A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3. The star chromatic index χ'_st(G) of a graph G is the smallest integer k such that G has a star k-edge-coloring. The list star chromatic index ch'_st(G) is defined analogously. The star edge coloring problem is known to be NP-complete, and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear. In this paper, we study, in contrast, the best linear upper bound for sparse graph classes. We show that for every ε > 0 there exists a constant c(ε) such that if mad(G) < 8/3-ε, then ch'_st(G) ≤ 3Δ/2 +c(ε) and the coefficient 3/2 of Δ is the best possible. The proof applies a newly developed coloring extension method by assigning color sets with different sizes.

Given integer k and a k-graph F, let t_k-1(n, F) be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F -factor. For integers k ≥ 3 and 0 ≤ l ≤ k-1, let Y_k,l be a k-graph with two edges that shares exactly l vertices. Han and Zhao (J. Combin. Theory Ser. A, (2015)) asked the following question:For all k ≥ 3, 0 ≤ l ≤ k-1 and sufficiently large n divisible by 2k -l, determine the exact value of t_k-1(n, Y_k,l). In this paper, we show that t_k-1(n, Y_k,l)=(n/2k -l) for k ≥ 3 and 1 ≤ l ≤ k-2, combining with two previously known results of Rödl, Ruciński and Szemerédi (J. Combin. Theory Ser. A, (2009)) and Gao, Han and Zhao (Combinatorics, Probability and Computing, (2019)), the question of Han and Zhao is solved completely.

We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain. A new sufficient condition, uniformly positive reach is introduced. Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach, the H² regularity of the solution of the Poisson equation is established. In particular, this includes all star-shaped domains whose closures are of positive reach, regardless if they are Lipschitz domains or non-Lipschitz domains. Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.

The first and second Zagreb eccentricity indices of graph G are defined as:E₁(G)=∑_vi∈V(G) ε_G(v_i)², E₂(G)=∑_vivj∈E(G) ε_G(v_i)ε_G(v_j) where ε_G(v_i) denotes the eccentricity of vertex v_i in G. The eccentric complexity C_ec(G) of G is the number of different eccentricities of vertices in G. In this paper we present some results on the comparison between (E₁(G)/n) and (E₂(G)/m) for any connected graphs G of order n with m edges, including general graphs and the graphs with given C_ec. Moreover, a Nordhaus-Gaddum type result C_ec(G) + C_ec(G) is determined with extremal graphs at which the upper and lower bounds are attained respectively.

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper. Using Fountain Theorem, one multiplicity result of periodic solutions is obtained, which improves some previous results.

The purpose of this paper is to show that for one-sided symbolic systems, there exists an uncountable distributionally scrambled set contained in the set of proper positive upper Banach density recurrent points.

This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. Especially, state-dependent delay is introduced into a nonlocal reaction-diffusion model. The conditions of Schauder's fixed point theorem are proved by constructing a reasonable set of functions Γ (see Section 2) and a pair of upper-lower solutions, so the existence of traveling wavefronts is established. The present study is continuation of a previous work that highlights the Laplacian diffusion.

In this article, we introduce a robust sparse test statistic which is based on the maximum type statistic. Both the limiting null distribution of the test statistic and the power of the test are analysed. It is shown that the test is particularly powerful against sparse alternatives. Numerical studies are carried out to examine the numerical performance of the test and to compare it with other tests available in the literature. The numerical results show that the test proposed significantly outperforms those tests in a range of settings, especially for sparse alternatives.

We classify all Leibniz conformal algebras of rank two.

For stochastic reaction-diffusion equations with Lévy noises and non-Lipschitz reaction terms, we prove that W₁H transportation cost inequalities hold for their invariant probability measures and for their process-level laws on the path space with respect to the L¹-metric. The proofs are based on the Galerkin approximations.

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively. The sets of limit points of those Gaussian random fields are obtained. The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.

This paper is concerned with one-dimensional derivative quintic nonlinear Schrödinger equation,
iu_t-u_xx + i(|u|⁴u)_x=0, x ∈ T.
The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu[J. Math. Phys., ₅₃, 102702 (2012)].

Combining the dual least action principle with Mountain-pass lemma, we obtain the existence of brake orbits for first-order convex Hamiltonian systems with particular anisotropic growth.

As a special case of experimental design, locating array is useful for locating interaction faults in component-based systems. In this paper, bipartite locating array is proposed to locate interaction faults between two specific groups. Such arrays are especially suitable for antagonism tests. Based on the definition of bipartite locating array, the lower bound on run sizes are established to measure the optimality for specific parameters. When a single fault is to be located, optimal bipartite locating arrays are proved to be equivalent to certain specific combinatorial configurations. As a result, approaches to constructing optimal bipartite locating arrays are proposed and some infinite classes of optimal bipartite locating arrays are obtained using the corresponding construction method.

Any bounded tile of the field Q_p of p-adic numbers is a compact open set up to a set of zero Haar measure. In this note, we present two simple and direct proofs of this fact.

The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums, and give several interesting identities and asymptotic formulae for them.

In this paper we provide some relationships between Catalan's constant and the ₃F₂ and ₄F₃ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for ₃F₂, a second identity related to ₄F₃ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.

In the current article, we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again. We also show that a just-infinite C^*-algebra is quasidiagonal if and only if it is inner quasidiagonal. Finally, we compute the topological free entropy dimension in just-infinite C^*-algebras.

In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided.

