We discuss some applications of an intrinsic multipication in the space of simple loops in a surface.

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over R n. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the FischerBurmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation

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We propose a new nonmonotone filter method to promote global and fast local convergence for sequential quadratic programming algorithms. Our method uses two filters: a standard, global g-filter for global convergence, and a local nonmonotone l-filter that allows us to establish fast local convergence. We show how to switch between the two filters efficiently, and we prove global and superlinear local convergence. A special feature of the proposed method is that it does not require second-order correction steps. We present preliminary numerical results comparing our implementation with a classical filter SQP method.

Local polynomial fitting has many exciting statistical properties which where established under i.i.d. setting. However, the need for nonlinea r time series modeling, constructing predictive intervals, understanding divergence of nonlinear time series requires the development of the theory of local polynomial fitting for dependent data. In this paper, we study the problem of estimating conditional mean functions and their derivatives via a local polynomial fit. The functions include conditional moments, conditional distribution as well as conditional density functions. Joint asymptotic normality for derivative estimation is established for both strongly mixing and mixing processes.

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All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using LittlewoodPaley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.

Local quasilikelihood estimation is a useful extension of local least squares methods, but its computational cost and algorithmic convergence problems make the procedure less appealing, particularly when it is iteratively used in methods such as the backfitting algorithm, crossvalidation and bootstrapping. A onestep local quasilikelihood estimator is introduced to overcome the computational drawbacks of the local quasilikelihood method. We demonstrate that as long as the initial estimators are reasonably good, the onestep estimator has the same asymptotic behaviour as the local quasilikelihood method. Our simulation shows that the onestep estimator performs at least as well as the local quasilikelihood method for a wide range of choices of bandwidths. A datadriven bandwidth selector is proposed for the onestep estimator based on the preasymptotic substitution method of Fan and Gijbels. It is

We develop a specification test for the transition density of a discretely-sampled continuous-time diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts assumed by the null hypothesis. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are critically investigated via simulation studies and are compared with other tests. Our approaches are illustrated by applications to implied volatility data and Treasury bill data.

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In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the E2 page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a π flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.

We propose the idea of building a secure hash using quadratic or higher degree multivariate polynomials over a finite field as the compression function. We analyze some security properties and potential feasibility, where the compression functions are randomly chosen high-degree polynomials, and show that under some plausible assumptions, high-degree polynomials as compression functions has good properties. Next, we propose to improve on the efficiency of the system by using some specially designed polynomials generated by a small number of random parameters, where the security of the system would then relies on stronger assumptions, and we give empirical evidence for the validity of using such polynomials.

Guy Bonneau has kindly pointed out two errors in [1]. The first is that the manifolds M (k) and M (k)/Z2 do not admit U (2)-invariant Einstein-Weyl structures for k 2; thus the last four entries in Table 4 (page 422) do not occur. The error is on pages 429430. The analysis there is correct, except that the critical line to consider in Lemma 7.5 and the subsequent calculation is = , instead of = 2/k, because of the constraint (D, ](equivalently, > ) occurring in the definition of . On this line, one has

Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and q sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, q sensitivity, has certain

In all the odd dimensions which allow Majorana spinors, we consider a gravitational Lagrangian possessing local Poincare invariance and given by the dimensional continuation of the Euler density in one dimension less. We show that the local supersymmetric extension of this Lagrangian requires the algebra to be the maximal extension of the N=1 super-Poincare algebra. By maximal, we mean that in the right hand side of the anticommutator of the Majorana super charge appear all the possible central charges. The resulting action defines a Chern-Simons gauge theory for the maximal extension of the super-Poincare algebra. In these dimensions, we address the same problem for the AdS invariant gravity and we derive its supersymmetric extension for the minimal super-AdS algebra. The connection between both models is realized at the algebraic level through an expansion of their corresponding Lie super algebras. Within a procedure consistent with the expansion of the algebras, the local supersymmetric extension of the Poincare invariant gravity Lagrangian is derived from the super AdS one.

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

The HFEv- signature scheme is one of the most studied multivariate schemes and one of the major candidates for the upcoming standardization of post-quantum digital signature schemes. In this paper, we propose three new attack strategies against HFEv-, each of them using the idea of projection. Especially our third attack is very effective and is, for some parameter sets, the most efficient known attack against HFEv-. Furthermore, our attack requires much less memory than direct and rank attacks. By our work, we therefore give new insights in the security of the HFEv- signature scheme and restrictions for the parameter choice of a possible future standardized HFEv- instance.

For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an Iitaka fibration of W such that the pluricanonical system |mK_W| defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

The geometry arising from Michelson &amp; Strominger's study of =4<i>B</i> supersymmetric quantum mechanics with superconformal <i>D</i>(2, 1; )-symmetry is a hyperKhler manifold with torsion (HKT) together with a special homothety. It is shown that different parameters are related via changes in potentials for the HKT target spaces. For 0, 1, we describe how each such HKT manifold <i>M</i> ^<i>4m</i> is derived from a space <i>N</i> ^<i>4m4</i> which is quaternionic Khler with torsion and carries an Abelian instanton.

The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (<i>GLR</i>) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using <i>GLR</i> tests. Anumber of open topics worthy of further study are given in a discussion section.

The goal of this chapter is an attempt to relate some ideas of Grothendieck in his Esquisse dun programme [10] and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss Teichmller theory, the mapping class groups, the SL (2, C) representation variety of surface groups, and Thurstons theory of measured laminations.

We study surface charges on a generic null boundary in three dimensional topological massive gravity (TMG). We construct the solution phase space which involves four independent functions over the two dimensional null boundary. One of these functions corresponds to the massive chiral propagating graviton mode of TMG. The other three correspond to three surface charges of the theory, two of which can always be made integrable, while the last one can become integrable only in the absence of the chiral massive graviton flux through the null boundary. As the null boundary symmetry algebra we obtain Heisenberg ⊕ Virasoro algebra with a central charge proportional to the gravitational Chern-Simons term of TMG. We also discuss that the flux of the chiral massive gravitons appears as the (Bondi) news through the null surface.

In this paper, we present an efficient attack on the multivariate Quadratic Quasigroups (MQQ) public key cryptosystem. Our cryptanalysis breaks the MQQ cryptosystem by solving a system of multivariate quadratic polynomial equations using both the MutantXL algorithm and the F4 algorithm. We present the experimental results that show that MQQ systems is broken up to size n equal to 300. Based on these results we show also that MutantXL solves MQQ systems with much less memory than the F4 algorithm implemented in Magma.

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C-algebras. The stronger one of the two is called property (T) and the weaker one is called property (T e). It is shown that all non-unital C-algebras do not have property (T)(neither do their unitalizations). Moreover, all non-unital -unital C-algebras do not have property (T e).