Search results for: sierpinski%E2%80%99s%20triangle

Authors: Lingge Tan, Hongpeng Xu, Jieun Yang, Maarten Hornikx

Abstract:

Acoustic diffusers are important components in enhancing the quality of room acoustics. This paper provides a type of modular diffuser based on the Sierpinski Triangle of the plane and combines it with fractal theory to expand the effective frequency range. In numerical calculations and full-scale model experiments, the effect of fractal design elements on normal-incidence diffusion coefficients is examined. It is demonstrated the reasonable times of iteration of modules is three, and the coverage density is 58.4% in the design frequency from 125Hz to 4kHz.

Keywords: acoustic diffuser, fractal, Sierpinski-triangle, diffusion coefficient

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Authors: Abhilash Sahu, Amit Priyadarshi

Abstract:

In this paper, we will discuss the existence of weak solutions of the Kirchhoff type boundary value problem on the Sierpinski gasket. Where S denotes the Sierpinski gasket in R² and S₀ is the intrinsic boundary of the Sierpinski gasket. M: R → R is a positive function and h: S × R → R is a suitable function which is a part of our main equation. ∆p denotes the p-Laplacian, where p > 1. First of all, we will define a weak solution for our problem and then we will show the existence of at least two solutions for the above problem under suitable conditions. There is no well-known concept of a generalized derivative of a function on a fractal domain. Recently, the notion of differential operators such as the Laplacian and the p-Laplacian on fractal domains has been defined. We recall the result first then we will address the above problem. In view of literature, Laplacian and p-Laplacian equations are studied extensively on regular domains (open connected domains) in contrast to fractal domains. In fractal domains, people have studied Laplacian equations more than p-Laplacian probably because in that case, the corresponding function space is reflexive and many minimax theorems which work for regular domains is applicable there which is not the case for the p-Laplacian. This motivates us to study equations involving p-Laplacian on the Sierpinski gasket. Problems on fractal domains lead to nonlinear models such as reaction-diffusion equations on fractals, problems on elastic fractal media and fluid flow through fractal regions etc. We have studied the above p-Laplacian equations on the Sierpinski gasket using fibering map technique on the Nehari manifold. Many authors have studied the Laplacian and p-Laplacian equations on regular domains using this Nehari manifold technique. In general Euler functional associated with such a problem is Frechet or Gateaux differentiable. So, a critical point becomes a solution to the problem. Also, the function space they consider is reflexive and hence we can extract a weakly convergent subsequence from a bounded sequence. But in our case neither the Euler functional is differentiable nor the function space is known to be reflexive. Overcoming these issues we are still able to prove the existence of at least two solutions of the given equation.

Keywords: Euler functional, p-Laplacian, p-energy, Sierpinski gasket, weak solution

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Authors: Wajdi Mohamed Ratemi

Abstract:

The very well-known stacked sets of numbers referred to as Pascal’s triangle present the coefficients of the binomial expansion of the form (x+y)n. This paper presents an approach (the Staircase Horizontal Vertical, SHV-method) to the generalization of planar Pascal’s triangle for polynomial expansion of the form (x+y+z+w+r+⋯)n. The presented generalization of Pascal’s triangle is different from other generalizations of Pascal’s triangles given in the literature. The coefficients of the generalized Pascal’s triangles, presented in this work, are generated by inspection, using embedded Pascal’s triangles. The coefficients of I-variables expansion are generated by horizontally laying out the Pascal’s elements of (I-1) variables expansion, in a staircase manner, and multiplying them with the relevant columns of vertically laid out classical Pascal’s elements, hence avoiding factorial calculations for generating the coefficients of the polynomial expansion. Furthermore, the classical Pascal’s triangle has some pattern built into it regarding its odd and even numbers. Such pattern is known as the Sierpinski’s triangle. In this study, a presentation of Sierpinski-like patterns of the generalized Pascal’s triangles is given. Applications related to those coefficients of the binomial expansion (Pascal’s triangle), or polynomial expansion (generalized Pascal’s triangles) can be in areas of combinatorics, and probabilities.

