THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

EFFECT OF β ON EFFECTIVE MULTIPLICATION FACTOR IN 1/fβ SPECTRUM RANDOM SYSTEM

We investigated the effect of β on effective multiplication factor(keff) in the 1/fβ spectrum random system. The random system was generated by the 1/fβ noise model. The model is a continuous space model based on the Randomized Weierstrass function and describes the component spatial distribution with a power spectrum of 1/fβ, where f and β are the frequency domain variable and the characteristic parameter related to randomness, respectively. In this work, the two-group Monte Carlo calculations were performed to obtain the keff for a simple cubic geometry that consisted of two materials (fuel burned to 12 GWd/t and concrete). A large number of replicas having different spatial distribution and characterized by the representative β values were generated using the model, and the distribution on keff was analyzed. We found the dependency on β of standard deviation, skewness, and kurtosis of keff distribution. This result is expected to help to predict the keff distribution due to the randomizing model.

Elliptic Curve Cryptography based Secure Data Communication and Enhance sensor Reliability in Wireless Sensor Network

Wireless Sensor Network (WSN) extends the advantages of small price, quick employment, and shared transaction medium, although it induces a lot of security and secrecy challenges. In this paper, the Elliptic Curve Cryptography based Secure Data Communication and Enhance sensor Reliability (SDER) in WSN. In this scheme, an Elliptic Curve Cryptography (ECC) Weierstrass function is used to verify the sensor reliability, and ECC cryptography technique is useful for providing the data security in the network. The simulation result demonstrates that the SDER reduces both the packet loss rate and the network delay.

On fractal properties of Weierstrass-type functions

In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {\mathcal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left(2\, \pi\,N_b^n\,x \right)$, where $\lambda$ and $N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and $ \lambda\,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.

Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function

In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by\[{\mathcal W}(x)= \sum_{n=0}^{+\infty} \lambda^n\,\cos \left ( 2\, \pi\,N_b^n\,x \right),\]where $\lambda$ and $N_b$ are two real numbers such that $0 <\lambda<1$, $N_b\,\in\,\N$ and $\lambda\,N_b >1$, using a sequence a graphs that approximate the studied one.