Expand all Collapse all | Results 76 - 100 of 208 |
76. CMB 2008 (vol 51 pp. 497)
Expected Norms of Zero-One Polynomials Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zero-one polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n :
\alf(1) = m \}$, whose elements correspond to the ${n \choose m}$
subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus
\cA_n$, whose elements are the zero-one polynomials of degree
exactly~$n$.
Many researchers have studied norms of polynomials with restricted
coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm
of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 +
a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$
and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k
:= \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$.
If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots +
z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the
number of times $k$ appears as a difference $\beta_i - \beta_j$. The
condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1
\le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1,
\ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of
pairs of elements are distinct).
In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in
\cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show
that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$
yields a new proof of the known result: if $m = o(n^{1/4})$ and
$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then
almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that
$\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.
Categories:11B83, 11C08, 30C10 |
77. CMB 2008 (vol 51 pp. 627)
Summation of Series over Bourget Functions In this paper we derive formulas for summation of series involving
J.~Bourget's generalization of Bessel functions of integer order, as
well as the analogous generalizations by H.~M.~Srivastava. These series are
expressed in terms of the Riemann $\z$ function and Dirichlet
functions $\eta$, $\la$, $\b$, and can be brought into closed form in
certain cases, which means that the infinite series are represented
by finite sums.
Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions Categories:33C10, 11M06, 65B10 |
78. CMB 2008 (vol 51 pp. 561)
Expansion of the Riemann $\Xi$ Function in Meixner--Pollaczek Polynomials In this article we study in detail the expansion of the Riemann
$\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit
formulas, recurrence relation and asymptotic expansion for the
coefficients and investigate the zeros of the partial sums.
Categories:41A10, 11M26, 33C45 |
79. CMB 2008 (vol 51 pp. 337)
Differences between Perfect Powers We apply the hypergeometric method of Thue and Siegel to prove
that if $a$ and $b$ are positive integers, then the inequality $
0 <| a^x - b^y | < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$
has at most a single solution in positive integers $x$ and $y$.
This essentially sharpens a classic result of LeVeque.
Categories:11D61, 11D45 |
80. CMB 2008 (vol 51 pp. 399)
Linear Equations with Small Prime and Almost Prime Solutions Let $b_1, b_2$ be any integers such that
$\gcd(b_1, b_2)=1$ and $c_1|b_1|<|b_2|\leq c_2|b_1|$, where
$c_1, c_2$ are any given positive constants. Let $n$ be any
integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote
any integer with no more than $k$ prime factors, counted according
to multiplicity. In this paper, for almost all $b_2$, we prove (i)
a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$
is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$
whenever both $b_i$ are positive, and (ii) a sharp upper bound for the
least solutions $p, m$ of the above equation whenever $b_i$ are
not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.
Keywords:sieve method, additive problem Categories:11P32, 11N36 |
81. CMB 2008 (vol 51 pp. 172)
Consecutive Large Gaps in Sequences Defined by Multiplicative Constraints In this paper we obtain quantitative results on the occurrence of
consecutive large gaps between $B$-free numbers, and consecutive
large gaps between nonzero Fourier coefficients of a class of
newforms without complex multiplication.
Keywords:$B$-free numbers, consecutive gaps Categories:11N25, 11B05 |
82. CMB 2008 (vol 51 pp. 57)
A Note on Integer Symmetric Matrices and Mahler's Measure We find a lower bound on the absolute value of the discriminant of
the minimal polynomial of an integral symmetric matrix and apply
this result to find a lower bound on Mahler's measure of related
polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure Categories:11C20, 11R06 |
83. CMB 2008 (vol 51 pp. 3)
84. CMB 2008 (vol 51 pp. 32)
On Linear Independence of a Certain Multivariate Infinite Product Let $q,m,M \ge 2$ be positive integers and
$r_1,r_2,\dots ,r_m$ be positive rationals and
consider the following $M$ multivariate infinite products
\[
F_i = \prod_{j=0}^\infty ( 1+q^{-(Mj+i)}r_1+q^{-2(Mj+i)}r_2+\dots +
q^{-m(Mj+i)}r_m)
\]
for $i=0,1,\dots ,M-1$.
In this article, we study the linear independence of these infinite products.
In particular, we obtain a lower bound for the dimension of the vector space
$\IQ F_0+\IQ F_1 +\dots + \IQ F_{M-1} + \IQ$ over $\IQ$ and show that
among these $M$ infinite products, $F_0, F_1,\dots ,F_{M-1}$, at least
$\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow
\infty$.
Category:11J72 |
85. CMB 2008 (vol 51 pp. 100)
Dynamical Zeta Function for Several Strictly Convex Obstacles The behavior of the dynamical zeta function $Z_D(s)$ related to
several strictly convex disjoint obstacles is similar to that of the
inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function
$\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing
only over primitive periodic rays. In this paper we examine the
analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re
s = s_2$, where $s_2$ is the abscissa of absolute convergence of the
series obtained by the second iterations of the primitive periodic
rays. We show that at least one of the functions $Z_D(s), \Pi(s)$
has a singularity at $s = s_2$.
Keywords:dynamical zeta function, periodic rays Categories:11M36, 58J50 |
86. CMB 2008 (vol 51 pp. 134)
Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups
coincides with the class of numerical semigroups having a Toms
decomposition.
