Singular Points of Real Quartic and Quintic Curves

There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for irreducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that cam be done nicely using Maple

Singular Points of Reducible Sextic Curves

There are 106 individual types of singular points for reducible complex sextic curves.</jats:p

Singular Points of Real Sextic Curves I

A complete classification of the individual types of singular points is given for irreducible real sextic curves. This classification is derived by using the computer algebra system Maple. There are 191 types of singular points for real irreducible sextic curves. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. A significant portion of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple

Singular Points of Reducible Sextic Curves

There are 106 individual types of singular points for reducible complex sextic curves.Comment: 11 page

BRCA1 and CtIP suppress long tract gene conversion between sister chromatids

BRCA1 controls early steps of the synthesis-dependent strand annealing (SDSA) pathway of homologous recombination, but has no known role following Rad51-mediated synapsis. Here we show that BRCA1 influences post-synaptic homologous recombination events, controlling the balance between short- (STGC) and long-tract gene conversion (LTGC) between sister chromatids. Brca1 mutant cells reveal a bias towards LTGC that is corrected by expression of wild type but not cancer-predisposing BRCA1 alleles. The LTGC bias is enhanced by depletion of CtIP but reversed by inhibition of 53BP1, implicating DNA end resection as a contributor to the STGC/LTGC balance. The impact of BRCA1/CtIP loss on the STGC/LTGC balance is abolished when the second (non-invading) end of the break is unable to support termination of STGC by homologous pairing (“annealing”). This suggests that BRCA1/CtIP-mediated processing of the second end of the break controls the annealing step that normally terminates SDSA, thereby suppressing the error-prone LTGC outcome

BRCA1 controls homologous recombination at Tus/Ter-stalled mammalian replication forks

Replication fork stalling can promote genomic instability, predisposing to cancer and other diseases1–3. Stalled replication forks may be processed by sister chromatid recombination (SCR), generating error-free or error-prone homologous recombination (HR) outcomes4–8. In mammalian cells, a long-standing hypothesis proposes that the major hereditary breast/ovarian cancer predisposition gene products, BRCA1 and BRCA2, control HR/SCR at stalled replication forks9. Although BRCA1 and BRCA2 affect replication fork processing10–12, direct evidence that BRCA genes regulate HR at stalled chromosomal replication forks is lacking due to a dearth of tools for studying this process. We report that the Escherichia coli Tus/Ter complex13–16 can be engineered to induce site-specific replication fork stalling and chromosomal HR/SCR in mammalian cells. Tus/Ter-induced HR entails processing of bidirectionally arrested forks. We find that the BRCA1 C-terminal tandem BRCT repeat and regions of BRCA1 encoded by exon 11—two BRCA1 elements implicated in tumor suppression—control Tus/Ter-induced HR. Inactivation of either BRCA1 or BRCA2 increases the absolute frequency of “long-tract” gene conversions at Tus/Ter-stalled forks—an outcome not observed in response to a restriction endonuclease-mediated chromosomal double strand break (DSB). Therefore, HR at stalled forks is regulated differently from HR at DSBs arising independently of a fork. We propose that aberrant long-tract HR at stalled replication forks contributes to genomic instability and breast/ovarian cancer predisposition in BRCA mutant cells

Improving the normalization of complex interventions: measure development based on normalization process theory (NoMAD): study protocol

&lt;b&gt;Background&lt;/b&gt; Understanding implementation processes is key to ensuring that complex interventions in healthcare are taken up in practice and thus maximize intended benefits for service provision and (ultimately) care to patients. Normalization Process Theory (NPT) provides a framework for understanding how a new intervention becomes part of normal practice. This study aims to develop and validate simple generic tools derived from NPT, to be used to improve the implementation of complex healthcare interventions.&lt;p&gt;&lt;/p&gt; &lt;b&gt;Objectives&lt;/b&gt; The objectives of this study are to: develop a set of NPT-based measures and formatively evaluate their use for identifying implementation problems and monitoring progress; conduct preliminary evaluation of these measures across a range of interventions and contexts, and identify factors that affect this process; explore the utility of these measures for predicting outcomes; and develop an online users’ manual for the measures.&lt;p&gt;&lt;/p&gt; &lt;b&gt;Methods&lt;/b&gt; A combination of qualitative (workshops, item development, user feedback, cognitive interviews) and quantitative (survey) methods will be used to develop NPT measures, and test the utility of the measures in six healthcare intervention settings.&lt;p&gt;&lt;/p&gt; &lt;b&gt;Discussion&lt;/b&gt; The measures developed in the study will be available for use by those involved in planning, implementing, and evaluating complex interventions in healthcare and have the potential to enhance the chances of their implementation, leading to sustained changes in working practices

A protective role for BRCA2 at stalled replication forks

The hereditary breast and ovarian cancer predisposition genes BRCA1 and BRCA2 account for the lion's share of heritable breast cancer risk in the human population. Loss of function of either gene results in defective homologous recombination (HR) and triggers genomic instability, accelerating breast tumorigenesis. A long-standing hypothesis proposes that BRCA1 and BRCA2 mediate HR following attempted replication across damaged DNA, ensuring error-free processing of the stalled replication fork. A recent paper describes a new replication fork protective function of BRCA2, which appears to collaborate with its HR function to suppress genomic instability