Expand all Collapse all | Results 1 - 14 of 14 |
1. CJM Online first
Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$-values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$-values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$-values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$-functions.
Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric series Categories:11F67, 33C20 |
2. CJM 2013 (vol 66 pp. 1382)
Weighted Carleson Measure Spaces Associated with Different Homogeneities In this paper, we introduce weighted Carleson measure spaces associated
with different homogeneities and prove that these spaces are the dual spaces
of weighted Hardy spaces studied in a forthcoming paper.
As an application, we establish
the boundedness of composition of two CalderÃ³n-Zygmund operators with
different homogeneities on the weighted Carleson measure spaces; this,
in particular, provides the weighted endpoint estimates for the operators
studied by Phong-Stein.
Keywords:composition of operators, weighted Carleson measure spaces, duality Categories:42B20, 42B35 |
3. CJM 2013 (vol 66 pp. 566)
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms |
Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of
MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.
Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence Categories:22E50, 11F70, 22E55, 22E35 |
4. CJM 2011 (vol 64 pp. 1075)
A Stochastic Difference Equation with Stationary Noise on Groups We consider the stochastic difference equation $$\eta _k = \xi _k
\phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$
where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued
random variables and $\eta _k$ are unknown $G$-valued random variables.
This equation was considered by Tsirelson and Yor on
one-dimensional torus. We consider the case when $\xi _k$ have a
common law $\mu$ and prove that if $G$ is a distal group and $\phi$
is a distal automorphism of $G$ and if the equation has a solution,
then extremal solutions of the equation are in one-one
correspondence with points on the coset space $K\backslash G$ for
some compact subgroup $K$ of $G$ such that $\mu$ is supported on
$Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide
a necessary and sufficient condition for the existence of solutions
to the equation.
Keywords:dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers Categories:60B15, 60G20 |
5. CJM 2011 (vol 64 pp. 961)
Densities of Short Uniform Random Walks We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic
to expect similar results for more than five steps. New results are also
presented concerning the moments of uniform random walks and, in particular,
their derivatives. Relations with Mahler measures are discussed.
Keywords:random walks, hypergeometric functions, Mahler measure Categories:60G50, 33C20, 34M25, 44A10 |
6. CJM 2011 (vol 63 pp. 648)
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps |
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional
self-similar measures that are defined by iterated function systems
with overlaps and satisfy a family of second-order self-similar
identities. As applications of our result we obtain the spectral dimension
of important measures such as the infinite Bernoulli convolution
associated with the golden ratio and convolutions of Cantor-type measures.
The main novelty of our result is that the iterated function systems
we consider are not post-critically finite and do not satisfy the
well-known open set condition.
Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75 |
7. CJM 2010 (vol 62 pp. 1419)
BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$.
Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu) Categories:42B25, 42B30, 47A30, 43A99 |
8. CJM 2010 (vol 62 pp. 827)
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces Categories:46E40, 42B25, 46B20 |
9. CJM 2009 (vol 61 pp. 124)
Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces Categories:28C10, 46B20, 54F65 |
10. CJM 2006 (vol 58 pp. 401)
On Pointwise Estimates of Positive Definite Functions With Given Support The following problem has been suggested by Paul Tur\' an. Let
$\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$
or in the torus $\TT^d$. Then, what is the largest possible value
of the integral of positive definite functions that are supported
in $\Omega$ and normalized with the value $1$ at the origin? From
this, Arestov, Berdysheva and Berens arrived at the analogous
pointwise extremal problem for intervals in $\RR$. That is, under
the same conditions and normalizations, the supremum of possible
function values at $z$ is to be found for any given point
$z\in\Omega$. However, it turns out that the problem for the real
line has already been solved by Boas and Kac, who gave several
proofs and also mentioned possible extensions to $\RR^d$ and to
non-convex domains as well.
Here we present another approach to the problem, giving the
solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we
elaborate on the fact that the problem is essentially
one-dimensional and investigate non-convex open domains as well.
We show that the extremal problems are equivalent to some more
familiar ones concerning trigonometric polynomials, and thus find
the extremal values for a few cases. An analysis of the
relationship between the problem for $\RR^d$ and that for $\TT^d$
is given, showing that the former case is just the limiting case
of the latter. Thus the hierarchy of difficulty is established, so
that extremal problems for trigonometric polynomials gain renewed
recognition.
Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems Categories:42B10, 26D15, 42A82, 42A05 |
11. CJM 2005 (vol 57 pp. 1139)
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set We prove that it is relatively consistent with $\ZFC$ that in any
perfect Polish space, for every nonmeager set $A$ there exists a
nowhere dense Cantor set $C$ such that $A\cap C$ is nonmeager in
$C$. We also examine variants of this result and establish a
measure theoretic analog.
Keywords:Property of Baire, Lebesgue measure,, Cantor set, oracle forcing Categories:03E35, 03E17, 03E50 |
12. CJM 2005 (vol 57 pp. 204)
On the Duality between Coalescing Brownian Motions A duality formula is found for coalescing Brownian motions on the
real line. It is shown that the joint distribution of a coalescing
Brownian motion can be determined by another coalescing Brownian
motion running backward. This duality is used to study a
measure-valued process arising as the high density limit of the
empirical measures of coalescing Brownian motions.
Keywords:coalescing Brownian motions, duality, martingale problem,, measure-valued processes Categories:60J65, 60G57 |
13. CJM 2003 (vol 55 pp. 3)
An Exactly Solved Model for Mutation, Recombination and Selection It is well known that rather general mutation-recombination models can be
solved algorithmically (though not in closed form) by means of Haldane
linearization. The price to be paid is that one has to work with a
multiple tensor product of the state space one started from.
Here, we present a relevant subclass of such models, in continuous time,
with independent mutation events at the sites, and crossover events
between them. It admits a closed solution of the corresponding
differential equation on the basis of the original state space, and
also closed expressions for the linkage disequilibria, derived by means
of M\"obius inversion. As an extra benefit, the approach can be extended
to a model with selection of additive type across sites. We also derive
a necessary and sufficient criterion for the mean fitness to be a Lyapunov
function and determine the asymptotic behaviour of the solutions.
Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, MÃ¶bius inversion Categories:92D10, 34L30, 37N30, 06A07, 60J25 |
14. CJM 1997 (vol 49 pp. 887)
Polynomials with $\{ 0, +1, -1\}$ coefficients and a root close to a given point For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,-1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,-1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha
-\beta| \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
|\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 |