Research
Research
Research
Research
82+ open-access research outputs.
Christoph, Dragani\'{c}, Gir\~{a}o, Hurley, Michel, and M\"{u}yesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph …
In this survey, we review recent developments in extending Hodge theory to differential forms with values in bundles equipped with singular metrics, based on joint work with Ya Deng, Christopher D. Ha…
Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we s…
We use contemporary mathematical notation to describe the method for determining the age of the ecclesiastical moon as mandated by pope Gregory XIII and elaborated in the book of Christopher Clavius \…
Generalising the Cameron--Erd\H{o}s conjecture to two dimensions, Elsholtz and Rackham conjectured that the number of sum-free subsets of $[n]^2$ is $2^{0.6n^2+O(n)}$. We prove their conjecture.…
We investigate the maximal number $N_h(m)$ of normally hyperbolic limit tori in three-dimensional polynomial vector fields of degree $m$, which extends the classical notion of Hilbert numbers to highe…
The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes t…
For any $\varepsilon \in (0,+\infty)$, consider the metric spaces $\mathbb{R} \times [0,\varepsilon]$ in the Euclidean plane named layers or strips. B. Baslaugh in 1998 found the minimal width $\varep…
The cycle space $\mathcal{C}(G)$ of a graph $G$ is defined as the linear space spanned by all cycles in $G$. For an integer $k\ge 3$, let $\mathcal{C}_k (G)$ denote the subspace of $\mathcal{C}(G)$ ge…
In their famous 1974 paper introducing the local lemma, Erd\H{o}s and Lov\'asz posed a question-later referred by Erd\H{o}s as one of his three favorite open problems: What is the minimum number of ed…
We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field $K$. In view of the Prelle-Singer theorem,…
In our earlier work with Christopher Skinner (J. Eur. Math. Soc 24 (2022), no. 2; DOI 10.4171/JEMS/1124; Arxiv 1706.00201), we constructed Euler systems for the 4-dimensional spin Galois representatio…
In the present article, we study the integral aspects of the Fourier transform of an abelian variety $A$ over a field $k$, using \'etale motivic cohomology, following the ideas and theory given by Moo…
We study the least gradient problem in bounded regions with Lipschitz boundary in the plane. We provide a set of conditions for the existence of solutions in non-convex simply connected regions. We as…
The paper presents various modifications of the Frank-Wolfe algorithm in the equilibrium traffic assignment problem. The Beckman model is used as a model for experiments. In this article, first of all…
In this paper an algebraic proof of Christoph's theorem is provided. This theorem from algebraic-geometry is about the existence of a finite automaton for computing coefficient of a series for an alge…
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound…
For any positive definite rational quadratic form $q$ of $n$ variables let $G(\mathbb{Q}^n, q)$ denote the graph with vertices $\mathbb{Q}^n$ and $x, y \in \mathbb{Q}^n$ connected iff $q(x - y) = 1$. …
Hilbert's 16th Problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree $m$, has been one of the most important driving forces for new developments in the…
We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and comprehensive textbook of practical geometry, whose first edition appeared in 1604. Our focus is on four par…
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