3-Calabi-Yau Algebras from Steiner Systems
... associative algebra in the way described below. We refer the reader to Bocklandt’s work in [Boc08] for a deeper approach on the subject. Fix k a ground field of characteristic zero; all tensor products are over k unless we say otherwise. Let V be a finite dimensional vector space and let X = { x1 , ...
... associative algebra in the way described below. We refer the reader to Bocklandt’s work in [Boc08] for a deeper approach on the subject. Fix k a ground field of characteristic zero; all tensor products are over k unless we say otherwise. Let V be a finite dimensional vector space and let X = { x1 , ...
Central limit theorems for linear statistics of heavy tailed random
... is also an exploding moments Wigner matrix, with Φ(λ) = p(e−iλ − 1) (the measure m of Lemma 1.3 is pδ1 ). In this case the fluctuations were already studied in [43]. The method of [43] can be adapted to study the fluctuations of linear statistics of Wigner matrices with exploding moments. Neverthele ...
... is also an exploding moments Wigner matrix, with Φ(λ) = p(e−iλ − 1) (the measure m of Lemma 1.3 is pδ1 ). In this case the fluctuations were already studied in [43]. The method of [43] can be adapted to study the fluctuations of linear statistics of Wigner matrices with exploding moments. Neverthele ...
ALTERNATING TRILINEAR FORMS AND GROUPS OF EXPONENT
... Then V has a subspace of codimension at most r(r — 1) on which every element of X vanishes. PROOF. We prove the lemma by induction on r, the case r = 1 being immediate. We may suppose that X contains a form Sj of rank precisely 2(r — 1). Then Ui = { u e F l S ^ u , ! ) ) = 0 for all veV) is of codim ...
... Then V has a subspace of codimension at most r(r — 1) on which every element of X vanishes. PROOF. We prove the lemma by induction on r, the case r = 1 being immediate. We may suppose that X contains a form Sj of rank precisely 2(r — 1). Then Ui = { u e F l S ^ u , ! ) ) = 0 for all veV) is of codim ...
Computing the sign or the value of the determinant of an integer
... In algebraic complexity—i.e. when counting the number of operations in an abstract domain—we refer to Strassen [52] and Bunch and Hopcroft [13] for the reduction of the problem of computing the determinant to matrix multiplication. Conversely, Strassen [53] and Bunch and Hopcroft [13] reduce matrix ...
... In algebraic complexity—i.e. when counting the number of operations in an abstract domain—we refer to Strassen [52] and Bunch and Hopcroft [13] for the reduction of the problem of computing the determinant to matrix multiplication. Conversely, Strassen [53] and Bunch and Hopcroft [13] reduce matrix ...
notes on the subspace theorem
... Roth’s Theorem is easy to prove if α ∈ C\R, or if α is a real quadratic number. For real algebraic numbers α of degree > 3, the proof of Roth’s Theorem is very difficult. In the formulation of the Subspace Theorem, we need some notions from linear algebra, which we recall below. Let n be an integer ...
... Roth’s Theorem is easy to prove if α ∈ C\R, or if α is a real quadratic number. For real algebraic numbers α of degree > 3, the proof of Roth’s Theorem is very difficult. In the formulation of the Subspace Theorem, we need some notions from linear algebra, which we recall below. Let n be an integer ...
On the existence of equiangular tight frames
... The focus of this paper is a geometric object called an equiangular tight frame (ETF). An ETF can be viewed as a set of unit vectors in a Hilbert space with the property that the absolute inner products between pairs of vectors are (i) identical and (ii) minimal. As a result, ETFs generalize the geo ...
... The focus of this paper is a geometric object called an equiangular tight frame (ETF). An ETF can be viewed as a set of unit vectors in a Hilbert space with the property that the absolute inner products between pairs of vectors are (i) identical and (ii) minimal. As a result, ETFs generalize the geo ...
Construction of Transition Matrices for Reversible Markov Chains
... I hereby certify that I am the sole author of this major paper and that no part of this major paper has been published or submitted for publication. I certify that, to the best of my knowledge, my major paper does not infringe upon anyone’s copyright nor violate any proprietary rights and that any i ...
... I hereby certify that I am the sole author of this major paper and that no part of this major paper has been published or submitted for publication. I certify that, to the best of my knowledge, my major paper does not infringe upon anyone’s copyright nor violate any proprietary rights and that any i ...
Fraction-free matrix factors: new forms for LU and QR factors
... Gaussian elimination and LU factoring have been greatly studied from the algorithmic point of view, but much less from the point view of the best output format. In this paper, we give new output formats for fraction free LU factoring and for QR factoring. The formats and the algorithms used to obtai ...
... Gaussian elimination and LU factoring have been greatly studied from the algorithmic point of view, but much less from the point view of the best output format. In this paper, we give new output formats for fraction free LU factoring and for QR factoring. The formats and the algorithms used to obtai ...
Systems of Equations
... A system of equations can be consistent or inconsistent. What does that mean? A system of equations [ A][ X ] = [C ] is consistent if there is a solution, and it is inconsistent if there is no solution. However, a consistent system of equations does not mean a unique solution, that is, a consistent ...
... A system of equations can be consistent or inconsistent. What does that mean? A system of equations [ A][ X ] = [C ] is consistent if there is a solution, and it is inconsistent if there is no solution. However, a consistent system of equations does not mean a unique solution, that is, a consistent ...
Lecture 5: Quotient group - CSE-IITK
... Exercise 11. Given a group G and a normal subgroup N . Say the set of cosets is called S and has composition operation (gH)(kH) = (gk)H. Show that, – Identity exists in this set. – Inverses exist in this set. – Associativity is satisfied. Since Closure is obvious we get that S is a group with respec ...
... Exercise 11. Given a group G and a normal subgroup N . Say the set of cosets is called S and has composition operation (gH)(kH) = (gk)H. Show that, – Identity exists in this set. – Inverses exist in this set. – Associativity is satisfied. Since Closure is obvious we get that S is a group with respec ...
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.