Continuous Data Assimilation with Stochastically Noisy Data

We analyze the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimension incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the the limit supremum, as the time tends to infinity, of the expected value of the $L^2$-norm and of the $H^1$ Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated

A connection between the Camassa-Holm equations and turbulent flows in channels and pipes

In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant $\alpha$ version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order $\alpha$ distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale $\alpha$ is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant $\alpha$ region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that $\alpha$ decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply $\alpha$ is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.Comment: tex file, 29 pages, 4 figures, Physics of Fluids (in press

Statistical Properties of Nonlinear Shell Models of Turbulence from Linear Advection Models: Rigorous Results

In a recent paper it was proposed that for some nonlinear shell models of turbulence one can construct a linear advection model for an auxiliary field such that the scaling exponents of all the structure functions of the linear and nonlinear fields coincide. The argument depended on an assumption of continuity of the solutions as a function of a parameter. The aim of this paper is to provide a rigorous proof for the validity of the assumption. In addition we clarify here when the swap of a nonlinear model by a linear one will not work.Comment: 7 pages, 4 figures, submitted to Nonlinearit

Pengembangan Perangkat Pembelajaran Matematika Berbasis Kurikulum 2013 dengan Model Penemuan Terbimbing pada Materi Lingkaran untuk Peserta Didik Kelas VIII Smp/mts

The background of this study is the restrict of mathematic learning instruments as supporting learning referring to Curriculum 2013. This study aimed to develop mathematics learning instruments with guided discovery model on circle's topic for 8th grade SMP/MTs. Learning instruments which will be developed in this study are syllabus, lesson plans, and student worksheet. This study method used is ADDIE model by Dick and Carey that consist of analysis, design, development, implementation and evaluation. Test instruments that used in this study are validation instrument that is validation sheets to evaluate the advisability of syllabus, lesson plans and student worksheet and then practicality instrument that is questionnaire which will be filled up by students to evaluate practicality of student worksheet. Learning instruments that has been developed is being validated by three validators and being tested in a small test group consist of five on June 9th–10th, 2016 in SMPN 1 Pekanbaru for student worksheet. Based on the data analysis, it can be conclude that mathematics learning instruments is very valid with mean score of syllabus is 4,49, for lesson plans is 4,31, and for student worksheet is 4,22. This student worksheet also valued as very practical with mean percentage of student responses is 4,87 with 97%

Pemantauan Kualitas Air Sekitar Kolam Limbah PPGN Secara Kimia Dan Radioaktivitas

Penelitian ini bertujuan untuk mengetahui secara dini kemungkinan adanya pelepasan polutan supaya tidak mencemari lingkungan. Kualitas air di sekitar kolam limbah PPGN dapat diketahui dengan menganalisis contoh dari 4 buah sumur kontrol sedalam 20 meter terletak pada keempat sisi kolam dan 2 (dua) buah sumur pembanding dengan jarak 50 meter dan 100 meter dari kolam limbah. Metode yang digunakan untuk mengukur kualitas air di sekitar kolam limbah adalah metode spektrofotometri. Pengukuran kandungan kimia air sumur kontrol dan air sumur pembanding dilakukan dengan Spektrophotometer Serapan Atom (AAS), kandungan U dengan UV-VIS Spektrophotometer, sedangkan pengukuran radioaktivitas dengan detektor a SPA-1 Eberline yang dihubungkan dengan alat pencacah Scaler Ludlum Model 1000. Hasil pengukuran tahun 2010 diperoleh: kandungan kimia pada sumur kontrol Ca (2.31-2.91) mg/l, Mg (0.22-0.34) mg/l, Fe (0.024-0.033) mg/l Ni (0.0028-0.030) mg/l, Zn (0.0019-0.025) mg/l, Cu (0.038-0.060) mg/l, Pb (0.003-0.041) mg/l, Mn (0.004-0.005) mg/l, U (0.03-0.04) Bq/l x 10-3. Pada sumur pembanding kandungan Ca (2.31-2.33) mg/l, Mg (0.25-0.27) mg/l, Fe (0.051-0.298) mg/l, Ni (0.003-0.004) mg/l, Zn (0.03-0.04) mg/l, Cu (0.004-0.004) mg/l, Pb (0.003-0.003) mg/l, Mn (0.005-0.021), U (0.025-0.028) Bq/l x 10-3 . Kandungan radioaktivitas sumur kontrol pada triwulan I (2.321-2.635).10-2 Bq/l, triwulan II (2.162-2.823).10-2 Bq/l, triwulan III . (2.424-2.931).10-2 Bq/l, triwulan IV (2.283-2.643).10-2 Bq/l. Sedangkan kandungan radioaktivitas sumur pembanding pada triwulan I (2.931-2.931).10-2 Bq/l, triwulan II (2.162-2.550).10-2 Bq/l, triwulan III. (2.931-2.931).10-2 Bq/l, triwulan IV (2.450-2.632).10-2 Bq/l. Dari data pengukuran menunjukkan tidak ada pelepasan polutan ke lingkungan. Berdasarkan evaluasi data di atas dengan menggunakan metoda Storet dan US-EPA [Environmental Protection] Agency maka kualitas air di sekitar kolam limbah PPGN–BATAN dinyatakan sebagai klasifikasi Kelas A [memenuhi baku mutu]. This research purpose is to monitor the water quality of soil around the waste pond through measurement of chemical constituents (Ca, Mg, Fe, Ni, Zn, Cu, Pb, Mn and U) and water radioactivity. The water quality around tailing pond can be identified by analized the water sample from 4 control wells as deep as 20 m located on the fourth side of the pool and 2 comparison wells with a distance of 50 m and 100 m from the tailing pond. The measurement of chemical constituents of controll well water and comparison well water was done by using Atomic Absorption Spectrophotometer (AAS). The measurement of Uranium content was done by using UV – VIS Spectrophotometer, whereas measurement of radioactivity was measured by Eberline SPA-1 α detector associated with a counter scalers Ludlum model 1000. Determination of the quality of well water was used Storet method. Measurement result obtained in 2010 : the chemical content water in the control wells; Ca (2.31 – 2.91) mg/l, Mg (0.22 – 0.34) mg/l, Fe (0.024 – 0.033) mg/l, Ni (0.0028 – 0.030) mg/l, Zn (0.0019 – 0.025) mg/l, Cu (0.038 – 0.060) mg/l, Pb (0.003 – 0.041) mg/l, Mn (0.004 – 0.005) mg/l, U (0.051 – 0.298) mg/l, Ni (0.003 – 0.004) mg/l, Zn (0.03 – 0.04) mg/l, Cu (0.004 – 0.004) mg/l, Pb (0.003 - 0.003), Mn (0.005 – 0.021) mg/l, and radioactivity of Uranium was (0.025. 10-3 – 0.028.10-3) Bq/l. The radioactivity of control wells in the first quarter (2.321 – 2.635) . 10-2 Bq/l, second quarter (2.162 – 2.823) . 10-2 Bq/l, third quarter (2.424 – 2.931).10-2 Bq/l, fourth quarter (2.283 – 2.643).10-2 Bq/l. The radioactivity of comparison well water in the firs quarter was (2.931 - 2.931). 10-2 Bq/l., second quarter (2.162 – 2.550).10-2 Bq/l, third quarter (2.931- 2.931).10-2, fourth quarter (2.450 – 2.632).10-2 Bq/l. This result showed that there are no pollutant release into the environment. Based on the evaluation result using Storet and US-EPA (Environmental Protection Agency) method, water quality around tailing pond of PPGN – BATAN is expressed as A in class classification (best)