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This paper compares the divisorial gonality of a finite graph $G$ to the divisorial gonality of the associated metric graph $\Gamma(G,\mathbb{1})$ with unit lengths. We show that $\text{dgon}(\Gamma(G,\mathbb{1}))$ is equal to the minimal divisorial gonality of all regular subdivisions of $G$, and we provide a class of graphs for which this number is strictly smaller than the divisorial gonality of $G$. This settles a conjecture of M. Baker in the negative., Comment: 15 pages, 4 figures. Changes: improved Lemma 4.4, added Proposition 5.3, changed open question 2

We show that some natural problems that are XNLP-hard (hence $$\textrm{W}[t]$$ -hard for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds, Minimum Maximum Outdegree, and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set. Our hardness claims beat existing results. The FPT algorithms use a new parameter “treebreadth”, associated to a weighted tree partition, as well as DP and ILP.

The aim was to provide evidence on systemically treated patients with advanced melanoma not represented in phase III trials to support clinical decision‐making. Analysis were performed on advanced melanoma patients diagnosed between 2014 and 2017 in the Netherlands, treated with immune‐ or targeted therapy, who met ≥1 trial exclusion criteria. These criteria were derived from the KEYNOTE‐006 and CHECKMATE‐067/‐066 phase III trials. Prognostic importance of factors associated with overall survival (OS) was assessed with the Kaplan‐Meier method, Cox models, predicted OS probabilities of prognostic subgroups and a conditional inference survival tree (CIST). A nationwide population‐based registry was used as data source. Of 2536 systemically treated patients with advanced melanoma, 1004 (40%) patients were ineligible for phase IIII trials. Ineligible patients had a poorer median OS (mOS) compared to eligible patients (8.8 vs 23 months). Eligibility criteria strongly associated with OS in systemically treated ineligible patients were Eastern Cooperative Oncology Group Performance Score (ECOG PS) ≥2, brain metastases (BM) and lactate dehydrogenase (LDH) of >500 U/L. Patients with ECOG PS of ≥2 with or without symptomatic BM had a predicted mOS of 6.5 and 11.3 months and a 3‐year survival probability of 9.3% and 23.6%, respectively. The CIST showed the strongest prognostic covariate for survival was LDH, followed by ECOG PS. The prognosis of patients with LDH of >500 U/L is poor, but long‐term survival is possible. The prognosis of ineligible patients with advanced melanoma in real‐world was very heterogeneous and highly dependent on LDH value, ECOG PS and symptomatic BM., What's new? By necessity, randomized controlled trials (RCTs) exclude many patients. However, these ineligible patients are often still treated with new systemic therapies on an individual basis. In this study, the authors examined how various subgroups of ineligible patients fared following treatment for advanced melanoma. They found that several criteria were strongly associated with prognosis in these patients, including lactate dehydrogenase (LDH) levels. These results should provide clinicians with a decision tree of prognostic factors to help guide treatment decisions.

In this article, a two-dimensional nonlinear boundary value problem which is strongly related to the well-known Gelfand–Bratu model is solved numerically. The numerical results are obtained by employing three different numerical strategies namely: finite difference based method, a Newton multigrid method and a nonlinear multigrid full approximation storage (FAS). We are able to handle the difficulty of unstable convergence behaviour by using MINRES method as a relaxation smoother in multigrid approach with an appropriate sinusoidal approximation as an initial guess. A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate an improvement for the values of λ ∈ (0, λc]. Numerical results illustrate the performance of the proposed numerical methods wherein FAS-MG method is shown to be the most efficient. Further, we present the numerical bifurcation behaviour for two-dimensional Gelfand-Bratu models and find new multiplicity of solutions in the case of a quadratic and cubic approximation of the nonlinear exponential term. Numerical experiments confirm the convergence of the solutions for different values of λ and prove the effectiveness of the nonlinear FAS-MG scheme.

