Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

Vacuum polarization radiative correction to the parity violating electron scattering on heavy nuclei

The effect of vacuum polarization on the parity violating asymmetry in the elastic electron-nucleus scattering is considered. Calculations are performed in the high-energy approximation with an exact account for the electric field of the nucleus. It is shown that the radiative correction to the parity violating asymmetry is logarithmically enhanced and the value of the correction is about -1%.Comment: 6 pages, 3 figures, REVTex

Bremsstrahlung and pair production processes at low energies, multi-differential cross section and polarization phenomena

Radiative electron-proton scattering is studied in peripheral kinematics, where the scattered electron and photon move close to the direction of the initial electron. Even in the case of unpolarized initial electron the photon may have a definite polarization. The differential cross sections with longitudinally or transversal polarized initial electron are calculated. The same phenomena are considered for the production of an electron-positron pair by the photon, where the final positron (electron) can be also polarized. Differential distributions for the case of polarized initial photon are given. Both cases of unscreened and completely screened atomic targets are considered.Comment: 15 pages, 6 figure

On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

Thermodynamic Geometric Stability of Quarkonia states

We compute exact thermodynamic geometric properties of the non-abelian quarkonium bound states from the consideration of one-loop strong coupling. From the general statistical principle, the intrinsic geometric nature of strongly coupled QCD is analyzed for the Columbic, rising and Regge rotating regimes. Without any approximation, we have obtained the non-linear mass effect for the Bloch-Nordsieck rotating strongly coupled quarkonia. For a range of physical parameters, we show in each cases that there exists a well-defined, non-degenerate, curved, intrinsic Riemannian manifold. As the gluons become softer and softer, we find in the limit of the Bloch-Nordsieck resummation that the strong coupling obtained from the Sudhakov form factor possesses exact local and global thermodynamic properties of the underlying mesons, kaons and $D_s$ particles.Comment: 45 pages, 17 figures, Keywords: Thermodynamic Geometry, Quarkonia, Massive Quarks, QCD Form Factor. PACS: 02.40.-k; 14.40.Pq; 12.40.Nn; 14.70.D

Evolution models for mass transportation problems

We present a survey on several mass transportation problems, in which a given mass dynamically moves from an initial configuration to a final one. The approach we consider is the one introduced by Benamou and Brenier in [5], where a suitable cost functional $F(\rho,v)$, depending on the density $\rho$ and on the velocity $v$ (which fulfill the continuity equation), has to be minimized. Acting on the functional $F$ various forms of mass transportation problems can be modeled, as for instance those presenting congestion effects, occurring in traffic simulations and in crowd motions, or concentration effects, which give rise to branched structures.Comment: 16 pages, 14 figures; Milan J. Math., (2012

Ramsey goodness of cycles

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) \geq (| G| - 1)(\chi (H) - 1) + \sigma (H), where \chi (H) is the chromatic number of H and \sigma (H) is the size of the smallest color class in a \chi (H)-coloring of H. A graph G is called H-good if R(G, H) = (| G| - 1)(\chi (H) - 1) + \sigma (H). The notion of Ramsey goodness was introduced by Burr and Erd\H os in 1983 and has been extensively studied since then. In this paper we show that if n \geq 1060| H| and \sigma (H) \geq \chi (H) 22, then the n-vertex cycle Cn is H-good. For graphs H with high \chi (H) and \sigma (H), this proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Linearly many rainbow trees in properly edge-coloured complete graphs

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. In this paper we discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there are 10−6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method we also show that every properly (n − 1)-edge-coloured Kn has n/9 − 6 edge-disjoint rainbow trees, giving further improvement on the Brualdi-Hollingsworth Conjectur

The oriented size Ramsey number of directed paths

An oriented graph is a directed graph with no bi-directed edges, i.e. if xy is an edge then yx is not an edge. The oriented size Ramsey number of an oriented graph H, denoted by \vec{r}(H), is the minimum m for which there exists an oriented graph G with m edges, such that every 2-colouring of G contains a monochromatic copy of H. In this paper we prove that the oriented size Ramsey number of the directed paths on n vertices satisfies \vec{r}(\vec{P}_{n}) = \Omega (n^{2} log n). This improves a lower bound by Ben-Eliezer, Krivelevich and Sudakov. It also matches an upper bound by Bucić and the authors, thus establishing an asymptotically tight bound on \vec{r}(\vec{P}_{n}). We also discuss how our methods can be used to improve the best known lower bound of the k-colour version of \vec{r}(\vec{P}_{n})