The orbifold Hochschild product for Fermat hypersurface

Let $G$ be an abelian group acting on a smooth algebraic variety $X$. We investigate the product structure and the bigrading on the cohomology of polyvector fields on the orbifold $[X/G]$, as introduced by C\u{a}ld\u{a}raru and Huang. In this paper we provide many new examples given by quotients of Fermat hypersurfaces, where the product is shown to be associative. This is expected due to the conjectural isomorphism at the level of algebras between the cohomology of polyvector fields and Hochschild cohomology of orbifolds. We prove this conjecture for Calabi-Yau Fermat hypersurface orbifold. We also show that for Calabi-Yau orbifolds, the multiplicative bigrading on the cohomology of polyvector fields agrees with what is expected in homological mirror symmetry.Comment: 35 pages. The results in this new version have been greatly improved by the second autho

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numerical experiments demonstrate that our control strategy is effective and sufficiently accurate

Quantized distributed Nash equilibrium seeking under DoS attacks: A quantized consensus based approach

This paper studies distributed Nash equilibrium (NE) seeking under Denial-of-Service (DoS) attacks and quantization. The players can only exchange information with their own direct neighbors. The transmitted information is subject to quantization and packet losses induced by malicious DoS attacks. We propose a quantized distributed NE seeking strategy based on the approach of dynamic quantized consensus. To solve the quantizer saturation problem caused by DoS attacks, the quantization mechanism is equipped to have zooming-in and holding capabilities, in which the holding capability is consistent with the results in quantized consensus under DoS. A sufficient condition on the number of quantizer levels is provided, under which the quantizers are free from saturation under DoS attacks. The proposed distributed quantized NE seeking strategy is shown to have the so-called maximum resilience to DoS attacks. Namely, if the bound characterizing the maximum resilience is violated, an attacker can deny all the transmissions and hence distributed NE seeking is impossible

Dynamic quantized consensus under DoS attacks: Towards a tight zooming-out factor

This paper deals with dynamic quantized consensus of dynamical agents in a general form under packet losses induced by Denial-of-Service (DoS) attacks. The communication channel has limited bandwidth and hence the transmitted signals over the network are subject to quantization. To deal with agent's output, an observer is implemented at each node. The state of the observer is quantized by a finite-level quantizer and then transmitted over the network. To solve the problem of quantizer overflow under malicious packet losses, a zooming-in and out dynamic quantization mechanism is designed. By the new quantized controller proposed in the paper, the zooming-out factor is lower bounded by the spectral radius of the agent's dynamic matrix. A sufficient condition of quantization range is provided under which the finite-level quantizer is free of overflow. A sufficient condition of tolerable DoS attacks for achieving consensus is also provided. At last, we study scalar dynamical agents as a special case and further tighten the zooming-out factor to a value smaller than the agent's dynamic parameter. Under such a zooming-out factor, it is possible to recover the level of tolerable DoS attacks to that of unquantized consensus, and the quantizer is free of overflow

Novel global asymptotic stability criteria for delayed cellular neural networks

This brief provides improved conditions for the existence of a unique equilibrium point and its global asymptotic stability of cellular neural networks with time delay. Both delay-dependent and delay-independent conditions are obtained by using more general Lyapunov-Krasovskii functionals. These conditions are expressed in terms of linear matrix inequalities, which can be checked easily by recently developed standard algorithms. Examples are provided to demonstrate the reduced conservatism of the proposed criteria by numerically comparing with those reported recently in the literature. © 2005 IEEE.published_or_final_versio

Robust H∞ filtering for uncertain 2-D continuous systems

This paper considers the problem of robust H∞ filtering for uncertain two-dimensional (2-D) continuous systems described by the Roesser state-space model. The parameter uncertainties are assumed to be norm-bounded in both the state and measurement equations. The purpose is the design of a 2-D continuous filter such that for all admissible uncertainties, the error system is asymptotically stable, and the H∞ norm of the transfer function, from the noise signal to the estimation error, is below a prespecified level. A sufficient condition for the existence of such filters is obtained in terms of a set of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H∞ filter is given. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed method. © 2005 IEEE.published_or_final_versio

On Two Factors Affecting the Efficiency of MILP Models in Automated Cryptanalyses

In recent years, mixed integer linear programming (MILP, in short) gradually becomes a popular tool of automated cryptanalyses in symmetric ciphers, which can be used to search differential characteristics and linear approximations with high probability/correlation. A key problem in the MILP method is how to build a proper model that can be solved efficiently in the MILP solvers like Gurobi or Cplex. It is known that a MILP problem is NP-hard, and the numbers of variables and inequalities are two important measures of its scale and time complexity. Whilst the solution space and the variables in many MILP models built for symmetric cryptanalyses are fixed without introducing dummy variables, the cardinality, i.e., the number of inequalities, is a main factor that might affect the runtime of MILP models. We notice that the norm of a MILP model, i.e., the maximal absolute value of all coefficients in its inequalities, is also an important factor affecting its runtime. In this work we will illustrate the effects of two parameters cardinality and norm of inequalities on the runtime of Gurobi by a large number of cryptanalysis experiments. Here we choose the popular MILP solver Gurobi and view it a black box, construct a large number of MILP models with different cardinalities or norms by means of differential analyses and impossible differential analyses for some classic block ciphers with SPN structure, and observe their runtimes in Gurobi. As a result, our experiments show that although minimizing the number of inequalities and the norm of coefficients might not always minimize the runtime, it is still a better choice in most situations

A Framework with Improved Heuristics to Optimize Low-Latency Implementations of Linear Layers

In recent years, lightweight cryptography has been a hot field in symmetric cryptography. One of the most crucial problems is to find low-latency implementations of linear layers. The current main heuristic search methods include the Boyar-Peralta (BP) algorithm with depth limit and the backward search. In this paper we firstly propose two improved BP algorithms with depth limit mainly by minimizing the Euclidean norm of the new distance vector instead of maximizing it in the tie-breaking process of the BP algorithm. They can significantly increase the potential for finding better results. Furthermore, we give a new framework that combines forward search with backward search to expand the search space of implementations, where the forward search is one of the two improved BP algorithms. In the new framework, we make a minor adjustment of the priority of rules in the backward search process to enable the exploration of a significantly larger search space. As results, we find better results for the most of matrices studied in previous works. For example, we find an implementation of AES MixColumns of depth 3 with 99 XOR gates, which represents a substantial reduction of 3 XOR gates compared to the existing record of 102 XOR gates