A system that is measured to be in a state |ni cannot simultaneously
... A system that is measured to be in a state |ni cannot simultaneously be measured to be in an orthogonal state |mi. The probabilities sum to unity because the system must be in some state. Since the density operator ⇢ is hermitian, it has a complete, orthonormal set of eigenvectors |ki all of which h ...
... A system that is measured to be in a state |ni cannot simultaneously be measured to be in an orthogonal state |mi. The probabilities sum to unity because the system must be in some state. Since the density operator ⇢ is hermitian, it has a complete, orthonormal set of eigenvectors |ki all of which h ...
![[2013 question paper]](http://s1.studyres.com/store/data/008844912_1-7494f7d6a29d900492173910cd64b54c-300x300.png)
[2013 question paper]
... (a) If G has nontrivial centre C, then G/C has trivial centre. (b) If G 6= 1, there exists a nontrivial homomorphism h : Z → G. (c) If |G| = p3 , for p a prime, then G is abelian. (d) If G is nonabelian, then it has a nontrivial automorphism. 3. Let C[0, 1] be the space of continuous real-valued fun ...
... (a) If G has nontrivial centre C, then G/C has trivial centre. (b) If G 6= 1, there exists a nontrivial homomorphism h : Z → G. (c) If |G| = p3 , for p a prime, then G is abelian. (d) If G is nonabelian, then it has a nontrivial automorphism. 3. Let C[0, 1] be the space of continuous real-valued fun ...
lecture18-lsi
... Each row and column of A gets mapped into the k-dimensional LSI space, by the SVD. Claim – this is not only the mapping with the best (Frobenius error) approximation to A, but in fact improves retrieval. A query q is also mapped into this space, by ...
... Each row and column of A gets mapped into the k-dimensional LSI space, by the SVD. Claim – this is not only the mapping with the best (Frobenius error) approximation to A, but in fact improves retrieval. A query q is also mapped into this space, by ...
first lecture - UC Davis Mathematics
... It’s easy to generalize the idea of representation theory, replacing the category Vect with any other category C we wish... • Representations in C are functors ρ : G → C. • ‘Intertwiners’ between reps are natural transformations. Let’s generalize this story to 2-groups! From now on, I’ll consider o ...
... It’s easy to generalize the idea of representation theory, replacing the category Vect with any other category C we wish... • Representations in C are functors ρ : G → C. • ‘Intertwiners’ between reps are natural transformations. Let’s generalize this story to 2-groups! From now on, I’ll consider o ...
EXPLORATION OF VARIOUS ITEMS IN LINEAR ALGEBRA
... 4.1. Maximal Eventown Solution. Now let us try and find a maximal solution to the Eventown problem, which happens to be quite simple. We know that there are n number of people in the town. If there is an even number of people, pair everyone with someone else, e.g. {1, 2}, {3, 4}, . . . , {n − 1, n}. ...
... 4.1. Maximal Eventown Solution. Now let us try and find a maximal solution to the Eventown problem, which happens to be quite simple. We know that there are n number of people in the town. If there is an even number of people, pair everyone with someone else, e.g. {1, 2}, {3, 4}, . . . , {n − 1, n}. ...
Lec 31: Inner products. An inner product on a vector space V
... (1) fails. Then A is not a matrix of inner product. Hence the formula (u, v) = uT Av (with symmetric A) defines an inner product in Rn , if and only if (u, u) = uT Au is positive for all nonzero u (so, (1) is satisfied). Such symmetric matrices A are called positive definite. Thus, positive definite ...
... (1) fails. Then A is not a matrix of inner product. Hence the formula (u, v) = uT Av (with symmetric A) defines an inner product in Rn , if and only if (u, u) = uT Au is positive for all nonzero u (so, (1) is satisfied). Such symmetric matrices A are called positive definite. Thus, positive definite ...
Solutions - Penn Math
... x2 if (and only if) d 6= 0. Inserting the values of x2 and x3 in the first equation, it can always be solved for x1 if (and only if) a 6= 0. Summary: An upper triangular matrix A is invertible if and only if none of its diagonal elements are 0. b) If A is invertible, is its inverse also upper triang ...
... x2 if (and only if) d 6= 0. Inserting the values of x2 and x3 in the first equation, it can always be solved for x1 if (and only if) a 6= 0. Summary: An upper triangular matrix A is invertible if and only if none of its diagonal elements are 0. b) If A is invertible, is its inverse also upper triang ...
Quasi-Shuffle Products
... Recently another pair of dual Hopf algebras has inspired much interest. The Hopf algebra Sym of noncommutative symmetric functions, introduced in [7], has as its graded dual the Hopf algebra of quasi-symmetric functions [5, 13]. In a recent paper of the author [12], the algebra of quasi-symmetric fu ...
