Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e., up to a specified combinatorial factor, the monotone Hurwitz number in genus g with ramification specified by a given partition is a polynomial indexed by g in the parts of the partition.Comment: 23 page

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil

Monotone Hurwitz numbers in genus zero

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic

Evidence of Odderon-exchange from scaling properties of elastic scattering at TeV energies

We study the scaling properties of the differential cross section of elastic proton-proton ($pp$) and proton-antiproton ($p\bar p$) collisions at high energies. We introduce a new scaling function, that scales -- within the experimental errors -- all the ISR data on elastic $pp$ scattering from $\sqrt{s} = 23.5$ to $62.5$ GeV to the same universal curve. We explore the scaling properties of the differential cross-sections of the elastic $pp$ and $p\bar p$ collisions in a limited TeV energy range. Rescaling the TOTEM $pp$ data from $\sqrt{s} = 7$ TeV to $2.76$ and $1.96$ TeV, and comparing it to D0 $p\bar p$ data at $1.96$ TeV, our results provide an evidence for a $t$-channel Odderon exchange at TeV energies, with a significance of at least 6.26$\sigma$. We complete this work with a model-dependent evaluation of the domain of validity of the new scaling and its violations. We find that the $H(x)$ scaling is valid, model dependently, within $200$ GeV $\leq \sqrt{s} \leq$ $8$ TeV, with a $-t$ range gradually narrowing with decreasing colliding energies.Comment: Accepted in EPJ C, with typos fixed, reorganized institutions updated, Appendix A, B, C, D, E added, 60 pages, 29 figures, 13 tables, Odderon significance: 6.26 sigma, conclusions unchange

Framing research at the tourism and terrorism nexus

Given the rising significance of terrorism and its implications on international tourism, future tourism research will certainly intensify its focus on this particular type of crisis. To assist future research in this area, this article analyzes the evolution of research at the tourism and terrorism nexus. On the one hand, a classical bibliometric analysis of journal articles on "tourism" and "terrorism" is conducted, thereby not narrowing the focus exclusively to journals from the fields of travel, tourism and hospitality. On the other hand, qualitative content analysis and quantitative multiple correspondence analysis are paired to map this detailed research area. For this purpose, the HOMALS (analysis by means of altering least square) procedure was used. The results of this analysis create a valuable overview of the current state of research in this, unfortunately, topical area of research. Obzirom na sve veći značaj i utjecaj terorizma na međunarodni turizam, buduća će se istraživanja u turizmu zasigurno pojačano usredotočiti na ovu posebnu vrstu krize. U ovom se članku analizira evolucija istraživanja na poveznici turizma i terorizma kako bi se potakla buduća istraživanja u ovom području. S jedne strane, provedena je klasična bibliometrijska analiza članaka u časopisima na temu „turizam“ i „terorizam“ a da se pri tomu fokus nije sužavao isključivo na časopise iz područja putovanja, turizma i ugostiteljstva. S druge strane, uparene su kvalitativna analiza sadržaja i kvanti-tativna višestruka korespondencijska analiza kako bi se opisalo ovo specifično područje istraživanja. U tu svrhu se rabila procedura HOMALS (analiza homogenosti metodom najmanjeg kvadrata). Ova je analiza rezultirala vrijednim pregledom trenutnog stanja istraživanja u ovom, nažalost, aktualnom istraživačkom području