In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition $(A)$ on the real superalgebras in consideration (this condition is a generalization of the classical relation $1 + i^2 = 0$ in $\mathbb{C}$). Under the condition $(A)$, we get an integral representation formula for the superdifferentiable functions. We deduce properties of the superdifferentiable functions: analyticity, a result of separated superdifferentiability, a Liouville theorem and a continuation theorem of Hartogs-Bochner type.

In this paper we will give a proof of the McMillan twist point theorem using geometry, potential theory and Ito's formula but not the Riemann mapping theorem.

We study isolated singularities of binary differential equations of degree $n$ which are totally real. This means that at any regular point, the associated algebraic equation of degree $n$ has exactly $n$ different real roots (this generalizes the so called positive quadratic differential forms when $n=2$). We introduce the concept of index for isolated singularities and generalize Poincaré-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the $n$-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics $D_1$, $D_2$ and $D_3$.

The inner automorphisms of a group $G$ can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of $G$ that can be extended, in a functorial manner, to all groups $H$ given with homomorphisms $G\to H.$ (Precise statement in $\S$1.) The group of such extended systems of automorphisms, unlike the group of inner automorphisms of $G$ itself, is always isomorphic to $G.$ A similar characterization holds for inner automorphisms of an associative algebra $R$ over a field $K;$ here the group of functorial systems of automorphisms is isomorphic to the group of units of $R$ modulo the units of $K.$

If one looks at the above functorial extendibility property for endomorphisms, rather than just automorphisms, then in the group case, the only additional example is the trivial endomorphism; but in the $K$-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises.

Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism'' in the literature, some overlapping the one introduced here, are noted; the concept of an inner derivation of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a "co-inner'' endomorphism is briefly examined. Several open questions are noted.

We construct a canonical Green current $T_f$ for every quasi-algebraically stable meromorphic self-map $f$ of $\mathbb{P}^k$ such that its first dynamical degree $\lambda_1(f)$ is a simple root of its characteristic polynomial and that $\lambda_1(f)>1.$ We establish a functional equation for $T_f$ and show that the support of $T_f$ is contained in the Julia set, which is thus non empty.

We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in \mathit{BMO}$ and $T$ a singular integral operator (such as the Hilbert or Riesz transforms), maps $L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the commutators require a "double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator $I_\alpha$ we find the sharp one-weight bound on $[b,I_\alpha]$, $b\in \mathit{BMO}$, in terms of the $A_{p,q}$ constant of the weight. We also prove sharp two-weight bounds for $[b,I_\alpha]$ analogous to those of singular integrals. We prove two-weight weak type inequalities for $[b,T]$ and $[b,I_\alpha]$ for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.

Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function defining the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non-tangential boundary values of the functions of the backward shift invariant subspace after having applied a co-analytic (truncated) Toeplitz operator. There appears to be a smoothing effect.

We show that for every compact set $A\subset {\mathbb R}^m$ of finite $\alpha$-dimensional packing premeasure $0<\alpha\leq m$, the lower limit of the normalized discrete minimum Riesz $s$-energy ($s>\alpha$) coincides with the outer measure of $A$ constructed from this limit by method I. The asymptotic behavior of the discrete minimum energy on compact subsets of a self-similar set $K$ satisfying the open set condition is also studied for $s$ greater than the Hausdorff dimension of $K$. In addition, similar problems are studied for the best-packing radius.

We generalise a result of R. Thomas to establish the non-vanishing of the first $\ell^2$ Betti number for a class of finitely generated groups.