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Infinitely many small negative energy periodic solutions for second order Hamiltonian systems without spectrum 0. Henrique F. Complete maximal spacelike submanifolds immersed in a locally symmetric semi-Riemannian space. Alberto L. Delgado and Mathew Timm. Spaces with Regular Nonabelian Self Covers. The descriptive complexity of the family of Banach spaces with the bounded approximation property. Topological entropy and periods of self-maps on compact manifolds.

Katherine Heller. Ju Myung Kim. Duality between the K 1 - and the K u1 -approximation properties. Nikita Kozin and Deepak Majeti. On the arithmetic of one del Pezzo surface over the field with three elements. Yuan Li. Complete order structures for completely bounded maps involving trace class operators. Christopher Ryan Loga. Contact type hypersurfaces and Legendre duality. Helmut Maier and Michael Th. Asymptotics for moments of certain cotangent sums for arbitrary exponents. The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one.

Habib Marzougui and Issam Naghmouchi. Andrea Medini and Lyubomyr Zdomskyy. A quasitopological modification of paratopological groups. Real hypersurfaces in the complex projective plane attaining equality in a basic inequality. Tkachenko and V. More reflections in small continuous images. Dan-Virgil Voiculescu. The approximation of analytic function defined by Laplace-Stieltjes transformations convergent in the left half-plane.

Characterizations of some spaces with maps to ordered topological vector spaces. Jianjun Zhang. On transcendental meromorphic solutions of certain types of nonlinear differential equations. Almost meshed locally connected continua without unique n-fold hyperspace suspension. Some properties of Zermelo navigation in pseudo-Finsler metrics under an arbitrary wind. Gehring's lemma for Orlicz functions in metric measure spaces and higher integrability for convex integral functionals. Further results on a class of starlike functions related to the Bernoulli lemniscate.

An inverse Ackermannian lower bound on the local unconditionality constant of the James space. Analytic properties and the asymptotic behavior of the area function of a Funk metric. Some geometrical properties of minimal graph on space forms with nonpositive curvature. Viewed times. At first I was also thought about it, but I thought, I may be wrong, as this question came to my exam paper.

Then use the fact that two finite dimensional vector spaces over the same field are isomorphic if and only if they have the same dimension. In fact you can even find out basis of these two spaces, it's not that difficult. Frank Lu Frank Lu 5, 10 10 silver badges 28 28 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Extrema of a quadratic forms, Vector and matrix differentiation.

Least squares theory and Gauss-Markoff theorem, Cochran's theorem and distribution of quadratic forms, test of single linear hypothesis and more than one hypothesis, ANOVA table, Confidence interval and regions, Power of F-test. Multiple comparisons and simultaneous confidence intervals.

Rao and P. Debasis Sengupta and S. Simple and multiple linear regression, Polynomial regression and orthogonal polynomials, Test of significance and confidence intervals for parameters. Residuals and their analysis for test of departure from the assumptions such as fitness of model, normality, homogeneity of variances, detection of outliers, Influential observations, Power transformation of dependent and independent variables.

Problem of multi-collinearity, ridge regression and principal component regression, subset selection of explanatory variables, Mallow's Cp statistic. Nonlinear regression, different methods for estimation Least squares and Maximum likelihood , Asymptotic properties of estimators. Douglas C. Montgomery, Elizabeth A.

Peck, G. Norman R. Rao, H. Toutenburg, Shalabh, and C. Principles of sample surveys; Simple, Stratified and unequal probability Sampling with and without replacement; ratio, product and regression method of estimation; systematic sampling; cluster and subsampling with equal unequal sizes; double sampling; sources of errors in surveys.

