Let R = (R,<,+, ·, . . . ) be a non-valuational weakly o-minimal real closed field, I a definable convex open subset of R and f : I → R a definable function. We prove that {x ∈ I : f ′(x) exists in R} is definable and f ′ is definable if f is differentiable.

We consider probability measures µ n , µ on a metric space X such that µ n weakly converges to µ. The following convergence lim n→∞ X f n (x)µ n (dx) = X f (x)µ(dx) is proved under some restrictions on real valued functions f n and f which are measurable, not necessarily continuous nor bounded.

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to bounded linear operator $\widehat {f} \mathcal {F}(M) {F}(N)$ their corresponding Lipschitz-free spaces. In this paper, we give necessary and sufficient condition for {f}$ compact terms of conditions on $f$ . This extends result by A. Jiménez-Vargas M. Villegas-Vallecillos the case non-separable...

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where F$ is the family subsets natural numbers containing arbitrarily long arithmetic progressions. prove several properties operators, we characterize those which are weakly mixing recurrent, show that there but not mixing.

exists. The function f is continuously differentiable when it is differentiable and f ′ is continuous. A k-times continuously differentiable function is C, and a continuous function is C. A V -valued function f is weakly C when for every λ ∈ V ∗ the scalar-valued function λ◦ f is C. This sense of weak differentiability of a function f does not refer to distributional derivatives, but to differe...

The aim of this paper is to introduce the concepts of fuzzy upper and fuzzy lower almost continuous, weakly continuous and almost weakly continuous multifunctions. Several characterizations and properties of these multifunctions along with their mutual relationships are established in $L$-fuzzy topological spaces

We utilize recent results of André and Gabber on the existence weakly functorial, integral perfectoid big Cohen–Macaulay (BCM) algebras to study singularities local rings in mixed characteristic. In particular, we introduce a characteristic BCM-variant rational/F-rational singularities, log terminal/F-regular multiplier/test ideals divisor pairs. prove number about these objects including restr...

In this paper, we introduce some new types of pairs of mappings (f , g) on G-metric spaces called G-weakly commuting of type Gf and G–R-weakly commuting of type Gf . We obtain also several common fixed point results by using the (E.A) property. © 2012 Elsevier Ltd. All rights reserved.

The communication complexity of F with unbounded error is the limit of the ε-error randomized complexity of F as ε → 1/2. Communication complexity with weakly bounded error is defined similarly but with an additive penalty term that depends on 1/2− ε . Explicit functions are known whose two-party communication complexity with unbounded error is exponentially smaller than with weakly bounded err...