A note on the Brück conjecture

In this paper, we firstly consider the Brück conjecture itself and show that it holds exactly for the entire function c(Ae^cz-a)+a{f(z)=\frac{1}{c}(Ae^{cz}-a)+a} , where A, a, c are nonzero constants. Then we give a necessary and sufficient condition that f(z) and f ′(z) share a finite value a CM for some special cases. Finally, we investigate two analogues of the Brück conjecture including the difference analogue of the Brück conjecture raised by Liu and Yang (Arch. Math. 92, 270–278 (2009)) and the shifted analogue of the Brück conjecture raised by Heittokangas et al. (J. Math. Anal. Appl. 355, 352–363 (2009)). And we give some necessary conditions when f(z) shares a finite value a CM with its difference operators or shifts.     

Three classes of Weitzenböck manifolds

A Weitzenböck manifold is a triplet defined by a differentiable manifold with a metric g of certain signature and a linear connection with zero curvature tensor and nonzero torsion tensor which is a metric connection with respect to g. The theory of such manifolds is called the “new theory of gravity”. We study properties of three classes of Weitzenböck manifolds and prove some vanishing thorems.     

Erratum to: A note on the Brück conjecture

In the article “A note on the Brück conjecture” (Arch. Math. 95 (2010), 257–268), the proofs of our results Theorem 1.4 and Theorem 1.6 mainly depend on Lemma 2.6, which is wrong. In this corrigendum, we give a correct proof of Theorem 1.4. Theorem 1.6 can be proved similarly; so we omit its proof.     

A note on the Schrödinger maximal function

It is shown that control of the Schrödinger maximal function sup_{0 <t<1} ?e^itΔf? for f ∈ H^s(Rⁿ) requires s ≥ n/2(n + 1).     

A note on the collapsing geometry of hyperkähler four manifolds

Science China Mathematics - We make some observations concerning the one-dimensional collapsing geometry of four-dimensional hyperkähler metrics.     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

A note on Hölder inequality

The well-known Hölder inequality is generalized and refined, a condition at which the equality holds is obtained.     

A note on “branch-and-price approach for the multi-skill project scheduling problem”

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721–1734, 2014) proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

A note on the illposedness for anisotropic nonlinear Schrödinger equation

In this short note, we show the illposedness of anisotropic Schroedinger equation in L^2 if the growth of nonlinearity is larger than a threshold power pc which is also the critical power for blowup, as Fibich, Ilan and Schochet have pointed out recently. The illposedness in anisotropic Sobolev space Hk,d-d^2s,s where 0 〈 s 〈 sc, sc =d/2-k/4-2/p-1, and the illposedness in Sobolev space of negative order H^s, s 〈 0 are also proved.     

A note on the Hamilton–Waterloo problem with C8-factors and Cm-factors

In this paper, we almost completely solve the Hamilton–Waterloo problem with $C_8$-factors and $C_m$-factors where the number of vertices is a multiple of $8m$.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

A note on the Ramanujan τ-function

Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that

A note on the α-quantile option

Some properties of a class of path-dependent options based on the α-quantiles of Brownian motion are discussed. In particular, it is shown that such options are well behaved in relation to standard options and comparatively cheaper than an equivalent class of lookback options.