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Books on the topic 'Algebraic normal form'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 January 2023

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1

Manichev, Vladimir, Valentina Glazkova, and Кузьмина Анастасия. Numerical methods. The authentic and exact solution of the differential and algebraic equations in SAE systems of SAPR. ru: INFRA-M Academic Publishing LLC., 2016. http://dx.doi.org/10.12737/13138.

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In the manual classical numerical methods are considered and algorithms for the decision of systems of the ordinary differential equations (ODE), nonlinear and linear algebraic equations (NAU and LAU), and also ways of ensuring reliability and demanded accuracy of results of the decision. Ideas, which still not are stated are reflected in textbooks on calculus mathematics, namely: decision systems the ODE without reduction to a normal form of Cauchy resolved rather derivative, and refusal from any numerical an equivalent - nykh of transformations of the initial equations of mathematical models and is- the hodnykh of data because such transformations can change properties of models at a variation of coefficients in corresponding urav- neniyakh. It is intended for students, graduate students and teachers of higher education institutions in the direction of preparation "Informatics and computer facilities". The grant will also be useful for engineers and scientists on the corresponding specialties.

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2

Aleksani︠a︡n, A. A. Dizʺi︠u︡nktivnye normalʹnye formy nad lineĭnymi funkt︠s︡ii︠a︡mi: Teorii︠a︡ i prilozhenii︠a︡. Erevan: Izd-vo Erevanskogo universiteta, 1990.

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3

Zbigniew, Hajto, ed. Algebraic groups and differential Galois theory. Providence, R.I: American Mathematical Society, 2011.

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4

Ninul, Anatolij Sergeevič. Tenzornaja trigonometrija: Teorija i prilozenija / Theory and Applications /. Moscow, Russia: Mir Publisher, 2004.

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5

Ninul, Anatolij Sergeevič. Tensor Trigonometry. Moscow, Russia: Fizmatlit Publisher, 2021.

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6

Dufour, Jean-Paul, and Nguyen Tien Zung. Poisson Structures and Their Normal Forms. Springer London, Limited, 2006.

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7

Bokut, Leonid, Yuqun Chen, and Kyriakos Kalorkoti. Grobner-Shirshov Bases: Normal Forms, Combinatorial and Decision Problems in Algebra. World Scientific Publishing Co Pte Ltd, 2018.

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8

M¨uhlherr, Bernhard, Holger P. Petersson, and Richard M. Weiss. Quadratic Forms of Type E6, E7 and E8. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691166902.003.0008.

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This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

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9

Poisson Structures and Their Normal Forms (Progress in Mathematics). Birkhauser, 2005.

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10

Silva, Rapti Manohara De. Two spectral theorems: The Jordan canonical form for linear operators and the spectral theorem for normal operators. 1988.

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11

Lukas, Andre. The Oxford Linear Algebra for Scientists. Oxford University PressOxford, 2022. http://dx.doi.org/10.1093/oso/9780198844914.001.0001.

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Abstract This book provides a introduction into linear algebra which covers the mathematical set-up as well as applications to science. After the introductory material on sets, functions, groups and fields, the basic features of vector spaces are developed, including linear independence, bases, dimension, vector subspaces and linear maps. Practical methods for calculating with dot, cross and triple products are introduced early on. The theory of linear maps and their relation to matrices is developed in detail, culminating in the rank theorem. Algorithmic methods bases on row reduction and determinants are discussed an applied to computing the rank and the inverse of matrices and to solve systems of linear equations. Eigenvalues and eigenvectors and the application to diagonalising linear maps, as well as scalar products and unitary linear maps are covered in detail. Advanced topics included are the Jordon normal form, normal linear maps, the singular value decomposition, bi-linear and sesqui-linear forms, duality and tensors. The book also included short expositions of diverse scientific applications of linear algebra, including to internet search, classical mechanics, graph theory, cryptography, coding theory, data compression, special relativity, quantum mechanics and quantum computing.

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12

E, Tournier, ed. Computer algebra and differential equations. Cambridge: Cambridge University Press, 1994.

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13

Mann, Peter. Near-Equilibrium Oscillations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0012.

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In this chapter, the theory of near-equilibrium oscillations is developed and normal mode analysis is performed. This topic requires a little bit of linear algebra when dealing with matrices, as well as an understanding of differential equations. The chapter explores small perturbations (small nudges or tiny shifts) to a stable equilibrium point in configuration space and introduces the characteristic equation. Interdisciplinary examples are then investigated, including a surface science example in which the bond frequencies of surface adsorbates are calculated, an example in which the motion of atoms in a triatomic molecule is examined and an example in which the molecular physics of atomic force microscopy is analysed. The properties of the eigenvalue problem for small oscillations are also investigated.

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14

Haesemeyer, Christian, and Charles A. Weibel. The Norm Residue Theorem in Motivic Cohomology. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691191041.001.0001.

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This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

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15

Rajeev, S. G. Fluid Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.001.0001.

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Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using conformal transformations, viscous flows such as Couette and Stokes flow around a sphere, shocks in the Burgers equation. Prandtl’s boundary layer theory and the Blasius solution are presented. Rayleigh–Taylor instability is studied in analogy with the inverted pendulum, with a digression on Kapitza’s stabilization. The possibility of transients in a linearly stable system with a non-normal operator is studied using an example by Trefethen et al. The integrable models (KdV, Hasimoto’s vortex soliton) and their hamiltonian formalism are studied. Delving into deeper mathematics, geodesics on Lie groups are studied: first using the Lie algebra and then using Milnor’s approach to the curvature of the Lie group. Arnold’s deep idea that Euler’s equations are the geodesic equations on the diffeomorphism group is then explained and its curvature calculated. The next three chapters are an introduction to numerical methods: spectral methods based on Chebychev functions for ODEs, their application by Orszag to solve the Orr–Sommerfeld equation, finite difference methods for elementary PDEs, the Magnus formula and its application to geometric integrators for ODEs. Two appendices give an introduction to dynamical systems: Arnold’s cat map, homoclinic points, Smale’s horse shoe, Hausdorff dimension of the invariant set, Aref ’s example of chaotic advection. The last appendix introduces renormalization: Ising model on a Cayley tree and Feigenbaum’s theory of period doubling.

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16

Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments. Moscow, Russia: Fizmatlit Publisher, 2009.

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