In this paper, we classify 3-dimensional Lagrangian Willmore submanifolds of the nearly kaehler 6-sphere $S^6(1)$ with constant scalar curvature.

Let $K$ be a field of characteristic zero and $A_{2}:=A_{2}(K)$ the 2$nd$–Weyl algebra over $K$. We establish a close connection between the maximal left ideals of $A_{2}$ and the simple derivations of $K[X_{1},X_{2}]$.

MAIN THEOREM. Let$d = \partial_1 + \beta\partial_2$be a simple derivation of$K[X_1, X_2]$with$\beta\in K[X_1, X_2]$. Then, there exists$\gamma\in K[X_1, X_2]$such that$d + \gamma$generates a maximal left ideal of$A_2$. More precisely, the following is true:

1. $\deg_{X_2} (\beta) \geq 2$ or $\deg_{X_2} (\beta) = 1$ and $\deg_{X_1} (\partial_2(\beta)) \geq 1$;

2. $d + gX_2$generates a maximal left ideal of$A_2$if$g \in K[X_1]\\{0\}$is such that

1. (a) $g \in -(\partial^2_2(\beta)/2)\mathbb{N}$when$\deg_{X_2} (\beta) \geq 2$,

2. (b) $\deg_{X_1} (g) < \deg_{X_1} (\partial_2(\beta))$when$\deg_{X_2} (\beta) = 1$.

As applications, we obtain large families of concrete examples of cyclic maximal left ideals of $A_2$; such examples have been rather scarce so far.

Pointing out the difference between the Discrete Nonlinear Schrödinger equation with the classical power law nonlinearity – for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice – we prove either global existence or nonexistence, in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of “blow-up” sign), under suitable conditions on the initial data, and sometimes on the dimension of the lattice. The results consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts are remarked. We also make a short comment on the existence of excitation thresholds, for forced solutions of damped and parametrically driven Klein-Gordon lattices.

In this paper, we construct a generalized degree theory of Browder-Petryshyn or Petryshyn type for a class of semilinear operator equations involving a Fredholm type mapping with infinite dimensional kernel.

Let $(e_n)$ be the canonical basis of the predual of the Lorentz sequence space $d_{*}(w,1).$ We consider the restriction operator $R$ associated to the basis $(e_i)$ from some Banach space of analytic functions into the complex sequence space and we characterize the ranges of $R.$

Let $\Delta$ be one of the dual polar spaces $DW(5,q)$ or $DH(5,q^2)$. We consider a class of subspaces of $\Delta$, each member of which carries the structure of a near hexagon, and classify all these subspaces. Using this classification, we determine all hyperplanes of $DW(5,q)$ without ovoidal quads.

We establish some upper and lower bounds for the number of rational points of Prym varieties defined over finite fields. They are better than the usual Weil bounds valid for any abelian varieties defined over such fields.

We prove that Mazur's functional characterization for one-sided estimates can be restricted to smaller classes of functionals in the case in which the functions under consideration are continuous. We apply this result to stability problems for dynamical systems in $l^\infty$, and in the Banach space of all selfadjoint operators on a Hilbert space.

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ a subgroup of $G$ and s$\theta\in \hbox{\rm Irr}(H)$ an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $G$ induced by $\theta$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p\,{+}\,1}{2}$ distinct irreducible constituents.

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree $n$ over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever $n \geq 9$. It is also known to be true when $n \leq 3$. We show that there exists a primitive polynomial of any degree $n\geq 4$ over any finite field with its second coefficient (i.e., that of $x^{n-2}$) arbitrarily prescribed. In particular, this establishes the HMPC when $n=4$. The lone exception is the absence of a primitive polynomial of the form $x^4+a_1x^3 +x^2+a_3x+1$ over the binary field. For $n \geq 6$ we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.

It is an open question whether every strongly locally $\varphi$-symmetric contact metric space is a $(\kappa,\mu)$-space. We show that the answer is positive for locally homogeneous contact metric manifolds.

A hereditarily indecomposable asymptotic $\ell_2$ Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W. T. Gowers.

The existence of multiple positive solutions is presented for the singular Dirichlet boundary value problems \[\left\{\begin{array}{@{}ll} x^{\prime\prime}+\Phi(t)\,f(t,x(t),|x'(t)|)=0,\\[3pt] x(0)=0,\ \ x(1)=0, \end{array}\right.\] using the fixed point index; here $f$ may be singular at $x=0$ and $x'=0$.

Let $R$ be a ring. An $R$-module $M$ is called a (weak) duo module provided every (direct summand) submodule of $M$ is fully invariant. It is proved that if $R$ is a commutative domain with field of fractions $K$ then a torsion-free uniform $R$-module is a duo module if and only if every element $k$ in $K$ such that $kM$ is contained in $M$ belongs to $R$. Moreover every non-zero finitely generated torsion-free duo $R$-module is uniform. In addition, if $R$ is a Dedekind domain then a torsion $R$-module is a duo module if and only if it is a weak duo module and this occurs precisely when the $P$-primary component of $M$ is uniform for every maximal ideal $P$ of $R$.

Let $B(H)$ denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space $H$. Let $I$ be a two-sided ideal in $B(H)$. For operators $A, B$ and $X \in B(H)$, we say that $X$intertwines$A$and$B$modulo$I$ if $AX - XB \in I$. It is easy to see that if $X$ intertwines $A$ and $B$ modulo $I$, then it intertwines $A^{n}$ and $B^{n}$ modulo $I$ for every integer $n > $1. However, the converse is not true. In this paper, sufficient conditions on the operators $A$ and $B$ are given so that any operator $X$ which intertwines certain powers of $A$ and $B$ modulo $I$ also intertwines $A$ and $B$ modulo $J$ for some two-sided ideal $J \supseteq I$.

We define a subset $\mathcal Z$ of $(1,+\infty)$ with the property that for each $\alpha \in {\mathcal Z}$ there is a nonzero real number $\xi = \xi(\alpha)$ such that the integral parts $[\xi \alpha^n]$ are even for all $n \in \mathbb{N}$. A result of Tijdeman implies that each number greater than or equal to 3 belongs to $\mathcal{Z}$. However, Mahler's question on whether the number 3/2 belongs to $\mathcal{Z}$ or not remains open. We prove that the set ${\mathcal S}:=(1,+\infty) \textbackslash {\mathcal Z}$ is nonempty and find explicitly some numbers in ${\mathcal Z} \cap$ (5/4,3) and in ${\mathcal S} \cap (1,2)$.

The classical maximum principle is utilized to obtain maximum principles for functionals which are defined on solutions of fourth, sixth and eighth-order elliptic equations. The principles derived lead to uniqueness results.

Answering a question of Hoffmann and of Kambites, an example is exhibited of a finitely generated semigroup $S$ such that $S$ embeds in a group and $S$ is not automatic, but the universal group of $S$ is automatic.

We determine for all $d$ and $p$ the maximal derived length of a soluble subgroup of the multiplicative group of a division ring of finite degree $d$ and characteristic $p\,\ge\,0$ to within one.

A proper submodule $N$ of an $R$-module $M$ is called a weakly prime submodule, if for each submodule $K$ of $M$ and elements $a, b$ of $R$, $abK \subseteq N$, implies that $aK\subseteq N$ or $bK\subseteq N$. In this paper we will study weakly prime submodules and we shall compare weakly prime submodules with prime submodules.