O. Geil, F. Özbudak, and D. Ruano give a construction of a sequence of length (q - 1)(q² - 1) with high nonlinear complexity by using a function on a Hermitian curve over F_q² with the pole divisor (q - 1)P_∞ + (q - 1)Q, where P_∞ is the point at infinity, Q is a rational point with the order of its orbit is q² - 1 and q is a prime power. They give lower bounds on the k-th ordered nonlinear complexities N^k(s) and L^k(s) on the Hermitian curve. In this work, we examine the lower bounds on N^k(s) and L^k(s) using all possible pairs of rational points that can be selected on a Hermitian curve over F_q². In particular, we improve the bounds on N^k(s) and L^k(s) obtained by O. Geil, F. Özbudak, and D. Ruano.

A Cayley map 𝓜 = CM(G, X, p) is t-balanced if p(x)^-1 = p^t(x^-1) for all x ∈ X. Conder et al. classified the regular anti-balanced Cayley maps on abelian groups and Kwak et al. classified the regular t-balanced Cayley maps on dihedral groups and dicyclic groups. Classifications of regular t-balanced Cayley maps on semi-dihedral groups and cyclic groups were done by Oh and the first author, respectively. In this paper, we classify the regular t-balanced Cayley maps on ℤ₂-extensions of a cyclic 2-group.

For the right-angled Artin group action on the extension graph, it is known that the minimal asymptotic translation length is bounded above by 2 provided that the defining graph has diameter at least 3. In this paper, we show that the same result holds without any assumption. This is done by exploring some graph theoretic properties of biconnected graphs, i.e., connected graphs whose complement is also connected.

In this paper, we obtain a result that a U-statistic U_n based on associated random variables and a kernel k(x₁, . . . , x_m) which is a bounded variance function can be written as the difference of two strong demimartingales. Moreover, we establish Chow type minimal inequalities for strong demimartingales and obtain Marshall type minimal inequalities as well as a Value-at-Risk bound for (min_1≤k≤n S_k).

Let R be a commutative ring with nonzero identity. An R-module M is said to be 𝜙-P-flat if, for any s ∈ R ∖ Nil(R) and any x ∈ M such that sx = 0, we have x ∈ (0 : s)M. An R-module M is said to be nonnil-P-injective if, for any a ∈ R ∖ Nil(R), every homomorphism from Ra to M extends to a homomorphism from R to M. Then R is said to be a nonnil-P-coherent ring (resp., 𝜙-PF, nonnil-PP-ring) if, for any a ∈ R ∖ Nil(R), Ra is a finitely presented (resp., flat, projective) module. In this paper, we study nonnil-P-coherent rings, 𝜙-PF-rings, and nonnil-PP-rings using 𝜙-P-flat and nonnil-P-injective modules.

In this paper, the notions of iw-split modules and iw-split dimension are introduced, and some equivalent characterizations of these notions are given. With the help of iw-split modules and iw-split dimensions, new characterizations of DW rings, semi-simple rings, and Dedekind domains are given. More precisely, it is shown that R is a DW ring if and only if every iw-split module is an injective module; while R is a semi-simple ring if and only if every R-module is an iw-split module; and R is a Dedekind domain if and only if every factor module of an iw-split module is iw-split.

In this paper, we define a generalized integral transform 𝓣_α on function space. We then establish some fundamental formulas for the integral transform 𝓣_α involving the convolution product and the first variation. Furthermore, we establish some relationships among the integral transform, the convolution product and the first variation. Finally, we state an application with some examples.

A cyclic complementary extension of a finite group A is a finite group G which contains A as a subgroup and contains a cyclic subgroup C such that A∩C={1_G} and G = AC. A skew morphism of a finite group A is a permutation 𝜑 on A such that 𝜑(1_A) = 1_A and 𝜑(xy)=𝜑(x)𝜑^π(x)(y) for all x, y ∈ A, where π : A → ℤ_|𝜑| is a power function associated with 𝜑. For a positive multiple n of the order |𝜑| of 𝜑, if there exists an extended power function Π : A → ℤ_n of 𝜑, then 𝜑 and Π can be used to construct a cyclic complementary extension of A. In this paper, we use this approach to construct and classify the cyclic complementary extensions of the generalized quaternion groups corresponding to automorphisms.

