Related papers: On the crossing estimates for simple conformal loo…

We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of $\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$ commute…

We show that by cutting off the vertices and then the edges of neighborly cubical polytopes, one obtains simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log^{3/2}n)$. This…

We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$…

A competition model on $\N^{2}$ between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability $6-8\log2$.…

A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in the star of a vertex. A $4$-conic complex is simply connected. We prove that an $8$-conic complex is $2$-connected. In general a $(2n+1)$-conic…

A doubling covering $\U$ of a complex $n$-dimensional manifold $Y$ consists of analytic functions $\psi_j:B_1\to Y$, each function being analytically extendable, as a mapping to $Y$, to a four times larger concentric ball $B_4$. Main result…

We determine the algebraic structure of the multiplicative loops for locally compact $2$-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of…

In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $\theta_n(p)$ is the probability that there is a path…

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp\left[-\Omega(n^\gamma)\right]$ for some…

Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and T\'oth (A…

Given $\omega\ge 1$, let $Z^2_{(\omega)}$ be the graph with vertex set $Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most $\omega$ in the other. (Thus $Z^2_{(1)}$ is precisely $Z^2$.) Let…

We study the crossing number of links that are formed by edges of a triangulation T of the 3-sphere with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n^2. In general, this bound can not…

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course…

We report a combined theoretical and experimental investigation of the superconducting state in the quasi-two-dimensional organic superconductor $\kappa$-(ET)$_2$Cu[N(CN)$_2$]Br. Applying spin-fluctuation theory to a low-energy…

Continuing the investigation for the number of crossing limit cycles of nonsmooth Li\'enard systems in [Nonlinearity 21(2008), 2121-2142] for the case of a unique equilibrium, in this paper we consider the case of any number of equilibria.…

We consider the measure on multiple chordal Schramm-Loewner evolution ($SLE_\kappa$) curves. We establish a derivative estimate and use it to give a direct proof that the partition function is $C^2$ if $\kappa<4$.

We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar ${\cal N}=4$ super-Yang-Mills theory from recent…

In this note we investigate the existence of frames of exponentials for $L^2(\Omega)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $\Omega \subset \widehat{G}$ with respect to a uniform lattice…

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary…

In 1985, Razborov discovered a proof that the monotone circuit complexity of the clique problem is super-polynomial. Alon and Boppana improved the result into exponential lower bound exp(\Omega(n / \log n)^{1/3})) of a monotone circuit C to…