Let G be a nilpotent group, and 1=ζ0G < ζ1G < … < ζcG=G be the upper central series of G. End(ζiG/ζi-1G) is an endomorphism ring of an abelian group ζiG/ζi-1G, naturally, End(ζiG/ζi-1G) is a Lie ring. Suppose that α1, α2,…, αn are automorphisms of G. Denote by α1i, α2i,…, αni the automorphisms induced by α1, α2,…, αn on ζiG/ζi-1G, then α1i, α2i,…, αni generate a Lie subring of End(ζiG/ζi-1G). If Lie ring generated by α1i, α2i,…, αni is completely solvable, then automorphism subgroup generated by α1, α2,…, αn has good nilpotence. In addition, we investigate analogous problems of the lower central series of G and obtain the similar arguments.
Let p ≥ 5, n ≥ 0. Then (i1i0)*(hn) ∈ ExtA1,pnq(H*K, Zp) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element in πpnq-1K. Based on this result, we consider the convergence of the product involving the third Greek letter family element and expand the filtration s + 1 of the nontrivial element in π*S. In other words, we prove that shn ∈ ExtAs+1,t(Zp, Zp) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element γsξn in πt-s-1S, for p + 1 < s + 1 < 2p, where p ≥ 7, n ≥ 3, t=pnq + sp2q + (s -1)pq + (s -2)q + s -3, q=2(p -1).
We acquire a sufficient condition to guarantee the nesting property of the dilated cubes associated with an expansive matrix A. Second, we obtain a covering theorem about the dilated cubes. Finally, we discuss the relationship between the dilated cubes and Christ-dyadic cubes associated with an expansive matrix A.
Let H and K be Hilbert spaces. For given T ∈ B(H, K), the set UT={U ∈ B(H, K):U is a partial isometry and T=U(T*T)1/2} is characterized, and then it is also given that a characterization of TU={T ∈ B(H, K):N (T)=N (U), R(T)=R(U), T=U(T*T) 2/1 } for some partial isometry U ∈ B(H, K). In addition, as applications of the main results, some related results are obtained.
We study the mean value distribution of the difference of an integer and its m-th power over unions of short intervals, and give some asymptotic formulas. For details, let p be an odd prime, 1 ≤ H ≤ p, 0 < δ ≤ 1 be any fixed real number, and m ≥ 2 be integers. Let I(j) be disjoint subintervals of (0, p), 1 ≤ j ≤ J, satisfying H/2 ≤ |I(j)| ≤ H, and let (y)p denote the non-negative least residue of y modulo p.Define I=Uj=1J I(j), and let χ be the Dirichlet character modulo p. We prove that
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Let μ=(μi)i ≥ 0 be a measure on Z+:={0, 1, 2,...}, and p > 1. Consider the following p-Dirichlet form Dp(f)=∑i=0∞ μibi(fi-fi+1)(fip-1-fi+1p-1), f ≥ 0, where (bi)i ≥ 0 is a positive sequence. The purpose of this paper is to obtain upper and lower bounds for the first eigenvalue of p-Dirichlet form Dp(f) λ0,p=inf{Dp(f):‖f‖p=1, f ≥ 0 and has compact support}, where ‖f‖p=(∑i=0∞μifip)1/p.
We construct a double-kernel estimator of conditional distribution function by the local linear approach for left-truncated and dependent data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators are also established. Finite-sample performance of the estimator is investigated via simulation, and is better than the general kernel estimation in bias and adaptation of edge effects.
We consider the optimal time of merger for two first-line insurers with investment and proportional reinsurance policies. The risk processes of the two insurers are modeled by drifted Brownian motions and the objective is to maximize the survival probability of the two insurers. The safety loadings and variation coefficients of insurers play an important role in deciding whether to merge. If they decide to merge, the cost of the merger and the relation of survival probabilities before and after merger play an important role in deciding the merging time. We divide the problem into two cases. In both cases, we can get the optimal strategies and the value functions finally.
We determine the Ptolemy constant when norm is a mean of two norms by means of their corresponding continuous convex functions. The new results which not only contain the previous results, but also give some new results in some concrete Banach spaces.
By using Gauss sums and invariant factors of Smith normal form of degree matrices, we give the estimates for the exponential sums of generalized diagonal polynomials over finite fields, which improves the Deligne-Weil's type estimates on such exponentials sums.
A cost-effective sampling design is desirable in large cohort studies due to the cost of measurement on expensive covariates. An outcome-dependent sampling (ODS) design is such a biased-sampling scheme which can improve efficiency by allowing researchers to oversample in the regions of most information. We study hypothesis testing in linear regression model under the ODS design. We propose a likelihood ratio statistic and a Wald statistic for testing the significance of the model, and a U statistic for testing the significance of regression parameter by applying a semiparametric empirical profile-likelihood method. We establish the asymptotic theory for the proposed test statistics. We conduct simulation studies to assess the finite-sample performance of the proposed tests. We illustrate the application of the proposed methods with a real data example.
Let M1 be a finite von Neumann algebra with a faithful normal trace τ1 and let M1o={a ∈ M,τ1(a)=0}. We prove that, if there is a sequence {uk:k ∈ M } of orthogonal unitaries in M1o, then for any finite von Neumann algebra M2(≠C) with a faithful normal trace τ2, the tracial free product (M1, τ1) * (M2, τ2) is a type Ⅱ1 factor. As a corollary, we obtain that, if there is a von Neumann subalgebra N of M1 such that N has no minimal projection, then for any finite von Neumann algebra M2(≠ C) with a faithful normal trace τ2, the tracial free product (M1, τ1) * (M2, τ2) is a type Ⅱ1 factor.
We first obtain the refined estimates of all homogeneous expansions for a subclass of starlike mappings of order α on the unit ball in complex Banach spaces, especially the estimates are all sharp if these mappings are k-fold symmetric mappings. We next establish the refined estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in Cn. In particular, the estimates are also all sharp if these mappings are k-fold symmetric mappings. It is shown that we have proved a weak version of the Bieberbach conjecture for starlike mappings of order α in the case of several complex variables, and the derived results reduce to the classical results in the case of one complex variable.
Let p > 0, s ≥ 0, q > max{-n-1, -s -1}. In this paper, the authors discuss an equivalent characterization and a decomposition of the F(p,q,s) space on the unit ball. The results as follows:(1) f ∈ F(p,q,s) if and only if f ∈ H(B) and Ip=supa∈B∫B|Rα,γf(z)|p(1 -|z|2)q+pγ-p(1 -|?a(z)|2)sdv(z) < ∞, where α > -1 and γ > max{0,1 -(q +s+1)/p,1 -(q + n+1)/p}. (2) If {dk} ∈ ∫p, then there exists sequence {wk} ⊂B such that f(z)=∑k=1∞(dk(1-|wk|2)t+1)/((1-<z,wk>)t+((q+n+1)/p)) (z ∈ B) in F(p,q,s), where t > max{1 -1/p,0}(q + n + 1) + max{1/p,1}s -1.
We concern the relations between the voriticity of the velocity field and the global well-posedness of the 3D incompressible magneto-micropolar fluid equations. We improve and extend the result due to Constantin and Fefferman to a quite complete incompressible fluid systems. Moreover, our results have inclusion relation to micropolar fluid and MHD equations.