Quadratic Poisson brackets compatible with an algebra structure

Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.Comment: 6 page

Quadratic Poisson brackets and Drinfeld theory for associative algebras

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.Comment: 16 pages, latex, no figure

Quadratic Poisson brackets and Drinfel'd theory for associative algebras

Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Lie structure. A way to quantize such structures is suggested.Comment: latex, no figures

Towards a framework to evaluate the ‘total’ performance of buildings

Internationally, buildings are a major contributor to carbon emissions. Despite significant advances in the technology and construction of energy-efficient buildings, in many cases a performance gap between designed and actual performance exists. While much research has investigated the drivers of the building energy performance gap – both static and transient– there has been considerably less research into the total performance gap, defined here as performance gaps in building energy use, occupant satisfaction and Indoor Environmental Quality parameters such as thermal comfort and air quality which may impact on occupant health and wellbeing. This paper presents a meta-analysis of building performance data from buildings in the UK and China – selected due to their contrasting development environments – which illustrate the presence of and complexities of evaluating total performance gaps in both countries. The data demonstrate the need for (1) high end-use, spatial granularity and temporal resolution data for both energy and Indoor Environmental Quality, and (2) developing methodologies that allow meaningful comparisons between buildings internationally to facilitate learning from successful building design, construction methodologies and policy environments internationally. Using performance data from a UK building, a potential forward path is illustrated with the objective of developing a framework to evaluate total building performance. Practical application: While much research has examined building energy performance gaps, Indoor Environmental Quality and occupant satisfaction gaps are rarely included despite their relationship to energy. We use a meta-analysis of energy, indoor environmental quality, and occupant satisfaction data from buildings in the UK and China to illustrating the presence of and complexities of evaluating total performance gaps for buildings in the two countries, and the need for high resolution dynamic buildings data and novel methodologies for comparison between buildings across different contexts. Illustrative case studies are used to demonstrate potential future directions for evaluating ‘total’ building performance

Connected components of spaces of Morse functions with fixed critical points

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\in F$ in $F$. By means of the winding number introduced by Reinhart (1960), a surjection $\pi_0(F)\to{\mathbb Z}^{p+r-1}$ is constructed. In particular, $|\pi_0(F)|=\infty$, and the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point, does not preserve $F_f$. Let $\mathscr D$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, ${\mathscr D}^0$ be the connected component of ${\rm id}_M$ in $\mathscr D$, and ${\mathscr D}_f\subset{\mathscr D}$ the set of diffeomorphisms preserving $F_f$. Let ${\mathscr H}_f$ be the subgroup of ${\mathscr D}_f$ generated by ${\mathscr D}^0$ and all diffeomorphisms $h\in{\mathscr D}$ which preserve some functions $f_1\in F_f$, and let ${\mathscr H}_f^{\rm abs}$ be its subgroup generated ${\mathscr D}^0$ and the Dehn twists about the components of level curves of functions $f_1\in F_f$. We prove that ${\mathscr H}_f^{\rm abs}\subsetneq{\mathscr D}_f$ if $q\ge2$, and construct an epimorphism ${\mathscr D}_f/{\mathscr H}_f^{\rm abs}\to{\mathbb Z}_2^{q-1}$, by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\mu:\pi_1(K)\to{\mathscr D}_f/{\mathscr H}_f$ and finite generating sets for the groups ${\mathscr D}_f/{\mathscr D}^0$ and ${\mathscr D}_f/{\mathscr H}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.Comment: 12 pages with 2 figures, in Russian, to be published in Vestnik Moskov. Univ., a typo in theorem 1 is correcte

Connecdenn, a novel DENN domain-containing protein of neuronal clathrin-coated vesicles functioning in synaptic vesicle endocytosis

Clathrin-coated vesicles (CCVs) are responsible for the endocytosis of multiple cargo, including synaptic vesicle membranes. We now describe a new CCV protein, termed connecdenn, that contains an N-terminal DENN (differentially expressed in neoplastic versus normal cells) domain, a poorly characterized protein module found in multiple proteins of unrelated function and a C-terminal peptide motif domain harboring three distinct motifs for binding the α-ear of the clathrin adaptor protein 2 (AP-2). Connecdenn coimmunoprecipitates and partially colocalizes with AP-2, and nuclear magnetic resonance and peptide competition studies reveal that all three α-ear-binding motifs contribute to AP-2 interactions. In addition, connecdenn contains multiple Src homology 3 (SH3) domain-binding motifs and coimmunoprecipitates with the synaptic SH3 domain proteins intersectin and endophilin A1. Interestingly, connecdenn is enriched on neuronal CCVs and is present in the presynaptic compartment of neurons. Moreover, connecdenn has a uniquely stable association with CCV membranes because it resists extraction with Tris and high-salt buffers, unlike most other CCV proteins, but it is not detected on purified synaptic vesicles. Together, these observations suggest that connecdenn functions on the endocytic limb of the synaptic vesicle cycle. Accordingly, disruption of connecdenn interactions with its binding partners through overexpression of the C-terminal peptide motif domain or knock down of connecdenn through lentiviral delivery of small hairpin RNA both lead to defects in synaptic vesicle endocytosis in cultured hippocampal neurons. Thus, we identified connecdenn as a component of the endocytic machinery functioning in synaptic vesicle endocytosis, providing the first evidence of a role for a DENN domain-containing protein in endocytosis

Elements of mathematics in problems. Through olympiads and circles to profession

This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and theories. The materials can be used by pupils and students for self-study, and by teachers. This is an abridged pre-copyedit version of the published book submitted with the permission of the publisher. Each included individual material is self-contained and ready-for-use. Solutions to problems are not included intentionally. This collection consolidates updates of several arXiv submissions, e.g., arXiv:1305.2598.Comment: Edited by A. Skopenkov, M. Skopenkov, A. Zaslavsky. In Russian. Optimized for printing on A5 paper. Moscow Center for Continuous Mathematical Education, 2018, 592pp (in Russian

Gravitational wave background from sub-luminous GRBs: prospects for second and third generation detectors

We assess the detection prospects of a gravitational wave background associated with sub-luminous gamma-ray bursts (SL-GRBs). We assume that the central engines of a significant proportion of these bursts are provided by newly born magnetars and consider two plausible GW emission mechanisms. Firstly, the deformation-induced triaxial GW emission from a newly born magnetar. Secondly, the onset of a secular bar-mode instability, associated with the long lived plateau observed in the X-ray afterglows of many gamma-ray bursts (Corsi & Meszaros 2009a). With regards to detectability, we find that the onset of a secular instability is the most optimistic scenario: under the hypothesis that SL-GRBs associated with secularly unstable magnetars occur at a rate of (48; 80)Gpc^{-3}yr^{-1} or greater, cross-correlation of data from two Einstein Telescopes (ETs) could detect the GW background associated to this signal with a signal-to-noise ratio of 3 or greater after 1 year of observation. Assuming neutron star spindown results purely from triaxial GW emissions, we find that rates of around (130;350)Gpc^{-3}yr^{-1} will be required by ET to detect the resulting GW background. We show that a background signal from secular instabilities could potentially mask a primordial GW background signal in the frequency range where ET is most sen- sitive. Finally, we show how accounting for cosmic metallicity evolution can increase the predicted signal-to-noise ratio for background signals associated with SL-GRBs.Comment: Accepted by MNRA