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This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in written and oral presentation.   The three options for 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginni This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in written and oral presentation.   The three options for 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginni

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.241 examines linear, discrete- and continuous-time, and multi-input-output systems in control and related areas. Least squares and matrix perturbation problems are considered. Topics covered include: state-space models, modes, stability, controllability, observability, transfer function matrices, poles and zeros, minimality, internal stability of interconnected systems, feedback compensators, state feedback, optimal regulation, observers, observer-based compensators, measures of control performance, and robustness issues using singular values of transfer functions. Nonlinear systems are also introduced. 6.241 examines linear, discrete- and continuous-time, and multi-input-output systems in control and related areas. Least squares and matrix perturbation problems are considered. Topics covered include: state-space models, modes, stability, controllability, observability, transfer function matrices, poles and zeros, minimality, internal stability of interconnected systems, feedback compensators, state feedback, optimal regulation, observers, observer-based compensators, measures of control performance, and robustness issues using singular values of transfer functions. Nonlinear systems are also introduced.

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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The course is directed at making scientifically sensible deductions from the combination of observations with dynamics and kinematics represented, generically, as "models". There are two overlapping central themeslinear "inverse" methods and data "assimilation" including regression, singular value decomposition, objective mapping, non-stationary models and data, Kalman filters, adjoint methods ("assimilation") etc.standard time series analysis, including basic statistics, Fourier methods, spectra, coherence, filtering, etc. The course is directed at making scientifically sensible deductions from the combination of observations with dynamics and kinematics represented, generically, as "models". There are two overlapping central themeslinear "inverse" methods and data "assimilation" including regression, singular value decomposition, objective mapping, non-stationary models and data, Kalman filters, adjoint methods ("assimilation") etc.standard time series analysis, including basic statistics, Fourier methods, spectra, coherence, filtering, etc.

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This is a format to be used as a basis for a participant-centred, developmental approach to Peer Observation. It is being used as part of C@N-DO, the CPD Scheme at the University of Northampton, which is accredited by the HEA, however it is designed to be relevant beyond that context. It could be used independently by individuals, or in a facilitated way by academic developers or others who are in a support role, and who may be guiding a peer observation process as part of a CPD Scheme or taught professional development course. The form is designed to be completed electronically, with all sections being expanded accruing to need. Stage 1 is designed to support staff in identifying their personal agenda for observation, to devise key areas to be used as the basis for the observation itself

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This photograph shows a 74" reflector built by Grubb Parsons in Newcastle upon Tyne on display for the Festival of Britain, 1951 (TWAM ref. DS.GP). The telescope was manufactured for the Mount Stromlo Observatory near Canberra, Australia. This photograph is taken from the Grubb Parsons Ltd collection at Tyne & Wear Archives. The records of Grubb Parsons Ltd, Newcastle upon Tyne, England, consist of 65 linear metres (213 linear feet) of files, plans, photographs and glass plate negatives relating to this internationally renowned firm's manufacture of precision telescopic instruments. The original Business was founded in the early nineteenth century by Thomas Grubb, in 1925 the company was acquired by Sir Charles Parsons and continued to manufacture Telescopic and Astronomical instruments until 1985. This image is taken from a large collection that documents the work of Grubb Parsons Ltd at their workshop in Walkergate, Newcastle upon Tyne. It was here that Grubb Parsons Ltd manufactured Telescopic and Astronomical equipment for companies and observatories world wide. Their equipment was designed and built for use and research across the Globe, to name only a few of these locations Grubb Parsons Ltd supplied to the UK, Switzerland, Denmark, Egypt, South Africa, Greece, Australia, Japan, India, Hawaii, Poland, Chile, Canada, France and Spain. (Copyright) We're happy for you to share these digital images within the spirit of The Commons. Please cite 'Tyne & Wear Archives & Museums' when reusing. Certain restrictions on high quality reproductions and commercial use of the original physical version apply though; if you're unsure please email archives@twmuseums.org.uk

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This is a format to be used as a basis for a participant-centred, developmental approach to Peer Observation. It is being used as part of C@N-DO, the CPD Scheme at the University of Northampton, which is accredited by the HEA, however it is designed to be relevant beyond that context. It could be used independently by individuals, or in a facilitated way by academic developers or others who are in a support role, and who may be guiding a peer observation process as part of a CPD Scheme or taught professional development course. The form is designed to be completed electronically, with all sections being expanded accruing to need. Stage 1 is designed to support staff in identifying their personal agenda for observation, to devise key areas to be used as the basis for the observation itself

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Reference: DS.GP.1919/2473 This photograph documents the Pretoria 61FT Turret built for the Pretoria Observatory by Grubb Parsons in Newcastle upon Tyne. This photograph is taken from the Grubb Parsons Ltd collection at Tyne & Wear Archives. The records of Grubb Parsons Ltd, Newcastle upon Tyne, England, consist of 65 linear metres (213 linear feet) of files, plans, photographs and glass plate negatives relating to this internationally renowned firm's manufacture of precision telescopic instruments. The original Business was founded in the early nineteenth century by Thomas Grubb, in 1925 the company was acquired by Sir Charles Parsons and continued to manufacture Telescopic and Astronomical instruments until 1985. This Glass Lantern Slide is taken from a large collection that documents the work of Grubb Parsons Ltd at their workshop in Walkergate, Newcastle upon Tyne. It was here that Grubb Parsons Ltd manufactured Telescopic and Astronomical equipment for companies and observatories world wide. Their equipment was designed and built for use and research across the Globe, to name only a few of these locations Grubb Parsons Ltd supplied to the UK, Switzerland, Denmark, Egypt, South Africa, Greece, Australia, Japan, India, Hawaii, Poland, Chile, Canada, France and Spain. (Copyright) We're happy for you to share these digital images within the spirit of The Commons. Please cite 'Tyne & Wear Archives & Museums' when reusing. Certain restrictions on high quality reproductions and commercial use of the original physical version apply though; if you're unsure please email archives@twmuseums.org.uk

