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A general theory of efficient estimation for ergodic diffusion processes sampled at high frequency with an infinite time horizon is presented. High frequency sampling is common in many applications, with finance as a prominent example. The theory is formulated in term of approximate martingale estimating functions and covers a large class of estimators including most of the previously proposed estimators for diffusion processes. Easily checked conditions ensuring that an estimating function is an approximate martingale are derived, and general conditions ensuring consistency and asymptotic normality of estimators are given. Most importantly, simple conditions are given that ensure rate optimality and efficiency. Rate optimal estimators of parameters in the diffusion coefficient converge faster than estimators of drift coefficient parameters because they take advantage of the information in the quadratic variation. The conditions facilitate the choice among the multitude of estimators that have been proposed for diffusion models. Optimal martingale estimating functions in the sense of Godambe and Heyde and their high frequency approximations are, under weak conditions, shown to satisfy the conditions for rate optimality and efficiency. This provides a natural feasible method of constructing explicit rate optimal and efficient estimating functions by solving a linear equation.

This paper investigates the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time $T$, considering the scenario where both the time horizon $T$ and the number of particles $N$ tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction function. Additionally, we conduct a thorough analysis of the transform of the associated invariant density as a complex function, providing essential insights for our main results.

It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility condition (inclusion of kernels) of the mixed scheme and on local constants related to compact embeddings, which are often known explicitly. Applications include scalar second-order elliptic operators, linear elasticity, and the Steklov eigenvalue problem.

The use of alternative operations in differential cryptanalysis, or alternative notions of differentials, are lately receiving increasing attention. Recently, Civino et al. managed to design a block cipher which is secure w.r.t. classical differential cryptanalysis performed using XOR-differentials, but weaker with respect to the attack based on an alternative difference operation acting on the first s-box of the block. We extend this result to parallel alternative operations, i.e. acting on each s-box of the block. First, we recall the mathematical framework needed to define and use such operations. After that, we perform some differential experiments against a toy cipher and compare the effectiveness of the attack w.r.t. the one that uses XOR-differentials.

We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying $Z_{q}$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by L\'evy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of L\'evy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.

Time-optimal control of a multi-rotor remains an open problem due to the under-actuation and nonlinearity of its dynamics, which make it difficult to solve this problem directly. In this paper, the time-optimal control problem of the multi-rotor is studied. Firstly, a thrust limit optimal decomposition method is proposed, which can reasonably decompose the limited thrust into three directions according to the current state and the target state. As a result, the thrust limit constraint is decomposed as a linear constraint. With the linear constraint and decoupled dynamics, a time-optimal guidance trajectory can be obtained. Then, a cost function is defined based on the time-optimal guidance trajectory, which has a quadratic form and can be used to evaluate the time-optimal performance of the system outputs. Finally, based on the cost function, the time-optimal control problem is reformulated as an MPC (Model Predictive Control) problem. The experimental results demonstrate the feasibility and validity of the proposed methods.

We consider the surface Stokes equation on a smooth closed hypersurface in three-dimensional space. For discretization of this problem a generalization of the surface finite element method (SFEM) of Dziuk-Elliott combined with a Hood-Taylor pair of finite element spaces has been used in the literature. We call this method Hood-Taylor-SFEM. This method uses a penalty technique to weakly satisfy the tangentiality constraint. In this paper we present a discretization error analysis of this method resulting in optimal discretization error bounds in an energy norm. We also address linear algebra aspects related to (pre)conditioning of the system matrix.

Far-field speech recognition is a challenging task that conventionally uses signal processing beamforming to attack noise and interference problem. But the performance has been found usually limited due to heavy reliance on environmental assumption. In this paper, we propose a unified multichannel far-field speech recognition system that combines the neural beamforming and transformer-based Listen, Spell, Attend (LAS) speech recognition system, which extends the end-to-end speech recognition system further to include speech enhancement. Such framework is then jointly trained to optimize the final objective of interest. Specifically, factored complex linear projection (fCLP) has been adopted to form the neural beamforming. Several pooling strategies to combine look directions are then compared in order to find the optimal approach. Moreover, information of the source direction is also integrated in the beamforming to explore the usefulness of source direction as a prior, which is usually available especially in multi-modality scenario. Experiments on different microphone array geometry are conducted to evaluate the robustness against spacing variance of microphone array. Large in-house databases are used to evaluate the effectiveness of the proposed framework and the proposed method achieve 19.26\% improvement when compared with a strong baseline.

Sequential optimization methods are often confronted with the curse of dimensionality in high-dimensional spaces. Current approaches under the Gaussian process framework are still burdened by the computational complexity of tracking Gaussian process posteriors and need to partition the optimization problem into small regions to ensure exploration or assume an underlying low-dimensional structure. With the idea of transiting the candidate points towards more promising positions, we propose a new method based on Markov Chain Monte Carlo to efficiently sample from an approximated posterior. We provide theoretical guarantees of its convergence in the Gaussian process Thompson sampling setting. We also show experimentally that both the Metropolis-Hastings and the Langevin Dynamics version of our algorithm outperform state-of-the-art methods in high-dimensional sequential optimization and reinforcement learning benchmarks.