The effect of low-dose ionizing radiation exposure on leukemia incidence remains poorly understood. Possible dose-response curves for various forms of leukemia are largely based on cohorts of atomic bomb survivors. Animal studies can contribute to an improved understanding of radiation-induced acute myeloid leukemia (rAML) in humans. In male CBA/H mice, incidence of rAML can be described by a two-hit model involving a radiation-induced deletion with Sfpi1 gene copy loss and a point mutation in the remaining Sfpi1 allele. In the present study (historical) mouse data were used and these processes were translated into a mathematical model to study photon-induced low-dose AML incidence in male CBA/H mice following acute exposure. Numerical model solutions for low-dose rAML incidence and diagnosis times could respectively be approximated with a model linear-quadratic in radiation dose and a normal cumulative distribution function. Interestingly, the low-dose incidence was found to be proportional to the modeled number of cells carrying the Sfpi1 deletion present per mouse following exposure. After making only model-derived high-dose rAML estimates available to extrapolate from, the linear-quadratic model could be used to approximate low-dose rAML incidence calculated with our mouse model. The accuracy in estimating low-dose rAML incidence when extrapolating from a linear model using a low-dose effectiveness factor was found to depend on whether a data transformation was used in the curve fitting procedure.

In this article we develop algorithms for data assimilation based upon a computational time dependent stable/unstable splitting. Our particular method is based upon shadowing refinement and synchronization techniques and is motivated by work on assimilation in the unstable subspace [Carrassi et al., Chaos, 18 (2008), 023112; Trevisan, D'Isidoro, and Talagrand, Q. J. R. Meteorol. Soc., 136 (2010), pp. 487-496; Palatella, Carrassi, and Trevisan, J. Phys. A, 46 (2013), 254020] and pseudo-orbit data assimilation [Judd and Smith, Phys. D, 151 (2001), pp. 125-141; Judd et al., J. Atmos. Sci., 65 (2008), pp. 1749-1772; Du and Smith, J. Atmos. Sci., 71 (2014), pp. 469-482]. The algorithm utilizes time dependent projections onto the nonstable subspace determined by employing computational techniques for Lyapunov exponents/vectors. The method is extended to parameter estimation without changing the problem dynamics and we address techniques for adapting the method when (as is commonly the case) observations are not available in the full model state space. We use a combination of analysis and numerical experiments (with the Lorenz 63 and Lorenz 96 models) to illustrate the efficacy of the techniques and show that the results compare favorably with other variational techniques.

The future opportunities for high-density QCD studies with ion and proton beams at the LHC are presented. Four major scientific goals are identified: the characterisation of the macroscopic long wavelength Quark-Gluon Plasma (QGP) properties with unprecedented precision, the investigation of the microscopic parton dynamics underlying QGP properties, the development of a unified picture of particle production and QCD dynamics from small (pp) to large (nucleus--nucleus) systems, the exploration of parton densities in nuclei in a broad ($x$, $Q^2$) kinematic range and the search for the possible onset of parton saturation. In order to address these scientific goals, high-luminosity Pb-Pb and p-Pb programmes are considered as priorities for Runs 3 and 4, complemented by high-multiplicity studies in pp collisions and a short run with oxygen ions. High-luminosity runs with intermediate-mass nuclei, for example Ar or Kr, are considered as an appealing case for extending the heavy-ion programme at the LHC beyond Run 4. The potential of the High-Energy LHC to probe QCD matter with newly-available observables, at twice larger center-of-mass energies than the LHC, is investigated.

The replicator equation for a two-person symmetric game, which has an interval of the real line as strategy space, is extended with a mutation term. Assuming that the distribution of the strategies has a continuous density, a partial differential equation for this density is derived. The equation is analysed for two examples. A connection is made with the canonical equation from adaptive dynamics and the continuous stable strategy criterion.

