Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure

Formal deformations, contractions and moduli spaces of Lie algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions.Comment: 27 page

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs

Threefold Flops via Matrix Factorization

The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate "partially resolved" deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2.Comment: v2: minor change

Algebras and universal quantum computations with higher dimensional systems

Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It is shown next, how for quantum computation with qubits can be used two-dimensional analog of this Cayley-Weyl matrix algebras, i.e. Clifford algebras, and discussed well known applications to product operator formalism in NMR, Jordan-Wigner construction in fermionic quantum computations. It is introduced universal set of quantum gates for higher dimensional system (``qudit''), as some generalization of these models. Finally it is briefly mentioned possible application of such algebraic methods to design of quantum processors (programmable gates arrays) and discussed generalization to quantum computation with continuous variables.Comment: 12 pages, LaTeXe, Was prepared for QI2002, Moscow, 1-4.1

Stratification of moduli spaces of Lie algebras, similar matrices and bilinear forms

In this paper, the authors apply a stratification of moduli spaces of complex Lie algebras to analyzing the moduli spaces of nxn matrices under scalar similarity and bilinear forms under the cogredient action. For similar matrices, we give a complete description of a stratification of the space by some very simple projective orbifolds of the form P^n/G, where G is a subgroup of the symmetric group sigma_{n+1} acting on P^n by permuting the projective coordinates. For bilinear forms, we give a similar stratification up to dimension 4