Characterizing Level-set Families of Harmonic Functions

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects

Differential geometric prolongations of solution equations

This thesis is a study in the field of partial differential equations on differentiable manifolds. In particular non-linear evolution equations with solution solutions are studied by means of differential geometric tools and methods. Differential geometric prolongation technique is applied to the A.K.N.S. system as a unifying system for known 2-dimension solutions. Solution properties are studied in this differential geometric set up. The results are used to obtain a possible model for n-dimensional solutions

Stellar Differential Rotation and Coronal Timescales

We investigate the timescales of evolution of stellar coronae in response to surface differential rotation and diffusion. To quantify this we study both the formation time and lifetime of a magnetic flux rope in a decaying bipolar active region. We apply a magnetic flux transport model to prescribe the evolution of the stellar photospheric field, and use this to drive the evolution of the coronal magnetic field via a magnetofrictional technique. Increasing the differential rotation (i.e. decreasing the equator-pole lap time) decreases the flux rope formation time. We find that the formation time is dependent upon the geometric mean of the lap time and the surface diffusion timescale. In contrast, the lifetime of flux ropes are proportional to the lap time. With this, flux ropes on stars with a differential rotation of more than eight times the solar value have a lifetime of less than two days. As a consequence, we propose that features such as solar-like quiescent prominences may not be easily observable on such stars, as the lifetimes of the flux ropes which host the cool plasma are very short. We conclude that such high differential rotation stars may have very dynamical coronae

Analisis dan Implementasi Geometric Differential Evolution Dalam Studi Kasus Travelling Salesman Problem

ABSTRAKSI: Travelling Salesman Problem adalah suatu permasalahan kombinatorial yang melibatkan beberapa titik yang disebut dengan node yang saling terhubung dan memiliki jarak berbeda antara satu node ke node lain. Kasus ini sering dijadikan sebagai sebuah benchmark terhadap performansi dari suatu algoritma. Tujuan dari penyelesaian kasus Travelling Salesman Problem adalah mencari jarak terpendek yang dibutuhkan untuk mengunjungi semua node tanpa harus mengunjungi sebuah node lebih dari satu kali. Geometric Differential Evolution merupakan algoritma pencarian solusi optimal yang bekerja berdasarkan pergerakan dari kandidat-kandidat solusi yang direpresentasikan dalam bentuk vektor. Geometric Differential Evolution menggunakan pendekatan yang sedikit berbeda dengan Differential Evolution. Pada Differential Evolution , digunakan differential mutation dan discrete recombination untuk menggerakan vektor kandidat solusi, sedangkan, pada Geometric Differential Evolution, digunakan convex combination dan extension ray. Pada tugas akhir ini, akan dianalisis performansi dari Geometric Differential Evolution dalam penyelesaian kasus Travelling Salesman Problem berdasarkan jarak terpendek yang diperoleh. Untuk mengetahui performansi maksimum dari Geometric Differential Evolution, akan dianalisis juga mengenai pengaturan parameter yang paling optimal untuk Geometric Differential Evolution.Kata Kunci : Geometric Differential Evolution, convex combination, extension ray, Travelling Salesman Problem.ABSTRACT: Travelling Salesman Problem is a combinatorial problem that involves several points, called by nodes, that are connected each other and have different distances between one node to another. Travelling Salesman Problme usually used as a benchmark for the performance of an algorithm. The goal of Travelling Salesman Problem optimization is to find the shortest disance to visit all of the nodes without visiting one nodes more than once. Geometric Differential Evolution is an algorithm that works on the movements of some vector that represent the solution candidates. Geometric Differential Evolution uses a slightly different approach to a conventional Differential Evolution. In Differential Evolution, the vectors moved by differential mutation and discrete recombination, whereas, in Geometric Differential Evolution, the vectors movement is controlled by mutation and recombination that uses convex combination and extension ray. In this research, the performance of Geometric Differential Evolution in solving Travelling Salesman Problem will be analysed by the minimum route that produced by system. To analyse the maximum performance of Geometric Differential Evolution, the best parameter settings will be analysed too.Keyword: geometric differential evolution, convex combination, extension ray, travelling salesman proble

Studies on gluon evolution and geometrical scaling in kinematic constrained unitarized BFKL equation: application to high-precision HERA DIS data

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution $f(x,k_T^2)$ towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions ($F_2$ and $F_L$) as well as double differential reduced cross section $(\sigma_r)$. The validity of our theory in the low $Q^2$ transition region is established by studying virtual photon-proton cross section in light of HERA data