Support τ\tau-Tilting Modules under Split-by-Nilpotent Extensions

Let $\Gamma$ be a split extension of a finite-dimensional algebra $\Lambda$ by a nilpotent bimodule $_\Lambda E_\Lambda$, and let $(T,P)$ be a pair in $\mod\Lambda$ with $P$ projective. We prove that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is a support $\tau$-tilting pair in $\mod \Gamma$ if and only if $(T,P)$ is a support $\tau$-tilting pair in $\mod \Lambda$ and \Hom_\Lambda(T\otimes_\Lambda E,\tau T_\Lambda)=0=\Hom_\Lambda(P,T\otimes_\Lambda E). As applications, we obtain a necessary and sufficient condition such that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is support $\tau$-tilting pair for a cluster-tilted algebra $\Gamma$ corresponding to a tilted algebra $\Lambda$; and we also get that if $T_1,T_2\in\mod\Lambda$ such that $T_1\otimes_\Lambda \Gamma$ and $T_2\otimes_\Lambda \Gamma$ are support $\tau$-tilting $\Gamma$-modules, then $T_1\otimes_\Lambda \Gamma$ is a left mutation of $T_2\otimes_\Lambda \Gamma$ if and only if $T_1$ is a left mutation of $T_2$

Achieving an Efficient and Fair Equilibrium Through Taxation

It is well known that a game equilibrium can be far from efficient or fair, due to the misalignment between individual and social objectives. The focus of this paper is to design a new mechanism framework that induces an efficient and fair equilibrium in a general class of games. To achieve this goal, we propose a taxation framework, which first imposes a tax on each player based on the perceived payoff (income), and then redistributes the collected tax to other players properly. By turning the tax rate, this framework spans the continuum space between strategic interactions (of selfish players) and altruistic interactions (of unselfish players), hence provides rich modeling possibilities. The key challenge in the design of this framework is the proper taxing rule (i.e., the tax exemption and tax rate) that induces the desired equilibrium in a wide range of games. First, we propose a flat tax rate (i.e., a single tax rate for all players), which is necessary and sufficient for achieving an efficient equilibrium in any static strategic game with common knowledge. Then, we provide several tax exemption rules that achieve some typical fairness criterions (such as the Max-min fairness) at the equilibrium. We further illustrate the implementation of the framework in the game of Prisoners' Dilemma.Comment: This manuscript serves as the technical report for the paper with the same title published in APCC 201

Detecting Online Hate Speech Using Context Aware Models

In the wake of a polarizing election, the cyber world is laden with hate speech. Context accompanying a hate speech text is useful for identifying hate speech, which however has been largely overlooked in existing datasets and hate speech detection models. In this paper, we provide an annotated corpus of hate speech with context information well kept. Then we propose two types of hate speech detection models that incorporate context information, a logistic regression model with context features and a neural network model with learning components for context. Our evaluation shows that both models outperform a strong baseline by around 3% to 4% in F1 score and combining these two models further improve the performance by another 7% in F1 score.Comment: Published in RANLP 201

On the generalized resolvent of linear pencils in Banach spaces

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $\lambda\rightarrow T-\lambda S$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area

Silting Modules over Triangular Matrix Rings

Let $\Lambda,\Gamma$ be rings and $R=\left(\begin{array}{cc}\Lambda & 0 \\ M & \Gamma\end{array}\right)$ the triangular matrix ring with $M$ a $(\Gamma,\Lambda)$-bimodule. Let $X$ be a right $\Lambda$-module and $Y$ a right $\Gamma$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is a silting right $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are silting modules and $Y\otimes_\Gamma M$ is generated by $X$. Furthermore, we prove that if $\Lambda$ and $\Gamma$ are finite dimensional algebras over an algebraically closed field and $X_{\Lambda}$ and $Y_{\Gamma}$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is a support $\tau$-tilting $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are support $\tau$-tilting modules, \Hom_\Lambda(Y\otimes_\Gamma M,\tau X)=0 and \Hom_\Lambda(e\Lambda, Y\otimes_\Gamma M)=0 with $e$ the maximal idempotent such that \Hom_\Lambda(e\Lambda, X)=0.Comment: 17 pages, accepted for publication in Taiwanese Journal of Mathematic

Majorana zero modes in the hopping-modulated one-dimensional pp-wave superconducting model

