In this paper we bound character sums of the shape \[ \sum_{n \leq N} \chi_1(\lfloor f(n) \rfloor) \chi_2(\lfloor f(n+l) \rfloor), \] where $\chi_1$ and $\chi_2$ are non-principal multiplicative characters modulo a prime $p$, $f(x)$ is a real-valued, twice-differentiable function satisfying a certain condition on $f''(x)$, and $l$ is a positive integer. As an immediate application, we obtain some distribution properties of consecutive quadratic residues and consecutive primitive roots in Piatetski–Shapiro sequences $\lfloor n^c \rfloor$ with $c \in (1,4/3)$.

Let $S(t,f) := \pi^{-1} \arg L(1/2+it,f)$, where $f$ is a holomorphic Hecke cusp form for $\operatorname{SL}_2(\mathbb{Z})$ of weight $k$. We establish an asymptotic formula for the moments of $S(t,f)$.

The well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we show a generalized fundamental identity for arbitrary Dedekind domains. As an application, we show that any Japanese Dedekind domain is excellent.

This paper is concerned with studying the relaxed gradient-based iterative method based on tensor format to solve the Sylvester tensor equation. From the information given by the previous steps, we further develop a modified relaxed gradient-based iterative method which converges faster than the method above. Under some suitable conditions, we prove that the introduced methods are convergent to the unique solution for any initial tensor. At last, we provide some numerical examples to show that our methods perform much better than the GI algorithm proposed by Chen and Lu (Math. Probl. Eng. 2013) both in the number of iteration steps and the elapsed CPU time.

In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Fréchet–Kolmogorov–Riesz–Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with $\operatorname{CMO}$ functions, which is quite different from the non-compactness properties of these dyadic operators. These results are similar to those for Calderón–Zygmund singular integral operators.

In this paper, we obtain the $L^p(\mathbb{R}^2)$-boundedness of the maximal functions \[ M_{I,\gamma} f(x_1,x_2) := \sup_{j \in \mathbb{Z}} \sup_{\varepsilon > 0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} |f(x_1-t, x_2 - 2^j \gamma(t))| \, \mathrm{d}t \] and \[ M_{II,\gamma} f(x_1,x_2) := \sup_{i,j \in \mathbb{Z}} \sup_{\varepsilon > 0} \frac{1}{2\varepsilon} \int_{-\varepsilon}^{\varepsilon} |f(x_1 - 2^i t, x_2 - 2^j \gamma(t))| \, \mathrm{d}t, \] where $p \in (1,\infty]$ and $\gamma$ is a convex curve satisfying some suitable curvature conditions.

This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Laplace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The proof mechanism can also be applied to tensor-valued martingales and tensor-based Azuma, Hoeffding and McDiarmid inequalities are established.

In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier–Stokes equations on the non-compact manifolds with negative Ricci curvature tensor. In our strategy, we combine the dispersive and smoothing estimates for Stokes semigroups and Massera-type theorem to establish the existence and uniqueness of the time periodic mild solution to Stokes equation on Riemannian manifolds. Then using fixed point arguments, we can pass to semilinear equations to obtain the existence and uniqueness of the periodic solution to the imcompressible Navier–Stokes equations under the action of a periodic external force. The stability of the solution is also proved by using the cone inequality.

In this paper, a class of infinite horizon optimal control problems is established, where the state equation is given by a stochastic delay evolution equation (SDEE), and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). Firstly, we extend the form of Itô formula. After that, we establish the priori estimate for the solution to ABSEEs, and then the existence and uniqueness results of ABSEEs on infinite horizon are obtained. Finally, we establish necessary and sufficient conditions of stochastic maximum principle for infinite horizon optimal control problem in the form of Pontryagin's maximum principle.