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Sukhatme, B. V Sukhatme, S. Sukhatme and C. Introduction to ODE; Existence and uniqueness of solution; Continuity and differentiability of solution w. Introduction to PDEs, First order quasilinear and nonlinear equations; Higher order equations and classifications; Solution of wave equations, Duhamel's principle and applications; Existence and uniqueness of solutions; BVPs for Laplace's and Poisson's equations, Green's function, Maximum principle for the Laplace equation; Heat equation, Maximum principle for the heat equation, Uniqueness of solutions of IVPs for heat conduction equation.

Brief review of distribution theory of uni-dimensional random variables. Multi-dimensional random variables random vectors : Joint, marginal, and conditional distribution functions; Independence; Moments and moment generating function; Conditional mean and conditional variance; Some examples of conditional expectations useful in Rao-Blackwellization; Discrete and absolutely continuous random variables distributions ; Multinomial and multivariate normal distributions.

Distribution of functions of random variables including order statistics; Properties of random vectors which are equal in distribution; Exchangeable random variables and their properties. Reference material s :. Dudewicz and S. Rohatgi, A. What is a model? What is Mathematical modelling? Role of Mathematics in problem solving; Transformation of Physical model to Mathematical model with some illustrations of real world problems; Mathematical formulation, Dimensional analysis, Scaling, Sensitivity analysis, Validation, Simulation, Some case studies with analysis such as exponential growth and decay models, population models, Traffic flow models, Optimization models.

Murthy, N. Page and E. Sean Bohun, S.

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McCollum and T. Integral Transforms: Fourier and Hankel Transforms with their inverse transforms properties, convolution theorem and application to solve differential equation. Perturbation Methods: Perturbation theory, Regular perturbation theory, Singular perturbation theory, Asymptotic matching. Calculus of Variation: Introduction, Variational problems with functional containing first order derivatives and Euler equations. Functional containing higher order derivatives and several independent variables.

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Variational problem with moving boundaries. Boundaries with constraints. Higher order necessary conditions, Weierstrass function, Legendre and Jacobi's condition. Existence of solutions of variational problems. Rayleigh-Ritz method, statement of Ekelands variational principle; Self adjoint, normal and unitary operators; Banach algebras. Linear least squares problems, existence and uniqueness, sensitivity and conditioning, orthogonalization methods, SVD, Optimization, existence and uniqueness, sensitivity and conditioning, Newton's method, Unconstrained Optimization, Steepest descent, Conjugate gradient method, Constrained optimization optional , Numerical solution to ODE, IVP: Euler's method, One step and linear multistep methods, Stiff differential equations, boundary value problems, Numerical solution to PDEs, review of second order PDEs: hyerbolic, parabolic and elliptic PDEs, Time dependent problems, Time independent problems Reference material s :.

Simulation of random variables from discrete, continuous, multivariate distributions and stochastic processes, Monte Carlo methods. Regression analysis, scatterplot, residual analysis. Graphical representation of multivariate data, Cluster analysis, Principal component analysis for dimension reduction. Hastie, R. Tibshirani and M. Gilks, S. Richardson, D. Analysis of completely randomized design, randomized block design, Latin squares design; Split plot, 2 and 3-factorial designs with total and partial confounding, two way non-orthogonal experiment, BIBD, PBIBD; Analysis of covariance, missing plot techniques; First and second order response surface designs.

Sahai and M. Group families; Principle of invariance and equivariant estimators- location family, scale family, location-scale family; General Principle of equivariance; Minimum risk equivariant estimators under location scale and location-scale families; Bayesian estimation; prior distributions; posterior distribution; Bayes estimators; limit of Bayes estimators; hierarchical Bayes estimators; Generalized Bayes estimators; highest posterior density credible regions; Minimax estimators and their relationships with Bayes estimators; admissibility; Invariance in hypothesis testing; Review of convergence in probability and convergence in distributions; consistent estimators; Consistent and Asymptotic Normal CAN estimators; BAN estimator; asymptotic relative efficiency ARE ; Limiting risk efficiency LRE ; Limiting risk deficiency LRD; CRLB and asymptotically efficient estimator; large sample properties of MLE.

