We use the quantile residual lifetime models to analyze the length-biased data that are often encountered in observational studies. Ignoring sampling bias may lead to substantial estimation bias and fallacious inference. We consider a conditional log-linear regression model on the residual lifetimes at a fixed time point under right-censored and length-biased data for both covariate-independent censoring and covariate-dependent censoring. Consistency and asymptotically normalities of the regression estimators are established. Simmulation studies are performed to assess finite sample properties of the regression parameter estimator. Finally, we analyze the Oscar real data by the proposed method.

It is well known that a domain R is a Prüfer domain if and only if every divisible module is FP-injective; if and only if every h-divisible module is FP-injective. In this paper, we introduce the concept of Gorenstein FP-injective modules, and show that a domain R is a Gorenstein Prüfer domain if and only if every divisible module is Gorenstein FP-injective; if and only if every h-divisible module is Gorenstein FPinjective.

By means of the way of real analysis and the weight functions, introducing some parameters and intermediate variables, a few equivalent statements of a Hilberttype integral inequality with the general nonhomogeneous kernel in the whole plane are obtained. The constant factor is proved to be the best possible. As applications, a few equivalent statements of a Hilbert-type integral inequality with the general homogeneous kernel in the whole plane are deduced. We also consider some particular cases, the operator expressions and a few examples.

This paper addresses a class of dilation-and-modulation (MD) systems in the space L²(R₊) of square integrable functions defined on the right half real line R₊. In practice, the time variable cannot be negative. L²(R₊) models the causal signal space, but it admits no wavelet and Gabor systems due to R₊ being not a group under addition. We study the dilation-and-modulation systems in L²(R₊) generated by characteristic functions. We introduce the notion of MD-frame sets in R₊. Using "dilation-equivalence" and "cardinality function" methods we characterize MD-Bessel and complete sets; obtain two sufficient conditions for MD-Riesz basis sets; and prove that an arbitrary finite and measurable decomposition of an MD-Riesz basis set leads to an MD-frame set.

We study the optimal investment and optimal reinsurance problem for an insurer under the criterion of mean-variance. The insurer's risk process is modeled by a compound Poisson process and the insurer can invest in a risk-free asset and a risky asset whose price follows a jump-diffusion process. In addition, the insurer can purchase new business (such as reinsurance). The controls (investment and reinsurance strategies) are constrained to take nonnegative values due to nonnegative new business and no-shorting constraint of the risky asset. We control the risk by the new Basel regulation and use the stochastic linear-quadratic (LQ) control theory to derive the optimal value and the optimal strategy. The corresponding Hamilton-Jacobi-Bellman (HJB) equation no longer has a classical solution. With the framework of viscosity solution, we give a new verification theorem, and then the efficient strategy (optimal investment strategy and optimal reinsurance strategy) and the efficient frontier are derived explicitly.

Let X be a subcategory of an abelian category A. We proceed by generalizing the homotopy resolutions of complexes to the relative version, which is important basis making the relative derived category operational. We prove that every bounded above complex has a dg X resolution. Furthermore, we also show that the existence of resolutions for any unbounded complex when A=R-Mod and X is a particular subcategory. Finally, we establish a colocalization sequence of the homotopy category K(A) involving the relative derived category D_X (A) under some condition.

Let X^H={X^Ht, t ∈ R₊} be a subfractional Brownian motion in R^d. We establish sharp Hölder conditions and tail probability estimates for the local times of X^H in one-parameter case. We also give the existence and the L²-representation for the local time of X^H in multi-parameter case.

We study geometric and algebraic transience for discrete-time Markov chains on countable state spaces. Criteria are presented based on the moment of the last exit time for some state and the existence of solution for some equation. Moreover, we apply the results to investigating the stochastic stability of Geom/G/1 queueing models.

We propose a novel nonparametric estimator of the quantile difference based on the length-biased data subject to potential right censoring. In order to improve efficiency, the new estimator incorporates the auxiliary information inherent in the prevalent sampling design. And it has a simple expression, which is easy to compute. Moreover, the consistency and asymptotic normality of this quantile difference estimator are established using the empirical process theory and the asymptotic variance can be obtained consistently via a resampling method. We also demonstrate that the proposed estimator exhibits satisfactory performance with finite sample size through the Monte-Carlo studies and an analysis of a real data example about the Alzheimer's disease.