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let (Mⁿ⁺¹, F) be a Randers-Minkowski space. If (Mⁿ, F) is a hypersurface of (Mⁿ⁺¹, F) with constant flag curvature K=1, then we can prove that M is Riemannian. (2) Let (Mⁿ⁺¹, F) be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of (Mⁿ⁺¹, F) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of[Proc. Amer. Math. Soc., 120, 1223-1229 (1994)] from the Riemannian case to the Randers space.

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

We study the existence of solutions for the following class of nonlinear Schrödinger equations -Δ_Nu + V (x)u=K(x)f(u) in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s) has exponential critical growth. The approaches used here are based on a version of the Trudinger-Moser inequality and a minimax theorem.

In this paper, we study complex symmetric C₀-semigroups on the Bergman space A²(C₊) of the right half-plane C₊. In contrast to the classical case, we prove that the only involutive composition operator on A²(C₊) is the identity operator, and the class of J-symmetric composition operators does not coincide with the class of normal composition operators. In addition, we divide semigroups {ψ_t} of linear fractional self-maps of C₊ into two classes. We show that the associated composition operator semigroup {T_t} is strongly continuous and identify its infinitesimal generator. As an application, we characterize J_σ-symmetric C₀-semigroups of composition operators on A²(C₊).

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation. Then by using this result we prove that the essential growth bound of the C₀-semigroup generated by the operator is 0, the C₀-semigroup is not compact, not eventually compact, even not quasi-compact. Moreover, we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution. In addition, we obtain the spectral radius and essential spectral radius of the C₀-semigroup. Lastly, we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.

In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of C₀-semigroups and C-regularized groups. We show that a diskcyclic C₀-semigroup exists on a complex topological vector space X if and only if dim(X)=1 or dim(X)=∞ and we prove that diskcyclicity and disk transitivity of C₀-semigroups (resp C-regularized groups) are equivalent.

In this paper, assuming there is a fiberwise Morse function, we extend Bismut-Lott index theorem to the complex valued case.

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.

Let f and g be functions in Fock-Sobolev space F^2,m. In this paper, we completely characterize the boundedness and compactness of H_f T_g.

In this paper we extend the Fibonacci-like maps to a wider class with the so-called "bounded combinatorics". The Fibonacci-like renormalization operator R is defined and we show that the orbit of each map from this class converges to a universal limit under iterates of R.

Let (M, g) be a compact Kähler manifold and (E, F) be a holomorphic Finsler vector bundle of rank r ≥ 2 over M. In this paper, we prove that there exists a Kähler metric φ defined on the projective bundle P (E) of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for φ having positive scalar curvature is obtained, and a sufficient condition for φ having positive Ricci curvature is established.

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.

The aim of this paper is to define an extension of the controllability and observability for linear quaternion-valued systems (QVS). Some criteria for controllability and observability are derived, and the minimum norm control and duality theorem are also investigated. Compared with real-valued or complex-valued linear systems, it is shown that the classical Caylay-Hamilton Theorem as well as Popov-Belevitch-Hautus (PBH) type controllability and observability test do not hold for linear QVS. Hence, a modified PBH type necessary condition is studied for the controllability and observability, respectively. Finally, some examples are given to illustrate the effectiveness of the obtained results.

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups, we finally obtain the structure and invariants of the group G.

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé-Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel-Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.

A collector samples coupons with replacement from a pool containing g uniform groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur (while coupons of different groups have different probabilities to occur). For each j=1,..., g, let T_j be the number of trials needed to detect Group j, namely to collect all M_j coupons belonging to it at least once. We first derive formulas for the probabilities P {T₁ < … < T_g} and P {T₁=Λ_j=1^g T_j}. After that, without severe loss of generality, we restrict ourselves to the case g=2 and compute the asymptotics of P {T₁ < T₂} as the number of coupons grows to infinity in a certain manner. Then, we focus on T:=T₁ ∨ T₂, i.e. the number of trials needed to collect all coupons of the pool (at least once), and determine the asymptotics of E[T] and V[T], as well as the limiting distribution of T (appropriately normalized) as the number of coupons becomes large.

In this article, the authors introduce the concept of shadowable points for set-valued dynamical systems, the pointwise version of the shadowing property, and prove that a set-valued dynamical system has the shadowing property iff every point in the phase space is shadowable; every chain transitive set-valued dynamical system has either the shadowing property or no shadowable points; and for a set-valued dynamical system there exists a shadowable point iff there exists a minimal shadowable point. In the end, it is proved that a set-valued dynamical system with the shadowing property is totally transitive iff it is mixing and iff it has the specification property.

A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stability α, α ∈ (0, 2]. It is shown that when the model is stationary, the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution, which avoids the difficulty of Roknossadati and Zarepour (2010) in deriving their limiting distribution for an M-estimate. On the contrary, we show that when the model is not stationary, the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour. Furthermore, a Wald test statistic is proposed to consider the test for a linear restriction on the parameter, and it is shown that under a local alternative, the Wald statistic has a non-central chisquared distribution. Simulations and a real data example are also reported to assess the performance of the proposed method.

Pil?niak and Wo?niak put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree Δ admits an NSD total (Δ+3)-coloring in 2015. In 2016, Qu et al. showed that the list version of the conjecture holds for any planar graph with Δ ≥ 13. In this paper, we prove that any planar graph with Δ ≥ 7 but without 6-cycles satisfies the list version of the conjecture.

In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in R₃ by using the idea of central projection transformation, which sets up a bridge connecting the vector field X(x) in R₃ with the planar vector fields. We have proved that the cubic vector field in R₃ can have two isolated closed orbits or one closed orbit on the invariant cone. As an application of this result, we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones, and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.

This is a corrigendum to our paper, "On a class of prime entire functions", published in Acta Math. Sin., Engl. Ser., 25, 1647-1652 (2009).