Implementing Sparse Matrices for Graph Algorithms
... We argue that Assumption 2 is justified when the fast memory under consideration is either the L1 or L2 cache. Out-of-order CPUs can generally hide memory latencies from L1 cache misses, but not L2 cache misses [Hennessy & Patterson 2006]. Therefore, it is more reasonable to treat the L2 cache as the ...
... We argue that Assumption 2 is justified when the fast memory under consideration is either the L1 or L2 cache. Out-of-order CPUs can generally hide memory latencies from L1 cache misses, but not L2 cache misses [Hennessy & Patterson 2006]. Therefore, it is more reasonable to treat the L2 cache as the ...
Solvable Groups
... hei = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that, for each i, the subgroup Hi is normal in Hi+1 and the quotient group Hi+1 /Hi is Abelian. An Abelian group G is solvable; the chain of subgroups hei ⊂ G satisfies the definition. Also, the symmetric groups S3 and S4 are solvable by considering the chains hei ...
... hei = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that, for each i, the subgroup Hi is normal in Hi+1 and the quotient group Hi+1 /Hi is Abelian. An Abelian group G is solvable; the chain of subgroups hei ⊂ G satisfies the definition. Also, the symmetric groups S3 and S4 are solvable by considering the chains hei ...
LINEAR ALGEBRA TEXTBOOK LINK
... A set is a collection of things called elements of the set. For example, the set of integers, the collection of signed whole numbers such as 1,2,-4, and so on. This set, whose existence will be assumed, is denoted by Z. Other sets could be the set of people in a family or the set of donuts in a disp ...
... A set is a collection of things called elements of the set. For example, the set of integers, the collection of signed whole numbers such as 1,2,-4, and so on. This set, whose existence will be assumed, is denoted by Z. Other sets could be the set of people in a family or the set of donuts in a disp ...
Linear Algebra Notes
... 2. Questions of Understanding: I don’t get it! Admitting that you do not understand something is an act requiring utmost courage. But if you don’t, it is likely that many others in the audience also don’t. On the same vein, if you feel you can explain a point to an inquiring classmate, I will allow ...
... 2. Questions of Understanding: I don’t get it! Admitting that you do not understand something is an act requiring utmost courage. But if you don’t, it is likely that many others in the audience also don’t. On the same vein, if you feel you can explain a point to an inquiring classmate, I will allow ...
0 Stratification of globally defined Mackey functors
... Highest weight categories The simple GDMFs are parametrized by ...
... Highest weight categories The simple GDMFs are parametrized by ...
Explicit tensors - Computational Complexity
... What does it mean that “a field can be represented over {0, 1}∗ ”? Traditional complexity classes like P or NP are defined over some fixed alphabet, so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and ration ...
... What does it mean that “a field can be represented over {0, 1}∗ ”? Traditional complexity classes like P or NP are defined over some fixed alphabet, so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and ration ...
The discriminant
... Gl(n)A of invertible n ⇥ n-matrices with coefficients in A, and its determinant belongs therefore to groups A⇤ of units. The case of a number field In the all-important case of a number field the ring of integers A is a free Z-module, and the determinant of a transition matrix between two basises equa ...
... Gl(n)A of invertible n ⇥ n-matrices with coefficients in A, and its determinant belongs therefore to groups A⇤ of units. The case of a number field In the all-important case of a number field the ring of integers A is a free Z-module, and the determinant of a transition matrix between two basises equa ...
Subgroups of Finite Index in Profinite Groups
... Proof. Suppose that H and K are normal subgroups of G with G/H and G/K belonging to N l−1 . We will show by induction on l that G/(H ∩ K) belongs to N l−1 . If l = 1, then G is nilpotent, and therefore G/(H ∩ K) is a quotient of a nilpotent group. As factor groups of nilpotent groups are nilpotent, ...
... Proof. Suppose that H and K are normal subgroups of G with G/H and G/K belonging to N l−1 . We will show by induction on l that G/(H ∩ K) belongs to N l−1 . If l = 1, then G is nilpotent, and therefore G/(H ∩ K) is a quotient of a nilpotent group. As factor groups of nilpotent groups are nilpotent, ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.