Information complexity of the AND function in the two-Party, and multiparty settings

In a recent breakthrough paper [M. Braverman, A. Garg, D. Pankratov, and O. Weinstein, From information to exact communication, STOC'13] Braverman et al. developed a local characterization for the zero-error information complexity in the two party model, and used it to compute the exact internal and external information complexity of the 2-bit AND function, which was then applied to determine the exact asymptotic of randomized communication complexity of the set disjointness problem. In this article, we extend their results on AND function to the multi-party number-in-hand model by proving that the generalization of their protocol has optimal internal and external information cost for certain distributions. Our proof has new components, and in particular it fixes some minor gaps in the proof of Braverman et al

On computational complexity of Siegel Julia sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.Comment: Updated version, to appear in Commun. Math. Phy

On the Positivity Problem for Simple Linear Recurrence Sequences

Given a linear recurrence sequence (LRS) over the integers, the Positivity Problem} asks whether all terms of the sequence are positive. We show that, for simple LRS (those whose characteristic polynomial has no repeated roots) of order 9 or less, Positivity is decidable, with complexity in the Counting Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277

Intersection cohomology of Drinfeld's compactifications

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.Comment: An erratum adde

A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P^1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra U(g,e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of arXiv:math/0401409 when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of certain shifted Yangians.Comment: minor change

The Cutaneous Microcirculation

The cutaneous microcirculation is organized as two horizontal plexuses. One is situated 1–1.5 mm below the skin surface and the other is at the dermal–subcutaneous junction. Ascending arterioles and descending venules are paired as they connect the two plexuses. From the upper layer, arterial capillaries rise to form the dermal papillary loops that represent the nutritive component of the skin circulation. There are sphincter-like smooth muscle cells at the point where the ascending arterioles divide to form the arteriolar component of the upper horizontal plexus. At the dermal–subcutaneous junction, there are collecting veins with two cusped valves that are oriented to prevent the retrograde flow of blood. Laser Doppler flowmetry has demonstrated vasomotion of red cell flux localized to the sites of ascending arterioles. The simultaneous recording by laser Doppler flowmetry of red cell flux and the concentration of moving red blood cells from individual sites allows one to construct topographic maps of these two values. These two maps, based on initial studies using correlative skin biopsies, can define 1 mm3 volumes of skin that are predominantly arteriolar in composition, venular in composition, or essentially devoid of all microvascular elements. The electron and light microscopic features that define the microvascular segments, when coupled with that ability of laser Doppler flowmetry to define the predominant microvascular segments under the probe, allow one to study both the mechanisms of normal physiologic states and the pathogenetic mechanisms underlying pathologic skin disorders in which the microvasculature plays a predominant role