366,304 research outputs found
A nested sequence of projectors and corresponding braid matrices : (1) Odd dimensions
A basis of projectors, each an matrix with constant
elements, is implemented to construct a class of braid matrices
, being the spectral parameter. Only odd values of
are considered here. Our ansatz for the projectors appearing
in the spectral decomposition of leads to exponentials
as the coefficient of . The sums and
differences of such exponentials on the diagonal and the antidiagonal
respectively provide the nonzero elements of . One
element at the center is normalized to unity. A class of supplementary
constraints imposed by the braid equation leaves free
parameters . The diagonalizer of is presented for
all . Transfer matrices and operators corresponding
to our are studied. Our diagonalizer signals specific
combinations of the components of the operators that lead to a quadratic
algebra of constant matrices. The -dependence factors
out for such combinations. is developed in a power series in
. The basic difference arising for even dimensions is made explicit.
Some special features of our are discussed in a concluding
section.Comment: latex file, 32 page
Smallest eigenvalues of Hankel matrices for exponential weights
AbstractWe obtain the rate of decay of the smallest eigenvalue of the Hankel matrices ∫Itj+kW2(t)dtj,k=0n for a general class of even exponential weights W2=exp(−2Q) on an interval I. More precise asymptotics for more special weights have been obtained by many authors
A proof of global attractivity of a class of switching systems using a non-Lyapunov approach
A sufficient condition for the existence of a Lyapunov function of the form V(x)= xTpx, P=PT > 0, P ∈ Rnxn, for the stable linear time invariant systems x = Aix, Ai ∈ Rnxn, Ai ∈ A =∆ {A1,...,Am}, is that the matrices Ai are Hurwitz, and that a non-singular matrix T exists, such that TAiT-1, i ∈ {1,...,m}, is upper triangular (Mori, Mori & Kuroe 1996, Mori, Mori & Kuroe 1997, Liberzon, Hespanha &
Morse 1998, Shorten & Narendra 1998b). The existence of such a function referred to as a common quadratic Lyapunov function (CQLF) is sufficient to guarantee the exponential stability of the switching system x = A(t)x, A(t)∈ A. In this paper we investigate the stability properties of a related class of switching systems. We consider sets of matrices A, where no single matrix T exists that simultaneously transforms each Ai ∈ A to upper triangular form, but where a set of non-singular matrices Tij exist
such that the matrices TijAiTij-1,TijAjTij-1} i, j ∈ are upper triangular. We show that for a special class of such systems the origin of the switching system x = A(t)x, A(t) ∈ A, is globally attractive. A novel technique is developed to derive this result and the applicability of this technique to more general systems is discussed
towards the end of the paper
Generalization of the convex-hull-and-line traveling salesman problem
Two instances of the traveling salesman problem, on the same node set (1,2 n} but with different cost matrices C and C, are equivalent iff there exist {a, hi: -1, n} such that for any 1 _i, j _n, j, C(i, j) C(i,j) q-a -t-bj [7]. One ofthe well-solved special cases of the traveling salesman problem (TSP) is the convex-hull-and-line TSP. We extend the solution scheme for this class of TSP given in [9] to a more general class which is closed with respect to the above equivalence relation. The cost matrix in our general class is a certain composition of Kalmanson matrices. This gives a new, non-trivial solvable case of TSP
On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings
In recent work of T. Cassidy and the author, a notion of complete
intersection was defined for (non-commutative) regular skew polynomial rings,
defining it using both algebraic and geometric tools, where the commutative
definition is a special case. In this article, we extend the definition to a
larger class of algebras that contains regular graded skew Clifford algebras,
the coordinate ring of quantum matrices and homogenizations of universal
enveloping algebras. Regular algebras are often considered to be
non-commutative analogues of polynomial rings, so the results herein support
that viewpoint.Comment: This paper replaces the preprint "Defining the Notion of Complete
Intersection for Regular Graded Skew Clifford Algebras", and also has a
paragraph written correctly that is garbled by the publisher in the published
version (paragraph after Example 3.8
The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality
The validity of the von-Neumann inequality for commuting - tuples of
matrices remains open for . We give a partial answer to
this question, which is used to obtain a necessary condition for the
Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc
In the special case of (which follows from Ando's theorem as well), this
necessary condition is made explicit. An alternative approach to the
Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of
adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence,
a class of polynomials are isolated for which a complete solution to the
Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural
generalization of the Hankel operators on the Hardy space of
then becomes apparent. Many of our results remain valid for any however, the computations are somewhat cumbersome for and are
omitted. The inequality , where
is the complex Grothendieck constant and
is due to Varopoulos. Here the
supremum is taken over all complex polynomials in variables of degree
at most and commuting - tuples of
contractions. We show that obtaining a slight improvement in the inequality of Varopoulos.
We show that the normed linear space has no isometric
embedding into complex matrices for any and
discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore
on 20th July, 201
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