We studied the structure of 3-Lie algebras with involutive derivations, and proved that if A is an m-dimensional 3-Lie algebra with an involutive derivation, then there exists a compatible 3-pre-Lie algebra and a local cocycle 3-Lie bialgebraic structure on the 2m-dimensional semi-direct product 3-Lie algebra A_ad^* A^*. By means of involutive derivations, we constructed the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra A_ad^* A^*, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras.

Case-cohort design is a well-known cost-effective design and has been widely used in survival analysis. Many statistical methods have been developed to estimate the covariates effects on the survival time based on case-cohort data. However, little work has focused on checking the proportional hazards model assumptions with case-cohort data. In this article, we propose a class of test statistics through the asymptotically mean-zero processes for testing the proportional hazards assumption with case-cohort data. Re-sampling scheme is proposed to approximate the asymptotic distribution of the test statistics. Simulation studies are conducted to evaluate the finite sample performances of the proposed method and a data set from the National Wilm's Tumor Study Group is analyzed to illustrate the proposed method.

We prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space. Our results include the relevant conclusions of predecessors as special cases, and refine B. Biane's work as a corollary.

Let p₁, p₂, p₃ be diverse odd primes, and c > 1 be integer. We obtain all nonnegative integer solutions(x, y, z) on the Pell equations x²-(c²-1)y²=y²-2p₁p₂p₃z²=1. It generalizes the previous work of Keskin (2017) and Cipu (2018).

Let f, g be two nonconstant meromorphic functions, let a be a nonzero finite complex number, and let n ≥ 5 be a positive integer. If[f(z)]ⁿ and[g(z)]ⁿ share a CM, f(z) and g(z) share ∞ CM, and N₁₎(r, f)=S(r, f), then either f(z) ≡ tg(z), where tⁿ=1, or f(z)g(z) ≡ t, where tⁿ=a². This improves some unicity results concerning derivatives and differences of meromorphic functions.

It is well known that adding "long edges (shortcuts)" to a regularly constructed graph will make the resulted model a small world. Recently,[Internet Mathematics, DOI:10.1080/15427951, 2015.101208] indicated that, among all long edges, those edges with length proportional to the diameter of the regularly constructed graph may play the key role. In this paper, we modify the original Newman-Watts small world by adding only long special edges to the d (d ≥ 1)-dimensional lattice torus (with size n^d) according to[Internet Mathematics, DOI:10.1080/15427951, 2015.101208], and show that both the diameter of the modified model and the mixing time of random walk on it grow polynomially fast in log n.

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non identical exponentiated Pareto (EP) distributed random variables with doubly Type-II censored scheme. Different point estimations and interval estimations are proposed. The point estimators obtained are uniformly minimum variance unbiased estimators (UMVUE) and maximum likelihood estimators (MLE). The interval estimations obtained are the exact, approximate and bootstrap confidence intervals. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.

We study the following fractional Schrödinger-Kirchhoff equations with sign-changing potential function:

where s ∈ (0,1), p ∈[2, ∞), q ∈ (1, p), a, b > 0, λ, μ > 0 are positive constants, and by some appropriate assumptions on V, f, g, we use the fountain theorem to obtain the existence of infinitely many high energy solutions for the above system.

Let U=Tri(A, M, B) be a triangular algebra, and {φ_n}_n∈N:U→U be a sequence of linear maps. In this paper, we prove that if {φ_n}_n∈N satisfies φ_n([U, V]_ξ)=Σ_i+j=n φ_i(U)φ_j(V)-ξφ_i(V)φ_j(U) for any U, V ∈ U with U?V=P being the standard idempotent, then {φ_n}_n∈N is a higher derivation, where φ₀=id is the identity map, U?V=UV+VU is the Jordan product and[U, V]_ξ=UV-ξVU is the ξ-Lie product.

We discuss the fundamental solution for m-th powers of the sub-Laplacian on the quaternionic Heisenberg group, This result is the extension of the conclusion on the Heisenberg group. We use the representation theory of nilpotent Lie groups of step two to analyze the associated m-th powers of the sub-Laplacian on the quaternionic Heisenberg group and to construct its fundamental solution.

