We study knots in 3d Chern-Simons theory with complex gauge group SL(N,C), in the context of its relation with 3d N=2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2,0) theory, which is compactified on a 3-manifold M^. We identify such defects in various corners of the 3d-3d correspondence, namely in 3d SL(N,C) Chern-Simons theory, in 3d N=2 theory, in 5d N=2 super Yang-Mills theory, and in the M-theory holographic dual. We can make quantitative checks of the 3d-3d correspondence by computing partition functions at each of these theories. This Letter is a companion to a longer paper, which contains more details and more results.

Value-at-risk measures the worst loss to be expected of a portfolio over a given time horizon at a given confidence level. Calculation of VaR frequently involves estimating the volatility of return processes and quantiles of standardized returns. In this paper, several semiparametric techniques are introduced to estimate the volatilities. In addition, both parametric and nonparametric techniques are proposed to estimate the quantiles of standardized return processes. The newly proposed techniques also have the flexibility to adapt automatically to the changes in the dynamics of market prices over time. The combination of newly proposed techniques for estimating volatility and standardized quantiles yields several new techniques for evaluating multiple period VaR. The performance of the newly proposed VaR estimators is evaluated and compared with some of existing methods. Our simulation results and empirical

We propose a primal-dual continuation approach for the capacitated multi-facility Weber problem (CMFWP) based on its nonlinear second-order cone program (SOCP) reformulation. The main idea of the approach is to reformulate the CMFWP as a nonlinear SOCP with a nonconvex objective function, and then introduce a logarithmic barrier term and a quadratic proximal term into the objective to construct a sequence of convexified subproblems. By this, this class of nondifferentiable and nonconvex optimization problems is converted into the solution of a sequence of nonlinear convex SOCPs. In this paper, we employ the semismooth Newton method proposed in Kanzow etal. (SIAM Journal of Optimization 20:297320, 2009) to solve the KKT system of the resulting convex SOCPs. Preliminary numerical results are reported for eighteen test instances, which indicate that the continuation approach is

We present an alternating direction dual augmented Lagrangian method for solving semidefinite programming (SDP) problems in standard form. At each iteration, our basic algorithm minimizes the augmented Lagrangian function for the dual SDP problem sequentially, first with respect to the dual variables corresponding to the linear constraints, and then with respect to the dual slack variables, while in each minimization keeping the other variables fixed, and then finally it updates the Lagrange multipliers (i.e., primal variables). Convergence is proved by using a fixed-point argument. For SDPs with inequality constraints and positivity constraints, our algorithm is extended to separately minimize the dual augmented Lagrangian function over four sets of variables. Numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems demonstrate that our

Let X be a locally compact Hausdorff space, and T be a disjointness preserving bounded linear operator on C0 (X). We give a sufficient and necessary condition of T being power compact. Moreover, a description of eigenvalues and eigenfunctions of T is given.