Judul: Penentuan Sebaran Akuifer Dengan Metode Tahanan Jenis (Resistivity Method) Di Kota Tangerang Selatan, Provinsi Banten
Abstrak: Water is a basic requirement for humans being. The population growth can result a higher water demand. Surface water quality is declining due to human activities that make the usage of groundwater increased. The excessive exploration of groundwater can result soil subsidence, it is necessary to study the characteristics of groundwater. This study aimed to identify the lithology of soil, the position and thickness of aquifer, and to analyze aquifer distribution at the research area. The resistivity method was used to to identivity the lithology of soil. Analysis results showed that the depth of unconfined aquifer at the research area was 3,00-44,73 m below soil surface with a thickness of 2-12 m. Litology of soil was a clay and tufaan sand. The depth of confined aquifer was 80-130 m below soil surface with a thickness of more than 75 m. Groundwater flow patterns at the research location headed toward the north.
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Judul: Prediksi Potensi Cadangan Airtanah Menggunakan Persamaan Darcy di Kota Tangerang Selatan, Provinsi Banten
Abstrak: Water is very important for human beings. Sources of freshwater that can be used is only 3% of the total water availability on earth. One of fresh water’s sources that can be used is groundwater. The objective of this study was to determine the capacity of groundwater storages (discharge) in aquifers at South Tangerang city using the Darcy equation. The method of research was include several stages, such as the preparation phase, data collection, data processing, data analysis, and presentation of results. Data were obtained from geoelectric measurements using Schlumberger software, and were used to analyse the type and depth of rock layers. Beside, the data were also used to determined the position of confined and unconfined aquifer. The flownet of confined and unconfined aquifer indicated the scheme of groundwater’s flow and was used to decide the cross section of the flow. Darcy equation required another parameter called hydraulic conductivity. In this research, the aquifer was dominated by sand, so the hydraulic conductivity for sand (12 m/day) was used in calculation. As the result groundwater storages on unconfined aquifer was 3551,50 m3/day, while on confined aquifer was 16864.11 m3/day.
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Judul: The Construction of Greedy SVP LLL Algorithm
Abstrak: A lattice is a set of all integer linear combination of a set of linearly independent vectors in ℝ𝑛���. The independent vectors are called bases of lattice. Any lattice can be generated from many bases, and these bases have the same cardinality. In the lattice, the most fundamental and renowned problem is the Shortest Vector Problem (SVP). SVP is a tracking problem of the shortest nonzero vector in a lattice with equivalent bases. In two dimensions, SVP problem has resolved exactly by Gauss’ algorithm. When the lattice dimension is higher than two, A.K. Lenstra, H.W. Lenstra and L. Lovasz gave a LLL algorithm to compute approximation of nonzero shortest vector in the lattice. Reduced bases solution obtained from LLL algorithm still be an approximation and has polynomial running time of arbitrary dimension which large enough. In 1994, Schnoor and Euchner modified this LLL algorithm as a new variant which later was named deep insertion LLL algorithm. This algorithm is a modified version of the LLL algorithm to the terms of the exchange step to improve the accuracy of output of LLL reduced bases and applying it to the subset sum problem. The purpose of this research are: (1) Reconstruct the LLL algorithm and deep insertion LLL algorithm using a geometric approach. (2) Construct a greedy SVP LLL algorithm. (3) Implement the three algorithms in a symbolic programming language, and evaluate them experimentally. A bases ℬ=[𝐛���1,𝐛���2,…,𝐛���𝑛���] in ℝ𝑚��� is called reduced LLL with parameter 𝛿��� if satisfies: (a) |𝜇���𝑗���𝑖���|≤12, for every integer 𝑖���,𝑗��� with 1≤𝑖���<𝑗���<𝑛���. (b)𝛿���‖𝜋���𝑗���(𝐛���𝑗���)‖2≤‖𝜋���𝑗���(𝐛���𝑗���+1)‖2, for 𝑗���=1,2,…,𝑛���−1,where 𝛿��� is a reduced parameter of real numbers with 14<𝛿���<1. The first requirement is the reduced basis 𝛿��� must “nearly orthogonal” and in its computation case, this requirement can be reached out by using the Gram-Schmidt’s orthogonalization. While the second requirement is called exchange step, or used to called as Lovasz condition. By using both of these requirements, then they establish the LLL algorithm. Then, Schnoor and Euchner modify the LLL algorithm based on the requirements of the exchange step called the deep insertion LLL algorithm, where the exchange step is based on the comparison of the projected vector in orthogonal complement [𝐛���1,𝐛���2,…,𝐛���𝑘���−1]⊥ after j-th reduction. To construct greedy SVP LLL algorithm, the requirements of the exchange step is based on purely by comparing norm of lattice 𝐛���𝑗��� with norm lattice 𝐛���𝑖��� for 𝑖���=1,2,3…,𝑗���−1. The process of insertion reduced vector after the exchange step also has done greedily. Then, we calculate the number of arithmetic operation in greedy SVP LLL algorithm for further analysis. The result of our exprimentation for three algorithms on the certain matrix size shows that the running time of the three algorithms increases propotional to the size of the matrix. The result also shows that by using 𝛿���=34 for the LLL algorithm and the deep insertion LLL algorithm, and the greedy SVP LLL algorithm which is a new variant made by using no parameter of 𝛿���, outperform of the other of two previous algorithm in terms of speed with the same output.
Keyword: LLL algorithm, LLL deep insertion algorithm, greedy SVP LLL algorithm