On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms $\rho: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further identify the cohomological support variety of a finite-dimensional $G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. We then discuss how our methods, when combined with recently-announced results by Benson, Iyengar, Krause, and Pevtsova, can be applied to extend the homeomorphism $\mathcal{N}_r(G) \cong |G|$ to arbitrary infinitesimal unipotent supergroup schemes.Comment: Fixed some algebra misidentifications, primarily in Sections 1.3 and 3.3. Simplified the proof of Proposition 3.3.

Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition

Hierarchical uncertainty quantification can reduce the computational cost of stochastic circuit simulation by employing spectral methods at different levels. This paper presents an efficient framework to simulate hierarchically some challenging stochastic circuits/systems that include high-dimensional subsystems. Due to the high parameter dimensionality, it is challenging to both extract surrogate models at the low level of the design hierarchy and to handle them in the high-level simulation. In this paper, we develop an efficient ANOVA-based stochastic circuit/MEMS simulator to extract efficiently the surrogate models at the low level. In order to avoid the curse of dimensionality, we employ tensor-train decomposition at the high level to construct the basis functions and Gauss quadrature points. As a demonstration, we verify our algorithm on a stochastic oscillator with four MEMS capacitors and 184 random parameters. This challenging example is simulated efficiently by our simulator at the cost of only 10 minutes in MATLAB on a regular personal computer.Comment: 14 pages (IEEE double column), 11 figure, accepted by IEEE Trans CAD of Integrated Circuits and System

Stochastic Testing Simulator for Integrated Circuits and MEMS: Hierarchical and Sparse Techniques

Process variations are a major concern in today's chip design since they can significantly degrade chip performance. To predict such degradation, existing circuit and MEMS simulators rely on Monte Carlo algorithms, which are typically too slow. Therefore, novel fast stochastic simulators are highly desired. This paper first reviews our recently developed stochastic testing simulator that can achieve speedup factors of hundreds to thousands over Monte Carlo. Then, we develop a fast hierarchical stochastic spectral simulator to simulate a complex circuit or system consisting of several blocks. We further present a fast simulation approach based on anchored ANOVA (analysis of variance) for some design problems with many process variations. This approach can reduce the simulation cost and can identify which variation sources have strong impacts on the circuit's performance. The simulation results of some circuit and MEMS examples are reported to show the effectiveness of our simulatorComment: Accepted to IEEE Custom Integrated Circuits Conference in June 2014. arXiv admin note: text overlap with arXiv:1407.302

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(\lambda))$ as an $L_J$-module where $L(\lambda)$ is a simple $G$-module with high weight $\lambda$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.Comment: 18 pages. Some proofs streamlined over previous version. Additional details added to some proofs in Section

ABCD of Beta Ensembles and Topological Strings

We study beta-ensembles with Bn, Cn, and Dn eigenvalue measure and their relation with refined topological strings. Our results generalize the familiar connections between local topological strings and matrix models leading to An measure, and illustrate that all those classical eigenvalue ensembles, and their topological string counterparts, are related one to another via various deformations and specializations, quantum shifts and discrete quotients. We review the solution of the Gaussian models via Macdonald identities, and interpret them as conifold theories. The interpolation between the various models is plainly apparent in this case. For general polynomial potential, we calculate the partition function in the multi-cut phase in a perturbative fashion, beyond tree-level in the large-N limit. The relation to refined topological string orientifolds on the corresponding local geometry is discussed along the way.Comment: 33 pages, 1 figur

RepSeq-A database of amino acid repeats present in lower eukaryotic pathogens

BACKGROUND Amino acid repeat-containing proteins have a broad range of functions and their identification is of relevance to many experimental biologists. In human-infective protozoan parasites (such as the Kinetoplastid and Plasmodium species), they are implicated in immune evasion and have been shown to influence virulence and pathogenicity. RepSeq http://repseq.gugbe.com is a new database of amino acid repeat-containing proteins found in lower eukaryotic pathogens. The RepSeq database is accessed via a web-based application which also provides links to related online tools and databases for further analyses. RESULTS The RepSeq algorithm typically identifies more than 98% of repeat-containing proteins and is capable of identifying both perfect and mismatch repeats. The proportion of proteins that contain repeat elements varies greatly between different families and even species (3 - 35% of the total protein content). The most common motif type is the Sequence Repeat Region (SRR) - a repeated motif containing multiple different amino acid types. Proteins containing Single Amino Acid Repeats (SAARs) and Di-Peptide Repeats (DPRs) typically account for 0.5 - 1.0% of the total protein number. Notable exceptions are P. falciparum and D. discoideum, in which 33.67% and 34.28% respectively of the predicted proteomes consist of repeat-containing proteins. These numbers are due to large insertions of low complexity single and multi-codon repeat regions. CONCLUSION The RepSeq database provides a repository for repeat-containing proteins found in parasitic protozoa. The database allows for both individual and cross-species proteome analyses and also allows users to upload sequences of interest for analysis by the RepSeq algorithm. Identification of repeat-containing proteins provides researchers with a defined subset of proteins which can be analysed by expression profiling and functional characterisation, thereby facilitating study of pathogenicity and virulence factors in the parasitic protozoa. While primarily designed for kinetoplastid work, the RepSeq algorithm and database retain full functionality when used to analyse other species

Population genetics of trypanosoma brucei rhodesiense: clonality and diversity within and between foci

African trypanosomes are unusual among pathogenic protozoa in that they can undergo their complete morphological life cycle in the tsetse fly vector with mating as a non-obligatory part of this development. Trypanosoma brucei rhodesiense, which infects humans and livestock in East and Southern Africa, has classically been described as a host-range variant of the non-human infective Trypanosoma brucei that occurs as stable clonal lineages. We have examined T. b. rhodesiense populations from East (Uganda) and Southern (Malawi) Africa using a panel of microsatellite markers, incorporating both spatial and temporal analyses. Our data demonstrate that Ugandan T. b. rhodesiense existed as clonal populations, with a small number of highly related genotypes and substantial linkage disequilibrium between pairs of loci. However, these populations were not stable as the dominant genotypes changed and the genetic diversity also reduced over time. Thus these populations do not conform to one of the criteria for strict clonality, namely stability of predominant genotypes over time, and our results show that, in a period in the mid 1990s, the previously predominant genotypes were not detected but were replaced by a novel clonal population with limited genetic relationship to the original population present between 1970 and 1990. In contrast, the Malawi T. b. rhodesiense population demonstrated significantly greater diversity and evidence for frequent genetic exchange. Therefore, the population genetics of T. b. rhodesiense is more complex than previously described. This has important implications for the spread of the single copy T. b. rhodesiense gene that allows human infectivity, and therefore the epidemiology of the human disease, as well as suggesting that these parasites represent an important organism to study the influence of optional recombination upon population genetic dynamics