The representation category of a dihedral group is a symmetric semisimple monoidal category, so the Grothendieck ring of such a category is a commutative ring generated by finitely many elements. In this paper, the minimal generators of the Grothendieck ring are determined. Moreover, it is shown that the Grothendieck ring is isomorphic to a quotient of a polynomial ring.

p-adic hypergeometric functions are hypergeometric functions over finite fields analogous to the classical Gaussian hypergeometric functions, which have been found applications in diverse number theory problems. Let F_q be the finite field of q elements, λ ∈ F_q and n be a positive integer. This paper investigates the F_q-rational points on the Dwork hypersurface X_λⁿ:x₁ⁿ+x₂ⁿ+…+x_nⁿ=nλx₁x₂…x_n as well as its generalized form, and provides the formula for the number of the F_q-rational points in terms of a p-adic hypergeometric function when n and q(q-1) are coprime, which revises and improves the results given by Barman and Goodson et al.

The main purpose of this paper is using the properties of the trigonometric sums and the number of the congruence equation to study the computational problem of the one kind fourth power mean involving the four-term exponential sums, and give two interesting computational formulae for it.

In this paper, we perform a further investigation for a finite trigonometric sum considered by Hardy and Littlewood. By making use of some properties for the Chebyshev polynomials and M¨ obius function, we establish an interesting identity for the finite trigonometric sum of Hardy and Littlewood, by virtue of which an explicit asymptotic formula is also derived.

In this note, a structure of the Coleman automorphism group of a finite solvable group is given by using the projection limit property of the group. As an application, it is proved that the Coleman outer automorphism group of a dihedral group is either 1 or an elementary abelian 2-group.

The purpose of this article is to study representations of δ-BiHom-Jordan-Lie superalgebras. In particular, adjoint representations, trivial representations, deformations of δ-Bihom-Jordan-Lie superalgebras are studied in detail. Derivations of δ-BiHom-Jordan-Lie algebras are also discussed as an application.

We obtain new reverse Bonnesen-type inequalities for a surface of constant curvature by estimating the containment measure in integral geometry. These inequalities are generalizations and improvements of known Bottema inequality and new reverse Bonnesen-type inequalities in the Euclidean plane.

Let A be a factor von Neumann algebra. We prove that each nonlinear mixed ξ-Jordan triple derivable map φ:A → A is an additive *-derivation and φ(ξA)=ξφ(A) for all A ∈ A with ξ≠ 0, -1

In this paper, we generalize Vogt theorem in G-n-normed spaces:A mapping between two G-n-normed spaces which preserves ρ-gauge distance is affine.

This paper is concerned with the Cauchy problem of the non-resistive magnetohydrodynamics equations in R^d for d=2, 3. The local well-posedness in Sobolev space H^s-1×H^s for s > d/2 is obtained by establishing a commutator estimate.

Denote by p_n(A₁, …, A_n) the (n-1)-commutator of A₁, …, A_n. Assume that M is a von Neumann algebra, n ≥ 2 is any positive integer and L:M → M is a mapping. It is shown that, if M has no central summands of type I₁ and L satisfies L(p_n(A₁, …, A_n))=∑_k=1ⁿ p_n(A₁, …, A_k-1, L(A_k), A_k+1, …, A_n) for all A₁, A₂, …, A_n ∈ M with A₁A₂=0, then L(A)=φ(A) + f(A) for all A ∈ M, where φ:M → M and f:M → Z (M) (the center of M) are two mappings such that the restriction to P_iMP_j of φ is an additive derivation and f(p_n(A₁, A₂, …, A_n))=0 for all A₁, A₂, …, A_n ∈ P_iMP_j with A₁A₂=0 (1 ≤ i, j ≤ 2), P₁ ∈ M is a core-free projection and P₂=I -P₁; if M is a factor and n ≥ 3, then L satisfies L(p_n(A₁, A₂, …, A_n))=∑_k=1ⁿ p_n(A₁, …, A_k-1, L(A_k), A_k+1, …, A_n) for all A₁, A₂, …, A_n ∈ M with A₁A₂A₁=0 if and only if L(A)=φ(A) + h(A)I for all A ∈ M, where φ is an additive derivation on M and h is a functional of M such that h(p_n(A₁, A₂, …, A_n))=0 for all A₁, A₂, …, A_n ∈ M with A₁A₂A₁=0.

