Related papers: Adjacency preservers on invertible hermitian matri…
It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
This paper presents and explores a theory of \emph{multiholomorphic maps}. This group of ideas generalizes the theory of pseudoholomorphic curves in a direction suggested by consideration of the kinds of compatible geometric structures that…
A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is…
Spacetimes obtained by dimensional reduction along lattices containing a lightlike direction can admit semigroup extensions of their isometry groups. We show by concrete examples that such a semigroup can exhibit a natural order, which in…
This note generalizes a result contained in a previous paper [ J. Sanders, Circuit preserving edge maps II, J. Combin. Theory Ser. B 42 (1987), 146-155].
Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of curvature on the vertices $V$ via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix $A$ into another $B$ if and only if the vector of eigenvalues of $A$ majorizes that of $B$. In this work I characterize the…
Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics…
In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show…
We prove that given an $n$-dimensional integral current space and a $1$-Lipschitz map, from this space onto the $n$-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be…
In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…
We prove a first principle of preservation of multiplicity in difference geometry, paving the way for the development of a more general intersection theory. In particular, the fibres of a \sigma-finite morphism between difference curves are…
We study the moduli of the universal geometry of $d=4$ $N=1$ heterotic vacua. Universal geometry refers to a family of heterotic vacua fibered over the moduli space. The universal geometry mimics aspects of the original heterotic vacua, in…
The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…
For a fixed finite dimensional algebra $A$, we study representation embeddings of the form $mod(B)\rightarrow mod(A)$. Such an embedding is called homological, if it induces an isomorphism on all Ext-groups and weakly homological, if only…
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in…
The universal adjacency matrix $U$ of a graph $\Gamma$, with adjacency matrix $A$, is a linear combination of $A$, the diagonal matrix $D$ of vertex degrees, the identity matrix $I$, and the all-1 matrix $J$ with real coefficients, that is,…
We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean…