Surface salinity fields in the Arctic Ocean and statistical approaches to predicting anomalies and patterns

Significant salinity anomalies have been observed in the Arctic Ocean surface layer during the last decade. Using gridded data of winter salinity in the upper 50 m layer of the Arctic Ocean for the period 1950-1993 and 2007-2012, we investigated the inter-annual variability of the salinity fields, attempted to identify patterns and anomalies, and developed a statistical model for the prediction of surface layer salinity. The statistical model is based on linear regression equations linking the principal components with environmental factors, such as atmospheric circulation, river runoff, ice processes, and water exchange with neighboring oceans. Using this model, we obtained prognostic fields of the surface layer salinity for the winter period 2013-2014. The prognostic fields demonstrated the same tendencies of surface layer freshening that were observed previously. A phase portrait analysis involving the first two principal components exhibits a dramatic shift in behavior of the 2007-2012 data in comparison to earlier observations

Jets in Effective Theory: Summing Phase Space Logs

We demonstrate how to resum phase space logarithms in the Sterman-Weinberg (SW) dijet decay rate within the context of Soft Collinear Effective theory (SCET). An operator basis corresponding to two and three jet events is defined in SCET and renormalized. We obtain the RGE of the two and three jet operators and run the operators from the scale $\mu^2 = Q^2$ to the phase space scale $\mu^2_\delta = \delta^2 Q^2$. This phase space scale, where $\delta$ is the cone half angle of the jet, defines the angular region of the jet. At $\mu^2_{\delta}$ we determine the mixing of the three and two jet operators. We combine these results with the running of the two jet shape function, which we run down to an energy cut scale $\mu^2_{\beta}$. This defines the resumed SW dijet decay rate in the context of SCET. The approach outlined here demonstrates how to establish a jet definition in the context of SCET. This allows a program of systematically improving the theoretical precision of jet phenomenology to be carried out.Comment: 25 pages, 4 figures, V2: Typos fixed, writing clarified, detail on PSRG added. Matching onto jet definition changed to taking place at collinear scal

A simple shower and matching algorithm

We present a simple formalism for parton-shower Markov chains. As a first step towards more complete uncertainty bands, we incorporate a comprehensive exploration of the ambiguities inherent in such calculations. To reduce this uncertainty, we then introduce a matching formalism which allows a generated event sample to simultaneously reproduce any infrared safe distribution calculated at leading or next-to-leading order in perturbation theory, up to sub-leading corrections. To enable a more universal definition of perturbative calculations, we also propose a more general definition of the hadronization cutoff. Finally, we present an implementation of some of these ideas for final-state gluon showers, in a code dubbed VINCIA.Comment: 32 pages, 6 figure

Directed Ramsey number for trees

We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}). More than 40 years ago, Erd˝os and Kleitman conjectured that an s-saturated family of subsets of [n] has size at least (1 − 2 −(s−1))2n. It is easy to show that every s-saturated family has size at least 1 2 · 2 n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2n, for some fixed ε > 0, seems difficult. In this note, we prove such a result, showing that every s-saturated family of subsets of [n] has size at least (1 − 1/s)2n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on |F1| + . . . + |Fs| where F1, . . . , Fs are families of subsets of [n], such that there are no s pairwise disjoint sets, one from each family Fi , and furthermore no set can be added to any of the families while preserving this property. We show that |F1| + . . . + |Fs| ≥ (s − 1) · 2 n, which is tight e.g. by taking F1 to be empty, and letting the remaining families be the families of all subsets of [n]

Large cliques and independent sets all over the place

We study the following question raised by Erd\H{o}s and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of size $\log n$? We construct examples showing that $m$ is at most $2^{2^{(\log\log n)^{1/2+o(1)}}}$ obtaining a twofold sub-polynomial improvement over the upper bound of about $\sqrt{n}$ coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size $\log n$ in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.Comment: 12 page

Clique minors in graphs with a forbidden subgraph

The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on $n$ vertices of independence number $\alpha(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/\alpha(G)$. Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that $G$ is $H$-free for some bipartite graph $H$ then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvo\v{r}\'ak and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph $H$, answering a question of Dvo\v{r}\'ak and Yepremyan. In particular, we show that any $K_s$-free graph has a clique minor of order $c_s(n/\alpha(G))^{1+\frac{1}{10(s-2) }}$, for some constant $c_s$ depending only on $s$. The exponent in this result is tight up to a constant factor in front of the $\frac{1}{s-2}$ term.Comment: 11 pages, 1 figur

3‐Color bipartite Ramsey number of cycles and paths

The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gy´arf´as and Lehel, who determined the 2-colour Ramsey number of paths. In this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even) cycles