We use the large sieve inequality with sparse sets of moduli to prove a new estimate for exponential sums over primes. Subsequently, we apply this estimate to establish new results on the binary Goldbach problem where the primes are restricted to given arithmetic progressions.

We consider the oscillatory hyper Hilbert transform {{\displaystyle H}}_{\gamma ,\alpha ,\beta }f(x)={\displaystyle {\int }_{\mathrm{0}}^{\infty }f(x-\Gamma (t))}{e}^{it-\beta }{t}^{-(\mathrm{1}+\alpha )}\mathrm{d}t, where Γ(t) = (t, γ(t)) in {?}^{\mathrm{2}} is a general curve. When γ is convex, we give a simple condition on γ such that H_γ,_α,_β is bounded on L² when \beta \ge \mathrm{3}\alpha ,\beta >\mathrm{0}. As a corollary, under this condition, we obtain the L^p-boundedness of H_γ,_α,_β when . When Γ is a general nonconvex curve, we give some more complicated conditions on γ such that H_γ,_α,_β is bounded on L². As an application, we construct a class of strictly convex curves along which H_γ,_α,_β is bounded on L² only if β>2α>0.

Key distribution patterns (KDPs) are finite incidence structures satisfying a certain property which makes them widely used in minimizing the key storage and ensuring the security of communication between users in a large network. We construct a new KDP using t-design and combine two ω-KDPs to give new (ω−1)-KDPs, which provide secure communication in a large network and minimize the amount of key storage.

The anti-forcing number of a perfect matching M of a graph G is the minimal number of edges not in M whose removal makes M a unique perfect matching of the resulting graph. The anti-forcing spectrum of G is the set of anti-forcing numbers over all perfect matchings of G: In this paper, we prove that the anti-forcing spectrum of any cata-condensed hexagonal system is continuous, that is, it is a finite set of consecutive integers.

Let G = (V,A) be a digraph and k\ge \mathrm{1} an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V \ D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γk(G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G_B(n, d) and generalized Kautz digraphs G_K(n, d) are good candidates for interconnection networks. Denote {{\displaystyle \Delta }}_k:={({\displaystyle {\sum }_{j=\mathrm{0}}^k{d}^j})}^{-\mathrm{1}}. F. Tian and J. Xu showed that ?n{{\displaystyle \Delta }}_k?\le {{\displaystyle \gamma }}_k({{\displaystyle G}}_B(n,d))\le ?n/d^k? and ?n{{\displaystyle \Delta }}_k?\le {{\displaystyle \gamma }}_k({{\displaystyle G}}_K(n,d))\le ?n/d^k?. In this paper, we prove that every generalized de Bruijn digraph G_B(n, d) has the distance kdomination number ?n{{\displaystyle \Delta }}_k? or ?n{{\displaystyle \Delta }}_k?+\mathrm{1}, and the distance k-domination number of every generalized Kautz digraph G_K(n, d) bounded above by ?n/d^{k-\mathrm{1}}+d^k?. Additionally, we present various sufficient conditions for {{\displaystyle \gamma }}_k({{\displaystyle G}}_B(n,d))=?n{{\displaystyle \Delta }}_k? and {{\displaystyle \gamma }}_k({{\displaystyle G}}_K(n,d))=?n{{\displaystyle \Delta }}_k?.

We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.

We study the Cesàro means related to the divisor function. We show that the DDT Theorem holds over square-free numbers in short interval, which generalizes some results established by Deshouillers-Dress-Tenenbaum and by Cui-Wu.

We classify the family of pentavalent vertex-transitive graphs \Gamma with diameter 2. Suppose that the automorphism group of \Gamma is transitive on the set of ordered distance 2 vertex pairs. Then we show that either \Gamma is distance-transitive or \Gamma is one of \overline{{{\displaystyle C}}_{\mathrm{8}}},{{\displaystyle \mathrm{K}}}_{\mathrm{5}}{{\displaystyle \square \mathrm{K}}}_{\mathrm{2}},{{\displaystyle C}}_{\mathrm{5}}[{{\displaystyle \mathrm{K}}}_{\mathrm{2}}],\overline{{{\displaystyle \mathrm{2}C}}_{\mathrm{4}}},\mathrm{or}{{\displaystyle \mathrm{K}}}_{\mathrm{3}}{{\displaystyle \square \mathrm{K}}}_{\mathrm{4}} .

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

\{\begin{array}{l}{(-\Delta )}^{\alpha }{{\displaystyle u}}_i(x)={{\displaystyle f}}_i(u),\hspace{1em}x\in {?}^n,i=\mathrm{1},\mathrm{2},\mathrm{...},m,\\ u(x)=(u_{\mathrm{1}}(x),u_{\mathrm{2}}(x),\mathrm{...},u_m(x)),\end{array}

Where \mathrm{0}⟨\alpha ⟨\mathrm{1},n⟩\mathrm{2},{{\displaystyle f}}_i(u),1≤i≤m , are real-valued nonnegative functions of homogeneous degree p_i≥0 and nondecreasing with respect to the independent variables u₁, u₂, . . . , u_m. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x₀ if {{\displaystyle p}}_i=(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha ) for each 1≤i≤m; and the only nonnegative solution of this system is u ≡ 0 if \mathrm{1}⟨{{\displaystyle p}}_i⟨(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha ) for all 1≤i≤m.

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this paper, we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation whose solutions have nonzero backgrounds. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.

Let (M,F) be a Finsler manifold, and let TM₀ be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM₀,G) and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.

Let Gand Hbe two graphs. We say that G induces H if G has an induced subgraph isomorphic to H. A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T; there exists a function f_T; called binding function, depending only on T with the property that every graph G with chromatic number f_T(ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza conrmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer kand every tree T of radius r, every graph G with ω(G)≤k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14^r–1(r–1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r; every graph with ω(G)≤k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6^r–2) times.

Generalized constant ratio surfaces are defined by the property that the tangential component of the position vector is a principal direction on the surfaces. In this work, we study these class of surfaces in the 3-dimensional Minkowski space {\mathbb{L}}^{\mathrm{3}}. We achieve a complete classification of spacelike generalized constant ratio surfaces in {\mathbb{L}}^{\mathrm{3}}.

Two-grid mixed finite element method is proposed based on backward Euler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space two-grid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.

We consider a perturbed compound Poisson risk model with randomized dividend-decision times. Different from the classical barrier dividend strategy, the insurance company makes decision on whether or not paying off dividends at some discrete time points (called dividend-decision times). Assume that at each dividend-decision time, if the surplus is larger than a barrier b>0, the excess value will be paid off as dividends. Under such a dividend strategy, we study how to compute the moments of the total discounted dividend payments paid off before ruin.