Montel type theorem for CR mappings between strongly pseudoconvex psuedo-Hermitian CR manifolds satisfying curvature bound conditions is proved. Pre-compactness relies on recently introduced Schwarz lemma for CR mappings between pseudo-Hermitian manifolds. Smoothness of the limit CR map is proved using CR and holomorphic extensions of CR functions to higher dimensional q-concave CR manifolds.

In this paper we revisit the Cauchy problems for the DiracKlein-Gordon system (DKG) with pseudoscalar source and resonant interaction. In the case of scalar source Candy and Herr [4-6] showed the linear scattering for small data or large data with initial data of partial smallness associated with charge conjugation. We apply their arguments to the partial smallness problem of (DKG) with pseudoscalar source and resonant interaction. We get a linear scattering result by exploiting null structure induced by chiral operator and angular regularity.

Let h and k be positive integers such that h ≤ k. Let A = {a₀, a₁, . . . , a_k-1} be a nonempty finite set of k integers. The h-fold sumset, denoted by hA, is a set of integers that can be expressed as a sum of h elements (not necessarily distinct) of A. The restricted h-fold sumset, denoted by h^A, is a set of integers that can be expressed as a sum of h distinct elements of A. The characterization of the underlying set for small deviation from the minimum size of the sumset is called an extended inverse problem. Freiman studied such a problem and proved a theorem for 2A, which is known as Freiman's 3k - 4 theorem. Very recently, Tang and Xing, and Mohan and Pandey studied some more extended inverse problems for the sumset hA, where h ≥ 2. In this article, we prove some extended inverse theorems for sumsets 2^A, 3^A and 4^A. In particular, we classify the set(s) A for which |2^A| = 2k - 2, |2^A| = 2k - 1, and |2^A| = 2k. Furthermore, we also classify set(s) A when |3^A| = 3k - 7, |3^A| = 3k - 6, and |4^A| = 4k - 14.

This paper is devoted to studying the long-time behavior for the three-dimensional stochastic globally modified Navier-Stokes equations driven by a linear multiplicative white noise on some unbounded domains 𝓞. By using the Ornstein-Uhlenbeck process, we first transfer the original equation to a random dynamical system, and then prove the existence of pullback attractors as well as the upper semicontinuity of the attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, the asymptotic compactness of the solutions is proved by Ball's idea of energy equations. The periodicity of the attractors is also obtained when the deterministic non-autonomous external terms are periodic in time. Our results extend and generalize some existing results.

In this paper, we investigate the rigidity problem of compact submanifolds with parallel mean curvature in a locally symmetric Riemannian manifold. Firstly, we prove a Yau type rigidity theorem for compact hypersurfaces with parallel mean curvature in a locally symmetric Riemannian manifold. Then we extend the theorem to the case of higher codimension.

In this paper, we investigate the n-D Tropical climate model. We prove two Liouville-type theorems of smooth solutions to stationary tropical climate model under the suitable integrability assumption. We also establish a energy conservation criteria in terms of the 𝜔, ∇𝜐, ∇𝜃 for global weak solutions to evolutionary inviscid tropical climate model.

In this paper, we prove that a ring R is n-coherent and self-strongly FP_n-injective if and only if (${\mathcal{SFI}}_n$, ${\mathcal{SFI}}^{\bot}_n$) is a (perfect) cotorsion pair, if and only if every R-module has an epimorphic ${\mathcal{SFI}}_n$-cover, where ${\mathcal{SFI}}_n$ denotes the class of strongly FP_n-injective modules. In particular, we show that R is a coherent and self-strongly FP-injective ring, if and only if every R-module has an epimorphic ${\mathcal{SFI}}$-cover, which gives an affirmative answer to the question raised by Li-Guan-Ouyang.