Related papers: Effective Topological Degree Computation Based on …
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
We investigate average gradient degree of normal random polynomials of fixed algebraic degree n. In particular, for polynomials of two variables, asymptotics of the average gradient degree for large values of n is determined.
Let $G$ be a graph and let $m_{ij}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The {\em $M$-polynomial} of $G$ is introduced with $\displaystyle{M(G;x,y) = \sum_{i\le j} m_{ij}(G)x^iy^j}$.…
This monograph elucidates and extends many theorems and conjectures in analytic number theory and algebraic asymptotic analysis via the natural notion of "degree" and a more general notion that we call "logexponential degree." Specifically,…
We give a simple, computationally efficient, and node-differentially-private algorithm for estimating the parameter of an Erdos-Renyi graph---that is, estimating p in a G(n,p)---with near-optimal accuracy. Our algorithm nearly matches the…
Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
We use tools from applied topology for feature selection on time series data. We develop a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding…
Gradient-free optimizers allow for tackling problems regardless of the smoothness or differentiability of their objective function, but they require many more iterations to converge when compared to gradient-based algorithms. This has made…
Inferring topological characteristics of complex networks from observed data is critical to understand the dynamical behavior of networked systems, ranging from the Internet and the World Wide Web to biological networks and social networks.…
Topological correctness plays a critical role in many image segmentation tasks, yet most networks are trained using pixel-wise loss functions, such as Dice, neglecting topological accuracy. Existing topology-aware methods often lack robust…
Given a countable structure $\mathcal{A}$, the degree spectrum of $\mathcal{A}$ is the set of all Turing degrees which can compute an isomorphic copy of $\mathcal{A}$. One of the major programs in computable structure theory is to determine…
Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and…
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional…
Topological indices play a significant role in mathematical chemistry. Given a graph $\mathcal{G}$ with vertex set $\mathcal{V}=\{1,2,\dots,n\}$ and edge set $\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer…
This paper considers finite-automata based algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this…
In this article, we generalize the arithmetic degree and its related theory to dynamical systems defined over an arbitrary field $\mathbf{k}$ of characteristic $0$. We first consider a dynamical system $(X,f)$ over a finitely generated…
Topological degree theory is a useful tool for studying systems of differential equations. In this work, a biological model is considered. Specifically, we prove the existence of positive T-periodic solutions of a system of delay…