A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.

3. Thinking continuously

Many topologists might choose to describe their subject as the study of continuity. There are continuous and discontinuous functions in our everyday routines. ‘Thinking continuously’ aims to provide a more rigorous sense of what continuity entails for real-valued functions of a real variable. It focuses on functions having a single numerical input and a single numerical output. The properties of continuous functions are considered and the boundedness theorem and intermediate value theorem are also explained.

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝn → X.

On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia

If R is a ring of subsets of a set ? and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily ? of R is called a Nikod?m set for ba (R) if each set {?? : ???} in ba (R) which is pointwise bounded on ? is norm-bounded in ba (R). If the whole ring R is a Nikod?m set, R is said to have property (N), which means that R satisfies the Nikod?m-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck?s property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikod?m sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N)?(wN) holds.

A boundedness theorem for nearby slopes of holonomic -modules

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic${\mathcal{D}}$-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections${\mathcal{E}}_{t}$parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the${\mathcal{E}}_{t}$. This generalizes the regularity of the Gauss–Manin connection proved by Griffiths, Katz and Deligne.

Bidual Spaces and Reflexivity of Real Normed Spaces

Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].