Serre’s modularity conjecture (I)

This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases \(p\not=2\) and odd conductor, and p=2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi: 10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin (Invent. Math., doi: 10.1007/s00222-009-0207-5, 2009).     

A note “On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups”

We point out some gaps in the paper (Monatsh Math doi: 10.1007/s00605-016-0877-1, 2016) by Kong and Guo, and give new brief proofs of the main results.     

Erratum: A new tabu search heuristic for the open vehicle routing problem

Correction to: Journal of the Operational Research Society (2005) 56, 267–274. doi:10.1057/palgrave.jors.2601817     

$$C_0$$-semigroups associated with uniquely ergodic Kantorovich modifications of operators

We extend to the context of \(L^p\) spaces and \(C_0\)-semigroups of operators our previous results from Heilmann and Ra?a (Positivity 21:897–910, 2017.  https://doi.org/10.1007/s11117-016-0441-1), concerning the eigenstructure and iterates of uniquely ergodic Kantorovich modifications of linking operators.     

On the second boundary value problem for a class of fully nonlinear flows II

This article is a continuation of an earlier work (Huang and Ye in Int Math Res Not, 2017.  https://doi.org/10.1093/imrn/rnx278), where the long time existence and convergence for some special cases of parabolic type special Lagrangian equations were given. The long time existence and convergence of the flow are obtained for all cases in this article. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs.     

Bakry–Émery curvature and diameter bounds on graphs

We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J für die reine und angewandte Mathematik (Crelle’s J), 2017,  https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry–Émery curvature functions of graphs, 2016).     

Super duality and crystal bases for quantum ortho-symplectic superalgebras II

Let \(\mathcal {O}^\mathrm{int}_q(m|n)\) be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type B, C, D introduced in Kwon (Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076). It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type B, C, D from a viewpoint of super duality. Continuing the previous work on type B and C (Kwon in Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076), we classify the irreducible modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\) and prove the existence and uniqueness of their crystal bases in case of type D. A new combinatorial model of classical crystals of type D is introduced, whose super analog gives a realization of crystals for the highest weight modules in \(\mathcal {O}^\mathrm{int}_q(m|n)\).     

Deformations of Charged Axially Symmetric Initial Data and the Mass–Angular Momentum–Charge Inequality

We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi: 10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.     

Large time behavior and Lyapunov functionals for a nonlocal differential equation

A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016). Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016).     

Borwein–Preiss vector variational principle

This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi: 10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of \(\varepsilon \)-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function.     

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

Error bounds for mixed integer nonlinear optimization problems

We introduce a-posteriori and a-priori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the NLP relaxation of a mixed-integer nonlinear optimization problem. Our analysis mainly bases on the construction of a tractable approximation of the so-called grid relaxation retract. Under appropriate Lipschitz assumptions on the defining functions, we thereby generalize and slightly improve results for the mixed-integer linear case from Stein (Mathematical Programming, 2015, doi: 10.1007/s10107-015-0872-7). In particular, we identify cases in which the optimality and feasibility errors tend to zero at an at least linear rate for increasingly refined meshes.     

Sample selection models for count data in R

We provide a detailed hands-on tutorial for the R package SemiParSampleSel (version 1.5). The package implements selection models for count responses fitted by penalized maximum likelihood estimation. The approach can deal with non-random sample selection, flexible covariate effects, heterogeneous selection mechanisms and varying distributional parameters. We provide an overview of the theoretical background and then demonstrate how SemiParSampleSel can be used to fit interpretable models of different complexity. We use data from the German Socio-Economic Panel survey (SOEP v28, 2012. doi: 10.5684/soep.v28) throughout the tutorial.     

Some new families of partial difference sets in finite fields

In this paper we present some new cyclotomic families of partial difference sets. The argument rests on a general procedure for constructing cyclotomic difference sets or partial difference sets in Galois domains due to Ott (Des Codes Cryptogr, doi:10.1007/s10623-015-0082-6, 2015). Definitions and various properties of partial difference sets can be found for instance in Ma (Des Codes Cryptogr 4:221–261, 1994).     

A remark on the result of Radu Miculescu and Alexandru Mihail

In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi: 10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.     

Fatou Components of Attracting Skew-Products

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.     

The linear complexity of a class of binary sequences with optimal autocorrelation

Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period 4p with optimal autocorrelation was proposed by interleaving four suitable Ding–Helleseth–Lam sequences (Des. Codes Cryptogr.,  https://doi.org/10.1007/s10623-017-0398-5), where p is an odd prime with \(p \equiv 1(\bmod 4)\). The objective of this paper is to determine the minimal polynomial and the linear complexity of this class of binary optimal sequences via a sequence polynomial approach. It turns out that this class of sequences has quite good linear complexity.     

RBFOpt: an open-source library for black-box optimization with costly function evaluations

We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the solution methodology implemented in the open-source library RBFOpt, available on COIN-OR. The algorithm is based on the Radial Basis Function method originally proposed by Gutmann (J Glob Optim 19:201–227, 2001.  https://doi.org/10.1023/A:1011255519438), which builds and iteratively refines a surrogate model of the unknown objective function. The two main methodological contributions of this paper are an approach to exploit a noisy but less expensive oracle to accelerate convergence to the optimum of the exact oracle, and the introduction of an automatic model selection phase during the optimization process. Numerical experiments show that RBFOpt is highly competitive on a test set of continuous and mixed-integer nonlinear unconstrained problems taken from the literature: it outperforms the open-source solvers included in our comparison by a large amount, and performs slightly better than a commercial solver. Our empirical evaluation provides insight on which parameterizations of the algorithm are the most effective in practice. The software reviewed as part of this submission was given the Digital Object Identifier (DOI)  https://doi.org/10.5281/zenodo.597767.     

The Continuity Method to Deform Cone Angle

The continuity method is used to deform the cone angle of a weak conical Kähler–Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson’s Openness Theorem on deforming cone angle Donaldson (Essays in Mathematics and Its Applications, 2012) by combining it with the regularity result of Guenancia–P?un (arXiv:1307.6375 2013) and Chen–Wang (arXiv:1405.1201 2014). This continuity method uses relatively less regularity of the metric (only weak conical Kähler–Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a weak conical Kähler–Einstein metric along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).     

Learning customized and optimized lists of rules with mathematical programming

We introduce a mathematical programming approach to building rule lists, which are a type of interpretable, nonlinear, and logical machine learning classifier involving IF-THEN rules. Unlike traditional decision tree algorithms like CART and C5.0, this method does not use greedy splitting and pruning. Instead, it aims to fully optimize a combination of accuracy and sparsity, obeying user-defined constraints. This method is useful for producing non-black-box predictive models, and has the benefit of a clear user-defined tradeoff between training accuracy and sparsity. The flexible framework of mathematical programming allows users to create customized models with a provable guarantee of optimality. The software reviewed as part of this submission was given the DOI (Digital Object Identifier)  https://doi.org/10.5281/zenodo.1344142.