Linear Algebra I
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Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Orthogonal Complements (Revised Version)
Orthogonal Complements (Revised Version) Math 108A: May 19, 2010 John Douglas Moore 1 The dot product You will recall that the dot product was discussed in earlier calculus courses. If n x = (x1: : : : ; xn) and y = (y1: : : : ; yn) are elements of R , we define their dot product by x · y = x1y1 + ··· + xnyn: The dot product satisfies several key axioms: 1. it is symmetric: x · y = y · x; 2. it is bilinear: (ax + x0) · y = a(x · y) + x0 · y; 3. and it is positive-definite: x · x ≥ 0 and x · x = 0 if and only if x = 0. The dot product is an example of an inner product on the vector space V = Rn over R; inner products will be treated thoroughly in Chapter 6 of [1]. Recall that the length of an element x 2 Rn is defined by p jxj = x · x: Note that the length of an element x 2 Rn is always nonnegative. Cauchy-Schwarz Theorem. If x 6= 0 and y 6= 0, then x · y −1 ≤ ≤ 1: (1) jxjjyj Sketch of proof: If v is any element of Rn, then v · v ≥ 0. Hence (x(y · y) − y(x · y)) · (x(y · y) − y(x · y)) ≥ 0: Expanding using the axioms for dot product yields (x · x)(y · y)2 − 2(x · y)2(y · y) + (x · y)2(y · y) ≥ 0 or (x · x)(y · y)2 ≥ (x · y)2(y · y): 1 Dividing by y · y, we obtain (x · y)2 jxj2jyj2 ≥ (x · y)2 or ≤ 1; jxj2jyj2 and (1) follows by taking the square root. -
Does Geometric Algebra Provide a Loophole to Bell's Theorem?
Discussion Does Geometric Algebra provide a loophole to Bell’s Theorem? Richard David Gill 1 1 Leiden University, Faculty of Science, Mathematical Institute; [email protected] Version October 30, 2019 submitted to Entropy Abstract: Geometric Algebra, championed by David Hestenes as a universal language for physics, was used as a framework for the quantum mechanics of interacting qubits by Chris Doran, Anthony Lasenby and others. Independently of this, Joy Christian in 2007 claimed to have refuted Bell’s theorem with a local realistic model of the singlet correlations by taking account of the geometry of space as expressed through Geometric Algebra. A series of papers culminated in a book Christian (2014). The present paper first explores Geometric Algebra as a tool for quantum information and explains why it did not live up to its early promise. In summary, whereas the mapping between 3D geometry and the mathematics of one qubit is already familiar, Doran and Lasenby’s ingenious extension to a system of entangled qubits does not yield new insight but just reproduces standard QI computations in a clumsy way. The tensor product of two Clifford algebras is not a Clifford algebra. The dimension is too large, an ad hoc fix is needed, several are possible. I further analyse two of Christian’s earliest, shortest, least technical, and most accessible works (Christian 2007, 2011), exposing conceptual and algebraic errors. Since 2015, when the first version of this paper was posted to arXiv, Christian has published ambitious extensions of his theory in RSOS (Royal Society - Open Source), arXiv:1806.02392, and in IEEE Access, arXiv:1405.2355. -
5. Polynomial Rings Let R Be a Commutative Ring. a Polynomial of Degree N in an Indeterminate (Or Variable) X with Coefficients
5. polynomial rings Let R be a commutative ring. A polynomial of degree n in an indeterminate (or variable) x with coefficients in R is an expression of the form f = f(x)=a + a x + + a xn, 0 1 ··· n where a0, ,an R and an =0.Wesaythata0, ,an are the coefficients of f and n ··· ∈ ̸ ··· that anx is the highest degree term of f. A polynomial is determined by its coeffiecients. m i n i Two polynomials f = i=0 aix and g = i=1 bix are equal if m = n and ai = bi for i =0, 1, ,n. ! ! Degree··· of a polynomial is a non-negative integer. The polynomials of degree zero are just the elements of R, these are called constant polynomials, or simply constants. Let R[x] denote the set of all polynomials in the variable x with coefficients in R. The addition and 2 multiplication of polynomials are defined in the usual manner: If f(x)=a0 + a1x + a2x + a x3 + and g(x)=b + b x + b x2 + b x3 + are two elements of R[x], then their sum is 3 ··· 0 1 2 3 ··· f(x)+g(x)=(a + b )+(a + b )x +(a + b )x2 +(a + b )x3 + 0 0 1 1 2 2 3 3 ··· and their product is defined by f(x) g(x)=a b +(a b + a b )x +(a b + a b + a b )x2 +(a b + a b + a b + a b )x3 + · 0 0 0 1 1 0 0 2 1 1 2 0 0 3 1 2 2 1 3 0 ··· Let 1 denote the constant polynomial 1 and 0 denote the constant polynomial zero. -
