A mechanical model of Brownian motion for one massive particle including low energy light particles in dimension d ≥ 3

We provide a connection between Brownian motion and a classical Newton mechanical system in dimension d ≥ 3 {d\geq 3} . This paper is an extension of [S. Liang, A mechanical model of Brownian motion for one massive particle including slow light particles, J. Stat. Phys. 170 2018, 2, 286–350]. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random Newton mechanical principles, via interaction potentials, without any assumption requiring that the initial energies of the environmental particles should be restricted to be “high enough”. We prove that, as in the high-dimensional case, the position/velocity process of the massive particle converges to a diffusion process when the mass of the environmental particles converges to 0, while the density and the velocities of them go to infinity.

Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm

Quantum information processing enhances human’s power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams–Lloyd’s two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.

Communication-Efficient Modeling with Penalized Quantile Regression for Distributed Data

In order to deal with high-dimensional distributed data, this article develops a novel and communication-efficient approach for sparse and high-dimensional data with the penalized quantile regression. In each round, the proposed method only requires the master machine to deal with a sparse penalized quantile regression which could be realized fastly by proximal alternating direction method of multipliers (ADMM) algorithm and the other worker machines to compute the subgradient on local data. The advantage of the proximal ADMM algorithm is that it could make every parameter of iteration to have closed formula even in high-dimensional case, which greatly improves the speed of calculation. As for the communication efficiency, the proposed method does not sacrifice any statistical accuracy and provably improves the estimation error obtained by centralized method, provided the penalty levels are chosen properly. Moreover, the asymptotic properties of the proposed estimation and the convergence of the algorithm are convincible. Especially, it presents extensive experiments on both the numerical simulations and the HIV drug resistance data analysis, which all confirm the significant efficiency of our proposed method in quantile regression for distributed data by comparative and empirical analysis.

Covariance matrix filtering with bootstrapped hierarchies

Cleaning covariance matrices is a highly non-trivial problem, yet of central importance in the statistical inference of dependence between objects. We propose here a probabilistic hierarchical clustering method, named Bootstrapped Average Hierarchical Clustering (BAHC), that is particularly effective in the high-dimensional case, i.e., when there are more objects than features. When applied to DNA microarray, our method yields distinct hierarchical structures that cannot be accounted for by usual hierarchical clustering. We then use global minimum-variance risk management to test our method and find that BAHC leads to significantly smaller realized risk compared to state-of-the-art linear and nonlinear filtering methods in the high-dimensional case. Spectral decomposition shows that BAHC better captures the persistence of the dependence structure between asset price returns in the calibration and the test periods.

An Inexact Projected Gradient Method for Sparsity-Constrained Quadratic Measurements Regression

In this paper, we employ the sparsity-constrained least squares method to reconstruct sparse signals from the noisy measurements in high-dimensional case, and derive the existence of the optimal solution under certain conditions. We propose an inexact sparse-projected gradient method for numerical computation and discuss its convergence. Moreover, we present numerical results to demonstrate the efficiency of the proposed method.

On the center-stable manifolds for some fractional differential equations of Caputo type

This paper is devoted to study the existence of center-stable manifolds for some planar fractional differential equations of Caputo type with relaxation factor. After giving some necessary estimation for Mittag–Leffler functions, some existence results for center-stable manifolds are established under the mild conditions by virtue of a suitable Lyapunov–Perron operator. Moreover, an explicit example is given to illustrate the above result. Finally, high-dimensional case is considered.

Three Solutions for a Singular Quasilinear Elliptic Problem

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.