Casella : Theory of Point Estimation, Springer.

Order statistics, Run tests, Goodness of fit tests, rank order statistics, sign test and signed rank test. General two sample problems, Mann Whitney test, Linear rank tests for location and scale problem, k-sample problem, Measures of association, Power and asymptotic relative efficiency, Concepts of jack knifing, Bootstrap methods. Randles and D. Computer arithmetic. Vector and matrix norms. Condition number of a matrix and its applications.

Singular value decomposition of a matrix and its applications. Linear least squares problem. Householder matrices and their applications. Numerical methods for matrix eigenvalue problem. Numerical methods for systems and control. Mixed methods, Iterative Techniques. Preliminaries: Introduction to algorithms; Analysing algorithms: space and time complexity; growth of functions; summations; recurrences; sets, etc. Greedy Algorithms: General characteristics; Graphs: minimum spanning tree; The knapsack problem; scheduling. Divide and Conquer: Binary search; Sorting: sorting by merging, quicksort.

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HJM, Vol. 41 - 45 ( - )

Dynamic Programming: Elements of dynamic programming; The principle of optimality; The knapsack problem; Shortest paths; Chained matrix multiplication. Number Theoretic Algorithms: Greatest common divisor; Modular arithmetic; Solving modular linear equations. Introduction to cryptography. Computational Geometry: Line segment properties; Intersection of any pair of segments; Finding the convex hull; Finding the closest pair of points. Introduction to Data Mining; supervised and un-supervised data mining, virtuous cycle.

Dimension Reduction and Visualization Techniques; Chernoff faces, principal component analysis. Feature extraction; multidimensional scaling. Cluster Analysis: hierarchical and non-hierarchical techniques. Density estimation techniques; parametric and Kernel density estimation approaches.

Statistical Modelling; design, estimation and inferential aspects of multiple regression, Kernel regression techniques. Tree based methods; Classification and Regression Trees. Neural Networks; multi-layer perceptron, feed-forward and recurrent networks, supervised ANN model building using back-propagation algorithm, ANN model for classification. Genetic algorithms, neuro-genetic models. Self-organizing Maps. Project - I Click to collapse. Project - II Click to collapse. Elementary mathematical models; Role of mathematics in problem solving; Concepts of mathematical modelling; System approach; formulation, Analyses of models; Sensitivity analysis, Simulation approach; Pitfalls in modelling, Illustrations.

Fields: Definition and examples, Irreducibility Criterions, Prime Subfield, Algebraic and transcendental elements and extensions, Splitting field of a polynomial. Existence and uniqueness of algebraic closure. Finite fields, Normal and separable extensions, Inseparable and purely inseparable extensions. Simple extensions and the theorem of primitive elements, Perfect fields.

Galois Extension and Galois groups. Fundamental theorem of Galois Theory. Applications of Galois Theory: Roots of unity and cyclotomic polynomials, Wedderburn's and Dirichlet's theorem. Cyclic and abelian extensions, Fundamental Theorem of Algebra, Polynomials solvable by radicals, Symmetric functions, Ruler and compass constructions. Inverse Galois Problem. Time permitting : Simple transcendental extension and Luroth's theorem. Infinite Galois Extension and Krull's theorem. Murthy, K. Ramanathan, C. Seshadri, U.

Banach Spaces - Lec02 - Frederic Schuller

Shukla and R. Review of basic notions of Rings and Modules, Noetherian and Artinian Modules, Exactness of Hom and tensor, Localization of rings and modules, Primary decomposition theorem, Integral extensions, Noether normalization and Hilbert Nullstellensatz, Going up and Going down theorem, Discrete Valuation rings and Dedekind domains, Invertible modules, Modules over Dedekind domain, Krull dimension of a ring, Hilbert polynomial and dimension theory for Noetherian local ring, Height of a prime ideal, Krull's principal ideal theorem and height theorem.

