. "Analytical geometry and linear algebra"@en . . "5" . "not provided" . . "Presential"@en . "TRUE" . . "Linear algebra"@en . . "5" . "no data" . . "Presential"@en . "TRUE" . . "Geometry and linear algebra"@en . . "12" . "Description is not available" . . "Presential"@en . "TRUE" . . "Multivariate mathematics applied"@en . . "6" . "Contents:\nlinear algebra: matrices, eigenvalues and eigenvectors;\ncomplex numbers;\nordinary differential equations: separation of variables and variation of constants; systems of linear differential equations; systems of non-linear differential equations and classification of steady states;\nnumerical methods for ordinary differential equations: difference quotients and the Euler method; systems of differential equations; trapezoidal rule and Runge-Kutta; discretization errors; error propagation, stability and stiffness;\nintegration in two or three dimensions: limits of integration; coordinate systems and the Jacobian;\nintroduction to partial differential equations: flow models, diffusion and convection; boundary and initial conditions; steady states;\nvector fields: flow fields and force fields; the gradient and the laws of Fick, Fourier and Darcy; the potential function; divergence and the Laplace operator;\nFourier series for partial differential equations: separation of variables and the Sturm-Liouville problem; boundary value problems and Fourier series;\nuse of computer software.\nLearning outcomes:\nAfter successful completion of this course students are expected to be able to:\nexplain and apply concepts, methods and techniques from linear algebra, calculus, vector calculus and numerical mathematics;\napply mathematical knowledge, insights and methods to solve problems in the technological sciences using a systematic approach;\ncritically reflect upon the results;\ncorrectly report mathematical reasoning and argumentation;\ninterpret and evaluate the results in terms of the (physical, chemical, biological) problem that was modelled mathematically;\nuse mathematical software (Maple) in elaborating mathematical models." . . "Presential"@en . "TRUE" . . "Linear algebra in the mathematical and physical sciences"@en . . "no data" . "no data" . . "Presential"@en . "TRUE" . . "Algebra and geometry"@en . . "4" . "to get students familiar with basic concepts of linear algebra and with some elements of 3-d analytic geometry; to introduce fundamental abstract definitions of linear spaces, algebraic bases, linear mappings and to reinterpret earlier material from this abstract point of view. After completing the course students will know basic concepts of linear algebra and 3-d analytic\n geometry .They will also see them in the deeper abstract setting of linear spaces and linear mappings. Thus they will be prepared for other mathematical courses where some algebraic background is required." . . "Presential"@en . "TRUE" . . "Symmetry groups"@en . . "6" . "• Introduction: Introduction and definitions; Group theoretical notations\n• Representations: Reducible, irreducibele, equivalent representations; Orthogonality\n• relations for matrix elements and characters ; Reduction of representations and\n• character tables; Restricted and induced representations\n• Basis functions and supplements to representations: Transformation of functions and\n• operators; Basis functions and eigenfunctions; Restricted representations and\n• symmetry lowering; Projection operators; Direct product of groups and\n• representations; Selection rules\n• Twodimensional rotation and rotation reflection group: Introduction and\n• twodimensional rotation group; Twodimensional rotation-reflection group\n• Three dimensional rotation group and the group SU(2): Introduction - symmetry and\n• conservation laws; The group SU(2) and SO(3) - SU(2) homomorphism; Reduction of\n• direct product representations - addition of angular momenta; Wigner-Eckart theorem\n• Applications of group theory - capita selecta: Vibrational problems; Inversion\n• symmetry ; Translation symmetry of crystalline solids - space groups - band\n• structure; Time reversal symmetry - Kramers' theorem; Spin and double groups -\n• crystal field theory\nFinal competences: \n1 Being able to recognise symmetry present in physical systems.\r\n2 Being able to make use of the symmetry present in physical systems in a creative\r\nway.\n3 To master the mathematical methods of representation theory and to be able to\r\napply them in practical situations.\r\n4 To understand and to be able to predict the degeneracy of eigenvalues and the\r\nbehaviour under symmetry.\r\n5 To be able to identify the features and properties of physical systems related to\r\nsymmetry.\r\n6 To know and to be able to evaluate the possibilities and limitations of group theory.\r\n7 To have enough understanding with respect to group theoretical information in order\r\nto use and evaluate literature data in a correct way.