Box-counting measure of metric spaces

In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Bara\'nski carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent

On highly regular strongly regular graphs

In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the $7$-vertex condition fulfills an even stronger condition, $(3,7)$-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$-vertex condition. This strengthens and generalizes previous results by Reichard.Comment: 29 page

Curve counting, instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.Comment: 79 pages, 3 figures; v2: typos corrected, reference added, new summary section included; Final version to appear in Journal of Geometry and Physic

Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of unextendible mutually unbiased bases. We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields F d We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture

Generalized wordlength patterns and strength

Xu and Wu (2001) defined the \emph{generalized wordlength pattern} $(A_1, ..., A_k)$ of an arbitrary fractional factorial design (or orthogonal array) on $k$ factors. They gave a coding-theoretic proof of the property that the design has strength $t$ if and only if $A_1 = ... = A_t = 0$. The quantities $A_i$ are defined in terms of characters of cyclic groups, and so one might seek a direct character-theoretic proof of this result. We give such a proof, in which the specific group structure (such as cyclicity) plays essentially no role. Nonabelian groups can be used if the counting function of the design satisfies one assumption, as illustrated by a couple of examples