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This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that…

We study the $\Gamma$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of…

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our…

A covering of a digraph $D$ by Hamilton cycles is a collection of directed Hamilton cycles (not necessarily edge-disjoint) that together cover all the edges of $D$. We prove that for $1/2 \geq p\geq \frac{\log^{20} n}{n}$, the random…

Let $m$ be an odd positive integer and $D_m(\mathcal {A})$ be the $m$-periodic derived category of a finitary hereditary abelian category $\mathcal {A}$. In this note, we prove that there is an embedding of algebras from the derived Hall…

Suppose $\alpha$ is a rotationally symmetric norm on $L^{\infty}\left(\mathbb{T}\right) $ and $\beta$ is a "nice" norm on $L^{\infty}\left(\Omega,\mu \right) $ where $\mu$ is a $\sigma$-finite measure on $\Omega$. We prove a version of…

We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near…

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $\delta_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $\delta_2(H) \geq 4n/5 + o(n)$…

The work deals with the qualification of semidiscrete hyperbolic type equations. We study a class of equations of the form $$\frac{du_{n+1}}{dx}=f\left(\frac{du_{n}}{dx},u_{n+1},u_{n}\right),$$ here the unknown function $u_n(x)$ depends on…

We study the signed Bernoulli convolution $$\nu_\beta^{(n)}=*_{j=1}^n \left (\frac12\delta_{\beta^{-j}}-\frac12\delta_{-\beta^{-j}}\right ),\ n\ge 1$$ where $\beta>1$ satisfies $$\beta^m=\beta^{m-1}+\cdots+\beta+1$$ for some integer $m\ge…

We show that the cycle relation between Dehn twists about curves in a circuit detects whether the circuit bounds an embedded disc. This is done by determining the isomorphism type of the group generated by said Dehn twists for various…

Let $(X,d,T )$ be a topological dynamical system with specification property. For $ \alpha\in \mathbb R^+$ and any $x_0\in X$, define $$ \mathbf D^{x_0}_\alpha :=\Big\{x\in X: \lim\limits_{\epsilon\to…

Subgraph and homomorphism counting are fundamental algorithmic problems. Given a constant-sized pattern graph $H$ and a large input graph $G$, we wish to count the number of $H$-homomorphisms/subgraphs in $G$. Given the massive sizes of…

Let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\cR$ be the formal power series ring $k[[x_1, \ldots, x_n]]$. If $M$ is a $\D$-module over $R$, then $\cR \otimes_R M$ is naturally a…

DNA graph has important contribution in completing the computational step of DNA sequencing process. Using $(\alpha,k)$-labeling, several families of digraphs have characterized as DNA graphs. Dicycles and dipaths are DNA graphs, rooted…

The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a…

The Hamiltonian cycle problem in digraph is mapped into a matching cover bipartite graph. Based on this mapping, it is proved that determining existence a Hamiltonian cycle in graph is $O(n^3)$.

In this article we prove existence, uniqueness and regularity for the singular equation \begin{eqnarray*} \begin{cases} |\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^{\alpha}u+p(x)u^{-\gamma}=0 \ \mbox{ in } \ \Omega\\ u>0 \…

For a finite alphabet $A$ define by $d_1(x,y):=\limsup_{n\to\infty}\frac{1}{2n+1}\#\{|i|\le n: x_i\neq y_i\}$ the Besicovitch pseudo-metric on $A^{\mathbb Z}$. It is well known that a closed subshift of $A^{\mathbb Z}$ has finite covering…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…