\r\n8 To be able to apply group theoretical knowledge in other scientific domains as, e.g.,\r\nAtomic and Molecular Physics, Solid State Physics, Subatomic Physics, Chemistry,\r\nSpectroscopy, etc." . . "Presential"@en . "FALSE" . . "Linear algebra"@en . . "no data" . "no data" . . "no data"@en . "TRUE" . . "Linear algebra in geodesy"@en . . "4" . "The course covers those aspects of linear algebra which are needed in teaching basic geodetic theories and methods. The assumed effect of the course is a good skill in the use of vector-matrix calculus in solving basic geodetic problems; a good skill in interpretation with the use of methods and concepts of linear algebra of the solution of geodetic problems, like the least-squares approximation or the\nnetwork solutions. We focus attention on the application of matrix factorization methods, such as those using triangular matrices (A=LU, A=LDU’, PA=LU, A=P1L1U1), those associated with matrix diagonalization associated with the eigenvalue problem (A=SΛS-1, A=QΛQT) and those following from the orthogonalization procedures (A=QR)." . . "Presential"@en . "TRUE" . . "Algebra"@en . . "6" . "Objectives and Contextualisation\nThis is a basic introduction to linear algebra, emphasizing the most functional and instrumental aspects of linear techniques.\nA fundamental objective is to achieve an agile and efficient transition between the three following levels of knowledge:\n\nAbstract knowledge of mathematical concepts related to linear phenomena.\nDeepened knowledge of the same concept from its practical manipulation \"by hand\".\nDeepened knowledge of the same concept from its practical manipulation with a computer.\nThe most important fundamental objective is to learn to design efficient strategies to apply specific techniques to solve\ncomplex problems.\n\n\nCompetences\nElectronic Engineering for Telecommunication\nCommunication\nDevelop personal work habits.\nLearn new methods and technologies, building on basic technological knowledge, to be able to adapt to new situations.\nPerform measurements, calculations, estimations, valuations, analyses, studies, reports, task-scheduling and other similar work in the field of telecommunication systems\nWork in a team.\nTelecommunication Systems Engineering\nDevelop personal work habits.\nDevelop thinking habits.\nLearn new methods and technologies, building on basic technological knowledge, to be able to adapt to new situations.\nLearning Outcomes\nAnalyse measurements in the area of engineering, using statistical tools to extract and understand information.\nAnalyse measures in the area of engineering, using statistical tools to extract and understand information.\nApply, in the problems that arise in engineering, knowledge about linear algebra, geometry, differential geometry, differential and integral calculus, differential and partial derivative equations, numerical methods, numerical algorithms, statistics and optimisation.\nApply, to the problems that arise in engineering, knowledge of linear algebra, geometry, differential geometry, differential and integral calculus, differential and partial derivative equations, numerical methods, numerical algorithms, statistics and optimisation.\nCommunicate efficiently, orally and in writing, knowledge, results and skills, both professionally and to non-expert audiences.\nDevelop scientific thinking.\nDevelop the capacity for analysis and synthesis.\nManage available time and resources.\nModel systems and analyse their features.\nResolve the mathematical problems that can arise in engineering.\nWork autonomously.\nWork cooperatively.\n\nContent\nMatrices\nMatrices. Operations with marices. Special matrices: symmetric, Toeplitz, circulant, invertible, hermitian, orthogonal.\nElemental transformations by rows. Gauss-Jordan's normal form of a matrix. Rank of a matrix. Invertibility and calculation of inverse matrices.\nSystems of linear equations and linear varieties. Gauss method. Direction and dimension of linear varieties. Rouché's Theorem.\nVector Spaces\nDefinition of vector space and examples. Linear combinations of vectors. Subspaces. Generating systems.\nLinear maps. Matrix associated to a linear map. Composition of linear maps. Kernel and Image of a linear map. Isomorphisms.\nLinear dependence of vectors. Linear dependence criterion.\nBases, dimensions and coordinates. Working with coordinates. Base changes.\nDiagonalization of matrices and inner products.\nDeterminant of a square matrix. Properties of the determinant.\nEigenvalues and eigenvectors of a square matrix. Diagonalization criteria.\nApplications of diagonalisation: calculation of matrix powers and resolution of systems of linear differential equations with constant coefficients." . . "Presential"@en . "TRUE" . . "Linear algebra and analytical geometry"@en . . "9.5" . "Description in Bulgarian" . . "Presential"@en . "TRUE" . . "Single mathematics a"@en . . "20.0" . "## Prerequisites\n* Normally, A level Mathematics at Grade A or better, or equivalent.