... Recently another pair of dual Hopf algebras has inspired much interest. The Hopf algebra Sym of noncommutative symmetric functions, introduced in [7], has as its graded dual the Hopf algebra of quasi-symmetric functions [5, 13]. In a recent paper of the author [12], the algebra of quasi-symmetric fu ...
Chapter 8
... every element in the first matrix is equal to every corresponding element in the second. This means the two matrices must be the same size. • Matrix addition and subtraction are performed by adding or subtracting the corresponding elements. This requires that the two matrices be the same size. • Sca ...
... every element in the first matrix is equal to every corresponding element in the second. This means the two matrices must be the same size. • Matrix addition and subtraction are performed by adding or subtracting the corresponding elements. This requires that the two matrices be the same size. • Sca ...
Chapter 9 Lie Groups as Spin Groups
... where B is a bivector (section 1.10.2) called the generator of R or R, and the minus sign can usually be eliminated by a change in the definition of B. Thus every bivector determines a unique rotation. The bivector generators of a spin or rotation group form a Lie algebra under the commutator product ...
... where B is a bivector (section 1.10.2) called the generator of R or R, and the minus sign can usually be eliminated by a change in the definition of B. Thus every bivector determines a unique rotation. The bivector generators of a spin or rotation group form a Lie algebra under the commutator product ...
10.2. (continued) As we did in Example 5, we may compose any two
... 10.3. Classification by fixed points Main theorem. Any isometry of the plane is one of the ones in the table, namely: the identity, a rotation, a translation, a reflection, or a glide reflection. Let us first record two consequences of this theorem. Corollary 1. The composition of any two rotations, ...
... 10.3. Classification by fixed points Main theorem. Any isometry of the plane is one of the ones in the table, namely: the identity, a rotation, a translation, a reflection, or a glide reflection. Let us first record two consequences of this theorem. Corollary 1. The composition of any two rotations, ...
D. Applebaum SMOOTHNESS OF DENSITIES ON COMPACT LIE
... for all X,Y ∈ g. A maximal torus T in G is a maximal commutative subgroup of G. Its dimension r is called the rank of G. Here are some key facts about maximal tori. • Any σ ∈ G lies on some maximal torus. • Any two maximal tori are conjugate. Let t be the Lie algebra of T. Then it is a maximal abeli ...
... for all X,Y ∈ g. A maximal torus T in G is a maximal commutative subgroup of G. Its dimension r is called the rank of G. Here are some key facts about maximal tori. • Any σ ∈ G lies on some maximal torus. • Any two maximal tori are conjugate. Let t be the Lie algebra of T. Then it is a maximal abeli ...
Sequentially m-Barrelled Algebras
... A locally convex algebra E is called an m-barrelled algebra if every m-barrel (closed, circled, m-convex and absorbing set) is a neighbourhood of 0 in E [5]. A locally convex space is called a barrelled space (sequentially barrelled space) if every barrel, i.e. closed, circled, convex, absorbing set ...
... A locally convex algebra E is called an m-barrelled algebra if every m-barrel (closed, circled, m-convex and absorbing set) is a neighbourhood of 0 in E [5]. A locally convex space is called a barrelled space (sequentially barrelled space) if every barrel, i.e. closed, circled, convex, absorbing set ...
Bose, R.C. and J.N. Srivastava; (1963)Multidimensional partially balanced designs and their analysis, with applications to partially balanced factorial fractions."
... part of the analysis will however be discussed here; for once the best linear unbiased esti.ms.tes of the parameters have been obtained, the sum. of squares etc. and the analysis of variance table can be easily computed using standard methods discussed, for example, in [1, 6]. Consider the model (1) ...
... part of the analysis will however be discussed here; for once the best linear unbiased esti.ms.tes of the parameters have been obtained, the sum. of squares etc. and the analysis of variance table can be easily computed using standard methods discussed, for example, in [1, 6]. Consider the model (1) ...
Algebra 2 - Web Maths!
... Creating Expressions from Words (5 mins) Function Machines (easy) Function Machines (more complex) click 'Machines' to load Visualising Expressions using Area ...
... Creating Expressions from Words (5 mins) Function Machines (easy) Function Machines (more complex) click 'Machines' to load Visualising Expressions using Area ...
Eigenvalue perturbation theory of classes of structured
... that have rank one and are structure preserving. We also present results on the behavior of Jordan structures under rank one structured perturbations for rather general classes of structured matrices, both real and complex, that cover many particular cases and support the perturbation theory develop ...
... that have rank one and are structure preserving. We also present results on the behavior of Jordan structures under rank one structured perturbations for rather general classes of structured matrices, both real and complex, that cover many particular cases and support the perturbation theory develop ...
Linear Algebra In Dirac Notation
... of the momentum variable p. It is convenient to employ the Dirac symbol |ψi, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars, which are complex numbers, a ...