Categories:20M14, 11D75 |
87. CMB 2007 (vol 50 pp. 486)
Higher-Dimensional Modular\\Calabi--Yau Manifolds We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their
modularity.
Categories:14G10, 14J32, 11G40 |
88. CMB 2007 (vol 50 pp. 594)
Ramification des groupes abÃ©liens d'automorphismes des corps $\mathbb F_q(\!(X)\!)$ Soit $q$ une puissance d'un nombre premier
$p$. Dans cette note on \'etablit la g\'en\'eralisation suivante
d'un th\'eor\`eme de Wintenberger : tout sous-groupe ab\'elien
ferm\'e du groupe des $\mathbb F_q$-auto\-morphismes continus du corps
des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration
de ramification est un groupe filtr\'e isomorphe au groupe de Galois
d'une extension ab\'elienne d'un corps local {\`a} corps
r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification
de l'extension en num\'erotation inf\'erieure.
Category:11S15 |
89. CMB 2007 (vol 50 pp. 399)
Expansions in Complex Bases Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely
studied in the last
forty years. They turned out to be relevant in
various domains of mathematics, such as the theory of finite
automata, number
theory, fractals or dynamical systems.
Several results were extended by Dar\'oczy and K\'atai
for expansions
in complex bases. We introduce an adaptation of the so-called greedy
algorithm to the complex case, and we
generalize one of their main theorems.
Keywords:non-integer bases, greedy expansions, beta-expansions Categories:11A67, 11A63, 11B85 |
90. CMB 2007 (vol 50 pp. 334)
Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields We obtain Hauptmoduls of genus zero congruence
subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is
a prime and $w_p$ is the Atkin--Lehner involution. We then use the
Hauptmoduls, along with modular functions on $\Gamma_1(p)$
to construct families of cyclic extensions of quadratic number
fields. Further examples of cyclic extension of bi-quadratic and
tri-quadratic number fields are also given.
Categories:11F03, 11G16, 11R20 |
91. CMB 2007 (vol 50 pp. 409)
Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields We show that, for most of the elliptic curves $\E$ over a prime finite
field
$\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number
field containing the endomorphism ring of $\E$ over $\F_p$
is sufficiently large.
We also obtain an asymptotic formula for the number of distinct
quadratic number fields generated by the endomorphism rings
of all elliptic curves over $\F_p$.
Categories:11G20, 11N32, 11R11 |
92. CMB 2007 (vol 50 pp. 196)
Plane Quartic Twists of $X(5,3)$ Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genus-three case $N=5$.
Categories:11F03, 11F80, 14G05 |
93. CMB 2007 (vol 50 pp. 313)
On Cauchy--Liouville--Mirimanoff Polynomials Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
Cauchy--Liouville--Mirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are \emph{conjecturally}
also necessary for irreducibility.
Categories:11G30, 11R09, 12D05, 12E10 |
94. CMB 2007 (vol 50 pp. 284)
Second Order Mock Theta Functions In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$-series, mock theta function, Mordell integral Categories:11B65, 33D15 |
95. CMB 2007 (vol 50 pp. 234)
A Remark on a Modular Analogue of the Sato--Tate Conjecture The original Sato--Tate Conjecture concerns the angle distribution
of the eigenvalues arising from non-CM elliptic curves. In this paper,
we formulate a modular analogue of the Sato--Tate Conjecture and prove
that the angles arising from non-CM holomorphic Hecke
eigenforms with non-trivial central characters are not distributed
with respect to the Sate--Tate measure
for non-CM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
Keywords:$L$-functions, Elliptic curves, Sato--Tate Categories:11F03, 11F25 |
96. CMB 2007 (vol 50 pp. 215)
Elliptic $K3$ Surfaces with Geometric Mordell--Weil Rank $15$ We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$-surfaces with geometric Mordell--Weil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 |
97. CMB 2007 (vol 50 pp. 191)
Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjecture Categories:11R04, 11R06, 11R09, 11R33, 11D09 |
98. CMB 2007 (vol 50 pp. 11)
van der Pol Expansions of L-Series We provide concise series representations for various
L-series integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying L-series.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series Categories:11M35, 11M41, 30B50 |
99. CMB 2007 (vol 50 pp. 71)
Polynomials for Kloosterman Sums Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$
denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman
sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq
m$, $(d,m)=1$,
satisfy the polynomial
$$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1}
x^{\phi(m)-1} + \dots + c_{\phi(m)},$$
where the sum and product are taken over a complete system of reduced residues
modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely,
$$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$
where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$
of reduced residues modulo $m$. Questions concerning the explicit
determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their
beginning coefficients), their reducibility over the rational field
${\bf Q}$ and duplication among the factors are studied. The treatment
is similar to what has been done for period polynomials for finite
fields.
Categories:11L05, 11T24 |
100. CMB 2007 (vol 50 pp. 158)
A Note on Giuga's Conjecture Let $G(X)$ denote the number of positive composite integers $n$
satisfying $\sum_{j=1}^{n-1}j^{n-1}\equiv -1 \tmod{n}$.
Then $G(X)\ll X^{1/2}\log X$ for sufficiently large $X$.
Category:11A51 |