1. What is the cardinality of the following sets
... 3. Does A ∩ C = B ∩ C imply A = B prove your answer. 4. Show (A − B) − C ⊂ A − C. 5. Use symbolic notation to write the definition of A ⊂ B. 6. Is the function f : Z → N defined by f (x) = x2 − x one to one? Justify your answer. 7. Is the function f : students in CS247 → eyecolor defined choosing th ...
... 3. Does A ∩ C = B ∩ C imply A = B prove your answer. 4. Show (A − B) − C ⊂ A − C. 5. Use symbolic notation to write the definition of A ⊂ B. 6. Is the function f : Z → N defined by f (x) = x2 − x one to one? Justify your answer. 7. Is the function f : students in CS247 → eyecolor defined choosing th ...
Some Notes on Compact Lie Groups
... obeying relations i2 = j 2 = k 2 = −1, ij = −ji = k (hence jk = −kj = i and ki = −ik = j). We define the conjugation x 7→ x in H by x 1 + ix2 + jx3 + kx4 7→ x1 − ix2 − jx3 − kx4 . We P have x · y = y · x and |x|2 = xx = xx = 4i=1 (xi )2 . In particular, a non-zero element x has the inverse x/|x|2 . ...
... obeying relations i2 = j 2 = k 2 = −1, ij = −ji = k (hence jk = −kj = i and ki = −ik = j). We define the conjugation x 7→ x in H by x 1 + ix2 + jx3 + kx4 7→ x1 − ix2 − jx3 − kx4 . We P have x · y = y · x and |x|2 = xx = xx = 4i=1 (xi )2 . In particular, a non-zero element x has the inverse x/|x|2 . ...
Groups
... • Associative: a(bc) = (ab)c • Identity: 1 G, 1a = a1 = a, aG • Inverse: a-1 G, a-1a = aa-1 = 1, a G ...
... • Associative: a(bc) = (ab)c • Identity: 1 G, 1a = a1 = a, aG • Inverse: a-1 G, a-1a = aa-1 = 1, a G ...
Neitzke: What is a BPS state?
... where X is a Riemannian six-manifold. Because of the factor E3,1 here this is a quantum system of the type we have been discussing, with a Hilbert space acted on by ISO(3, 1). Moreover, if we choose X to be a Calabi-Yau threefold, then the system is supersymmetric. Now let us consider the space of ...
... where X is a Riemannian six-manifold. Because of the factor E3,1 here this is a quantum system of the type we have been discussing, with a Hilbert space acted on by ISO(3, 1). Moreover, if we choose X to be a Calabi-Yau threefold, then the system is supersymmetric. Now let us consider the space of ...
4. Transition Matrices for Markov Chains. Expectation Operators. Let
... 4. Transition Matrices for Markov Chains. Expectation Operators. Let us consider a system that at any given time can be in one of a finite number of states. We shall identify the states by {1, 2, . . . , N }. The state of the system at time n will be denoted by xn . The system is ’noisy’ so that xn ...
... 4. Transition Matrices for Markov Chains. Expectation Operators. Let us consider a system that at any given time can be in one of a finite number of states. We shall identify the states by {1, 2, . . . , N }. The state of the system at time n will be denoted by xn . The system is ’noisy’ so that xn ...
Problem Set 7
... the Weyl group W . Give an explicit construction of the irreducible representations of G, compute their characters, and use the Weyl integration formula to show that they are orthonormal. Problem 3: For G = SU (3), explicitly define an infinite sequence of irreducible representations on spaces of ho ...
... the Weyl group W . Give an explicit construction of the irreducible representations of G, compute their characters, and use the Weyl integration formula to show that they are orthonormal. Problem 3: For G = SU (3), explicitly define an infinite sequence of irreducible representations on spaces of ho ...
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
... 11. Show that a non empty subset W of a vector space V over a field F is a subspace of V if and only if W is closed under addition and scalar multiplication. 12. Express (1, -2, 5) as a linear combination of the vectors { (1, 1, 1), (1, 2, 3), (2, -2, 1)} in R3. 13. Let v1, v2, . . . , vr be element ...
