Expand all Collapse all | Results 1 - 12 of 12 |
1. CJM Online first
Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized
KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$-energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hler-Einstein metrics with
semi-positive twisting $(1, 1)$-form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$-energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality.
Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequality Categories:53C55, 32W20 |
2. CJM 2011 (vol 64 pp. 935)
The H and K Families of Mock Theta Functions In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions.
He observed that as $q$ radially approaches any root of unity $\zeta$ at which
$F(q)$ has an exponential singularity, there is a $\theta$-function
$T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have
been found that possess this property. These functions are related to
a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some
rational number $r$. For this reason we refer to $H$ as a ``universal'' mock
$\theta$-function. Modular transformations of $H$ give rise to the functions
$K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost
notebook. We prove various linear relations between these functions using
Appell-Lerch sums (also called generalized Lambert series). Some relations
(mock theta ``conjectures'') involving mock $\theta$-functions
of even order and $H$ are listed.
Keywords:mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert series Categories:11B65, 33D15 |
3. CJM 2011 (vol 64 pp. 805)
Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 |
4. CJM 2011 (vol 63 pp. 1284)
Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0
\operatorname{mod} 5$ for the partition
function. We provide a method to find all simple
congruences of this type in the coefficients of the inverse of a
modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper
half plane. This is applied to answer open questions about the
(non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored $F$-partitions.
Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crank Categories:11F33, 11P83 |
5. CJM 2009 (vol 61 pp. 534)
Girsanov Transformations for Non-Symmetric Diffusions Let $X$ be a diffusion process, which is assumed to be
associated with a (non-symmetric) strongly local Dirichlet form
$(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For
$u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet
space, we investigate some properties of the Girsanov transformed
process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of
$X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$.
We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to
be in duality with respect to the measure $e^{2u}m$. We also
construct a counterexample, which shows that this condition may
not be satisfied and hence $(Y,\widehat{Y})$ may not be dual
processes. Then we present a sufficient condition under which $Y$
is associated with a semi-Dirichlet form. Moreover, we give an
explicit representation of the semi-Dirichlet form.
Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroup Categories:60J45, 31C25, 60J57 |
6. CJM 2008 (vol 60 pp. 457)
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are
generalizations of post-crit8cally finite self-similar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local self-similarity, and allow countably many cells
connected at each junction point.
In particular, we consider post-critically infinite fractals.
We prove that if Kigami's resistance form
satisfies certain assumptions, then there exists a weak Riemannian metric
such that the energy can be expressed as the integral of the norm squared
of a weak gradient with respect to an energy measure.
Furthermore, we prove that if such a set can be homeomorphically represented
in harmonic coordinates, then for smooth functions the weak gradient can be
replaced by the usual gradient.
We also prove a simple formula for the energy measure Laplacian in harmonic
coordinates.
Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18 |
7. CJM 2004 (vol 56 pp. 1068)
Regular Embeddings of Generalized Hexagons We classify the generalized hexagons which are laxly
embedded in projective space such that the embedding is flat and
polarized. Besides the standard examples related to the hexagons
defined over the algebraic groups of type $\ssG_2$, $^3\ssD_4$ and
$^6\ssD_4$ (and occurring in projective dimensions $5,6,7$), we
find new examples in unbounded dimension related to the mixed
groups of type $\ssG_2$.
Keywords:Moufang generalized hexagons, embeddings, mixed hexagons, classical, hexagons Categories:51E12, 51A45 |
8. CJM 2004 (vol 56 pp. 293)
Structure of modules induced from simple modules with minimal annihilator We study the structure of generalized Verma modules over a
semi-simple complex finite-dimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
Bernstein-Gelfand-Gelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of Harish-Chandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.
Keywords:parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categories Categories:17B10, 22E47 |
9. CJM 2001 (vol 53 pp. 1174)
A Generalized Variational Principle We prove a strong variant of the Borwein-Preiss variational principle, and
show that on Asplund spaces, Stegall's variational principle follows
from it via a generalized Smulyan test. Applications are discussed.
Keywords:variational principle, strong minimizer, generalized Smulyan test, Asplund space, dimple point, porosity Category:49J52 |
10. CJM 1998 (vol 50 pp. 210)
Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$ In this paper, we determine when two simple generalized Cartan
type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss
the relationship between the Jacobian conjecture and the generalized
Cartan type $W$ Lie algebras.
Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture Categories:17B40, 17B65, 17B56, 17B68 |
11. CJM 1997 (vol 49 pp. 798)
Boundedness of solutions of parabolic equations with anisotropic growth conditions In this paper, we consider the parabolic equation
with anisotropic growth conditions, and obtain some criteria on
boundedness of solutions, which generalize the corresponding results
for the isotropic case.
Keywords:Parabolic equation, anisotropic growth conditions, generalized, solution, boundness Categories:35K57, 35K99. |
12. CJM 1997 (vol 49 pp. 468)
Fine spectra and limit laws I. First-order laws Using Feferman-Vaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a first-order law.
The condition presented is best possible in the sense that if it is
violated then one can find an admissible class with the same fine
spectrum which does not have a first-order law. We present three
conditions for verifying that the above property actually holds.
The first condition is that the count function of an admissible class
has regular variation with a certain uniformity of convergence. This
applies to a wide range of admissible classes, including those
satisfying Knopfmacher's Axiom A, and those satisfying Bateman
and Diamond's condition.
The second condition is similar to the first condition, but designed
to handle the discrete case, {\it i.e.}, when the sizes of the structures
in an admissible class $K$ are all powers of a single integer. It applies
when either the class of indecomposables or the whole class satisfies
Knopfmacher's Axiom A$^\#$.
The third condition is also for the discrete case, when there is a
uniform bound on the number of $K$-indecomposables of any given size.
Keywords:First order limit laws, generalized number theory Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |