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... compare with quantum theory is that in quantum theory, we need to work in a state space of fixed dimension. There are two aspects to the Boolean circuit: • The values taken by a finite set of variables at any given time. These correspond to the classical “state” of the system. • The gates themselves ...
... compare with quantum theory is that in quantum theory, we need to work in a state space of fixed dimension. There are two aspects to the Boolean circuit: • The values taken by a finite set of variables at any given time. These correspond to the classical “state” of the system. • The gates themselves ...
Lecture notes 3
... The orbit OrbG (x) of a transformation group G acting on X is the set of points attainable from x by a transformation, that is OrbG (x) = {y ∈ X : ∃ g ∈ G : φg (x) = y}. Clearly G acts transitively on each of its orbits. If G acts transitively on X, then X = OrbG (x) ∀x ∈ X. One may think of the set ...
... The orbit OrbG (x) of a transformation group G acting on X is the set of points attainable from x by a transformation, that is OrbG (x) = {y ∈ X : ∃ g ∈ G : φg (x) = y}. Clearly G acts transitively on each of its orbits. If G acts transitively on X, then X = OrbG (x) ∀x ∈ X. One may think of the set ...
Week 3 - people.bath.ac.uk
... This finishes the proof. 2 Definition. Let G be a group with a subgroup H. The number of left cosets of H in G is called the index of H in G and is denoted [G : H]. Remark. Suppose that G is finite. Recall from the proof of Lagrange’s Theorem that we get a partition of G into a union of pairwise dis ...
... This finishes the proof. 2 Definition. Let G be a group with a subgroup H. The number of left cosets of H in G is called the index of H in G and is denoted [G : H]. Remark. Suppose that G is finite. Recall from the proof of Lagrange’s Theorem that we get a partition of G into a union of pairwise dis ...
Definition of a quotient group. Let N ¢ G and consider as before the
... We are now going to see that any group can be thought of as a group of permutations. Theorem 4.1 (Cayley). Any group G is isomorphic to a subgroup of Sym (G). Proof For a ∈ G, let Ta ∈ Sym (G) be the permutation of G that arises from left multiplication by a. So Ta (x) = ax. Consider the map Φ : G → ...
... We are now going to see that any group can be thought of as a group of permutations. Theorem 4.1 (Cayley). Any group G is isomorphic to a subgroup of Sym (G). Proof For a ∈ G, let Ta ∈ Sym (G) be the permutation of G that arises from left multiplication by a. So Ta (x) = ax. Consider the map Φ : G → ...
Unit 2 - Irene McCormack Catholic College
... 2.3.7 define the imaginary number i as a root of the equation x2=−1 2.3.8 represent complex numbers in the form a+bi where a and b are the real and imaginary parts 2.3.9 determine and use complex conjugates 2.3.10 perform complex-number arithmetic: addition, subtraction, multiplication and division. ...
... 2.3.7 define the imaginary number i as a root of the equation x2=−1 2.3.8 represent complex numbers in the form a+bi where a and b are the real and imaginary parts 2.3.9 determine and use complex conjugates 2.3.10 perform complex-number arithmetic: addition, subtraction, multiplication and division. ...
Using Matrices to Perform Geometric Transformations
... by a number, called a scalar. In this example the number being multiplied by is 2. ...
... by a number, called a scalar. In this example the number being multiplied by is 2. ...
Full text
... and Their Applications, Pisa? Italy, July 25-29, 1988.' edited by G.E. Bergtim, A.N. Phllippou and A.F. Horadam This volume contains a selection of papers presented at the Third International Conference on Fibonacci Numbers and Their Applications. The topics covered include number patterns, linear r ...
... and Their Applications, Pisa? Italy, July 25-29, 1988.' edited by G.E. Bergtim, A.N. Phllippou and A.F. Horadam This volume contains a selection of papers presented at the Third International Conference on Fibonacci Numbers and Their Applications. The topics covered include number patterns, linear r ...
Solutions 8 - D-MATH
... 1. We start by eliminating non-simple groups of order < 60. First, recall that pgroups (pn , n > 1) have non-trivial center and that the center of a group is a normal subgroup. Recall from exercise sheet 7, that there are no simple groups of order pq or p2 q. The candidates left are then the groups ...
... 1. We start by eliminating non-simple groups of order < 60. First, recall that pgroups (pn , n > 1) have non-trivial center and that the center of a group is a normal subgroup. Recall from exercise sheet 7, that there are no simple groups of order pq or p2 q. The candidates left are then the groups ...
ALGEBRAIC OBJECTS 1. Binary Operators Let A be a set. A
... Note that these results are in B because Q itself is closed under addition and multiplication. Therefore a1 a2 + 2b1 b2 ∈ Q, and so forth. ¤ Exercise 1. In each case, we define a binary operation ∗ on R. Determine if ∗ is commutative and/or associative, find an identity if it exists, and find any in ...