Signing a Linear Subspace: Signature Schemes for Network Coding
Signing a Linear Subspace: Signature Schemes for Network Coding Dan Boneh1?, David Freeman1?? Jonathan Katz2???, and Brent Waters3† 1 Stanford University, {dabo,dfreeman}@cs.stanford.edu 2 University of Maryland, [email protected] 3 University of Texas at Austin, [email protected]. Abstract. Network coding offers increased throughput and improved robustness to random faults in completely decentralized networks. In contrast to traditional routing schemes, however, network coding requires intermediate nodes to modify data packets en route; for this reason, standard signature schemes are inapplicable and it is a challenge to provide resilience to tampering by malicious nodes. Here, we propose two signature schemes that can be used in conjunction with network coding to prevent malicious modification of data. In particular, our schemes can be viewed as signing linear subspaces in the sense that a signature σ on V authenticates exactly those vectors in V . Our first scheme is homomorphic and has better performance, with both public key size and per-packet overhead being constant. Our second scheme does not rely on random oracles and uses weaker assumptions. We also prove a lower bound on the length of signatures for linear subspaces showing that both of our schemes are essentially optimal in this regard. 1 Introduction Network coding [1, 25] refers to a general class of routing mechanisms where, in contrast to tra- ditional “store-and-forward” routing, intermediate nodes modify data packets in transit. Network coding has been shown to offer a number of advantages with respect to traditional routing, the most well-known of which is the possibility of increased throughput in certain network topologies (see, e.g., [21] for measurements of the improvement network coding gives even for unicast traffic). -
Quantification of Stability in Systems of Nonlinear Ordinary Differential Equations Jason Terry
University of New Mexico UNM Digital Repository Mathematics & Statistics ETDs Electronic Theses and Dissertations 2-14-2014 Quantification of Stability in Systems of Nonlinear Ordinary Differential Equations Jason Terry Follow this and additional works at: https://digitalrepository.unm.edu/math_etds Recommended Citation Terry, Jason. "Quantification of Stability in Systems of Nonlinear Ordinary Differential Equations." (2014). https://digitalrepository.unm.edu/math_etds/48 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Mathematics & Statistics ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Candidate Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: , Chairperson Quantification of Stability in Systems of Nonlinear Ordinary Differential Equations by Jason Terry B.A., Mathematics, California State University Fresno, 2003 B.S., Computer Science, California State University Fresno, 2003 M.A., Interdiscipline Studies, California State University Fresno, 2005 M.S., Applied Mathematics, University of New Mexico, 2009 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Mathematics The University of New Mexico Albuquerque, New Mexico December, 2013 c 2013, Jason Terry iii Dedication To my mom. iv Acknowledgments I would like -
Lecture 6: Linear Codes 1 Vector Spaces 2 Linear Subspaces 3