Time permitting : Artinian local rings and structure theorem of Artinian rings, Tensor products and multilinear forms, Exterior and Symmetric Algebra, Direct and Inverse system of modules. Fourier transform on , and theory, Complex interpolation, theory, Paley-Wiener theorem, Wiener-Tauberian theorem, Hilbert transform, Maximal function, real interpolation,Riesz transform, transference principle, Multipliers and Fourier Stieljes transform, Calderon-Zygmund singular integrals, Littlewood-Paley theory.

Differentiable manifolds; Tangent space. Direct inverse saturation theorems, Applications. Introduction to Banach algebras: Gelfand Transform, commutative Banach algebras, Gelfand-Naimark Theorem, Spectral Theorem for normal operators and its applications to operators on a Hilbert space. The theory of Fredholm operators: spectral theory of compact operators Fredholm Alternative.

Operator matrices: Invariant and Reducing subspaces, the theory of ideals of compact operators if time permits.

Banach Algebras and Spectral theory, Locally compact groups, Basic representation theory, Analysis on Locally compact abelian group, Analysis on compact groups, Group -algebra and structure of dual space. Reference materials. Amsterdam, Definitions and first examples.


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  • Local Algebra (Springer Monographs in Mathematics).
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Classical Lie algebras. Ideals and homomorphisms. Nilpotent Lie algebras. Engel's theorem. Solvable Lie algebras. Lie's theorem. Jordan Chevalley Decomposition. Radical and semisimplicity. The Killing form and Cartan's criterion. The structure of semisimple Lie algebras. Complete reducibility and Weyls theorem.


  • On Banach spaces, nonisomorphic to their Cartesian squares.
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Representation theory of the Lie algebra sl 2. Total subalgebras and root systems. Integrality properties. Simple Lie algebras and irreducible root systems. Curvature, Bianchi Identity, Sectional curvature. William M. Recalling definition of homotopy and fundamental group. Van Kampen theorem. Free groups and Free product of groups. Fundamental groups of sphere, Wedgeof circles. CW-complexes definition and examples. Statement of classification of compact surfaces and polygonal representation. Computation of fundamental groups of compact surfaces.

Homotopy extension property of a CW pair. Examples from Hatcher. Covering spaces, Path lifting, homotopy lifting, general lifting. Examples of covering of wedge of circles, circles. Universal cover and existence of covering, classification of covering by subgroups of fundamental groups. Deck transformation,action of fundamental group on Universal cover, normal covering. Applications: Subgroup of free group is free, Cayley complexes. Homology: Singular homology; Theory long exact, homotopy invariance and statement of excision, Mayer vietoris and computation for sphere, statement of Universal Coefficients Theorem Applications: Degree of sphere and applications, homology and fundamental groups.

Pre-requisite: None Only for Ph. Volterra and Fredholm integral equations, Resolvent Kernels. Operator equations, Fredholm theory, Hilbert Schmidt theory.

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Nonlinear integral equations, Singular integral equation. Elements of operator theory and Hilbert spaces; Introduction to the theory of distributions. Sobolev Spaces: Imbedding and compactness theorems, Fractional spaces and elements of trace theory. Applications to elliptic equations or parabolic equations. Basic results on system of linear equations, Cauchy-Binet formula for computing determinant, Rank factorization of a singular matrix, Vector Spaces associated with a matrix, Different types of generalized inverses, Moore-Penrose Inverse.

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Spectral Theorem of symmetric matrices, algebraic and geometric multiplicities, characteristic and minimal polynomials, Courant-Fischer Theorem, Interlacing theorems for eigenvalues, Quadratic forms, Positive definite matrices and its characterizations. Picard's theorem, Boundedness of solutions, Omega limit points of bounded trajectories. LaSalle's invariance principle; Stability via Lyapanov's indirect method, Converse Lyapanov functions, Sublevel sets of Lyapanov functions, Stability via Lyapanov's direct method, Converse Lyapanov's theorems, Brokett's theorem, Applications to control system; Stable and unstable manifolds of equilibria, Stable manifold theorem, Hartman Grobman theorem, Examples and applications, Center manifold theorem, Center manifold theorem, Normal form theory, Examples and applications to nonlinear systems and control; Poincare map, and stability theorems for periodic orbits; Elementary Bifurcation theory.

