Representation Theory of Twisted Group Double

This text collects useful results concerning the quasi-Hopf algebra \D . We give a review of issues related to its use in conformal theories and physical mathematics. Existence of such algebras based on 3-cocycles with values in ${R} / {Z}$ which mimic for finite groups Chern-Simons terms of gauge theories, open wide perspectives in the so called "classification program". The modularisation theorem proved for quasi-Hopf algebras by two authors some years ago makes the computation of topological invariants possible. An updated, although partial, bibliography of recent developments is provided.Comment: 15 pages, no figur

Near-linear convergence of the Random Osborne algorithm for Matrix Balancing

We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most numerical software packages. However, its theoretical properties are not well understood. Here, we show that a simple random variant of Osborne's algorithm converges in near-linear time in the input sparsity. Specifically, it balances $K\in\mathbb{R}_{\geq 0}^{n\times n}$ after $O(m\epsilon^{-2}\log\kappa)$ arithmetic operations, where $m$ is the number of nonzeros in $K$, $\epsilon$ is the $\ell_1$ accuracy, and $\kappa=\sum_{ij}K_{ij}/(\min_{ij:K_{ij}\neq 0}K_{ij})$ measures the conditioning of $K$. Previous work had established near-linear runtimes either only for $\ell_2$ accuracy (a weaker criterion which is less relevant for applications), or through an entirely different algorithm based on (currently) impractical Laplacian solvers. We further show that if the graph with adjacency matrix $K$ is moderately connected--e.g., if $K$ has at least one positive row/column pair--then Osborne's algorithm initially converges exponentially fast, yielding an improved runtime $O(m\epsilon^{-1}\log\kappa)$. We also address numerical precision by showing that these runtime bounds still hold when using $O(\log(n\kappa/\epsilon))$-bit numbers. Our results are established through an intuitive potential argument that leverages a convex optimization perspective of Osborne's algorithm, and relates the per-iteration progress to the current imbalance as measured in Hellinger distance. Unlike previous analyses, we critically exploit log-convexity of the potential. Our analysis extends to other variants of Osborne's algorithm: along the way, we establish significantly improved runtime bounds for cyclic, greedy, and parallelized variants.Comment: v2: Fixed minor typos. Modified title for clarity. Corrected statement of Thm 6.1; this does not affect our main result

Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedule

Can we accelerate convergence of gradient descent without changing the algorithm -- just by carefully choosing stepsizes? Surprisingly, we show that the answer is yes. Our proposed Silver Stepsize Schedule optimizes strongly convex functions in $k^{\log_{\rho} 2} \approx k^{0.7864}$ iterations, where $\rho=1+\sqrt{2}$ is the silver ratio and $k$ is the condition number. This is intermediate between the textbook unaccelerated rate $k$ and the accelerated rate $\sqrt{k}$ due to Nesterov in 1983. The non-strongly convex setting is conceptually identical, and standard black-box reductions imply an analogous accelerated rate $\varepsilon^{-\log_{\rho} 2} \approx \varepsilon^{-0.7864}$. We conjecture and provide partial evidence that these rates are optimal among all possible stepsize schedules. The Silver Stepsize Schedule is constructed recursively in a fully explicit way. It is non-monotonic, fractal-like, and approximately periodic of period $k^{\log_{\rho} 2}$. This leads to a phase transition in the convergence rate: initially super-exponential (acceleration regime), then exponential (saturation regime).Comment: 7 figure

Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization

We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an $\epsilon$-minimizer in $O(\epsilon^{-\log_{\rho} 2}) = O(\epsilon^{-0.7864})$ iterations, where $\rho = 1+\sqrt{2}$ is the silver ratio. This is intermediate between the textbook unaccelerated rate $O(\epsilon^{-1})$ and the accelerated rate $O(\epsilon^{-1/2})$ due to Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal: the $i$-th stepsize is $1+\rho^{v(i)-1}$ where $v(i)$ is the $2$-adic valuation of $i$. The design and analysis are conceptually identical to the strongly convex setting in Part I, but simplify remarkably in this specific setting.Comment: 10 pages, 3 figure

