We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_{g}$-ideal if and only if it is a maximally elliptic singularity of degree $1$.

We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.

We study modules for the divided power algebra $\mathbf{D}$ in a single variable over a commutative Noetherian ring $\mathbf{K}$. Our first result states that $\mathbf{D}$ is a coherent ring. In fact, we show that there is a theory of Gröbner bases for finitely generated ideals, and so computations with finitely presented $\mathbf{D}$-modules are in principle algorithmic. We go on to determine much about the structure of finitely presented $\mathbf{D}$-modules, such as: existence of certain nice resolutions, computation of the Grothendieck group, results about injective dimension, and how they interact with torsion modules. Our results apply not just to the classical divided power algebra, but to its $q$-variant as well, and even to a much broader class of algebras we introduce called “generalized divided power algebras.” On the other hand, we show that the divided power algebra in two variables over $\mathbf{Z}_{p}$ is not coherent.

We prove the central limit theorem from the definition of weak convergence using the Haar basis, calculus, and elementary probability, and we estimate the rate of convergence off the tails. The use of the Haar basis pinpoints the role of $L^{2}([0,1])$ in the CLT as well as the assumption of finite variance.

Let $F$ be a homogeneous polynomial in $n$ variables of degree $d$ over a field $K$. Let $A(F)$ be the associated Artinian graded $K$-algebra. If $B\subset A(F)$ is a subalgebra of $A(F)$ which is Gorenstein with the same socle degree as $A(F)$, we describe the Macaulay dual generator for $B$ in terms of $F$. Furthermore when $n=d$, we give necessary and sufficient conditions on the polynomial $F$ for $A(F)$ to be a complete intersection.

Let $\mathcal{Q}$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\mathcal{Q}$ and quotients of the path algebra $K\mathcal{Q}$. We introduce the homological heart of $\mathcal{Q}$ which is a particularly nice convex subquiver of $\mathcal{Q}$. For any algebra of the form $K\mathcal{Q}/I$, the algebra associated to $K\mathcal{Q}/I$ and the homological heart have similar homological properties. We give an application showing that the finitistic dimension conjecture need only be proved for algebras with path connected quivers.

We investigate the relationship between Ricci soliton structures and homothetic vector fields, especially Killing vector fields. More precisely, we present some characterizations for Ricci solitons endowed with Killing vector fields of constant norm as well as homothetic vector fields. In particular, we relate different Ricci soliton structures on a Riemannian manifolds in order to deduce some uniqueness results.

A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^{2}$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_{1},\ldots,d_{s})$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_{s}$ points of $\mathbb{X}$. In particular, we show that for all integers $m\gg0$, the number of such lines is precisely the value of $\Delta\mathbf{H}_{m\mathbb{X}}(md_{s}-1)$. Here, $\Delta\mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.

A connected compact subset $E$ of $\mathbb{R}^{N}$ is said to be a strict Whitney set if there exists a real-valued $C^{1}$ function $f$ on $\mathbb{R}^{N}$ with $\nabla f|_{E}\equiv 0$ such that $f$ is constant on no non-empty relatively open subsets of $E$. We prove that each self-similar arc of Hausdorff dimension $s>1$ in $\mathbb{R}^{N}$ is a strict Whitney set with criticality $s$. We also study a special kind of self-similar arcs, which we call “regular” self-similar arcs. We obtain necessary and sufficient conditions for a regular self-similar arc $\Lambda $ to be a $t$-quasi-arc, and for the Hausdorff measure function on $\Lambda $ to be a strict Whitney function. We prove that if a regular self-similar arc has “minimal corner angle” $\theta_{\min }>0$, then it is a 1-quasi-arc and hence its Hausdorff measure function is a strict Whitney function. We provide an example of a one-parameter family of regular self-similar arcs with various features. For some values of the parameter $\tau $, the Hausdorff measure function of the self-similar arc is a strict Whitney function on the arc, and hence the self-similar arc is an $s$-quasi-arc, where $s$ is the Hausdorff dimension of the arc. For each $t_{0}\ge 1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t$-quasi-arc for each $t>t_{0}$, but it is not a $t_{0}$-quasi-arc. For each $t_{0}>1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t_{0}$-quasi-arc, but it is a $t$-quasi-arc for no $t\in [1,t_{0})$.

Naimark’s problem asks whether a $C^{*}$-algebra that has only one irreducible $*$-representation up to unitary equivalence is isomorphic to the $C^{*}$-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable $C^{*}$-algebras and Type I $C^{*}$-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a $C^{*}$-algebra with $\aleph_{1}$ generators that is a counterexample to Naimark’s Problem. More precisely, they showed that the statement “There exists a counterexample to Naimark’s Problem that is generated by $\aleph_{1}$ elements.” is independent of the axioms of ZFC. Whether Naimark’s problem itself is independent of ZFC remains unknown. In this paper, we examine Naimark’s problem in the setting of graph $C^{*}$-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph $C^{*}$-algebras as well as for $C^{*}$-algebras of graphs in which each vertex emits a countable number of edges.

A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications for the case when branched covering surface-knots have a non-zero number of branch points, using chart moves involving black vertices.

In this paper, we study the complex simultaneous Waring rank for collections of monomials. For general collections, we provide a lower bound, whereas for special collections we provide a formula for the simultaneous Waring rank. Our approach is algebraic and combinatorial. We give an application to ranks of binomials and maximal simultaneous ranks. Moreover, we include an appendix of scripts written in the algebra software Macaulay2 to experiment with simultaneous ranks.

The purpose of this paper is to present several examples of non-autonomous basins of attraction that arise from sequences of automorphisms of $\mathbb{C}^{k}$. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of $\mathbb{C}^{2}$ of a prescribed form is biholomorphic to $\mathbb{C}^{2}$. This, in particular, provides a partial answer to a question raised in (A survey on non-autonomous basins in several complex variables (2013) Preprint) in connection with Bedford’s Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short $\mathbb{C}^{k}$’s with specified properties. First, we show that for $k\geq3$, there exist $(k-1)$ mutually disjoint Short $\mathbb{C}^{k}$’s in $\mathbb{C}^{k}$. Second, we construct a Short $\mathbb{C}^{k}$, large enough to accommodate a Fatou–Bieberbach domain, that avoids a given algebraic variety of codimension $2$. Lastly, we discuss examples of Short $\mathbb{C}^{k}$’s with (piece-wise) smooth boundaries.