... Note that these results are in B because Q itself is closed under addition and multiplication. Therefore a1 a2 + 2b1 b2 ∈ Q, and so forth. ¤ Exercise 1. In each case, we define a binary operation ∗ on R. Determine if ∗ is commutative and/or associative, find an identity if it exists, and find any in ...
The Geometric Realization of a Semi
... the distinct numbers t, and u I arranged in order. Set 6: = (wo, . . . , w,,,,). Then if p1 < . . . < p,-, are those integers p = 0, 1, . . . , n - 1 such that w,+, is not one of the t i , we have 6, = s,, . . . s,,- ,6: . Similarly 6; = s,, . . . s,,,-, 6; where the sets ( p L Jand ( v, J are disjo ...
... the distinct numbers t, and u I arranged in order. Set 6: = (wo, . . . , w,,,,). Then if p1 < . . . < p,-, are those integers p = 0, 1, . . . , n - 1 such that w,+, is not one of the t i , we have 6, = s,, . . . s,,- ,6: . Similarly 6; = s,, . . . s,,,-, 6; where the sets ( p L Jand ( v, J are disjo ...
IOSR Journal of Mathematics (IOSR-JM)
... unique invertible Hermitian matrix J with complex entries such that [x, y] = < x, Jy >, where <. , . > denotes the Euclidean inner product on ℂn , with an additional assumption on J, that is, J2 = I, to present the results with much algebraic ease. Thus an indefinite inner product space is a general ...
... unique invertible Hermitian matrix J with complex entries such that [x, y] = < x, Jy >, where <. , . > denotes the Euclidean inner product on ℂn , with an additional assumption on J, that is, J2 = I, to present the results with much algebraic ease. Thus an indefinite inner product space is a general ...
Solution 3 - D-MATH
... This implies that the group generated by f and g has the same multiplication table as S3 , hence the homomorphism defined by sending f to (12) and g to (123) gives an isomorphism of the two groups. 2. In the additive group Rm of vectors, let W be the set of solutions of a system of homogeneous linea ...
... This implies that the group generated by f and g has the same multiplication table as S3 , hence the homomorphism defined by sending f to (12) and g to (123) gives an isomorphism of the two groups. 2. In the additive group Rm of vectors, let W be the set of solutions of a system of homogeneous linea ...
Algebra Qualifying Exam January 2015
... non-zero prime ideal of R is maximal. (b) Give an example of a commutative ring R and a non-zero prime ideal I that is not maximal. (c) Let K a field which is NOT algebraically closed. Give an example of a maximal ideal of the ring R = K[X, Y ] which is NOT of the form (X − a, Y − b) with a, b ∈ K ( ...
... non-zero prime ideal of R is maximal. (b) Give an example of a commutative ring R and a non-zero prime ideal I that is not maximal. (c) Let K a field which is NOT algebraically closed. Give an example of a maximal ideal of the ring R = K[X, Y ] which is NOT of the form (X − a, Y − b) with a, b ∈ K ( ...
Jordan-Wigner Transform: Simulation of Fermionic Creation and
... The Jordan-Wigner transform allows us to take a system of interacting Fermions (second quantization, occupation notation), and map it into an equivalent model of interacting spins, which can then, in principle, be simulated using standard techniques in a quantum computer. This enables to use quantum ...
... The Jordan-Wigner transform allows us to take a system of interacting Fermions (second quantization, occupation notation), and map it into an equivalent model of interacting spins, which can then, in principle, be simulated using standard techniques in a quantum computer. This enables to use quantum ...
p-Groups - Brandeis
... saying that x commutes with every element of the group, i.e., x ∈ Z(P ) ∩ N .] However, we know that the number of fixed points is congruent modulo p to |N | which is congruent to 0. Therefore there must be at least p fixed points so |Z(P ) ∩ N | ≥ p. Corollary 3.2. Given a p-group P of order pk the ...
... saying that x commutes with every element of the group, i.e., x ∈ Z(P ) ∩ N .] However, we know that the number of fixed points is congruent modulo p to |N | which is congruent to 0. Therefore there must be at least p fixed points so |Z(P ) ∩ N | ≥ p. Corollary 3.2. Given a p-group P of order pk the ...
Solutions
... 2. By substituting x = 2 in the quotient ring Z[x]/(x2 − 7, x − 2) we obtain that it is the field Z/(3). So x − 2, and similarily x + 2 is a maximal ideal. These two maximal ideals are not associates. Because, assume that we have a + bx such that 2 + x = (2 − x)(a + bx) = 2a − 7b + x(2b − a). This i ...
