RESEARCH ANNOUNCEMENTS On the Fractional Calculus of a Type of Weierstrass Functions

1 IntroductionIn recent years, fractals have shown important applications in many fields. [1,2] and [3] have done some excellent initial and conclusion work on fractal and it's mathematical foundations. However, a fractal function: a type of Weierstrass functions defined byW(t) = (?), 0 < a < 1, A > 1, (1)because of it's special fractal properties, [1, 2, 4, 5] have given some detailed discussion about it's graph, fractal dimension, etc.Fractional calculus attract more and more attention recently. In this field, [6, 7] have done some important and detailed work. The fundamental conception is     

RESEARCH ANNOUNCEMENTS On a Generalization of the c-numerical Range

Let Cn×n be the set of　n×n complex matrices and (?)n　the set of　orthonormal　n-tuples ofvectors in Cn.For a vector c in Cn　and a matrix A in Cn×n,the c-numerical range of　A is theset     

RESEARCH ANNOUNCEMENTS—On a Generalization of the c—numerical Range

Let Cn×n be the set of n × n complex matrices and An the set of orthonormal n-tuples of vectors in Cn. For a vector c in Cn and a matrix A in Cn×n, the c-numerical range of A is the set Wc(A)={n∑i=1 Ci(Axi,xi):(x1,…xn)∈∧n} When c = (1,0,…,0), Wc(A) is reduced to the classical numerical range W(A) (see [1]). For the classical numerical range and its generalizations, one may see the survey article[2].     

RESEARCH ANNOUNCEMENTS──On the Maps of Deleted Product

ＬｅｔＭａｎｄＮｂｅｓｉｍｐｌｙｃｏｎｎｅｃｔｅｄｃｌｏｓｅｄｍａｎｉｆｏｌｄｓ，ｆ：Ｍ→Ｎｂｅａｈｏｍｏｔｏｐｙｅｑｕｉｖａｌｅｎｃｅ．Ｗｅｄｅｆｉｎｅｄａｎｉｓｏｍｏｒｐｈｉｓｍｏｆｃｏｈｏｍｏｌｏｇｙｓｙｓｔｅｍｆ：Ｈ（Ｎ（ｋ）；Ｇ）→Ｈ...     

RESEARCH ANNOUNCEMENTS——Einstein—Kahler Metric on Cartan—Hartogs Domain of the First Type

S. Y. Cheng and S. T. Yau showed in [CY] that any C2 bounded pseudoconvex domain in Cnhas a complete Einstein-Kahler metric with constant negative Ricci curvature. N. Mok and S. T.Yau[MY] have extended this result to arbitrary bounded pseudoconvex domain in Cn. CompleteEinstein-Kahler metric with Explicit form, however, is only known in the case of homogeneousdomain.     

RESEARCH ANNOUNCEMENTS On the Uniqueness of Limit Cycle for a Generalized Liénard System

ＲＥＳＥＡＲＣＨＡＮＮＯＵＮＣＥＭＥＮＴＳＯｎｔｈｅＵｎｉｑｕｅｎｅｓｏｆＬｉｍｉｔＣｙｃｌｅｆｏｒａＧｅｎｅｒａｌｉｚｅｄＬｉéｎａｒｄＳｙｓｔｅｍＨｅＱｉｍｉｎ（何启敏）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＳｕｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｓｕｚｈｏｕ，...     

RESEARCH ANNOUNCEMENTS：Morrey Regularity of the Solution to the Diri

To study the local regularity of solutions to second orderelliptic partial differential equations, Morrey in ［1］ introduced somefunction spaces, which are called the Morrey spaces today. Since then, manymathematicians have studied regularities of solutions to some kinds of secondorder elliptic equations in Morrey spaces.     

On the Pseudo—Primensee of Meromorphic Functions of Infinite Order

In this paper the pseudo-primeness of meromorphic functions of infinite order is disscossed in detail and quite a few result are obtained,which ae improvments of that of Ozawa.     

RESEARCH ANNOUNCEMENTS On the Least Pair of Quadratic Non-Residues (mod p)

For each odd prime p,let a(p) denote the least positive residue class n(mod p),so that both n and n+1 are quadratic non-residues.The previously best knownestimate for a(p) obtained by Elliott is as follows     

RESEARCH ANNOUNCEMENTS Einstein-Khler Metric on Cartan-Hartogs Domain of the First Type

S. Y. Cheng and S. T. Yau showed in [CY] that any C2 bounded pseudoconvex domain in C?has a complete Einstein-Kahler metric with constant negative Ricci curvature. N. Mok and S. T. Yau[MY] have extended this result to arbitrary bounded pseudoconvex domain in Cn. Complete Einstein-Kahler metric with Explicit form, however, is only known in the case of homogeneous domain.     

分形插值函数的分数阶微积分的分形维数

证明了线性分形插值函数的Riemann-Liouville分数阶微积分仍然是线性分形插值函数.在基于线性分形插值函数有关讨论的基础上,证明了线性分形插值函数的Box维数与Riemann-.Liouville分数阶微积分的阶之间成立着线性关系.文中给出的例子的图像和数值结果更进一步说明了这个结论.     

A Geometric Based Connection between Fractional Calculus and Fractal Functions

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann–Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann–Liouville f...     

Hausdorff Dimension of a Class of Weierstrass Functions

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.     

Weierstrass Functions Separate All Traces

Mathematical Notes -     

On Fractional Smoothness of Modulus of Functions

We consider the Nemytskii operators u→|u| and u→u~±in a bounded domain ? with C~2 boundary. We give elementary proofs of the boundedness in H~s(?) with 0 ≤ s 3/2.     

The Weierstrass Transform for a Class of Generalized Functions

The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken.     

On Generalized Linearity of Quadratic Fractional Functions

Quadratic fractional functions are proved to be quasilinear if and only if they are pseudo-linear. For these classes of functions, some characterizations are provided by means of the inertia of the quadratic form and the behavior of the gradient of the function itself. The study is then developed showing that generalized linear quadratic fractional functions share a particular structure. Therefore it is possible to suggest a sort of “canonical form” for those functions. A wider class of functions given by the sum of a quadratic fractional function and a linear one is also studied. In this case generalized linearity is characterized by means of simple conditions. Finally, it is deepened on the role played by generalized linear quadratic fractional functions in optimization problems.     

On Generalized Linearity of Quadratic Fractional Functions

Quadratic fractional functions are proved to be quasilinear if and only if they are pseudo-linear. For these classes of functions, some characterizations are provided by means of the inertia of the quadratic form and the behavior of the gradient of the function itself. The study is then developed showing that generalized linear quadratic fractional functions share a particular structure. Therefore it is possible to suggest a sort of canonical form for those functions. A wider class of functions given by the sum of a quadratic fractional function and a linear one is also studied. In this case generalized linearity is characterized by means of simple conditions. Finally, it is deepened on the role played by generalized linear quadratic fractional functions in optimization problems.     

一类分形函数的Weyl-Marchaud分数阶导数的Hausdorff维数

首先介绍广义Weierstrass型函数的Weyl-Marchaud分数阶导数,得到了带随机相位的广义Weierstrass型函数的Weyl-Marchaud分数阶导数图像的Hausdorff维数,证明了该分形函数图像的Hausdorff维数与Weyl-Marchaud分数阶导数的阶之间的线性关系.