In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})}
=1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
The minimal resolution of the degree four cyclic cover of the plane
branched along a GIT stable quartic is a $K3$ surface with a non
symplectic action of $\Z_4$. In this paper
we study the geometry of the one-dimensional family of $K3$ surfaces
associated to the locus of plane quartics with five nodes.
We use some classical estimates for polynomial roots to provide
several irreducibility criteria for polynomials with integer
coefficients that have one sufficiently large coefficient and take a
prime value.
In this paper, we give the $L^p$ boundedness for
a class of parabolic Littlewood--Paley $g$-function with its kernel
function $\Omega$ is in the Hardy space $H^1(S^{n-1})$.
We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
This paper introduces the concepts of
fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity
in intuitionistic fuzzy topological spaces. After defining the fundamental
concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological
spaces, we present intuitionistic fuzzy $\gamma$-open sets and
intuitionistic fuzzy $\gamma$-continuity and other results related
topological concepts.
This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewood-type theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
A ring $R$ is called {\it quasi-Baer} if the right
annihilator of every right ideal of $R$ is generated by an
idempotent as a right ideal. We investigate the quasi-Baer
property of skew group rings and fixed rings under a finite group
action on a semiprime ring and their applications to
$C^*$-algebras.
Various examples to illustrate and
delimit our results are provided.
We compute the minimal polynomials of the Ramanujan values $t_n$,
where $n\equiv 11 \mod 24$, using the Shimura reciprocity law.
These polynomials can be used for defining the Hilbert class field
of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ and have
much smaller coefficients than the Hilbert polynomials.
We say that a numerical semigroup is $d$-squashed if it can
be written in the form
$$ S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$
for $N,a_1,\dots,a_d$ positive integers with
$\gcd(a_1,\dots, a_d)=1$.
Rosales and Urbano have shown that a numerical semigroup is
2-squashed if and only if it is proportionally modular.
Recent works by Rosales et al. give a concrete example of a
numerical semigroup that cannot be written as an intersection of
$2$-squashed semigroups. We will show the existence of infinitely
many numerical semigroups that cannot be written as an
intersection of $2$-squashed semigroups. We also will prove the
same result for $3$-squashed semigroups. We conjecture that there
are numerical semigroups that cannot be written as the
intersection of $d$-squashed semigroups for any fixed $d$, and we
prove some partial results towards this conjecture.
We obtain new characterizations for Bergman spaces with standard
weights in terms of Lipschitz type conditions in the Euclidean,
hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we
prove optimal embedding theorems when an analytic function
on the unit disk is symmetrically lifted to the bidisk.
Let $f\in L_{2\pi }$ be a real-valued even function with its Fourier series $%
\frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let
$S_{n}(f,x) ,\;n\geq 1,$ be the $n$-th partial sum of the Fourier series. It
is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and
$\lim_{n\rightarrow \infty }a_{n}=0$, then%
\begin{equation*}
\lim_{n\rightarrow \infty }\Vert f-S_{n}(f)\Vert _{L}=0
\text{ if
and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0.
\end{equation*}%
We weaken the monotone condition in this classical result to the so-called
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in real-valued function space is presented as a
special case of the main result in this paper, which gives the $L^{1}$%
-convergence of a function $f\in L_{2\pi }$ in complex space. We also give
results on $L^{1}$-approximation of a function $f\in L_{2\pi }$ under the
MVBV condition.