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Reference: DS.GP.1919/3444 This photograph shows Mirror Tests using Grubb Parsons equipment for the San Fernando Observatory at some point in the early 20th Century. This photograph is taken from the Grubb Parsons Ltd collection at Tyne & Wear Archives. The records of Grubb Parsons Ltd, Newcastle upon Tyne, England, consist of 65 linear metres (213 linear feet) of files, plans, photographs and glass plate negatives relating to this internationally renowned firm's manufacture of precision telescopic instruments. The original Business was founded in the early nineteenth century by Thomas Grubb, in 1925 the company was acquired by Sir Charles Parsons and continued to manufacture Telescopic and Astronomical instruments until 1985. This Glass Lantern Slide is taken from a large collection that documents the work of Grubb Parsons Ltd at their workshop in Walkergate, Newcastle upon Tyne. It was here that Grubb Parsons Ltd manufactured Telescopic and Astronomical equipment for companies and observatories world wide. Their equipment was designed and built for use and research across the Globe, to name only a few of these locations Grubb Parsons Ltd supplied to the UK, Switzerland, Denmark, Egypt, South Africa, Greece, Australia, Japan, India, Hawaii, Poland, Chile, Canada, France and Spain. (Copyright) We're happy for you to share these digital images within the spirit of The Commons. Please cite 'Tyne & Wear Archives & Museums' when reusing. Certain restrictions on high quality reproductions and commercial use of the original physical version apply though; if you're unsure please email archives@twmuseums.org.uk

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Reference: DS.GP.1919/3232 This photograph documents construction of Observatory equipment by Grubb Parsons in Newcastle upon Tyne at some point in the mid 20th century. This photograph is taken from the Grubb Parsons Ltd collection at Tyne & Wear Archives. The records of Grubb Parsons Ltd, Newcastle upon Tyne, England, consist of 65 linear metres (213 linear feet) of files, plans, photographs and glass plate negatives relating to this internationally renowned firm's manufacture of precision telescopic instruments. The original Business was founded in the early nineteenth century by Thomas Grubb, in 1925 the company was acquired by Sir Charles Parsons and continued to manufacture Telescopic and Astronomical instruments until 1985. This Glass Lantern Slide is taken from a large collection that documents the work of Grubb Parsons Ltd at their workshop in Walkergate, Newcastle upon Tyne. It was here that Grubb Parsons Ltd manufactured Telescopic and Astronomical equipment for companies and observatories world wide. Their equipment was designed and built for use and research across the Globe, to name only a few of these locations Grubb Parsons Ltd supplied to the UK, Switzerland, Denmark, Egypt, South Africa, Greece, Australia, Japan, India, Hawaii, Poland, Chile, Canada, France and Spain. (Copyright) We're happy for you to share these digital images within the spirit of The Commons. Please cite 'Tyne & Wear Archives & Museums' when reusing. Certain restrictions on high quality reproductions and commercial use of the original physical version apply though; if you're unsure please email archives@twmuseums.org.uk

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This exercise sheet is designed to be used to train teachers in peer-to-peer observation. It outlines the phases of peer-to-peer observation and gives pointers on how to give feedback to colleagues.

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In this course, we will look at many important aspects of the circulation of the atmosphere and ocean, from length scales of meters to thousands of km and time scales ranging from seconds to years. We will assume familiarity with concepts covered in course 12.003 (Physics of the Fluid Earth). In the early stages of the present course, we will make somewhat greater use of math than did 12.003, but the math we will use is no more than that encountered in elementary electromagnetic field theory, for example. The focus of the course is on the physics of the phenomena which we will discuss. In this course, we will look at many important aspects of the circulation of the atmosphere and ocean, from length scales of meters to thousands of km and time scales ranging from seconds to years. We will assume familiarity with concepts covered in course 12.003 (Physics of the Fluid Earth). In the early stages of the present course, we will make somewhat greater use of math than did 12.003, but the math we will use is no more than that encountered in elementary electromagnetic field theory, for example. The focus of the course is on the physics of the phenomena which we will discuss.

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Forecasting is the ultimate form of model validation. But even if a perfect model is in hand, imperfect forecasts are likely. This course will cover the factors that limit our ability to produce good forecasts, will show how the quality of forecasts can be gauged a priori (predicting our ability to predict!), and will cover the state of the art in operational atmosphere and ocean forecasting systems. Forecasting is the ultimate form of model validation. But even if a perfect model is in hand, imperfect forecasts are likely. This course will cover the factors that limit our ability to produce good forecasts, will show how the quality of forecasts can be gauged a priori (predicting our ability to predict!), and will cover the state of the art in operational atmosphere and ocean forecasting systems.

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Information in this document and the tutor notes included within the PowerPoint presentations are intended to support the wider use and reuse of these Open Educational Resources.

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Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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The goal of this subject is to teach the fundamentals of control design and analysis using state-space methods. This includes both the practical and theoretical aspects of the topic. By the end of the course, students should be able to design controllers using state-space methods and evaluate whether these controllers are "robust," that is, if they are likely to work well in practice. The goal of this subject is to teach the fundamentals of control design and analysis using state-space methods. This includes both the practical and theoretical aspects of the topic. By the end of the course, students should be able to design controllers using state-space methods and evaluate whether these controllers are "robust," that is, if they are likely to work well in practice.

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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