Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value. In a previous paper, we had found URSCM of 7 points for the whole set of unbounded analytic functions inside an open disk. Here we show the existence of URSCM of 5 points for the same set of functions. We notice a characterization of BI-URSCM of 4 points (and infinity) for meromorphic functions in K and can find BI-URSCM for unbounded meromorphic functions with 9 points (and infinity). The method is based on the p-Adic Nevanlinna Second Main Theorem on 3 Small Functions applied to unbounded analytic and meromorphic functions inside an open disk and we show a more general result based upon the hypothesis of a finite symmetric difference on sets of zeros, counting multiplicities.

We describe a new general method for the computation of the group Aut(X) of self-homotopy equivalences of a space. It is based on the decomposition of Aut(X) induced by a factorization of X into a product of simpler spaces. Normally, such decompositions require assumptions (‘induced equivalence property’, ‘diagonalizability’), which are strongly restrictive and difficult to check. We derive computable homological criteria for an analogous assumption, called reducibility, and then show that these criteria are satisfied when the so-called atomic decomposition of the space is used. This essentially reduces the computation of Aut(X) to the computation of the group of self-equivalences of its atomic factors, and the computation of certain homotopy sets between those factors.

We give an asymptotic bound for the size of the n-generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.

In this paper we shall first show that if T is a class A(k) operator then its operator transform is hyponormal. Secondly we prove some spectral properties of T via . Finally we show that T has property (β).

We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.

Waves on a neutrally buoyant intrusion layer moving into otherwise stationary fluid are studied. There are two interfacial free surfaces, above and below the moving layer, and a train of waves is present. A small amplitude linearized theory shows that there are two different flow types, in which the two interfaces are either in phase or else move oppositely. The former flow type occurs at high phase speed and the latter is a low-speed solution. Nonlinear solutions are computed for large amplitude waves, using a spectral type numerical method. They extend the results of the linearized analysis, and reveal the presence of limiting flow types in some circumstances.

We present a theory that enables us to construct heteroclinic connections in closed form for $2\bf{u}_{xx}=W_{\bf u}({\bf u})$, where $x\in\mathbb{R},\;{\bf u}(x)\in \mathbb{R}^2$ and $W$ is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector Allen-Cahn equation ${\bf u}_t= 2\epsilon^2 \Delta {\bf u}-W_{\bf u}({\bf u})$ in two space dimensions with small $\epsilon>0$, show that between any fixed pair of phase regions, interfaces are partitioned into segments of different energy densities, where the proportions of the length of these segments are changing with time. Our results imply that for the case of triple-well potentials the usual Plateau angle conditions at the triple junction are generally violated.

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics (9(1998) 527–542) are presented.

In this paper, we analyse the asymptotic system corresponding to a thin film flow with two different (immiscible) fluids, from theoretical and numerical points of view. We also compare this model to the Elrod-Adams one, which is the reference model in tribology, when cavitation phenomena occur.

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $\mathbb{R}^{n+1}$, for $n\geq3$, which generalize the well known two-dimensional ‘Riemann minimal surfaces’. The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the two-dimensional case, they are not foliated by spheres.

Résumé Nous prouvons l’existence d’une famille à un paramètre d’hypersurfaces de $\mathbb{R}^{n+1}$, pour $n\geq 3$, qui sont minimales et qui généralisent les surfaces minimales de Riemann. Les hypersurfaces que nous obtenons sont des hypersurfaces complètes, simplement périodiques et qui ont une infinité de bouts hyperplans parallèles. Contrairement au cas des surfaces, i.e. $n=2$, ces hypersurfaces ne sont pas feuilletées par des sphères.

A ring $R$ with identity is called strongly clean if every element of $R$ is the sum of an idempotent and a unit that commute. For a commutative local ring $R$, $n=3,4$, and $m, k, s \in {\mathbb N}$ it is proved that ${\mathbb M}_n(R)$ is strongly clean if and only if ${\mathbb M}_n(R[[x]])$ is strongly clean if and only if ${\mathbb M}_n(R[[x_1, x_2, \ldots, x_m]])$ is strongly clean if and only if $ {\mathbb M}_n(\frac{R[x]}{(x^{k})})$ is strongly clean if and only if $ {\mathbb M}_n(\dfrac{R[x_{1}, x_{2}, \ldots , x_{s}]}{(x^{n_1}_{1}, x^{n_{2}}_{2}, \ldots , x^{n_{s}}_{s})}) $ is strongly clean if and only if ${\mathbb M}_n(R \propto R)$ is strongly clean where $ R\propto R=\{\scriptsize(\begin{array}{@{}c@{\quad}c@{}} a& b \\ 0& a \end{array}): a, b \in R \}$ is the trivial extension of $R$. This extends a result of J. Chen, X. Yang and Y. Zhou [$\mathbf{5}$] from $n=2$ to 3 and 4.

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 $R.$

If bounded linear operators $A$ and $B$ are each reguloid, and have the single valued extension property, then Weyl's theorem holds for all holomorphic functions of all operator matrices $M_{C}=\scriptsize\scriptsize(\begin{array}{@{}cc@{}}A&C\\0&B\end{array})$.

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.

We define “star reducible” Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan–Lusztig bases of these groups have a nice projection property to the Temperley–Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\mathbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families with two exceptional cases.

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.