For n = 2 or 3 and x\in {ℝ}^n, we study the oscillatory hyper Hilbert transform{{\displaystyle T}}_{\alpha ,\beta }f(x)={{\displaystyle \int }}_{ℝ}f(x-\Gamma (t,x)){\mathrm{e}}^{{-i\left|t\right|}^{-\beta }}{\left|t\right|}^{-\mathrm{1}-\alpha }\mathrm{d}talong an appropriate variable curve \mathrm{\Gamma }(t,x) in {ℝ}^n (namely, \mathrm{\Gamma }(t,x) is a curve in {ℝ}^n for each fixed x), where \alpha >\beta >\mathrm{0}. We obtain some L^p boundedness theorems of {{\displaystyle T}}_{\alpha ,\beta }, under some suitable conditions on \alphaand \beta. These results are extensions of some earlier theorems. However, {{\displaystyle T}}_{\alpha ,\beta }f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.

We study properties of a relation in *-rings, called the core-EP (pre)order which was introduced by H. Wang on the set of all n × n complex matrices [Linear Algebra Appl., 2016, 508: 289–300] and has been recently generalized by Y. Gao, J. Chen, and Y. Ke to *-rings [Filomat, 2018, 32: 3073–3085]. We present new characterizations of the core-EP order in *-rings with identity and introduce the notions of the dual core-EP decomposition and the dual core-EP order in-rings.

Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, {\mathcal{H}}^{p(\cdot )}({ℝ}^n), into Lebesgue spaces with variable exponent, L^{p(\cdot )}({ℝ}^n), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on L^{p(\cdot )}({ℝ}^n), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) by a known idea.

Let λ₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ₁/λ₂, λ₁/λ₃ are irrational and algebraic. Then there are in.nitely many solutions in primes p_j, j =1, 2, 3, 4, to the inequality . This improves the earlier result.

Let M be a 2n-dimensional smooth manifold associated with the structure of symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Let Q⊂Mbe a codimension 2 compact submanifold. We show some sufficient and necessary conditions on the existence of the structure of contact pair (α,β) on Q,which is a pair of 1-forms of constant classes whose characteristic foliations are transverse and complementary such that α and β restrict to contact forms on the leaves of the characteristic foliations of βand α,respectively. This is a generalization of the neighborhood theorem for contact-type hypersurfaces in symplectic topology.

The p-th moment and almost sure stability with general decay rate of the exact solutions of neutral stochastic differential delayed equations with Markov switching are investigated under given conditions. Two examples are provided to support the conclusions.

Let M be a 2n-dimensional closed unitary manifold with a Tⁿ⁻¹-action with only isolated fixed points. In this paper, we first prove that the equivariant cobordism class of a unitary Tⁿ⁻¹-manifold M is just determined by the equivariant Chern numbers c_{\omega }^{T^{n-\mathrm{1}}}[M],where ω= (i₁, i₂, ..., i₆) are the multi-indexes for all i₁, i₂, ..., i₆∈\mathbb{N}. Then we show that if Mdoes not bound equivariantly, then the number of fixed points is greater than or equal to \lceil n/\mathrm{6}\rceil +\mathrm{1}, where \lceil n/\mathrm{6}\rceil denotes the minimum integer no less than n/6.

An explicit and recursive representation is presented for moments of the first hitting times of birth-death processes on trees. Based on that, the criteria on ergodicity, strong ergodicity, and l-ergodicity of the processes as well as a necessary condition for exponential ergodicity are obtained.