We dedicate to study the compactness of commutators for fractional integral operators, Marcinkiewicz integrals and pseudo-differential operators with smooth symbols on the generalized Morrey spaces M_p,ω(Rⁿ). Notice the differences of dealing method respectively.

Let T be the singular integral operator with variable kernel and D^γ (0 ≤ γ ≤ 1) be the fractional differentiation operator. Denote T^* and T^# be the adjoint of T and the pseudo-adjoint of T respectively. In this paper, via the expansion of the spherical harmonical polynomials, the boundedness on ?^q,λ_ω (Rⁿ) is shown to hold for TD^γ-D^γT and (T^*-T^#)D^γ. Meanwhile, the authors also establish various weighted norm inequalities for the product T₁T₂ and the pseudo-product T₁°T₂.

In this paper, we discuss the computational problem of a hybrid power mean involving character sums of polynomials and two-term cubic exponential sums, by using analytic methods and the properties of two-term exponential sums and Dirichlet characters. Meanwhile, we obtain a sharp asymptotic formula.

The Lie algebra W (2, 2) is one kind of infinite-dimensional Lie algebras, which plays a key role in classification of vertex operator algebras generated by weight 2 vectors. Hom-Lie algebras are algebras with an algebra structure and a Lie algebra structure, both of which satisfy the Leibniz rule. This paper mainly determine all Hom-Lie structures on the Lie algebra W (2, 2). It is the main result that all Hom-Lie algebra structures are trivial on the Lie algebra W (2, 2), which will be helpful to the further researches on the Lie algebra W (2, 2).

In 2006, Schuster introduced the concept of radial Blaschke-Minkowski homomorphisms. Afterwards, Wang et al. extended this notion to L_p radial Blaschke-Minkowski homomorphisms. In this paper, we establish some inequalities of L_p dual geominimal surface areas for L_p radial Blaschke-Minkowski homomorphisms, including the Brunn-Minkowski type and monotonic inequalities. And we give an affirmative form and a negative form of Busemann-Petty problem for L_p radial Blaschke-Minkowski homomorphisms.

We establish the global well-posedness and analyticity of mild solution to the generalized three-dimensional incompressible Navier–Stokes equations for rotating fluids if the initial data are in Fourier–Herz spaces ?_q^1-2α (R³) under appropriate conditions for α and q. As corollaries, we also give the corresponding conclusions of the generalized Navier–Stokes equation.

By the use of the weight functions, the transfer formula and the technique of real analysis, an extended multidimensional half-discrete Hardy–Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given, which is an extension of a published result. Moreover, the equivalent forms, a few particular cases and the operator expressions with some examples are considered.

Let (X, d, μ) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let M_β,ρ,q be the fractional type Marcinkiewicz integral operator on (X, d, μ). In this paper, for β ∈ [0, ∞), ρ ∈ (0, ∞) and q ∈ (1, ∞), under the assumption that M_β,ρ,q is bounded on L²(μ), the authors prove that M_β,ρ,q is bounded from the weighted Lebesgue space L^p(w) into the weighted weak Lebesgue space L^{p, ∞}(w) and from the weighted Morrey space L^p,κ,η(ω) into the weighted weak Morrey space WL^p,κ,η(ω).

We introduce a new dimension for isometric linear actions of countable sofic groups on complex Banach spaces. This generalizes the Voiculescu dimension for isometric linear actions of countable amenable groups on complex Banach spaces, and answers a question of Gromov in the case of countable sofic groups.

How to find out the integral basis of the number field effectively is a problem that people have been thinking for a long time. This paper gives a simple method to find out the integral basis of the cubic field. In addition, people are interested in the existence of power integral basis in number field. There is a power integral basis in both the quadratic field and the cyclotomic field, but it is not clear for the cubic field. In this paper, we give the necessary and sufficient conditions for the existence of power integral basis in the cubic field, and give a complete answer for the case of the cubic field.