On Mixed Pressure-Velocity Regularity Criteria to the Navier-stokes Equations in Lorentz Spaces,Part Ⅱ:The Non-slip Boundary Value Problem

This paper is a continuation of the authors recent work [Beir?o da Veiga, H.and Yang, J., On mixed pressure-velocity regularity criteria to the Navier-Stokes equations in Lorentz spaces, Chin. Ann. Math., 42(1), 2021, 1–16], in which mixed pressure-velocity criteria in Lorentz spaces for Leray-Hopf weak solutions of the three-dimensional NavierStokes equations, in the whole space R~3 and in the periodic torus T~3, are established. The purpose of the present work is to extend the result of mentio...     

$$\pi $$π-Formulas and Gray code

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula

$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$

related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).

     

On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.

     

Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$

where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).

     

Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models

We study the problem of recovering an unknown signal \({\varvec{x}}\) given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator \(\hat{\varvec{x}}^\mathrm{L}\) and a spectral estimator \(\hat{\varvec{x}}^\mathrm{s}\). The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\). At the heart of our analysis is the exact characterization of the empirical joint distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\), given the limiting distribution of the signal \({\varvec{x}}\). When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form \(\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}\) and we derive the optimal combination coefficient. In order to establish the limiting distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\), we design and analyze an approximate message passing algorithm whose iterates give \(\hat{\varvec{x}}^\mathrm{L}\) and approach \(\hat{\varvec{x}}^\mathrm{s}\). Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.