Cambridge Primary Maths Worksheets Pdf
Aila Gilb
This course is based on a book KD is writing, "Brain Computation: A Hands-on Guidebook" using Jupyter notebook with Python codes.
The course will be in a "flipped learning" style; each week, students read a draft chapter and experiment with sample codes before the class.
In the first class of the week, they present what they have learned and raise questions.
In the second class of the week, they 1) present a paper in the reference list, 2) solve exercise problem(s), 3) make a new exercise problem and solve it, or 4) propose revisions in the chapter.
Toward the end of the course, students work on individual or group projects by picking any of the methods introduced in the course and apply that to a problem of their interest.
Students are assumed to be familiar with Python, as covered in the Computational Methods course in Term 1, and basic statistics, as covered in the Statistical Tests and Statistical Modeling courses in Term 2.
This course develops advanced mathematical techniques for application in the natural sciences. Particular emphasis will be placed on analytical and numerical, exact and approximate methods, for calculation of physical quantities. Examples and applications will be drawn from a variety of fields. The course will stress calculational approaches rather than rigorous proofs. There will be a heavy emphasis on analytic calculation skills, which will be developed via problem sets.
The course is aimed at students interested in modeling systems characterized by stochastic dynamics in different branches of science. Goals of the course are: to understand the most common stochastic processes (Markov chains, Master equations, Langevin equations); to learn important applications of stochastic processes in physics, biology and neuroscience; to acquire knowledge of simple analytical techniques to understand stochastic processes, and to be able to simulate discrete and continuous stochastic processes on a computer.
1) Basic concepts of probability theory. Discrete and continuous distributions, main properties. Moments and generating functions. Random number generators.
2) Definition of a stochastic process and classification of stochastic processes. Markov chains. Concept of ergodicity. Branching processes and Wright-Fisher model in population genetics.
3) Master equations, main properties and techniques of solution. Gillespie algorithm. Stochastic chemical kinetics.
4) Fokker-Planck equations and Langevin equations. Main methods of solution. Simulation of Langevin equations. Colloidal particles in physics.
5) First passage-time problems. Concept of absorbing state and main methods of solution. First passage times in integrate-and-fire neurons.
6) Element of stochastic thermodynamics. Work, heat, and entropy production of a stochastic trajectory. Fluctuation relations, Crooks and Jarzynski relations.
A geometrically oriented introduction to the calculus of vector and tensor fields on three-dimensional Euclidean point space, with applications to the kinematics of point masses, rigid bodies, and deformable bodies. Aside from conventional approaches based on working with Cartesian and curvilinear components, coordinate-free treatments of differentiation and integration will be presented. Connections with the classical differential geometry of curves and surfaces in three-dimensional Euclidean point space will also be established and discussed.
Many physical processes exhibit some form of nonlinear wave phenomena. However diverse they are (e.g. from engineering to finance), however small they are (e.g. from atomic to cosmic scales), they all emerge from hyperbolic partial differential equations (PDEs). This course explores aspects of hyperbolic PDEs leading to the formation of shocks and solitary waves, with a strong emphasis on systems of balance laws (e.g. mass, momentum, energy) owing to their prevailing nature in Nature. In addition to presenting key theoretical concepts, the course is designed to offer computational strategies to explore the rich and fascinating world of nonlinear wave phenomena.
By the end of this course, participants dealing with wave-like phenomena in their research field of interest should be able to identify components that can trigger front-like structures (e.g. shocks, solitons) and be able to explore their motion numerically. Whilst the course is aimed at graduate students with an engineering/physics background, biologists interested in wave phenomena in biological systems (e.g. neurones, arteries, cells) are also welcome. However, it is assumed that participants have prior knowledge of maths for engineers and physicists.
The target student has completed an engineering, physics or applied maths degree and is embarking on a PhD topic which involves in one way or another some nonlinear wave phenomena. The potential audience is therefore broad in terms of its interest but is expected to know the basic maths and physics from the aforementioned degrees. Although the course includes a large theoretical component it is designed to build a core knowledge of computational approaches to solve nonlinear waves. It is therefore not suited to students only interested in mathematical approaches, and could for example appeal more to the bio-physicist or quantum physicist looking into building a numerical model of wave-like phenomena in biology or physics.