\n\n## Corequisites\n* None.\n\n## Excluded Combination of Modules\n* Calculus I (Maths Hons) (MATH1081), Calculus (MATH1061), Linear Algebra I (Maths Hons) (MATH1091), Linear Algebra I\n (MATH1071), Mathematics for Engineers and Scientists (MATH1551)\nmay not be taken with or after this\nmodule.\n\n## Aims\n* This module has been designed to supply mathematics relevant to\nstudents of the physical sciences.\n\n## Content\n* Basic functions and elementary calculus: including\nstandard functions and their inverses, the Binomial Theorem, basic\nmethods for differentiation and integration.\n* Complex numbers: including addition, subtraction,\nmultiplication, division, complex conjugate, modulus, argument, Argand\ndiagram, de Moivre's theorem, circular and hyperbolic\nfunctions.\n* Single variable calculus: including discussion of real\nnumbers, rationals and irrationals, limits, continuity,\ndifferentiability, mean value theorem, L'Hopital's rule, summation of series,\nconvergence, Taylor's theorem.\n* Matrices and determinants: including determinants, rules\nfor manipulation, transpose, adjoint and inverse matrices,\nGaussian elimination, eigenvalues and eigenvectors,\n \n* Groups, axioms, non-abelian groups\n\n## Learning Outcomes\n* Subject-specific Knowledge: \n * By the end of the module students will: be able to solve a\nrange of predictable or less predictable problems in\nMathematics.\n * have an awareness of the basic concepts of theoretical\nmathematics in these areas.\n * have a broad knowledge and basic understanding of these\nsubjects demonstrated through one or more of the following topic\nareas: Elementary algebra.\n * Calculus.\n * Complex numbers.\n * Taylor's Theorem.\n * Linear equations and matrices.\n * Groups\n\n* Subject-specific Skills: \n\n* Key Skills: \n\n## Modes of Teaching, Learning and Assessment and how these contribute to\nthe learning outcomes of the module\n* Lectures demonstrate what is required to be learned and the\napplication of the theory to practical examples.\n* Initial diagnostic testing fills in gaps related to the wide\nvariety of syllabuses available at Mathematics A-level.\n* Tutorials provide the practice and support in applying the\nmethods to relevant situations as well as active engagement and feedback\nto the learning process.\n* Weekly coursework provides an opportunity for students\nto consolidate the learning of material as the module progresses (there\nare no higher level modules in the department of Mathematical Sciences which build on this module). It serves as a guide in the correct\ndevelopment of students' knowledge and skills, as well as an aid in developing their awareness of standards required.\n* The end-of-year written examination provides a substantial\ncomplementary assessment of the achievement of the student.\n\n## Teaching Methods and Learning Hours\n* Lectures: 63\n* Tutorials: 19\n* Support classes: 18\n* Preparation and Reading: 100\n* Total: 200\n\n## Summative Assessment\n* Examination: 90%\n * Written examination: 3 hours\n* Continuous Assessment: 10%\n * Fortnightly electronic assessments during the first 2 terms. Normally, each will consist of solving problems and will typically be one to two pages long. Students will have about one week to complete each assignment.\n\n## Formative Assessment: \n* 45 minute collection paper in the beginning of Epiphany term. Fortnightly formative assessment.\n\n## Attendance\n* Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University\n\nMore details in: https://apps.dur.ac.uk/faculty.handbook/2023/UG/module/MATH1561" . . "Presential"@en . "FALSE" . . "Single mathematics b"@en . . "20.0" . "#### Prerequisites\n\n* A level Mathematics at Grade A or better, or equivalent.\n\n#### Corequisites\n\n* Single Mathematics A (MATH1561).\n\n#### Excluded Combination of Modules\n\n* Mathematics for Engineers and Scientists (MATH1551) may not be taken with or after this module.\n\n#### Aims\n\n* This module has been designed to supply mathematics relevant to students of the physical sciences.\n\n#### Content\n\n* Vectors: including scalar and vector products, derivatives with respect to scalars, two-dimensional polar coordinates.\n* Ordinary differential equations: including first order, second order linear equations, complementary functions and particular integrals, simultaneous linear equations, applications.\n* Fourier analysis: including periodic functions, odd and even functions, complex form.\n* Functions of several variables: including elementary vector algebra (bases, components, scalar and vector products, lines and planes), partial differentiation, composite functions, change of variables, chain rule, Taylor expansions. Introductory complex analysis and vector calculus\n* Multiple integration: including double and triple integrals.\n* Introduction to probability: including sample space, events, conditional probability, Bayes' theorem, independent events, random variables, probability distributions, expectation and variance.