104 research outputs found
The Goldman-Rota identity and the Grassmann scheme
We inductively construct an explicit (common) orthogonal eigenbasis for the
elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a
constructive, linear algebraic interpretation of the Goldman-Rota recurrence
for the number of subspaces of a finite vector space. This interpretation shows
that the up operator on subspaces has an explicitly given recursive structure.
Using this we inductively construct an explicit orthogonal symmetric Jordan
basis with respect to the up operator and write down the singular values, i.e.,
the ratio of the lengths of the successive vectors in the Jordan chains. The
collection of all vectors in this basis of a fixed rank forms a (common)
orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the
Grassmann scheme. We also pose a bijective proof problem on the spanning trees
of the Grassmann graphs.Comment: 19 Page
Observation of chaotic beats in a driven memristive Chua's circuit
In this paper, a time varying resistive circuit realising the action of an
active three segment piecewise linear flux controlled memristor is proposed.
Using this as the nonlinearity, a driven Chua's circuit is implemented. The
phenomenon of chaotic beats in this circuit is observed for a suitable choice
of parameters. The memristor acts as a chaotically time varying resistor
(CTVR), switching between a less conductive OFF state and a more conductive ON
state. This chaotic switching is governed by the dynamics of the driven Chua's
circuit of which the memristor is an integral part. The occurrence of beats is
essentially due to the interaction of the memristor aided self oscillations of
the circuit and the external driving sinusoidal forcing. Upon slight
tuning/detuning of the frequencies of the memristor switching and that of the
external force, constructive and destructive interferences occur leading to
revivals and collapses in amplitudes of the circuit variables, which we refer
as chaotic beats. Numerical simulations and Multisim modelling as well as
statistical analyses have been carried out to observe as well as to understand
and verify the mechanism leading to chaotic beats.Comment: 30 pages, 16 figures; Submitted to IJB
Cones of closed alternating walks and trails
Consider a graph whose edges have been colored red and blue. Assign a
nonnegative real weight to every edge so that at every vertex, the sum of the
weights of the incident red edges equals the sum of the weights of the incident
blue edges. The set of all such assignments forms a convex polyhedral cone in
the edge space, called the \emph{alternating cone}. The integral (respectively,
) vectors in the alternating cone are sums of characteristic vectors
of closed alternating walks (respectively, trails). We study the basic
properties of the alternating cone, determine its dimension and extreme rays,
and relate its dimension to the majorization order on degree sequences. We
consider whether the alternating cone has integral vectors in a given box, and
use residual graph techniques to reduce this problem to searching for a closed
alternating trail through a given edge. The latter problem, called alternating
reachability, is solved in a companion paper along with related results.Comment: Minor rephrasing, new pictures, 14 page
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