We investigate the one-dimensional $p$-wave superconducting model with periodically modulated hopping and show that under time-reversal symmetry, the number of the Majorana zero modes (MZMs) strongly depends on the modulation period. If the modulation period is odd, there can be at most one MZM. However if the period is even, the number of the MZMs can be zero, one and two. In addition, the MZMs will disappear as the chemical potential varies. We derive the condition for the existence of the MZMs and show that the topological properties in this model are dramatically different from the one with periodically modulated potential

Zeroth-Order Stochastic Alternating Direction Method of Multipliers for Nonconvex Nonsmooth Optimization

Alternating direction method of multipliers (ADMM) is a popular optimization tool for the composite and constrained problems in machine learning. However, in many machine learning problems such as black-box attacks and bandit feedback, ADMM could fail because the explicit gradients of these problems are difficult or infeasible to obtain. Zeroth-order (gradient-free) methods can effectively solve these problems due to that the objective function values are only required in the optimization. Recently, though there exist a few zeroth-order ADMM methods, they build on the convexity of objective function. Clearly, these existing zeroth-order methods are limited in many applications. In the paper, thus, we propose a class of fast zeroth-order stochastic ADMM methods (i.e., ZO-SVRG-ADMM and ZO-SAGA-ADMM) for solving nonconvex problems with multiple nonsmooth penalties, based on the coordinate smoothing gradient estimator. Moreover, we prove that both the ZO-SVRG-ADMM and ZO-SAGA-ADMM have convergence rate of $O(1/T)$, where $T$ denotes the number of iterations. In particular, our methods not only reach the best convergence rate $O(1/T)$ for the nonconvex optimization, but also are able to effectively solve many complex machine learning problems with multiple regularized penalties and constraints. Finally, we conduct the experiments of black-box binary classification and structured adversarial attack on black-box deep neural network to validate the efficiency of our algorithms.Comment: To Appear in IJCAI 2019. Supplementary materials are adde

Efficient Characteristic Set Algorithms for Equation Solving in Finite Fields and Applications in Cryptanalysis

Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper and monic (or regular) triangular set is given. An improved zero decomposition algorithm which can be used to reduce the zero set of an equation system in general form to the union of zero sets of monic proper triangular sets is proposed. As a consequence, we can give an explicit formula for the number of solutions of an equation system. Bitsize complexity for the algorithm is given in the case of Boolean polynomials. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials are effectively controlled. The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations

Entropic Effects of Thermal Rippling on van der Waals Interactions between Monolayer Graphene and a Rigid Substrate

Graphene monolayer, with extremely low flexural stiffness, displays spontaneous rippling due to thermal fluctuations at a finite temperature. When a graphene membrane is placed on a solid substrate, the adhesive interactions between graphene and the substrate could considerably suppress thermal rippling. On the other hand, the statistical nature of thermal rippling adds an entropic contribution to the graphene-substrate interactions. In this paper we present a statistical mechanics analysis on thermal rippling of monolayer graphene supported on a rigid substrate, assuming a generic form of van der Waals interactions between graphene and substrate at T = 0 K. The rippling amplitude, the equilibrium average separation, and the average interaction energy are predicted simultaneously and compared with molecular dynamics (MD) simulations. While the amplitude of thermal rippling is reduced by adhesive interactions, the entropic contribution leads to an effective repulsion. As a result, the equilibrium average separation increases and the effective adhesion energy decreases with increasing temperature. Moreover, the effect of a biaxial pre-strain in graphene is considered, and a buckling instability is predicted at a critical compressive strain that depends on both the temperature and the adhesive interactions. Limited by the harmonic approximations, the theoretical predictions agree with MD simulations only for relatively small rippling amplitudes but can be extended to account for the anharmonic effects.Comment: 9 figures. Submitted for review on November 9, 201

Recognizing Explicit and Implicit Hate Speech Using a Weakly Supervised Two-path Bootstrapping Approach

In the wake of a polarizing election, social media is laden with hateful content. To address various limitations of supervised hate speech classification methods including corpus bias and huge cost of annotation, we propose a weakly supervised two-path bootstrapping approach for an online hate speech detection model leveraging large-scale unlabeled data. This system significantly outperforms hate speech detection systems that are trained in a supervised manner using manually annotated data. Applying this model on a large quantity of tweets collected before, after, and on election day reveals motivations and patterns of inflammatory language.Comment: Published in IJCNLP 201