... 2. By substituting x = 2 in the quotient ring Z[x]/(x2 − 7, x − 2) we obtain that it is the field Z/(3). So x − 2, and similarily x + 2 is a maximal ideal. These two maximal ideals are not associates. Because, assume that we have a + bx such that 2 + x = (2 − x)(a + bx) = 2a − 7b + x(2b − a). This i ...
Education - Denison University
... Contractive projections and operator spaces (with B. Russo), Trans. Amer. Math. Soc., Vol. 355, No.6, p. 2223-2262, (2003) Operator space characterizations of C*-algebras and ternary rings (with B. Russo), Pac. J. Math., Vol. 209, No. 2, (2003). Spectrum preserving maps on JBW*-triples, Archiv der M ...
... Contractive projections and operator spaces (with B. Russo), Trans. Amer. Math. Soc., Vol. 355, No.6, p. 2223-2262, (2003) Operator space characterizations of C*-algebras and ternary rings (with B. Russo), Pac. J. Math., Vol. 209, No. 2, (2003). Spectrum preserving maps on JBW*-triples, Archiv der M ...
On the q-exponential of matrix q-Lie algebras
... i,j=0 (the first index denotes the row) is defined by the formula X detq α ≡ sign(π)a0π(0) τ(a1π(1) ) . . . ξ (a n−1π(n−1) ). ...
... i,j=0 (the first index denotes the row) is defined by the formula X detq α ≡ sign(π)a0π(0) τ(a1π(1) ) . . . ξ (a n−1π(n−1) ). ...
Proceedings of the American Mathematical Society, 3, 1952, pp. 382
... orthog~nal.~ PROOF. The sufficiency is true by definition. By Lemma 3.2, if A has an inverse, both A and A-I are row and column consistent. Thus, using Lemma 4.1, it is sufficient to show AiknA jk = O = Akir\Akj for every i,j , k, i#j. We shall carry this out only for the first case as the other is ...
... orthog~nal.~ PROOF. The sufficiency is true by definition. By Lemma 3.2, if A has an inverse, both A and A-I are row and column consistent. Thus, using Lemma 4.1, it is sufficient to show AiknA jk = O = Akir\Akj for every i,j , k, i#j. We shall carry this out only for the first case as the other is ...
Lecture 1: Lie algebra cohomology
... We can take M = g with the adjoint representation # = ad. The groups H• (g; g) contain structural information about g. It can be shown, for example, that H1 (g; g) is the space of outer derivations, whereas H2 (g; g) is the space of nontrivial infinitesimal deformations. Similarly the obstructions t ...
... We can take M = g with the adjoint representation # = ad. The groups H• (g; g) contain structural information about g. It can be shown, for example, that H1 (g; g) is the space of outer derivations, whereas H2 (g; g) is the space of nontrivial infinitesimal deformations. Similarly the obstructions t ...
we defined the Poisson boundaries for semisimple Lie groups
... a sequence space, M is compact and metrizable. Moreover one sees quite easily that M is a Dynkin space for F2. This construction is essentially due to Dynkin and Malyutov [3] who use it to construct a Martin boundary for a class of harmonic functions on the free group, A similar construction may be ...
... a sequence space, M is compact and metrizable. Moreover one sees quite easily that M is a Dynkin space for F2. This construction is essentially due to Dynkin and Malyutov [3] who use it to construct a Martin boundary for a class of harmonic functions on the free group, A similar construction may be ...
Hurwitz`s Theorem
... for all x1 , . . . , xn , y1 , . . . , yn in F, where each zk is an F-bilinear function of the x’s and the y’s, then n = 1, 2, 4 or 8. Hurwitz’s original proof was stated for F = C, but the field of scalars only needs to be of characteristic not equal to 2 for his proof to work. Nothing would be los ...
... for all x1 , . . . , xn , y1 , . . . , yn in F, where each zk is an F-bilinear function of the x’s and the y’s, then n = 1, 2, 4 or 8. Hurwitz’s original proof was stated for F = C, but the field of scalars only needs to be of characteristic not equal to 2 for his proof to work. Nothing would be los ...
Notes
... A subset H ⊂ G is a subgroup if H is a group by using the binary operation of G, denoted H < G. Example 1.1.2. Let G = (R, +). The subset S = R − {0} ⊂ G is a group under multiplication. But it is not a subgroup of G because the operation on S is different from the one on G. Let K = Z. Then K is a s ...
... A subset H ⊂ G is a subgroup if H is a group by using the binary operation of G, denoted H < G. Example 1.1.2. Let G = (R, +). The subset S = R − {0} ⊂ G is a group under multiplication. But it is not a subgroup of G because the operation on S is different from the one on G. Let K = Z. Then K is a s ...