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We explicitly record the largely unnoticed property of a von Neumann regular ring R that every element a of R has a quasi-inverse a′ such that aa′ and a′a commute. We also show that this property can fail in a regular semigroup S, whence no purely multiplicative proof is available in regular rings.

This book develops the Weyr matrix canonical form, a largely unknown cousin of the Jordan form. It explores novel applications, including include matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry. Module theory and algebraic geometry are employed but with self-contained accounts.

We describe a MATLAB program that could produce a negative answer to the Gerstenhaber Problem by the construction of three commuting \\(n n\\) matrices \\(A,B,C\\) over a field \\(F\\) such that the subalgebra \\(F[A,B,C]\\) they generate has dimension greater than \\(n\\). This problem has remained open for nearly 60 years, following Gerstenhaber's surprising result (Annals Math.) that \\( F[A,B] n\\) for any two commuting matrices \\(A,B\\). The property fails for four or more commuting matrices. We also make the MATLAB files freely available.

We give a number of constructions where inverse limits seriously degrade properties of regular rings, such as unit-regularity, diagonalisation of matrices, and finite stable rank. This raises the possibility of using inverse limits to answer the long standing Separativity Problem (in the negative).

Levels of cancellativity in commutative monoids \\(M\\), determined by stable rank values in \\(Z_> 0 \\ınfty\\\) for elements of \\(M\\), are investigated. The behavior of the stable ranks of multiples \\(ka\\), for \\(k ın Z_> 0\\) and \\(a ın M\\), is determined. In the case of a refinement monoid \\(M\\), the possible stable rank values in archimedean components of \\(M\\) are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

The subclonal composition of human prostate tumours and their metastases has been mapped by whole-genome sequencing, thus establishing the evolutionary trees behind the development and spread of these cancers; an important observation was that metastases could be re-seeded multiple times, and spread from one tumour to another was frequently seen. Complexity in-built in tumour metastases Gunes Gundem et al . have mapped the subclonal composition of human prostate tumours and their metastases, thus establishing the evolutionary history behind the development and spread of these cancers. Importantly, they find that metastases could be re-seeded multiple times, and spread from one site of metastasis to another was frequently seen. This work sheds new light on the origin of the vast diversity of genetic and epigenetic alterations that can be seen within tumours and between primary tumours and metastases, and illustrates the clinical challenge of cancer therapy with targeted drugs. Cancers emerge from an ongoing Darwinian evolutionary process, often leading to multiple competing subclones within a single primary tumour 1 , 2 , 3 , 4 . This evolutionary process culminates in the formation of metastases, which is the cause of 90% of cancer-related deaths 5 . However, despite its clinical importance, little is known about the principles governing the dissemination of cancer cells to distant organs. Although the hypothesis that each metastasis originates from a single tumour cell is generally supported 6 , 7 , 8 , recent studies using mouse models of cancer demonstrated the existence of polyclonal seeding from and interclonal cooperation between multiple subclones 9 , 10 . Here we sought definitive evidence for the existence of polyclonal seeding in human malignancy and to establish the clonal relationship among different metastases in the context of androgen-deprived metastatic prostate cancer. Using whole-genome sequencing, we characterized multiple metastases arising from prostate tumours in ten patients. Integrated analyses of subclonal architecture revealed the patterns of metastatic spread in unprecedented detail. Metastasis-to-metastasis spread was found to be common, either through de novo monoclonal seeding of daughter metastases or, in five cases, through the transfer of multiple tumour clones between metastatic sites. Lesions affecting tumour suppressor genes usually occur as single events, whereas mutations in genes involved in androgen receptor signalling commonly involve multiple, convergent events in different metastases. Our results elucidate in detail the complex patterns of metastatic spread and further our understanding of the development of resistance to androgen-deprivation therapy in prostate cancer.

