Expand all Collapse all | Results 1 - 8 of 8 |
1. CJM 2012 (vol 64 pp. 1395)
Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form
\begin{align*}
\nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta
\\
u&=\varphi\text{ on }\partial \Theta.
\end{align*}
The principal part $\xi'P(x)\xi$ of the above equation is assumed to
be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that
may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using
techniques of functional analysis applied to the degenerate Sobolev
spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and
$QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in
previous works.
Sawyer and Wheeden give a regularity theory
for a subset of the class of equations dealt with here.
Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions Categories:35A01, 35A02, 35D30, 35J70, 35H20 |
2. CJM 2011 (vol 63 pp. 634)
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even
integral weight $k$ for the full modular group.
Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of
two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively.
In this paper we are able to show the following results about higher
moments of Fourier coefficients of holomorphic cusp forms.\newline
(i) For any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon}
\quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}.
\end{equation*}
(ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon};
\end{equation*}
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr);
\]
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr),
\]
where $P(x)$ is a polynomial of degree $3$.
Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function Categories:11F30, , , , 11F11, 11F66 |
3. CJM 2010 (vol 63 pp. 181)
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the Askey--Wilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$-ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$-Jacobi polynomials is given.
Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients Categories:33D45, 42C05 |
4. CJM 2009 (vol 61 pp. 762)
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring Let $(A,\m)$ be a Noetherian local ring with infinite residue
field and let $I$ be an ideal in $A$ and let $F(I) =
\bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$.
We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$
when the $a$-invariant of the associated graded ring $G(I)$ is negative.
Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone Categories:13A30, 13D40 |
5. CJM 2009 (vol 61 pp. 351)
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicity-free. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual Littlewood--Richardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Categories:14L30, 22E46 |
6. CJM 2008 (vol 60 pp. 685)
Closed and Exact Functions in the Context of Ginzburg--Landau Models For a general vector field we exhibit two Hilbert spaces, namely
the space of so called \emph{closed functions} and the space of \emph{exact functions}
and we calculate the codimension of the space of exact functions
inside the larger space of closed functions.
In particular we provide a new approach for the known cases:
the Glauber field and the second-order Ginzburg--Landau field
and for the case of the fourth-order Ginzburg--Landau field.
Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries Categories:42B05, 81Q50, 42A16 |
7. CJM 2005 (vol 57 pp. 1080)
The Gelfond--Schnirelman Method in Prime Number Theory The original Gelfond--Schnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshev-type lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$-function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomial-type weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 |
8. CJM 2001 (vol 53 pp. 33)
Merit Factors of Polynomials Formed by Jacobi Symbols We give explicit formulas for the $L_4$ norm (or equivalently for the
merit factors) of various sequences of polynomials related to the
polynomials
$$
f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n.
$$
where $(\frac{\cdot}{N})$ is the Jacobi symbol.
Two cases of particular interest are when $N = pq$ is a product of two
primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt,
Jensen and Jensen and of the authors.
This study arises from a number of conjectures of Erd\H{o}s,
Littlewood and others that concern the norms of polynomials with
$-1,1$ coefficients on the disc. The current best examples are of the
above form when $N$ is prime and it is natural to see what happens for
composite~$N$.
Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols Categories:11J54, 11B83, 12-04 |