On the ideals of equivariant tree models

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group based models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. The main novelty is our proof that this procedure yields the entire ideal, not just an ideal defining the model set-theoretically. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices.Comment: 23 pages. Greatly improved exposition, in part following suggestions by a referee--thanks! Also added exampl

Population-based mammography screening below age 50: balancing radiation-induced vs prevented breast cancer deaths

Introduction:Exposure to ionizing radiation at mammography screening may cause breast cancer. Because the radiation risk increases with lower exposure age, advancing the lower age limit may affect the balance between screening benefits and risks. The present study explores the benefit-risk ratio of screening before age 50.Methods:The benefits of biennial mammography screening, starting at various ages between 40 and 50, and continuing up to age 74 were examined using micro-simulation. In contrast with previous studies that commonly used excess relative risk models, we assessed the radiation risks using the latest BEIR-VII excess abso

Polarizations and Nullcone of Representations of Reductive Groups

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f_1,f_2,...,f_n be homogeneous invariant functions. Then the polarizations of f_1,f_2,...,f_n define the nullcone of k 0} h(t) x = 0 for all x in L. This is then applied to many examples. A surprising result is about the group SL(2,C) where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of $SL_2$ on the n-fold tensor product C^2 otimes C^2 otimes ... otimes C^2. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n <= 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title ``On the nullcone of representations of reductive groups'' is published in Pacific J. Math. {bf 224} (2006), 119--140.

Overdetection, overtreatment and costs in prostate-specific antigen screening for prostate cancer

Background:Prostate cancer screening with prostate-specific antigen (PSA) has shown to reduce prostate cancer mortality in the European Randomised study of Screening for Prostate Cancer (ERSPC) trial. Overdetection and overtreatment are substantial unfavourable side effects with consequent healthcare costs. In this study the effects of introducing widespread PSA screening is evaluated.Methods:The MISCAN model was used to simulate prostate cancer growth and detection in a simulated cohort of 100 000 men (European standard population) over 25 years. PSA screening from age 55 to 70 or 75, with 1, 2 and 4-year-intervals is simulated. Number of diagnoses, PSA tests, biopsies, treatments, deaths and corresponding costs for 100 000 men and for United Kingdom and United States are compared.Results:Without screening 2378 men per 100 000 were predicted to be diagnosed with prostate cancer compared with 4956 men after screening at 4-year intervals. By introducing screening, the costs would increase with 100% to \[euro]60 695 000. Overdetection is related to 39% of total costs (\[euro]23 669 000). Screening until age 75 is relatively most expensive because of the costs of overtreatment.Conclusion:Introduction of PSA screening will increase total healthcare costs for prostate cancer substantially, of which the actual screening costs will be a small part

Treatment of local-regional prostate cancer detected by PSA screening: Benefits and harms according to prognostic factors

Background:Men with screen-detected prostate cancer can choose to undergo immediate curative treatment or enter into an expectant management programme. We quantified how the benefits and harms of immediate treatment vary according to the prognostic factors of clinical T-stage, Gleason score, and patient age.Methods:A microsimulation model based on European Randomized Study of Screening for Prostate Cancer data was used to predict the benefits and harms of immediate treatment versus delayed treatment of local-regional prostate cancer in men aged 55-74 years. Benefits included life-years gained and reduced probability of death from prostate cancer. Harms included lead time and probability of overdiagnosis.Results:The ratio of mean lead time to mean life-years gained ranged from 1.8 to 31.2, and the additional number of treatments required per prostate cancer death prevented ranged from 0.3 to 11.6 across the different prognostic groups. Both harm-benefit ratios were lowest, most favourable, for men aged 55-59 years and diagnosed with moderate-risk prostate cancer. Ratios were high for men aged 70-74 years regardless of clinical T-stage and Gleason score.Conclusion:Men aged 55-59 years with moderate-risk prostate cancer are predicted to derive greatest benefit from immediate curative treatment. Immediate treatment is least favourable for men aged 70-74 years with either low-risk or high-risk prostate cancer

The overdiagnosis nightmare: a time for caution

Overdiagnosis (and overtreatment) of cancers not bound to become symptomatic during lifetime is an unavoidable drawback of mammography screening. The magnitude of overdiagnosis has been estimated to be in the range of 5-10%, and thus acceptable in view of screening benefits as to reduced mortality. In a recent research article in BMC Women's Health, Jørgensen, Zahl and Gøtzsche suggest that overdiagnosis may be as high as 33%, based on their analysis of breast cancer incidence in screened and non-screened areas in Denmark. Here we consider how reliable such analyses can be, why it might have been useful to adjust comparisons between screened and non-screened areas for early detection lead time, and what further evidence might be needed to build on or confirm these results

Cyclotomic Gaudin models: construction and Bethe ansatz

This is a pre-copyedited author produced PDF of an article accepted for publication in Communications in Mathematical Physics, Benoit, V and Young, C, 'Cyclotomic Gaudin models: construction and Bethe ansatz', Commun. Math. Phys. (2016) 343:971, first published on line March 24, 2016. The final publication is available at Springer via http://dx.doi.org/10.1007/s00220-016-2601-3 © Springer-Verlag Berlin Heidelberg 2016To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case $\sigma = \text{id}$. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.Peer reviewe

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.Comment: We removed the conjecture which appeared in previous versions: we give a counter-example. We fixed the proof of Lemma 2.2 (previously Remark 2.2). 16 page

On the geometry of the set of symmetric matrices with repeated eigenvalues

We investigate some geometric properties of the real algebraic variety \u394 of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart\u2013Young\u2013Mirsky-type theorem for the distance function from a generic matrix to points in \u394. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of \u394) and random matrix theory