... of the momentum variable p. It is convenient to employ the Dirac symbol |ψi, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars, which are complex numbers, a ...
Sample examinations Linear Algebra (201-NYC-05) Autumn 2010 1. Given
... and that A and B are inverses (e.g., because AB = In implies that B has linearly independent columns, so n 6 m, and likewise BA = Im implies that A has linearly independent columns, so m 6 n). ...
... and that A and B are inverses (e.g., because AB = In implies that B has linearly independent columns, so n 6 m, and likewise BA = Im implies that A has linearly independent columns, so m 6 n). ...
What we have met before: why individual students, mathematicians
... An algebraic expression 2x+1 does not have an ‘answe numbers. Later, by definition, a square is a rectangle. Different symbols can represent the same thing. ...
... An algebraic expression 2x+1 does not have an ‘answe numbers. Later, by definition, a square is a rectangle. Different symbols can represent the same thing. ...
Sum of Squares seminar- Homework 0.
... they are not always tight. It is often easier to compute tr(Ak )1/k than trying to compute kAk directly, and as k grows this yields a better and better estimate. Exercise 3. Let A be an n × n symmetric matrix. Prove that the following are equivalent: n ...
... they are not always tight. It is often easier to compute tr(Ak )1/k than trying to compute kAk directly, and as k grows this yields a better and better estimate. Exercise 3. Let A be an n × n symmetric matrix. Prove that the following are equivalent: n ...
LECTURE 16: REPRESENTATIONS OF QUIVERS Introduction
... of parameters describing the orbits of indecomposable representations equals 1 − 0 = 1. We have seen that this is indeed the case. We are not going to prove all statements of Kac’s theorem. We will only check that the dimension of any indecomposable representation is a root, we will prove part (b), ...
... of parameters describing the orbits of indecomposable representations equals 1 − 0 = 1. We have seen that this is indeed the case. We are not going to prove all statements of Kac’s theorem. We will only check that the dimension of any indecomposable representation is a root, we will prove part (b), ...
spectral theorem for self-adjoint compact operators on Hilbert spaces
... Now we use a sort of induction. Let Y be the completion of the sum of all the eigenspaces. Then Y is T -stable. Let Z = Y ⊥ . We claim that Z is also T -stable, and that on the Hilbert space Z the (restriction of) T is a compact operator. Indeed, for z ∈ Z and y ∈ Y , we have hT z, yi = hz, T yi = 0 ...
... Now we use a sort of induction. Let Y be the completion of the sum of all the eigenspaces. Then Y is T -stable. Let Z = Y ⊥ . We claim that Z is also T -stable, and that on the Hilbert space Z the (restriction of) T is a compact operator. Indeed, for z ∈ Z and y ∈ Y , we have hT z, yi = hz, T yi = 0 ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.
... If T is an operator on X, then show that (Tx, x) = 0 T 0. b) i) If x' is a bound linear functional on a Hilbert space X, prove that there is a unique y o X such that x'(x) = (x, yo ), V x X . ii) If M and N are closed linear subspaces of a Hilbert space H and if P and Q are projections on M ...
... If T is an operator on X, then show that (Tx, x) = 0 T 0. b) i) If x' is a bound linear functional on a Hilbert space X, prove that there is a unique y o X such that x'(x) = (x, yo ), V x X . ii) If M and N are closed linear subspaces of a Hilbert space H and if P and Q are projections on M ...
Representations of a finite group in positive characteristic
... book [Cur99], from where the proof of Brauer’s theorem given here (originally due to Brauer himself) has been adapted. The standard reference for results on non-semisimple algebras and modular representations is [CR62]. A picture of developments in the general theory of modular representations up to ...
... book [Cur99], from where the proof of Brauer’s theorem given here (originally due to Brauer himself) has been adapted. The standard reference for results on non-semisimple algebras and modular representations is [CR62]. A picture of developments in the general theory of modular representations up to ...
A proof of the Jordan normal form theorem
... Av = µv for some 0 6= v ∈ Ker(A − λ Id)k , then (A − λ Id)v = (µ − λ)v, and 0 = (A − λ Id)k v = (µ − λ)k v, so µ = λ), and the dimension of Im(A − λ Id)k is less than dim V (if λ is an eigenvalue of A), so we can apply the induction hypothesis for A acting on the vector space V 0 = Im(A − λ Id)k . T ...
... Av = µv for some 0 6= v ∈ Ker(A − λ Id)k , then (A − λ Id)v = (µ − λ)v, and 0 = (A − λ Id)k v = (µ − λ)k v, so µ = λ), and the dimension of Im(A − λ Id)k is less than dim V (if λ is an eigenvalue of A), so we can apply the induction hypothesis for A acting on the vector space V 0 = Im(A − λ Id)k . T ...