Keywords: pascal’s triangle, generalized pascal’s triangle, polynomial expansion, sierpinski’s triangle, combinatorics, probabilities

Procedia PDF Downloads 332

Authors: I. Zahraoui, J. Terhzaz, A. Errkik, El. H. Abdelmounim, A. Tajmouati, L. Abdellaoui, N. Ababssi, M. Latrach

Abstract:

This paper presents a novel design of a microstrip fractal antenna based on the use of Sierpinski triangle shape, it’s designed and simulated by using FR4 substrate in the operating frequency bands (GPS, WiMAX), the design is a fractal antenna with a modified ground structure. The proposed antenna is simulated and validated by using CST Microwave Studio Software, the simulated results presents good performances in term of radiation pattern and matching input impedance.

Keywords: dual-band antenna, fractal antenna, GPS band, modified ground structure, sierpinski triangle, WiMAX band

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Authors: Sanjeet Kumar Singh, Shanatanu Bhattacharaya

Abstract:

Design of high- efficiency low, frequency (<1000Hz) soundproof window or wall absorber which is transparent to airflow is presented. Due to the massive rise in human population and modernization, environmental noise has significantly risen globally. Prolonged noise exposure can cause severe physiological and psychological symptoms like nausea, headaches, fatigue, and insomnia. There has been continuous growth in building construction and infrastructure like offices, bus stops, and airports due to urban population. Generally, a ventilated window is used for getting fresh air into the room, but at the same time, unwanted noise comes along. Researchers used traditional approaches like noise barrier mats in front of the window or designed the entire window using sound-absorbing materials. However, this solution is not aesthetically pleasing, and at the same time, it's heavy and not adequate for low-frequency noise shielding. To address this challenge, we design a transparent hexagonal panel based on Sierpiński fractal triangle, which is aesthetically pleasing, demonstrates normal incident sound absorption coefficient more than 0.96 around 700 Hz and transmission loss around 23 dB while maintaining e air circulation through triangular cutout. Next, we present a concept of fabrication of large acoustic panel for large-scale applications, which lead to suppressing the urban noise pollution.

Keywords: acoustic metamaterials, noise, functional materials, ventilated

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Authors: Sanjeet Kumar Singh, Shantanu Bhatacharya

Abstract:

Design of high-efficiency low, frequency (<1000Hz) soundproof window or wall absorber which is transparent to airflow is presented. Due to the massive rise in human population and modernization, environmental noise has significantly risen globally. Prolonged noise exposure can cause severe physiological and psychological symptoms like nausea, headaches, fatigue, and insomnia. There has been continuous growth in building construction and infrastructure like offices, bus stops, and airports due to the urban population. Generally, a ventilated window is used for getting fresh air into the room, but at the same time, unwanted noise comes along. Researchers used traditional approaches like noise barrier mats in front of the window or designed the entire window using sound-absorbing materials. However, this solution is not aesthetically pleasing, and at the same time, it's heavy and not adequate for low-frequency noise shielding. To address this challenge, we design a transparent hexagonal panel based on the Sierpiński fractal triangle, which is aesthetically pleasing and demonstrates a normal incident sound absorption coefficient of more than 0.96 around 700 Hz and transmission loss of around 23 dB while maintaining e air circulation through the triangular cutout. Next, we present a concept of fabrication of large acoustic panels for large-scale applications, which leads to suppressing urban noise pollution.

Keywords: acoustic metamaterials, ventilation, urban noise pollution, noise control

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Authors: Grzegorz Sierpiński, Ireneusz Celiński, Marcin Staniek

Abstract:

The article discusses multi modal mobility in contemporary societies as a main planning and organization issue in the functioning of administrative bodies, a problem which really exists in the space of contemporary cities in terms of shaping modern transport systems. The article presents classification of available resources and initiatives undertaken for developing multi modal mobility. Solutions can be divided into three groups of measures–physical measures in the form of changes of the transport network infrastructure, organizational ones (including transport policy) and information measures. The latter ones include in particular direct support for people travelling in the transport network by providing information about ways of using available means of transport. A special measure contributing to this end is a trip planner. The article compares several selected planners. It includes a short description of the Green Travelling Project, which aims at developing a planner supporting environmentally friendly solutions in terms of transport network operation. The article summarizes preliminary findings of the project.

Keywords: mobility, modal split, multimodal trip, multimodal platforms, sustainable transport

Procedia PDF Downloads 375