Some geometric properties of Riemann's non-differentiable function

Intermittency of Riemann's non-differentiable function through the fourth-order flatness

Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, a…

Ostrowski type Inequalities for m- and (alpha,m)-geometrically convex functions via Riemann-Louville Fractional integrals

In this paper, some new inequalities of Ostrowski type established for the class of m- and (alpha,m)-geometrically convex functions which are generalizations of geometric convex functions.

Generalization of different type integral inequalities for (α,m)-convex functions via fractional integrals

In this paper, a general integral identity for a twice differentiable functions is derived. By using of this identity, the author establishes some new Hermite-Hadamard type and Simpson type inequalities for differentiable (α,m)-convex functions via Riemann Liouville fractional integral.

A change of variables theorem for the multidimensional Riemann integral

The most general change of variables theorem for the Riemann integral of functions of a single variable has been published in 1961 (by Kestelman). In this theorem, the substitution is made by an `indefinite integral', that is, by a function of the form (t\mapsto c+\int_a^tg=:G(t)) where (g) is Riemann integrable on ([a,b]) and (c) is any constant. We prove a multidimensional generalization of this theorem for the case where (G) is injective -- using the fact that the Riemann primitives are the same as those Lipschitz functions which are almost everywhere strongly differentiable in ((a,b)). We …

Characterization of non-differentiable points in a function by Fractional derivative of Jumarrie type

There are many functions which are continuous everywhere but not differentiable at some points, like in physical systems of ECG, EEG plots, and cracks pattern and for several other phenomena. Using classical calculus those functions cannot be characterized-especially at the non-differentiable points. To characterize those functions the concept of Fractional Derivative is used. From the analysis it is established that though those functions are unreachable at the non-differentiable points, in classical sense but can be characterized using Fractional derivative. In this paper we demonstrate use …

无处单调的黎曼可积导数且无局部极值

Nowhere monotone Riemann integrable derivative without local extrema

本文构造了一个无处单调且无局部极值的黎曼可积导数，并构造了可微函数 $G$，使其导数 $G'$ 黎曼可积且具有非拐点的孤立局部极值。

A Generalization of the GGR Conjecture

For each positive integer $n$, function $f$, and point $c$, the GGR Theorem states that $f$ is $n$ times Peano differentiable at $c$ if and only if $f$ is $n-1$ times Peano differentiable at $c$ and the following $n$-th generalized Riemann~derivatives of $f$ at $c$ exist: \[ \lim_{h\rightarrow 0}\frac 1{h^{n}}\sum_{i=0}^n(-1)^i\binom{n}{i}f(c+(n-i-k)h), \] for $k=0,\ldots,n-1$. The theorem has been recently proved in [AC2] and has been a conjecture by Ghinchev, Guerragio, and Rocca since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new se…

A class of nowhere differentiable functions satisfying some concavity type estimate

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this fun…

Nowhere differentiable functions of analytic type on products of finitely connected planar domains

Using Laurent decomposition and Mergelyan's theorem combined with Baire's category theorem, we prove generic nowhere differentiability on the distinguished boundary of functions of analytic type on products of planar domains bounded by finitely many disjoint Jordan curves. The parametrization of the boundaries are those induced by the natural Riemann maps.

Local geometric proof of Riemann Hypothesis

Riemann function $ξ(s)=u+iv, s=β+1/2+it$ has the important symmetry: $v=0$ if $β=0$. For $β>0$ we prove $|u|>0$ inside any root-interval $I_j=[t_j,t_{j+1}]$ and $v$ has opposite signs at two end-points of $I_j$. They imply local peak-valley structure and $||ξ||=|u|+|v/β|>0$ in $I_j$. Because each $t$ must lie in some $I_j$, then $||ξ||>0$ is valid for any $t$. By the equivalence $Re(\frac{ξ'}ξ)>0$ of Lagarias(1999), we show that RH implies the peak-valley structure,which may be the geometric model expected by Bombieri(2000).