Reidemeister-Turaev torsion.
Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sums Categories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25 |
46. CJM 2001 (vol 53 pp. 278)
Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson
Grassmannian give rise to Darboux transformations between the
solutions of the $\KP$ hierarchy corresponding to these planes. We
present a closed form of the operators that procure the transformation
and express them in the related geometric data. Further the
associated transformation on the level of $\tau$-functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 |
47. CJM 2001 (vol 53 pp. 73)
Stratification Theory from the Weighted Point of View In this paper, we investigate stratification theory in terms of the
defining equations of strata and maps (without tube systems), offering
a concrete approach to show that some given family is topologically
trivial. In this approach, we consider a weighted version of
$(w)$-regularity condition and Kuo's ratio test condition.
Categories:32B99, 14P25, 32Cxx, 58A35 |
48. CJM 2000 (vol 52 pp. 1235)
Representations with Weighted Frames and Framed Parabolic Bundles There is a well-known correspondence (due to Mehta and Seshadri in
the unitary case, and extended by Bhosle and Ramanathan to other
groups), between the symplectic variety $M_h$ of representations of
the fundamental group of a punctured Riemann surface into a compact
connected Lie group~$G$, with fixed conjugacy classes $h$ at the
punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles
on the unpunctured surface with a parabolic structure at the puncture
points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs
(representations of the fundamental group into $G$, ``weighted frame''
at the puncture points), and a corresponding complex variety ${\cal
P}$ of moduli of ``framed parabolic bundles'', which encompass
respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that
one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal
M}_h$ from ${\cal P}$ by a complex quotient. This allows us to
explain certain features of the toric geometry of the $\SU(2)$ moduli
spaces discussed by Jeffrey and Weitsman, by giving the actual toric
variety associated with their integrable system.
Categories:58F05, 14D20 |
49. CJM 2000 (vol 52 pp. 1057)
The Spectrum of an Infinite Graph In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.
Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum Categories:05C50, 58G25 |
50. CJM 2000 (vol 52 pp. 897)
Higher Order Scattering on Asymptotically Euclidean Manifolds We develop a scattering theory for perturbations of powers of the
Laplacian on asymptotically Euclidean manifolds. The (absolute)
scattering matrix is shown to be a Fourier integral operator
associated to the geodesic flow at time $\pi$ on the boundary.
Furthermore, it is shown that on $\Real^n$ the asymptotics of certain
short-range perturbations of $\Delta^k$ can be recovered from the
scattering matrix at a finite number of energies.
Keywords:scattering theory, conormal, Lagrangian Category:58G15 |