10,308 research outputs found
A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity
We give a short proof of the Cauchy-Binet determinantal formula using
multilinear algebra by first generalizing it to an identity {\em not} involving
determinants. By extending the formula to abstract Hilbert spaces we obtain, as
a corollary, a generalization of the classical Parseval identity.Comment: 9 pages, 2 diagram
On the equivalence of two fundamental theta identities
Two fundamental theta identities, a three-term identity due to Weierstrass
and a five-term identity due to Jacobi, both with products of four theta
functions as terms, are shown to be equivalent. One half of the equivalence was
already proved by R.J. Chapman in 1996. The history and usage of the two
identities, and some generalizations are also discussed.Comment: v3: 15 pages, minor errors corrected, references added, appendix on
four-term theta identities added, accepted by Analysis and Application
Proof of George Andrews's and David Robbins's q-TSPP Conjecture
The conjecture that the orbit-counting generating function for totally
symmetric plane partitions can be written as an explicit product formula, has
been stated independently by George Andrews and David Robbins around 1983. We
present a proof of this long-standing conjecture
Distance matrices of a tree: two more invariants, and in a unified framework
Graham-Pollak showed that for the distance matrix of a tree ,
det depends only on its number of edges. Several other variants of ,
including directed/multiplicative/- versions were studied, and always,
det depends only on the edge-data.
We introduce a general framework for bi-directed weighted trees, with
threefold significance. First, we improve on state-of-the-art for all known
variants, even in the classical Graham-Pollak case: we delete arbitrary pendant
nodes (and more general subsets) from the rows/columns of , and show these
minors do not depend on the tree-structure.
Second, our setting unifies all known variants (with entries in a commutative
ring). We further compute in closed form the inverse of , extending a result
of Graham-Lovasz [Adv. Math. 1978] and answering a question of Bapat-Lal-Pati
[Lin. Alg. Appl. 2006].
Third, we compute a second function of the matrix : the sum of all its
cofactors, cof. This was worked out in the simplest setting by
Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants.
We prove a stronger result, in our general setting, by computing cof for
minors as above, and showing these too depend only on the edge-data.
Finally, we show our setting is the 'most general possible', in that with
more freedom in the edgeweights, det and cof depend on the tree
structure. In a sense, this completes the study of the invariant det
(and cof) for trees with edge-data in a commutative ring.
Moreover: for a bi-directed graph we prove multiplicative
Graham-Hoffman-Hosoya type formulas for det, cof, . We
then show how this subsumes their 1978 result. The final section introduces and
computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type
result for our "most general" distance matrix .Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1
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