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 6: Linear Codes January 26, 2009 Lecturer: Atri Rudra Scribe: Steve Uurtamo 1 Vector Spaces A vector space V over a field F is an abelian group under “+” such that for every α ∈ F and every v ∈ V there is an element αv ∈ V , and such that: i) α(v1 + v2) = αv1 + αv2, for α ∈ F, v1, v2 ∈ V. ii) (α + β)v = αv + βv, for α, β ∈ F, v ∈ V. iii) α(βv) = (αβ)v for α, β ∈ F, v ∈ V. iv) 1v = v for all v ∈ V, where 1 is the unit element of F. We can think of the field F as being a set of “scalars” and the set V as a set of “vectors”. If the field F is a finite field, and our alphabet Σ has the same number of elements as F , we can associate strings from Σn with vectors in F n in the obvious way, and we can think of codes C as being subsets of F n. 2 Linear Subspaces Assume that we’re dealing with a vector space of dimension n, over a finite field with q elements. n We’ll denote this as: Fq . Linear subspaces of a vector space are simply subsets of the vector space that are closed under vector addition and scalar multiplication: n n In particular, S ⊆ Fq is a linear subspace of Fq if: i) For all v1, v2 ∈ S, v1 + v2 ∈ S. ii) For all α ∈ Fq, v ∈ S, αv ∈ S. Note that the vector space itself is a linear subspace, and that the zero vector is always an element of every linear subspace. -
Worksheet 1, for the MATLAB Course by Hans G. Feichtinger, Edinburgh, Jan
Worksheet 1, for the MATLAB course by Hans G. Feichtinger, Edinburgh, Jan. 9th Start MATLAB via: Start > Programs > School applications > Sci.+Eng. > Eng.+Electronics > MATLAB > R2007 (preferably). Generally speaking I suggest that you start your session by opening up a diary with the command: diary mydiary1.m and concluding the session with the command diary off. If you want to save your workspace you my want to call save today1.m in order to save all the current variable (names + values). Moreover, using the HELP command from MATLAB you can get help on more or less every MATLAB command. Try simply help inv or help plot. 1. Define an arbitrary (random) 3 × 3 matrix A, and check whether it is invertible. Calculate the inverse matrix. Then define an arbitrary right hand side vector b and determine the (unique) solution to the equation A ∗ x = b. Find in two different ways the solution to this problem, by either using the inverse matrix, or alternatively by applying Gauss elimination (i.e. the RREF command) to the extended system matrix [A; b]. In addition look at the output of the command rref([A,eye(3)]). What does it tell you? 2. Produce an \arbitrary" 7 × 7 matrix of rank 5. There are at east two simple ways to do this. Either by factorization, i.e. by obtaining it as a product of some 7 × 5 matrix with another random 5 × 7 matrix, or by purposely making two of the rows or columns linear dependent from the remaining ones. Maybe the second method is interesting (if you have time also try the first one): 3. -
Formal Power Series Rings, Inverse Limits, and I-Adic Completions of Rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely many variables over a ring R, and show that it is Noetherian when R is. But we begin with a definition in much greater generality. Let S be a commutative semigroup (which will have identity 1S = 1) written multi- plicatively. The semigroup ring of S with coefficients in R may be thought of as the free R-module with basis S, with multiplication defined by the rule h k X X 0 0 X X 0 ( risi)( rjsj) = ( rirj)s: i=1 j=1 s2S 0 sisj =s We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s 2 S, f(s1; s2) 2 S × S : s1s2 = sg is finite. Thus, each element of S has only finitely many factorizations as a product of two k1 kn elements. For example, we may take S to be the set of all monomials fx1 ··· xn : n (k1; : : : ; kn) 2 N g in n variables. For this chocie of S, the usual semigroup ring R[S] may be identified with the polynomial ring R[x1; : : : ; xn] in n indeterminates over R. We next construct a formal semigroup ring denoted R[[S]]: we may think of this ring formally as consisting of all functions from S to R, but we shall indicate elements of the P ring notationally as (possibly infinite) formal sums s2S rss, where the function corre- sponding to this formal sum maps s to rs for all s 2 S. -
The Method of Coalgebra: Exercises in Coinduction