... 11. Show that a non empty subset W of a vector space V over a field F is a subspace of V if and only if W is closed under addition and scalar multiplication. 12. Express (1, -2, 5) as a linear combination of the vectors { (1, 1, 1), (1, 2, 3), (2, -2, 1)} in R3. 13. Let v1, v2, . . . , vr be element ...
Physics 880K20 (Quantum Computing): Problem Set 1. David Stroud, instructor
... Show that the requirement that (A’, B’) = (A, B) requires that the square matrix U is unitary, i. e., that U† U = I, where I is the identity matrix. (n × n if A and B have n components). 2. Consider a system described by a time-independent Hamiltonian. The solution of the time-dependent Schrodinger ...
... Show that the requirement that (A’, B’) = (A, B) requires that the square matrix U is unitary, i. e., that U† U = I, where I is the identity matrix. (n × n if A and B have n components). 2. Consider a system described by a time-independent Hamiltonian. The solution of the time-dependent Schrodinger ...
To Download! - CBSE PORTAL
... Q. 3. Let ‘R’ be a reflexive relation on a finite set ‘A’ having ‘n’ elements and let there be ‘m’ ordered pairs in ‘R’. Then a. b. c. d. none of these. Q. 4. Let ‘X’ be a family of sets and ‘R’ be a relation defined by ‘A is disjoint from B’ .Then R is a. b. c. d. ...
... Q. 3. Let ‘R’ be a reflexive relation on a finite set ‘A’ having ‘n’ elements and let there be ‘m’ ordered pairs in ‘R’. Then a. b. c. d. none of these. Q. 4. Let ‘X’ be a family of sets and ‘R’ be a relation defined by ‘A is disjoint from B’ .Then R is a. b. c. d. ...
Homework 4
... 18) Let G be a finite group and K a field with charK =/ 2. Consider an KG module V with basis v1 , . . . , vn and let W = V ⊗ V . We define a linear map α∶ W → W by prescribing images of the basis vectors as α∶ vi ⊗ vj ↦ vj ⊗ vi and set WS = {w ∈ W ∣ w α = w} and WA = {w ∈ W ∣ w α = −w}. (These are ...
... 18) Let G be a finite group and K a field with charK =/ 2. Consider an KG module V with basis v1 , . . . , vn and let W = V ⊗ V . We define a linear map α∶ W → W by prescribing images of the basis vectors as α∶ vi ⊗ vj ↦ vj ⊗ vi and set WS = {w ∈ W ∣ w α = w} and WA = {w ∈ W ∣ w α = −w}. (These are ...
Lecture 3 Operator methods in quantum mechanics
... particular basis, e.g. for Ĥ = 2m , we can represent p̂ in spatial coordinate basis, p̂ = −i!∂x , or in the momentum basis, p̂ = p. Equally, it would be useful to work with a basis for the wavefunction, ψ, which is coordinate-independent. ...
... particular basis, e.g. for Ĥ = 2m , we can represent p̂ in spatial coordinate basis, p̂ = −i!∂x , or in the momentum basis, p̂ = p. Equally, it would be useful to work with a basis for the wavefunction, ψ, which is coordinate-independent. ...
..
... 4. Let V .be the vector space of n x n matrices over IR. Consider the linear operator S:V -t V given by S(A) = (1/2)(A + AT), where T denotes . - transpose. Compute the rank and nullity of S. 5. Give n x n matrices over e that ha~e the following properties: i. the characteristic polynomial is (x + 1 ...
... 4. Let V .be the vector space of n x n matrices over IR. Consider the linear operator S:V -t V given by S(A) = (1/2)(A + AT), where T denotes . - transpose. Compute the rank and nullity of S. 5. Give n x n matrices over e that ha~e the following properties: i. the characteristic polynomial is (x + 1 ...
Formula and function intro
... Comma ( , ) combines multiple cell ranges into one reference. (SUM(B5:B15,D5:D15)) Space ( ) is the intersection operator, which produces a common reference for both cell ranges. (B7:D7 C6:C8) ...
... Comma ( , ) combines multiple cell ranges into one reference. (SUM(B5:B15,D5:D15)) Space ( ) is the intersection operator, which produces a common reference for both cell ranges. (B7:D7 C6:C8) ...