We construct elliptic curves over the field $\mathbf{Q}(\sqrt{-3})$ with torsion group $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/3\mathbf{Z}$ and ranks equal to 7 and an elliptic curve over the same field with torsion group $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/6\mathbf{Z}$ and rank equal to 6.

In this note, we will prove two congruences involving broken 3-diamond partitions and broken 5-diamond partitions. The two congruences were conjectured by Peter Paule and Silviu Radu in 2009.

We extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions we construct different kinds of sheaves of multi-asymptotically developable functions, whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima.

In this paper, we mainly investigate the growth and the value distribution of meromorphic solutions of the linear difference equation \begin{equation*} a_{n}(z)f(z+n)+…+a_{1}(z)f(z+1)+a_{0}(z)f(z)=b(z), \end{equation*} where $a_{0}(z),a_{1}(z),\cdots,a_{n}(z),b(z)$ are entire functions such that $a_{0}(z)a_{n}(z)\not\equiv 0$. For a finite order meromorphic solution $f(z)$, some interesting results on the relation between $\rho=\rho(f)$ and $\lambda_{f}=\max\{\lambda(f),\lambda(1/f)\}$, are proved. And examples are provided for our results.

In this paper, we discuss the linearity of a sequence space $\Lambda_{p}(f)$, and the conditions such that $\ell_{1} = \Lambda_{1}(f)$ holds are characterized in term of the essential bounded variation of $f\in L_{1}(\mathbf{R})$, i.e. $\ell_{1} = \Lambda_{1}(f)$ if and only if $f\in BV(\mathbf{R})$.

In a preceding article~[7] the authors and Tran Ngoc Nam constructed a minimal injective resolution of the mod 2 cohomology of a Thom spectrum. A Segal conjecture type theorem for this spectrum was proved. In this paper one shows that the above mentioned resolutions can be realized topologically. In fact there exists a family of cofibrations inducing short exact sequences in mod 2 cohomology. The resolutions above are obtained by splicing together these short exact sequences. Thus the injective resolutions are realizable in the best possible sense. In fact our construction appears to be in some sense an injective closure of one of Takayasu. It strongly suggests that one can construct geometrically (not only homotopically) certain dual Brown-Gitler spectra.

In the present paper, transitivity of a linear operators acting on a reflexive Banach space with the weak topology is investigated. We show any bounded operator, transitive on an open bounded subset of $X$ with the weak topology, is weakly hypercyclic.