On Dijkgraaf-Witten Type Invariants

We explicitly construct a series of lattice models based upon the gauge group $Z_{p}$ which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link, or $1$-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher dimensional simplices.Comment: 18 page

The Drinfel'd Double and Twisting in Stringy Orbifold Theory

This paper exposes the fundamental role that the Drinfel'd double \dkg of the group ring of a finite group $G$ and its twists \dbkg, \beta \in Z^3(G,\uk) as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of \dkg--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of global quotient given by the inertia variety of a point with a $G$ action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full $K$--theory of the stack $[pt/G]$. Finally, we show how one can use the co-cycles $\beta$ above to twist a) the global orbifold $K$--theory of the inertia of a global quotient and more importantly b) the stacky $K$--theory of a global quotient $[X/G]$. This corresponds to twistings with a special type of 2--gerbe.Comment: 35 pages, no figure

A multi-modal data resource for investigating topographic heterogeneity in patient-derived xenograft tumors.

Patient-derived xenografts (PDXs) are an essential pre-clinical resource for investigating tumor biology. However, cellular heterogeneity within and across PDX tumors can strongly impact the interpretation of PDX studies. Here, we generated a multi-modal, large-scale dataset to investigate PDX heterogeneity in metastatic colorectal cancer (CRC) across tumor models, spatial scales and genomic, transcriptomic, proteomic and imaging assay modalities. To showcase this dataset, we present analysis to assess sources of PDX variation, including anatomical orientation within the implanted tumor, mouse contribution, and differences between replicate PDX tumors. A unique aspect of our dataset is deep characterization of intra-tumor heterogeneity via immunofluorescence imaging, which enables investigation of variation across multiple spatial scales, from subcellular to whole tumor levels. Our study provides a benchmark data resource to investigate PDX models of metastatic CRC and serves as a template for future, quantitative investigations of spatial heterogeneity within and across PDX tumor models

Unifying W-Algebras

We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying simple Lie algebra, the field content does not change with the rank of the Casimir algebra any more. This leads to identifications between the Casimir algebras themselves but also gives rise to new, `unifying' W-algebras. For example, the kth unitary minimal model of WA_n has a unifying W-algebra of type W(2,3,...,k^2 + 3 k + 1). These unifying W-algebras are non-freely generated on the quantum level and belong to a recently discovered class of W-algebras with infinitely, non-freely generated classical counterparts. Some of the identifications are indicated by level-rank-duality leading to a coset realization of these unifying W-algebras. Other unifying W-algebras are new, including e.g. algebras of type WD_{-n}. We point out that all unifying quantum W-algebras are finitely, but non-freely generated.Comment: 13 pages (plain TeX); BONN-TH-94-01, DFTT-15/9

Topological Change in Mean Convex Mean Curvature Flow

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m. We also prove that for each m from 1 to n, there is a nonempty open set of compact, mean convex regions K in R^(n+1) with smooth boundary for which the resulting mean curvature flow has a shrinking S^m x R^(n-m) singularity.Comment: 19 pages. This version includes a new section proving that certain kinds of mean curvature flow singularities persist under arbitrary small perturbations of the initial surface. Newest update (Oct 2013) fixes some bibliographic reference

Defect free global minima in Thomson's problem of charges on a sphere

Given $N$ unit points charges on the surface of a unit conducting sphere, what configuration of charges minimizes the Coulombic energy $\sum_{i>j=1}^N 1/r_{ij}$? Due to an exponential rise in good local minima, finding global minima for this problem, or even approaches to do so has proven extremely difficult. For \hbox{$N = 10(h^2+hk+k^2)+ 2$} recent theoretical work based on elasticity theory, and subsequent numerical work has shown, that for $N \sim >500$--1000 adding dislocation defects to a symmetric icosadeltahedral lattice lowers the energy. Here we show that in fact this approach holds for all $N$, and we give a complete or near complete catalogue of defect free global minima.Comment: Revisions in Tables and Reference