As efforts to control malaria are expanded across the world, understanding the role of transmission intensity in determining the burden of clinical malaria is crucial to the prediction and measurement of the effectiveness of interventions to reduce transmission. Furthermore, studies comparing several endemic sites led to speculation that as transmission decreases morbidity and mortality caused by severe malaria might increase. We aimed to assess the epidemiological characteristics of malaria in Kilifi, Kenya, during a period of decreasing transmission intensity. We analyse 18 years (1990–2007) of surveillance data from a paediatric ward in a malaria-endemic region of Kenya. The hospital has a catchment area of 250 000 people. Clinical data and blood-film results for more than 61 000 admissions are reported. Hospital admissions for malaria decreased from 18·43 per 1000 children in 2003 to 3·42 in 2007. Over 18 years of surveillance, the incidence of cerebral malaria initially increased; however, malaria mortality decreased overall because of a decrease in incidence of severe malarial anaemia since 1997 (4·75 to 0·37 per 1000 children) and improved survival among children admitted with non-severe malaria. Parasite prevalence, the mean age of children admitted with malaria, and the proportion of children with cerebral malaria began to change 10 years before hospitalisation for malaria started to fall. Sustained reduction in exposure to infection leads to changes in mean age and presentation of disease similar to those described in multisite studies. Changes in transmission might not lead to immediate reductions in incidence of clinical disease. However, longitudinal data do not indicate that reductions in transmission intensity lead to transient increases in morbidity and mortality. Wellcome Trust, Kenya Medical Research Institute.

Background To establish the feasibility of the dosimetric compliance criteria of the RTOG 1308 trial through testing against Intensity Modulation Radiation Therapy (IMRT) and Passive Scattering Proton Therapy (PSPT) plans. Methods Twenty-six lung IMRT and 26 proton PSPT plans were included in the study. Dose Volume Histograms (DVHs) for targets and normal structures were analyzed. The quality of IMRT plans was assessed using a knowledge-based engineering tool. Results Most of the RTOG 1308 dosimetric criteria were achieved. The deviation unacceptable rates were less than 10 % for most criteria; however, a deviation unacceptable rate of more than 20 % was computed for the planning target volume minimum dose compliance criterion. Dose parameters for the target volume were very close for the IMRT and PSPT plans. However, the PSPT plans led to lower dose values for normal structures. The dose parameters in which PSPT plans resulted in lower values than IMRT plans were: lung V 5Gy (%) (34.4 in PSPT and 47.2 in IMRT); maximum spinal cord dose (31.7 Gy in PSPT and 43.5 Gy in IMRT); heart V 5Gy (%) (19 in PSPT and 47 in IMRT); heart V 30Gy (%) (11 in PSPT and 19 in IMRT); heart V 45Gy (%) (7.8 in PSPT and 12.1 in IMRT); heart V 50% (Gy) (7.1 in PSPT and 9.8 in IMRT) and mean heart dose (7.7 Gy in PSPT and 14.9 Gy in IMRT). Conclusions The revised RTOG 1308 dosimetric compliance criteria are feasible and achievable.

Multiple myeloma is an incurable plasma cell malignancy with a complex and incompletely understood molecular pathogenesis. Here we use whole-exome sequencing, copy-number profiling and cytogenetics to analyse 84 myeloma samples. Most cases have a complex subclonal structure and show clusters of subclonal variants, including subclonal driver mutations. Serial sampling reveals diverse patterns of clonal evolution, including linear evolution, differential clonal response and branching evolution. Diverse processes contribute to the mutational repertoire, including kataegis and somatic hypermutation, and their relative contribution changes over time. We find heterogeneity of mutational spectrum across samples, with few recurrent genes. We identify new candidate genes, including truncations of SP140 , LTB, ROBO1 and clustered missense mutations in EGR1 . The myeloma genome is heterogeneous across the cohort, and exhibits diversity in clonal admixture and in dynamics of evolution, which may impact prognostic stratification, therapeutic approaches and assessment of disease response to treatment. Multiple myeloma is a malignant plasma cell disorder with a complex molecular pathogenesis. Here, the authors perform whole-exome sequencing, copy-number profiling and cytogenetic analysis in 84 myeloma samples and highlight the diversity and evolution of the mutational profile underlying the disease.