Related papers: A Partial Ordering on Slices of Planar Lagrangians
An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced…
The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive…
The countable condensation on a linear order $L$ is the equivalence relation $\sim_\omega$ defined by declaring $x \sim_\omega y$ when the set of points between $x$ and $y$ is countable. We characterize the linear orders $L$ that condense…
In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to…
We describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the \v Cech simplicial complex. We…
We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated, where $k_0$ is the initial…
The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…
A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex…
The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial…
We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II_1 factor,…
We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.
Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}$ to $G$ factors through the projection to some finite product $\mathbb Z^n$. Various authors have proposed…
In this paper, we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus…
A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also…
The following representation theorem is proven: A partially ordered commutative ring $R$ is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space $X$ if and only if $R$ is archimedean…
In a totally ordered set the notion of sorting a finite sequence is defined through a suitable permutation of the sequence's indices. In this paper we prove a simple formula that explicitly describes how the elements of a sequence are…
A positive integer $d$ is a floor quotient of $n$ if there is a positive integer $k$ such that $d = \lfloor n/k \rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal…
A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal Hopf algebra structure is defined on the vector space…
Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly…
As in [5], we study holomorphic maps of positive degree between compact complex manifolds, and prove that any holomorphic map of degree one from a compact complex manifold to itself is biholomorphic. This conclusion confirms that under a…