\n\n#### Learning Outcomes\n\nSubject-specific Knowledge:\n\n* By the end of the module students will: be able to solve a range of predictable or less predictable problems in Mathematics.\n* have an awareness of the basic concepts of theoretical mathematics in these areas.\n* have a broad knowledge and basic understanding of these subjects demonstrated through one or more of the following topic areas: Vectors.\n* Ordinary differential equations.\n* Fourier analysis.\n* Partial differentiation, multiple integrals.\n* Vector calculus.\n* Probability.\n\nSubject-specific Skills:\n\nKey Skills:\n\n#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module\n\n* Lectures demonstrate what is required to be learned and the application of the theory to practical examples.\n* Initial diagnostic testing fills in gaps related to the wide variety of syllabuses available at Mathematics A-level.\n* Tutorials provide the practice and support in applying the methods to relevant situations as well as active engagement and feedback to the learning process.\n* Weekly coursework provides an opportunity for students to consolidate the learning of material as the module progresses (there are no higher level modules in the department of Mathematical Sciences which build on this module). It serves as a guide in the correct development of students' knowledge and skills, as well as an aid in developing their awareness of standards required.\n* The end-of-year written examination provides a substantial complementary assessment of the achievement of the student." . . "Presential"@en . "FALSE" . . "Linear algebra"@en . . "20.0" . "#### Prerequisites\n\n* Normally, A level Mathematics at grade A or better and AS level Further Mathematics at grade A or better, or equivalent.\n\n#### Corequisites\n\n* Calculus I (MATH1061)\n\n#### Excluded Combination of Modules\n\n* Calculus I (Maths Hons) (MATH1081), Linear Algebra I (Maths Hons) (MATH1091), Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571) may not be taken with or after this module.\n\n#### Aims\n\n* This module is designed to follow on from, and reinforce, A level mathematics.\n* It will present students with a wide range of mathematics ideas in preparation for more demanding material later.\n* Aim: to give a utilitarian treatment of some important mathematical techniques in linear algebra.\n* Aim: to develop geometric awareness and familiarity with vector methods.\n\n#### Content\n\n* A range of topics are treated each at an elementary level to give a foundation of basic definitions, theorems and computational techniques.\n* A rigorous approach is expected.\n* Linear Algebra in n dimensions with concrete illustrations in 2 and 3 dimensions.\n* Vectors, matrices and determinants.\n* Vector spaces and linear mappings.\n* Diagonalisation, inner-product spaces and special polynomials.\n* Introduction to group theory.\n\n#### Learning Outcomes\n\nSubject-specific Knowledge:\n\n* By the end of the module students will: be able to solve a range of predictable or less predictable problems in Linear Algebra.\n* have an awareness of the basic concepts of theoretical mathematics in Linear Algebra.\n* have a broad knowledge and basic understanding of these subjects demonstrated through one of the following topic areas:\n* Vectors in Rn, matrices and determinants.\n* Vector spaces over R and linear mappings.\n* Diagonalisation and Jordan normal form.\n* Inner product spaces.\n* Introduction to groups.\n* Special polynomials.\n\nSubject-specific Skills:\n\n* Students will have basic mathematical skills in the following areas: Modelling, Spatial awareness, Abstract reasoning, Numeracy.\n\nKey Skills:\n\n#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module\n\n* Lectures demonstrate what is required to be learned and the application of the theory to practical examples.\n* Tutorials provide active engagement and feedback to the learning process.\n* Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills. They are also an aid in developing students' awareness of standards required.\n* Initial diagnostic testing and associated supplementary support classes fill in gaps related to the wide variety of syllabuses available at Mathematics A-level.\n* The examination provides a final assessment of the achievement of the student.\n\nMore details at: https://apps.dur.ac.uk/faculty.handbook/2023/UG/module/MATH1071" . . "Presential"@en . "FALSE" . . "Mathematics workshop"@en . . "20.0" . "#### Prerequisites\n\n* Mathematical Methods in Physics (PHYS2611) OR Analysis in Many Variables II (MATH2031).\n\n#### Corequisites\n\n* Foundations of Physics 3A (PHYS3621).\n\n#### Excluded Combination of Modules\n\n* None.\n\n#### Aims\n\n* This module is designed primarily for students studying Department of Physics or Natural Sciences degree programmes.\n* It builds on the Level 2 module Mathematical Methods in Physics (PHYS2611).