The Method of Coalgebra: exercises in coinduction Jan Rutten CWI & RU [email protected] Draft d.d. 8 July 2018 (comments are welcome) 2 Draft d.d. 8 July 2018 Contents 1 Introduction 7 1.1 The method of coalgebra . .7 1.2 History, roughly and briefly . .7 1.3 Exercises in coinduction . .8 1.4 Enhanced coinduction: algebra and coalgebra combined . .8 1.5 Universal coalgebra . .8 1.6 How to read this book . .8 1.7 Acknowledgements . .9 2 Categories { where coalgebra comes from 11 2.1 The basic definitions . 11 2.2 Category theory in slogans . 12 2.3 Discussion . 16 3 Algebras and coalgebras 19 3.1 Algebras . 19 3.2 Coalgebras . 21 3.3 Discussion . 23 4 Induction and coinduction 25 4.1 Inductive and coinductive definitions . 25 4.2 Proofs by induction and coinduction . 28 4.3 Discussion . 32 5 The method of coalgebra 33 5.1 Basic types of coalgebras . 34 5.2 Coalgebras, systems, automata ::: ....................... 34 6 Dynamical systems 37 6.1 Homomorphisms of dynamical systems . 38 6.2 On the behaviour of dynamical systems . 41 6.3 Discussion . 44 3 4 Draft d.d. 8 July 2018 7 Stream systems 45 7.1 Homomorphisms and bisimulations of stream systems . 46 7.2 The final system of streams . 52 7.3 Defining streams by coinduction . 54 7.4 Coinduction: the bisimulation proof method . 59 7.5 Moessner's Theorem . 66 7.6 The heart of the matter: circularity . 72 7.7 Discussion . 76 8 Deterministic automata 77 8.1 Basic definitions . 78 8.2 Homomorphisms and bisimulations of automata . -
NEW TRENDS in SYZYGIES: 18W5133
NEW TRENDS IN SYZYGIES: 18w5133 Giulio Caviglia (Purdue University, Jason McCullough (Iowa State University) June 24, 2018 – June 29, 2018 1 Overview of the Field Since the pioneering work of David Hilbert in the 1890s, syzygies have been a central area of research in commutative algebra. The idea is to approximate arbitrary modules by free modules. Let R be a Noetherian, commutative ring and let M be a finitely generated R-module. A free resolution is an exact sequence of free ∼ R-modules ···! Fi+1 ! Fi !··· F0 such that H0(F•) = M. Elements of Fi are called ith syzygies of M. When R is local or graded, we can choose a minimal free resolution of M, meaning that a basis for Fi is mapped onto a minimal set of generators of Ker(Fi−1 ! Fi−2). In this case, we get uniqueness of minimal free resolutions up to isomorphism of complexes. Therefore, invariants defined by minimal free resolutions yield invariants of the module being resolved. In most cases, minimal resolutions are infinite, but over regular rings, like polynomial and power series rings, all finitely generated modules have finite minimal free resolutions. Beyond the study of invariants of modules, syzygies have connections to algebraic geometry, algebraic topology, and combinatorics. Statements like the Total Rank Conjecture connect algebraic topology with free resolutions. Bounds on Castelnuovo-Mumford regularity and projective dimension, as with the Eisenbud- Goto Conjecture and Stillman’s Conjecture, have implications for the computational complexity of Grobner¨ bases and computational algebra systems. Green’s Conjecture provides a link between graded free resolutions and the geometry of canonical curves. -
3. Hilbert Spaces
3. Hilbert spaces In this section we examine a special type of Banach spaces. We start with some algebraic preliminaries. Definition. Let K be either R or C, and let Let X and Y be vector spaces over K. A map φ : X × Y → K is said to be K-sesquilinear, if • for every x ∈ X, then map φx : Y 3 y 7−→ φ(x, y) ∈ K is linear; • for every y ∈ Y, then map φy : X 3 y 7−→ φ(x, y) ∈ K is conjugate linear, i.e. the map X 3 x 7−→ φy(x) ∈ K is linear. In the case K = R the above properties are equivalent to the fact that φ is bilinear. Remark 3.1. Let X be a vector space over C, and let φ : X × X → C be a C-sesquilinear map. Then φ is completely determined by the map Qφ : X 3 x 7−→ φ(x, x) ∈ C. This can be see by computing, for k ∈ {0, 1, 2, 3} the quantity k k k k k k k Qφ(x + i y) = φ(x + i y, x + i y) = φ(x, x) + φ(x, i y) + φ(i y, x) + φ(i y, i y) = = φ(x, x) + ikφ(x, y) + i−kφ(y, x) + φ(y, y), which then gives 3 1 X (1) φ(x, y) = i−kQ (x + iky), ∀ x, y ∈ X. 4 φ k=0 The map Qφ : X → C is called the quadratic form determined by φ. The identity (1) is referred to as the Polarization Identity.