A Remark on the Weak Topology and the Degree

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More on the Strength of Engeler’s Lemma

We introduce semilattices equipped with a partial n-ary operation ρ. A useful separation lemma for such nontrivial complete partial ρ-semilattices is proved equivalent to PIT, the prime ideal theorem. The relation of various versions of the Lemma to each other and to PIT is also explored. Supported by the Ministry of Education of the Czech Republic under the project MSM0021622409.     

Strong Geodesic α-Preinvexity and Invariant α-Monotonicity on Riemannian Manifolds

In this article, we introduce and study a new class of generalized convex functions on Riemannian manifold, called strongly α-invex and strongly geodesic α-preinvex functions. Several kinds of invariant α-monotonicities on Riemannian manifold are introduced. We establish the relationships among the strong α-invexity, strong geodesic α-preinvexity and invariant α-monotonicities under suitable conditions. Various types of α-invexities for functions on Riemannian manifolds are introduced and relations among them are established.     

Weak Hopf Algebra in Yetter-Drinfeld Categories and Weak Biproducts

The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.     

Weak π-rings

In this paper we establish some characterizations for weak π-rings.     

Calderón-Zygmund-Type Operators on Weighted Weak Hardy Spaces over ℝn

We introduce certain Calderón-Zygmund-type operators and discuss their boundedness on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. The sharpness of some results is also investigated. Received December 1, 1998, Revised May 2, 1999, Accepted May 19, 1999     

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.     

Self-similar Solutions of the Navier–Stokes Equations on Weak Weighted Lorentz Spaces

In the present paper, we prove the existence of global solutions for the Navier–Stokes equations in Rnwhen the initial velocity belongs to the weighted weak Lorentz space Λn,∞(u) with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree-1. Also the uniqueness is discussed.     

More on stability of almost surjective ε-isometries of Banach spaces

Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε = 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the following sharp stability inequality of f:There is a surjective linear operator T:Y→X of norm one so that ||T f(x)-x||= 2ε,for all x∈X.As its application,we prove the following statements are equivalent for a standard ε-isometry f:X→Y:(i)lim inf_(t→∞) dist(ty,f(X))/|t|1/2,for all y∈S_Y;(ii)τ(f)≡sup_(y∈S_Y) lim inf_(t→∞) dist(ty,f(X))/|t|=0;(iii)there is a surjective linear isometry U:X→Y so that || f(x)-Ux||= 2ε,for all x∈X.This gives an affirmative answer to a question proposed by Vestfrid(2004,2015).     

Boundedness of Multiplication Operators on α-Bloch Spaces

In this note we study and give a full characterization of the boundeness of mul-tiplication operators on α-Bloch spaces.     

Deterministic Quadrature Formulas for SDEs Based on Simplified Weak Itô–Taylor Steps

We study the problem of approximating the expected value \({\mathbb E}f(X(1))\) of a function f of the solution X of a d-dimensional system of stochastic differential equations (SDE) at time point 1 based on finitely many evaluations of the coefficients of the SDE, the integrand f and their derivatives. We present a deterministic algorithm, which produces a quadrature rule by iteratively applying a Markov transition based on the distribution of a simplified weak Itô–Taylor step together with strategies to reduce the diameter and the size of the support of a discrete measure. We essentially assume that the coefficients of the SDE are s-times continuously differentiable and the diffusion coefficient satisfies a uniform non-degeneracy condition and that the integrand f is r-times continuously differentiable. In the case \(r \le (\lfloor s/2 \rfloor - 1) \cdot 2d/(d + 2)\), we almost achieve an error of order \(\min (r, s)/d\) in terms of the computational cost, which is optimal in a worst-case sense.     

More on an Erdős–Szekeres-Type Problem for Interior Points

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k≥1, let g(k) be the smallest integer such that every planar point set in general position with at least g(k) interior points has a convex subset of points with exactly k interior points of P. In this article, we prove that g(3)=9.     

More on the Laterally σ-Complete Reflection of an Archimedean Lattice-Ordered Group

The laterally -complete reflection is given a new, one-step construction, which, in some notable cases affords an explicit realization. For example, if X is a compact Hausdorff topological space, the laterally -complete reflection of C(X) consists of all real-valued functions on X which are countably Baire-piecewise continuous. This realization of the reflection also shows, easily, that it carries rings to rings and, for objects with a designated unit, that the singularity of the unit is preserved.     

More on generalized inverses of partitioned matrices with Banachiewicz–Schur forms

Necessary and sufficient conditions are derived for a 2-by-2 partitioned matrix to have {1}-, {1,2}-, {1,3}-, {1,4}-inverses and the Moore–Penrose inverse with Banachiewicz–Schur forms. As applications, the Banachiewicz–Schur forms of {1}-, {1,2}-, {1,3}-, {1,4}-inverses and the Moore–Penrose inverse of a 2-by-2 partitioned Hermitian matrix are also given.     

Weak Wiener–Ditkin sets

A weak form of Wiener–Ditkin set is introduced and some of its properties are studied. As a consequence, a weak form of C-set-S-set problem has come up.     

Boundedness of fractional integral operators on α-modulation spaces

In this paper, we obtain the boundedness of the fractional integral operators, the bilinear fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.     

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(?), using “local smoothing” estimates. L ²(?) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in $L^{2}(\mathbb{T})In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L²(\mathbbT)L^{2}(\mathbb{T}). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L ²(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in L²(\mathbbT)L^{2}(\mathbb{T}).     

Weak type inequality for the maximal operator of the (C, α) means of two-dimensionalWalsh-Fourier series

The main aim of this paper is to prove that the maximal operator σ _* ^α of the (C, α) means of the cubical partial sums of the two-dimensional Walsh-Fourier series is bounded from the Hardy space H _2/(2+α) to the space weak-L _2/(2+α).     

Boundedness for pseudodifferential operators on multivariate α-modulation spaces

The α-modulation spaces M ^s,α _p,q (R ^d ), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x,D) with symbol in the Hörmander class S ^b _ρ,0 extends to a bounded operator σ(x,D):M ^s,α _p,q (R ^d )→M ^s-b,α _p,q (R ^d ) provided 0≤α≤ρ≤1, and 1<p,q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S ^b _1,0 maps the Besov space B ^s _p,q (R ^d ) into B ^s-b _p,q (R ^d ).