Judul: Masalah residu kuadratik
Abstrak: Masalah residu kuadratik memuat pertanyaan apakah a adalah residu kuadratik modulo n atau bukan jika diberikan nlebih dari 3 integer ganjil dan a2x suatu integr. Jika ada solusi yang memenuhi X kuadrat = a(mod n), maka a adalah residu kuadratik modulo n. Hal yang sama juga berlaku untuk sebaliknya.
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Judul: Kuadrat Di Z/mz
Abstrak: Misal a, m E Z, (a, m) = I, m > 3. Masalah menentukan apakah kongruensi x2= a(mod m) terselesaikan (memiliki solusi) atau tidak adalah masalah masalah klasik yang cukup terkenal dalam teori bilangan. Masalah ini dapat diselesaikan dengan apa yang disebut hukum kebalikan kuadrat (quadratic reciprocity law) yang mengatakan bahwa kongruensi x2=p(mod q) dan x2 = q(mod p) keduanya terselesaikan atau keduanya tidak terselesaikan jika dan hanya jika satu dari p atau q kongruen 1 modulo 4. Inti utama dari tulisan ini adalah membuktikan hukum kebalikan kuadrat (HKK) dan menggunakannya untuk menentukan keterselesaian dari kongruensi x2 = a(mod m). Hingga saat ini telah diketahui 193 bukti yang berbeda untuk HKK dan sebahagian besar dari bukti-bukti itu menggunakan apa yang disebut dengan Lemma Gauss. Dalam tulisan, HKK juga dibuktikan dengan menggunakan Lemma Gauss; perbedaannya terletak pada metode-metode yang digunakan untuk membuktikan Lemma Gauss itu sendiri.
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Judul: Thermal Characteristics and Dynamics of Tropical Cyclone over Western North Pacific Ocean: A Case Study of Typhoon Songda (Chedeng) 2011
Abstrak: The frequent occasion of Tropical Cyclone (TC) over Western North Pacific (WNP) Ocean lies from 10˚N to 26˚N and 121˚E to 170˚E. Sources of TC development come from warm ocean temperature where latent heat is released during condensation processes and subsequently concentrated in the boundary layer. Relative humidity at 850-700 hPa is responsible to produce the most intense storm. Meanwhile, weak vertical wind shear between 850-200 hPa is one of the dynamical factors that lead to the development of TC. Pressure level of 850-200 hPa is chosen to minimize the thermodynamic effect in the lower troposphere. Typhoon Songda (Chedeng in the Philippines) or TSC lasted from May 18th to 30th, 2011 and strengthened into powerful Super Typhoon stage between May 24th and 27th. Long curving track and the exhibition of all sequences of TC life cycle, TSC is then considered as an ideal case of TC. The objective of the study with case of TSC is to investigate the role of thermal characteristics and dynamics on TC intensity. The research is carried out by using Advanced Research of Weather Research and Forecasting model (WRF ARW model) version 3.3. Two domains are employed in the model. The first domain situate from 02˚N to 40˚N and 118˚E to 149˚E and cover spatial resolution () of 33.1 km. The second domain situate from 09˚N to 27˚N and 119˚E to 132˚E with of 11 km. Timestep of the model () is 120 seconds. The model is supported by the following physical parameterization, i.e., Yonsei University Scheme (YSU) in the boundary layer, Kain-Fritsch scheme (KF) in the cumulus, and WRF Single-Moment 3-class scheme in the microphysics. Initial meteorological condition prior to the formation of the cyclone is generated using WRF ARW standard-release model. Sensitivity of the model output is obtained from a selected single-parameter change scenarios, namely, 10 % to 20 % decrease and increase of wind shear initial data at 850-200 hPa, and by 10 % to 20 % decrease of relative humidity initial data at 850-700 hPa. The result showed that the negative correlations between wind shear and intensity are found in wind shear scenarios. Wind shears of 10 ms-1 which lead to strengthening of the wind intensity in the early period of Typhoon development are found in wind shear scenarios. The effect of wind shear changes on wind intensity is statistically significant in the 20 % wind shear scenarios. During mature stage, weak wind intensity occurred when wind shear increase as given in the WRF standard-release and the 20 % decreasing scenario. Different situation is found in the 20 % increasing scenario where wind shear tend to weak and lead to strong wind intensity at this stage. Meanwhile, the reduction of moisture supply at 850-700 hPa resulting from 20 % decreasing relative humidity play a major role in the weakening of wind intensity and reduces mechanical energy of the cycle process as much as 300 J kg-1.
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