Narrow Results

We introduce a new class of entire functions ${ℰ}$ which consists of all $F_0\in {𝒪}(ℂ)$ for which there exists a sequence $(F_n)\in {𝒪}(ℂ)$ and a sequence $({\lambda }_n)\in ℂ$ satisfying $F_n(z)={\lambda }_{n+1}{e}^{F_{n+1}(z)}$ for all $n\ge 0$. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set $V\subset ℂ$ containing the origin and at least one more point is the set of singular values of some locally univalent function in ${ℰ}$, hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set $V\subset ℂ$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class ${ℰ}$ contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

In this paper, I give an estimate on the dimension of the singular set of a tangent cone at infinity of a stable minimal hypersurface. Namely, let Mⁿ ⊂ ℝⁿ⁺¹, n ≥ 2, be a complete orientable stable minimal immersion with bounded volume growth. Then n < 7 implies T_∞(M) is smooth, and n ≥ 7 implies the singular set of T_∞(M) has codimension at least seven.

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator–prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.

We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ¹⁺ⁿ, characterized by the fact that they admit a C¹ parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class C^k, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ¹⁺ⁿ is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.

Let $R$ be a commutative Noetherian ring, $d\ge 0$ be an integer and let $\sum$ denote the ideals of $R$ of dimension $\le d$. For an integer $i\ge 0$ and $R$-modules $M,N$, we define the $i$th cohomology module $H_d^i(M,N)={\underrightarrow{lim}}_{𝔞\in \sum }{\mathrm{\text{Ext}}}_R^i(M/𝔞M,N)$ and study some of its properties. Some vanishing results will be proved and the connection between the vanishing of this cohomology modules with the dimension of certain subsets of $\mathrm{\text{Supp}}(M)$ will be investigated.

In this note we study removable singularities for analytic functions in Hardy H^p spaces, in BMO and in locally Lipschitz spaces.