Review of topics in Functional Analysis and Sobolev Spaces, Mapping between Banach Spaces, Degree theory, Bifurcation theory, Variation method, Constrained critical points, deformation and Palais condition, Linking thorems, Mountain pass theorem and Ekeland's variation principle. Ambrosetti and A. Background from commutative algebra: Local rings, localizations, primary decomposition, Integral extensions, integral closures.

Algebraic geometry: Affine algebraic sets, Hilbert Nullstellansatz, Projective algebraic sets, projective Nullstellansatz, Affine varieties, structure sheaf, Prevarieties, varieties, morphisms, Affine and projective algebraic sets are varieties, dimension of varieties, products of varieties, images and fibers of morphisms, Tangent spaces, differential of a morphism, Smooth morphisms, smooth varieties, complete varieties. Algebraic Groups: Basic definitions and examples, Lie algebra of an algebraic group, Linear representations of algebraic groups, Affine algebraic groups are linear; connected projective algebraic groups are abelian varieties, rigidity of abelian varieties, quotients, Homogeneous spaces, Chevalley's theorem on algebraic groups without proof.

Quadratic forms: definition of quadratic forms and bilinear forms, equivalence of quadratic forms, local-global principle Hasse-Minkowski theorem , rational points on conics. Infinite Galois Theory: Profinite groups and profinite topology, Infinite Galois extensions and Galois group as profinite groups, absolute Galois groups, the fundamental theorem of Galois theory for infinite extensions , absolute Galois group of finite fields, Frobenius automorphism, absolute Galois group of and.

Geometry of curves over Affine Varieties and projective varieties, curves and function fields, divisors on curves, the Riemann-Roch theorem statement without proof , Elliptic curves over , Group law on elliptic curves, Weierstrass equations, action of the absolute Galois group of over points of elliptic curves, Weak Mordell-Weil Theorem, Mordell-Weil Theorem, Faltings' Theorem statement without proof. Joseph H. Silverman, The arithmetic of elliptic curves, GTM, vol.

Corrected reprint of the original. Harmonic analysis on : Fourier Transform, basic properties, inversion formula, Plancherel formula, Paley-Wiener theorem, Young's inequality. Tangent space of , inner product on the tangent spaces of , as a Riemannian symmetric space, geodesics on , horocycles. Iwasawa decomposition, Cartan decompostion of , Haar measures in these decompositions, unimodular group.

Laplace-Beltrami operator on and its eigenfunctions, the Helgason-Fourier transform, Radon transform, elementary spherical functions, spherical transform, Abel transform, assymptotics of elementary spherical functions. Gangolli, and V. Varadarajan: Harmonic analysis of spherical functions on real reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin, Helgason: Topics in harmonic analysis on homogeneous spaces, Progress in Mathematics, Helgason: Groups and geometric analysis, Integral geometry, invariant differential operators, and spherical functions.

Helgason: Geometric analysis on symmetric spaces, Second edition. Kunze and E. Stein: Uniformly bounded representations and harmonic analysis of the 2x2 real unimodular group, Amer. Brief review of simple and multiple linear regression along with the outlier detection methods. Basic idea of non-parametric regression.

Measures of robustness in different statistical problems e. Least squares and least absolute deviations in regression model; Least median squares and least trimmed squares estimators; Different statistical properties and the computational algorithms of the robust estimators of the location and the scale parameters for the univariate as well as the multivariate data ; Robust measure of association and robust testing of hypothesis problems.

