It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

Le radical R(A) d′un anneau A peut être caract érisé de plusieurs manières (Jacobson [1]). Rappelons seulement ici deux des caractérisations les plus importantes:

(1) R(A) est l′intersection de tous les idéaux à droite (à gauche) modulaires maximaux de A.

(2) R(A) est un idéal quasi - régulier contenant tout idéal à droite (á gauche) quasi - régulier.

L′objet de cet article est de donner quelques caractérisations du radical d′un anneau.

For the $n$-th order nonlinear differential equation, ${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k\,+\,j)$-point boundary conditions for $1\,\le \,j\,\le \,n\,-\,1$ and $1\,\le \,k\,\le \,n\,-\,j$. We define $(k;\,j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n\,-\,{{j}_{0}};\,{{j}_{0}})$-point unique solvability implies $(k;\,j)$-point unique solvability for $1\,\le \,j\,\le \,{{j}_{0}}$, and $1\,\le \,k\,\le \,n\,-\,j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\,\le \,k\,\le \,n\,-\,1$.

In this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves ${{\overline{M}}_{g,n}}$ Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the $\varphi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^{n}$ with all coordinates in the upper and lower half planes respectively, through a set in real space, $\mathbb{R}^{n}$. The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in $\mathbb{R}^{n}$ that are positively oriented with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.

A general procedure is introduced for identifying categories of Cauchy spaces which have completions which possess a certain convergence property P, and for constructing completion functors on such categories. These results are applied to obtain a characterization of Cauchy spaces which allow T3 completions, and to construct T3 completions for categories of Cauchy groups and Cauchy lattices.

We give an upper bound for the number of elliptic Carmichael numbers $n\,\le \,x$ that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non $\text{CM}$). We also discuss several possible further improvements.

The Arens extension of the triple product of an associative triple system is studied. Using a representation theorem for C*-ternary rings due to Zettl, it is shown that the second dual of a C*-ternary ring is itself a C*-ternary ring

We give a direct proof that wis an A∞+(g) weight if and only if w satisfies a one-sided, weighted reverse Hölder inequality.

For finite-dimensional subspaces of ℓ∞ over a non-archimedean valued base field K we study orthocomplementation as related to the Hahn-Banach property and strictness. As a corollary we obtain that, if K is not spherically complete, a closed hyperplane in c0 having the Hahn-Banach property is orthocomplemented (Theorem 2.1, Remark 2).

We prove that for every function $f\,:\,X\,\to \,Y$ , where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\,\in \,\overset{\sim }{\mathop{\mathcal{A}}}\,$ such that $f$ is Gâteaux differentiable at all $x\,\in \,S\left( f \right)\backslash A$, where $S\left( f \right)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde{C}\,;$ this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\,:\,X\,\to \,\mathbb{R}$ cone monotone, $g\,:\,X\,\to \,\mathbb{R}$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fréchet differentiable.

This article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

We study the behavior of the eigenvalues of a sublaplacian ${{\Delta }_{b}}$ on a compact strictly pseudoconvex $\text{CR}$ manifold $M$, as functions on the set ${{\mathcal{P}}_{+}}$ of positively oriented contact forms on $M$ by endowing ${{\mathcal{P}}_{+}}$ with a natural metric topology.

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies

1

say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there.

The Tarry-Escott problem in Diophantine analysis is the following: consider the system of equations

1

for the unknowns a1, …, ak, b1, …, bk, what is the smallest integer K = K(n) in the set of all k! s for which the system (1) possesses a non - trivial solution in integers ? By an integer we mean a rational integer and a solution is called trivial if the sets { a1, …, ak} and {b1, …, bk} are permutations of each other. By the Tarry-Escott problem one may also mean the problem of finding an actual solution of (1) or finding all such solutions but we shall be concerned with the bound K(n) only. However, the method given here is constructive, that is to say, with each estimate K for K it leads to finding actual non - trivial solutions of (1) with k = K1.

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{C}{{\mathbb{P}}^{2}}$ and a projection of the image curve froman appropriate point $p\in \mathbb{C}{{\mathbb{P}}^{2}}$ to the pencil of lines through $p$ . We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

Block decomposition of $L^{p}$ spaces with weighted Hausdorff content is established for $0<p\leqslant 1$ and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

The main result of this paper is the determination of all nonisomorphic local near-rings <N, +, •> with <N, +> =<C(p) × C(p), +> which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.

We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)