\n* It provides the mathematical tools appropriate to Level 3 physics students necessary to tackle a variety of physical problems.\n\n#### Content\n\n* The syllabus contains:\n* Vectors and matrices, Hilbert spaces, linear operators, matrices, eigenvalue problem, diagonalisation of matrices, co-ordinate transformations, tensor calculus.\n* Complex Analysis: functions of complex variables, differentiable functions, Cauchy-Riemann conditions, Harmonic functions, multiple valued functions and Riemann surfaces, branch points and cuts, complex integration, Cauchy's theorem, Taylor and Laurent series, poles and residues, residue theorem and definite integrals, residue theorem and series summation.\n* Calculus of Variations: Euler–Lagrange equations, classic variational problems, Lagrange multipliers.\n* Infinite series and convergence, asymptotic series. Integration, Gaussian and related integrals, gamma function.\n* Integral Transforms: Fourier series and transforms, convolution theorem, Parseval's relation, Wiener-Khinchin theorem. Momentum representation in quantum mechanics, Hilbert transform, sampling theorem, Laplace transform, inverse Laplace transform and Bromwich integral.\n\n#### Learning Outcomes\n\nSubject-specific Knowledge:\n\n* Having studied this module students will have knowledge of and an ability to use a range of mathematical methods needed to solve a wide array of physical problems.\n\nSubject-specific Skills:\n\nKey Skills:\n\n#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module\n\n* Teaching will be by two-hour workshops which are a mix of lectures and examples classes.\n* The lectures provide the means to give a concise, focussed presentation of the subject matter of the module. The lecture material will be explicitly linked to the contents of recommended textbooks for the module, thus making clear where students can begin private study.\n* When appropriate, the lectures will also be supported by the distribution of written material, or by information and relevant links online.\n* New material is immediately backed up by example classes which give students the chance to develop their theoretical understanding and problem solving skills.\n* Students will be able to obtain further help in their studies by approaching their lecturers, during the workshop sessions or at other mutually convenient times.\n* Student performance will be summatively assessed through two open-book examinations.\n* The example classes provide opportunities for feedback, for students to gauge their progress and for staff to monitor progress throughout the duration of the module.\n\nMore information at: https://apps.dur.ac.uk/faculty.handbook/2023/UG/module/PHYS3591" . . "Presential"@en . "FALSE" . . "Linear algebra"@en . . "6.0" . "Objectives\nMaster matrix calculus and methods for solving systems of linear equations. Learn about vector spaces and linear transformations. Study canonical forms of matrices, eigenvectors, eigenvalues and singular values. Study applications of the previous subjects.\n\nProgram\nGauss and Gauss-Jordan elimination applied to the solution of linear systems. Matrices, inverse matrices and determinants. Definition and examples of vector spaces. Linearly independent sets. Linear transformations. Nullspace (kernel) and range of a linear transformation. Solution space of a linear equation. Eigenvectors and eigenvalues. Algebraic and geometric multiplicity of an eigenvalue. Jordan canonical form. Applications (e.g. systems of linear ordinary differential equations with constant coefficients, stability of linear dynamical systems, Markov chains, PageRank algorithm). Inner product spaces. Gram-Schmidt orthogonalization. The least squares method. Spectral theorem. Orthogonal, unitary and hermitean linear transformations. Singular value decomposition of a linear transformation between euclidean spaces. Classification of quadratic forms.\n\nEvaluation Methodology\nExam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).\n\nCross-Competence Component\nThe UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.\n\nLaboratorial Component\nNot applicable.\n\nProgramming and Computing Component\nNot applicable.\n\nMore information at: https://fenix.tecnico.ulisboa.pt/cursos/lerc/disciplina-curricular/845953938489996" . . "Presential"@en . "TRUE" . . "Linear algebra for physicists"@en . . "7" . "no data" . . "Presential"@en . "TRUE" . . "Discrete structures"@en . . "20.0" . "https://portal.stir.ac.uk/calendar/calendar.jsp?modCode=MATU9S1&_gl=1*1rqncef*_ga*MTY1OTcwNzEyMS4xNjkyMDM2NjY3*_ga_ENJQ0W7S1M*MTY5MjAzNjY2Ny4xLjEuMTY5MjAzODY1MC4wLjAuMA.." . . "Presential"@en . "FALSE" . . "Discrete structures (matu9s1)"@en . . "20.0" . "https://portal.stir.ac.uk/calendar/calendar.jsp?modCode=MATU9S1&_gl=1*17xpbct*_ga*MTY1OTcwNzEyMS4xNjkyMDM2NjY3*_ga_ENJQ0W7S1M*MTY5MjAzNjY2Ny4xLjEuMTY5MjAzOTM5NC4wLjAuMA.." . . "Presential"@en . "FALSE" . . "Algebra"@en . . . . . . . . . . . . . . . . . . . .