Data-depth and the robust estimators based on the data-depth; Multivariate quantiles and its properties along with the computational algorithm; Possible extension of the depth-based and the quantile-based estimators for the functional data. Some applications of robust estimators e.

Ambrosetti and G. Prodi: A primer of Nonlinear Analysis, Cambridge studies in advanced mathematics. Malchiodi: Nonlinear Analysis and semilinear elliptic problems, Cambridge studies in advanced mathematics. Definition of manifolds and the fundamental ideas connected with them: Local coordinates, topological manifolds, differentiable manifolds, tangent spaces, vector fields, integral curves of vector fields and one-parameter group of local transformations, define manifold with boundary and orientation of a manifold.

Differential forms on differentiable manifolds: Differential forms on Euclidean n-spaces and on a general manifold, the exterior algebra, Interior product and Lie derivative, The Cartan formula and properties of Lie derivatives. Frobenius theorem. Applications of the de Rham theorem; Hopf invariant, the Massey product, cohomology of compact Lie groups, Mapping degree, Integral expression of the linking number by Gauss. The Hodge theorem and the Hodge decomposition of differential forms. Applications of the Hodge theorem. Indian edition Frank W. Warner: Foundations of differentiable manifolds and Lie groups, volume 94 of Graduate Texts in Mathematics.

Springer-Verlag, New York, Examples from nature and laboratory, Role of spatio-temporal models in biology. Linear Stability analysis: Formulation, Normal modes, Application to system of one, two and more variables. Spatial pattern formation: Reaction-diffusion system, Turing instability, Pattern formation in reaction-diffusion system. Chemotaxis: Introduction, Modelling chemotaxis, Linear and Nonlinear analysis.

Tumor modelling: Introduction, Models of tumor growth, Moving boundary problems, Response of immune system. Numerical simulation of Spatio-temporal model: Introduction, Finite-difference techniques, Monotone methods. Basic distribution theory, Moments of order statistics including recurrence relations, Bounds and approximations, Estimation of parameters, Life testing, Short cut procedures, Treatment of outliers, Asymptotic theory of extremes.

Estimation methods, Commonly encountered problems in estimation, Statistical inference, Multi-response nonlinear model, Asymptotic theory, Computational methods. Finite Fields, polynomial equations over finite fields; Chevalley-Warning theorem, Quadratic residue; law of quadratic reciprocity, p-adic numbers and p-adic integers, Quadratic forms over and.

Riemann Zeta function and Dirichlet L-functions, Dirichlet's theorem on primes in arithmetic progression, Modular forms for ; relation with elliptic curves. Results of convergence in almost sure sense and in probability, DCT, Basic inequalities, Conditional expectation, Methods of resampling. Introduction to discriminant analysis, Bayes' risk, and its properties.

Distance measures for density functions, and its relation with Bayes' risk. Empirical Bayes' risk and its convergence. Consistency results. Idea of curse of dimensionality, and the use of dimension reduction techniques like random projections principal component analysis, etc. Invariance results for models, Bisimulations, Finite model property, Translation into First order logic. Frame definability and Second order logic, Definable and undefinable properties. Completeness, Applications. Examples of some other modal systems and applications - Temporal logic with Since and Until, Multi-modal and Epistemic Logics.

Non-normal modal logics. Neighbourhood semantics. Blackburn, M. Approximation of functions, Numerical quadrature, Methods of numerical linear algebra, Numerical solutions of nonlinear systems and optimization, Numerical solution of ordinary and partial differential equations. A brief review of commutative algebra - localization, Noetherian rings and modules, integral extensions, Dedekind domains and discrete valuation ring, Spec of a ring. Number field, ring of integer, primes and ramifications, class group, finiteness of class number, Dirichlet's unit theorem, global fields, local fields, valuations Time permitting : Cyclotomic fields, zeta functions and L-functions, class number formula, adeles and ideles.

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