Distance matrices of a tree: two more invariants, and in a unified framework

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ 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 $D$, 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 $D$, 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 $D$: the sum of all its cofactors, cof$(D)$. 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$(D)$ and cof$(D)$ depend on the tree structure. In a sense, this completes the study of the invariant det$(D_T)$ (and cof$(D_T)$) for trees $T$ with edge-data in a commutative ring. Moreover: for a bi-directed graph $G$ we prove multiplicative Graham-Hoffman-Hosoya type formulas for det$(D_G)$, cof$(D_G)$, $D_G^{-1}$. 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 $D_T$.Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1

A Quantum Field Theoretical Representation of Euler-Zagier Sums

We establish a novel representation of arbitrary Euler-Zagier sums in terms of weighted vacuum graphs. This representation uses a toy quantum field theory with infinitely many propagators and interaction vertices. The propagators involve Bernoulli polynomials and Clausen functions to arbitrary orders. The Feynman integrals of this model can be decomposed in terms of an algebra of elementary vertex integrals whose structure we investigate. We derive a large class of relations between multiple zeta values, of arbitrary lengths and weights, using only a certain set of graphical manipulations on Feynman diagrams. Further uses and possible generalizations of the model are pointed out.Comment: Standard latex, 31 pages, 13 figures, final published versio

On qq-Analogs of Some Families of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

In recent years, there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can always recover the corresponding classical identities by taking $q\to 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their $q$-analogs proved by the authors recently. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman $\star$-elements $\zeta^{\star}(s_1,\dots,s_r)$ with $s_i\in\{2,3\}$ span the vector space generated by multiple zeta values over ${\mathbb Q}$.Comment: The abstract and references are updated. Two new theorems, Theorem 1.4, Theorem 5.4, and Corollary 1.6 are adde

Non-discursive knowledge and the construction of identity. Potters, potting and performance at the bronze age tell of Százhalombatta, Hungary

This article explores the relationship between the making of things and the making of people at the Bronze Age tell at Százhalombatta, Hungary. Focusing on potters and potting, we explore how the performance of non-discursive knowledge was critical to the construction of social categories. Potters literally came into being as potters through repeated bodily enactment of potting skills. Potters also gained their identity in the social sphere through the connection between their potting performance and their audience. We trace degrees of skill in the ceramic record to reveal the material articulation of non-discursive knowledge and consider the ramifications of the differential acquisition of non-discursive knowledge for the expression of different kinds of potter's identities. The creation of potters as a social category was essential to the ongoing creation of specific forms of material culture. We examine the implications of altered potters' performances and the role of non-discursive knowledge in the construction of social models of the Bronze Ag