We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some "volume" of the original graph; this can be seen as a substitute for graphs of the Li-Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Gamma of Gamma(1) that is linear in the index [Gamma(1):Gamma], with a constant that only depends on the residue field degree and the degree of the chosen "infinite" place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian., 45 pages; added a worked example in section 6, and expanded section 8

From his observations of the A.D. 1572 super-nova and the A.D. 1577 comet, Tycho Brahe concluded that such transient celestial objects are outside the Earth's atmosphere, and he quoted the 9th century A.D. Persian astrologer and astronomer Abu Mashar: Dixit Albumasar, Cometa supra Venerem visus fuit, i.e. that he had reported much earlier that comets were seen beyond Venus. However, even from a more detailed Latin translation, the observations and logic behind Abu Mashar's conclusion were not understandable. We present here the original Arabic text (MS Ankara, Saib 199) together with our translation and interpretation: Abu Mashar reported that he had observed Venus in (or projected onto) the tail of a comet and concluded that the comet was behind Venus, because he had observed the extinction of Venus due to the cometary tail to be negligible (light of Venus was unimpaired). He then concluded that the comet would be located behind Venus. He also mentioned that others had observed Jupiter and Saturn in cometary tails, so that those comets would even be located beyond those two outer planets - in the sphere of the stars. The dates of the observed close conjunctions were not mentioned; using known orbital elements for a few comets, we found a few close conjunctions between comets and planets from A.D. 770 to 868, but we cannot be sure regarding which conjunctions were reported. While the argument of Abu Mashar is not correct (as cometary tails are optically thin), parts of the conclusion - namely that comets are outside the Earth atmosphere and beyond the moon - is correct. This may have helped Tycho Brahe to come to his revolutionary conclusion., Comment: 18 pages plus figures (this is the submitted manuscript text 2015, accepted at JHA 2016)

We prove a convexity theorem for semisimple symmetric spaces G / H ${G/H}$ which generalizes an earlier theorem of the second named author to a setting without restrictions on the minimal parabolic subgroup involved. The new more general result specializes to Kostant’s non-linear convexity theorem for a real semisimple Lie group G in two ways, firstly by taking H maximal compact and secondly by viewing G as a symmetric space for G × G ${G\times G}$ .

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve., Comment: 17 pages, 4 figures

On March 19, 2010, mathematics lost one of its leading geometric analysts, Johannes Jisse Duistermaat. At age 67 he passed away, after a short illness following a renewed bout of lymphoma the doctors thought they had controlled. ‘Hans’, as Duistermaat was universally known among friends and colleagues, was not only a brilliant research mathematician and inspiring teacher, but also an accomplished chess player, very fond of several physical sports, and a devoted husband and (grand)father. Erik van den Ban and Johan Kolk look back on Duistermaat’s life and work in this contribution; it is followed by remembrances and surveys by some of his friends, students, and colleagues

We prove the following result on the distribution of Dedekind sums: limM→∞logMM∑c=1M1c∑Dmodc∗gS(d,c),dc=12π2∫−∞∞∫RZg(x,y) dy dx, for each compactly supported continuous function g on R × (R/Z). The proof uses Kuznetsov's sum formula in the modular case for varying real weight.

Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.UittrekselSnellius’ Fundamenta Arithmetica et Geometrica (1615) is veel meer dan de Latijnse vertaling van Ludolph van Ceulens Arithmetische en Geometrische Fondamenten. Omdat Willebrord Snellius de Nederlandse versie aanpaste en becommentarieerde in de Fundamenta, kan deze Latijnse editie gelezen worden als een dialoog tussen twee verschillende benaderingen van wiskunde in de vroeg-moderne tijd: Snellius’ humanistische aanpak en Van Ceulens rekenmeestersaanpak. In het bijzonder legt dit artikel sommige van hun uitspraken over het gebruik van getallen in de meetkunde onder de loep. Bovendien wordt Snellius’ gebruik van de Fundamenta als een instrument om zijn carrière te bevorderen uitgelegd.