Target Students: Students interested in solving partial differential equations numerically and in understating possibilities and limitations of numerical techniques. Students should have a general knowledge of partial differential equations.
This course aims to provide the mathematical and numerical tools to solve problems governed by partial differential equations. These techniques have wide application in many areas of physics, engineering, mechanics, and applied mathematics.
Students should have a general knowledge of partial differential equations. such as from the course B28.
A basic knowledge of Python, MATLAB or any other programming language is preferred but not essential.
Students will learn the basic structure theory of simple Lie algebras over the complex numbers, as well as the theory of their highest weight representations. This will develop students' understanding of these fundamental objects in algebra, and give them some hands-on experience computing representations and proving some powerful (and quite beautiful!) results.
Students will learn the basic structures of simple Lie algebras over the complex numbers, including classification, root systems, Cartan subalgebras, Cartan/triangular decomposition, Dynkin diagrams, Weyl groups, and the Killing form. We will develop a highest weight theory of representations, including Verma modules and enveloping algebras. We will end with Weyl's character formula for finite-dimensional simple modules.
Students wanting to learn graduate-level algebra, and especially representation theory. A solid grasp of linear algebra will be assumed, as well as an ability to understand and construct quite sophisticated mathematical proofs.
A solid grasp of undergraduate linear algebra, as well as experience following long proofs and constructing your own proofs. Students must be very comfortable with proofs in order to understand the material in this course and complete the homework questions adequately. If you are unsure, please discuss this further with your academic mentor. Some prior knowledge of the representation theory of finite groups will also be helpful when grappling with analogous results for Lie algebras, but it is not completely necessary.
Review of geometrical optics; wave properties of light and the wave equation; Helmholtz equation; wave optics, including Fresnel and Fraunhofer diffraction, transfer functions, coherence, auto and cross-correlation; Gaussian and non-Gaussian beam profiles; quantum optics and photon statistics; spin squeezing; applications of optics including fiber optics, laser resonators, laser amplifiers, non-linear optics, and optical trapping; quantum properties of light; interaction of photons and atoms.
This course covers quantum field theory. Due to recent developments, we organize it with emphasizing statistical field theory.
The renormalization group method, symmetry breaking, gauge field and string theory, random matrix theory are key ingredients.
This course covers essential concepts and recent advances in the design and synthesis of functional molecules used for understanding and controlling biological systems. Topics of this course include design and synthesis of small organic molecules, organic reactions, methods for controlling reaction pathways, asymmetric synthesis, mechanisms of catalysis and molecular recognition, and creation of designer proteins and peptides.
This course will be an introductory graduate level course to initiate students into the techniques of ultrafast spectroscopy. They will be introduced to the basic concepts underlying sub-picosecond phenomena in nature (ultrafast chemical processes, femtosecond electron dynamics in materials, etc.) and the tools used to study such phenomena (pump-probe spectroscopy, Terahertz Time Domain Spectroscopy, etc.).
Advanced course in Quantum Mechanics, based on recent theoretical and experimental advances. Evolution in Hilbert space and quantum bits; conditional quantum dynamics; quantum simulations; quantum Fourier transform and quantum search algorithms; ion-trap and NMR experiments; quantum noise and master equations; Hilbert space distances; Von Neumann entropy; Holevo bound; entanglement as a physical resource; quantum cryptography; lab: quantum eraser, interaction free measurement.
Advanced level course in atomic physics. Progress in laser control of atoms has led to the creation of Bose-Einstein condensates, ultrafast time and frequency standards and the ability to develop quantum technologies. In this course we will cover the essentials of atomic physics including resonance phenomena, atoms in electric and magnetic fields, and light-matter interactions. This leads to topics relevant in current research such as laser cooling and trapping.
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