Derived representation schemes and Nakajima quiver varieties

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $$S^4$$ S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.

Supersymmetry and Exceptional Points

The phenomenon of degeneracy of energy levels is often attributed either to an underlying (super)symmetry (SUSY), or to the presence of a Kato exceptional point (EP). In our paper a conceptual bridge between the two notions is proposed to be provided by the recent upgrade of the basic principles of quantum theory called, equivalently, PT − symmetric or three-Hilbert-space (3HS) or quasi-Hermitian formulation in the current physical literature. Although the original purpose of the 3HS approach laid in the mere simplification of technicalities, it is shown here to serve also as a natural theoretical link between the apparently remote concepts of EPs and SUSY. An explicit illustration of their close mutual interplay is provided by the description of infinitely many supersymmetric, mutually non-equivalent and EP-separated regularized spiked harmonic oscillators.

SMALL-WORLD EFFECT IN GEOGRAPHICAL ATTACHMENT NETWORKS

In this work, we use rigorous probabilistic methods to study the asymptotic degree distribution, clustering coefficient, and diameter of geographical attachment networks. As a type of small-world network model, these networks were first proposed in the physical literature, where they were analyzed only with heuristic arguments and computational simulations.

Continuous, not discrete: The mutual influence of digital and physical literature

The use of computational methods to develop innovative forms of storytelling and poetry has gained traction since the late 1980s. At the same time, legacy publishing has largely migrated to using digital workflows. Despite this possible convergence, the electronic literature community has generally defined their practice in opposition to print and traditional publishing practices more generally. Not only does this ignore a range of hybrid forms, but it also limits non-digital literature to print, rather than considering a range of physical literatures. In this article, I argue that it is more productive to consider physical and digital literature as convergent forms as both a historicizing process, and a way of identifying innovations. Case studies of William Gibson et al.’s Agrippa ( A Book of the Dead) and Christian Bök’s The Xenotext Project’s playful use of innovations in genetics demonstrate the productive tensions in the convergence between digital and physical literature.

On classical irregular q-difference equations

The primary aim of this paper is to (provide tools to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of certain differential equations that arise frequently in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.

ON THE CONTINUITY OF THE RESPONSE MAP AND AN ALTERNATIVE SHOOTING METHOD FOR THE HALF-SPACE GINZBURG–LANDAU MODEL

As a consequence of a rather complete analysis of the qualitative properties of the solutions of the Ginzburg–Landau equations, we prove, in this paper, both the continuity of a fundamental map σ, called response map in the physical literature on superconductors, and the convergence of an efficient algorithm for the computation of the graph of σ. The response map σ gives the intensity h of the external magnetic field for which the Ginzburg–Landau equations (in a half-space) have a solution such that the parameter order has a prescribed value at the boundary of the sample. Our study involves a shooting method on either one or the other unknown of the system; our algorithm has been introduced in Bolley–Helffer for small values of the Ginzburg–Landau parameter κ and extended in Bolley to any value of κ. Our preceding mathematical studies were not sufficient to prove the convergence, but a recent result (in Ref. 3) on the monotonicity of the solutions with respect to h, combined with a more extensive use of the properties of the solutions of the Ginzburg–Landau system, allow us to complete the proof and to get, as a by-product, the continuity of σ.

SCALING LIMITS FOR BEAM WAVE PROPAGATION IN ATMOSPHERIC TURBULENCE

We prove the convergence of the solutions of the parabolic wave equation to that of the Gaussian white-noise model widely used in the physical literature. The random medium is isotropic and is assumed to have integrable correlation coefficient in the propagation direction. We discuss the limits of vanishing inner scale and divergent outer scale of the turbulent medium.

Brownian Survival in a Clusterized Trapping Medium

The survival problem for a Brownian particle moving among random traps is considered in the case where the traps are correlated in a particular fashion being gathered in clusters. It is assumed that the clusters are statistically identical and independent of each other and are distributed in space according to a Poisson law. Mathematically, such a trapping medium is described via a Poisson cluster point process. We prove that the particle survival probability is increased at all times as compared to the case of noncorrelated (Poissonian) traps, which implies the slowdown of the trapping process. It is shown that this effect may be interpreted as the manifestation of the trap "attraction", thus supporting assertions on the qualitative influence of the trap "interaction" on the trapping rate claimed earlier in the physical literature. The long-time survival asymptotics (of Donsker–Varadhan type) is also derived. By way of appendix, FKG inequalities for certain functionals are proven and the limiting distribution for a Poisson cluster process, under clusters' scaling, is determined.

Radiocarbon Dating Lists and Their Use

“The Denbigh Flint Complex is Not Yet Dated,” (American Antiquity, Vol. 20, No. 4, pp. 375-6) by J. L. Giddings raises a number of general questions concerning the collection of samples and the listings and use of radiocarbon dates which are not confined to the Denbigh material. Giddings says he is a “little worried about our use of ‘lists’ of radiocarbon dates in view of the recent treatment of some dates from Cape Denbigh.” The publication of lists of dates has proved to be of considerable value to anyone who is interested in chronological problems. Primarily the lists facilitate reference. If the more than 2000 determinations were scattered in archaeological, geological, botanical, palaeobotanical and physical literature, it would be practically impossible to find them. For example, an archaeologist might never learn that a date on a level correlative with the Mesolithic but identified by pollen analysis had ever been determined.