Wall Crossings Research Articles

Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

We study multiple Bernoulli series associated to a sequence of vectors generating a lattice in a vector space. The associated multiple Bernoulli series is a periodic and locally polynomial function, and we give an explicit formula (called wall crossing formula) comparing the polynomial densities in two adjacent domains of polynomiality separated by a hyperplane. We also present a formula in the spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula for the Bernoulli series describing it as a superposition of convolution products of lower dimensional Bernoulli series and multisplines. The study of these series is motivated by the work of E. Witten, computing the symplectic volume of the moduli space of flat G-connections on a Riemann surface with one boundary component.

Quivers of sections on toric orbifolds

Starting from a collection of line bundles on a projective toric orbifold X , we introduce a stacky analogue of the classical linear series. Our first main result extends work of King by building moduli stacks of refined representations of labelled quivers. We associate one such stack to any collection of line bundles on X to obtain our notion of a stacky linear series; as in the classical case, X maps to the ambient stack by evaluating sections of line bundles in the collection. As a further application, we describe a finite sequence of GIT wall crossings between [ A n / G ] and G - Hilb ( A n ) for a finite abelian subgroup G ⊂ SL ( n , k ) where n ⩽ 3 .

In-plane cyclic behaviour of a new reinforced masonry system: Experimental results

This paper describes the behaviour under cyclic shear compression of an innovative reinforced masonry system, composed of horizontally perforated units, having common steel bars or prefabricated trusses as horizontal reinforcement. At the wall edges or crossings, confining columns for vertical reinforcement are built with vertically perforated units. Experimental tests to obtain information on the in-plane cyclic behaviour of the construction system were performed on masonry panels made of horizontally perforated units and on completed reinforced masonry walls. Tests on the entire system were repeated for two wall aspect ratios and two vertical stress levels, in order to force shear type and flexural behaviour. In particular, this paper presents: (a) results of shear compression tests in terms of strength, ductility parameters, energy dissipation, viscous damping and stiffness degradation, (b) strains and the effectiveness of reinforcement, (c) the influence of various parameters such as axial load, aspect ratio, and reinforcement type on the behaviour of the reinforced masonry walls, and (d) comparison of walls built with and without vertical reinforcement.

Hilbert schemes and stable pairs: GIT and derived category wall crossings

Nous montrons que le schéma de Hilbert de courbes et l'espace de modules de Le Potier de paires stables à support à une dimension, ont une construction GIT commune. Les deux espaces correspondents aux chambres de par et d'autre d'un mur dans l'espace de linéarisations GIT. Nous expliquons pourquoi cela ne suffit pas pour prouver la « conjecture de croisement de murs DT/PT » qui relie les invariants dérivés de ces espaces de modules quand la variété sous-jacente est un 3-fold. Nous donnons, ensuite, une introduction simple à une petite partie de la théorie de Joyce sur les croisements de murs de ce type, et nous nous en servons pour donner une brève démonstration d'une identité reliant les caractéristiques d'Euler de ces espaces de modules. Quand le 3-fold est de type Calabi-Yau, l'identité est le pendant, pour la caractéristique d'Euler, de la conjecture de croisement de murs DT/PT, mais dans le cas général elle s'avère être différente de celle-ci, comme nous l'expliquons.

Compressive behaviour of a new reinforced masonry system

This paper presents the behaviour in compression of an innovative reinforced masonry system. The system is made of horizontally perforated units, having common steel bars or prefabricated trusses as horizontal reinforcement. At the wall edges or crossings, confining columns for vertical reinforcement are built with vertically perforated units. Experimental tests, aimed at obtaining basic mechanical characterisation of the construction system, were performed on single constitutive elements i.e., confining columns and masonry panels made of horizontally perforated units, and on completed reinforced masonry walls. Non-linear numerical models, interpreting stress and strain distributions, were developed on the basis of the results. In particular, this paper presents: (a) results of compression tests on columns, masonry panels, and complete reinforced masonry system; (b) comparison of walls built with two types of horizontal reinforcement; (c) outcome of numerical models; and (d) effectiveness of various design equations to evaluate the compressive strength of the system.

Insider trading

PurposeThis paper aims to remind investment firms of the importance of policies, procedures, and supervisory controls to detect the misuse of material, non‐public information.Design/methodology/approachSummarizes a recent increase in regulatory concern over insider trading and suggests that firms review their “information wall” procedures.FindingsAt a minimum, a firm's information wall procedures should include the following elements: surveillance of employee trading; supervision of interdepartmental communications, including “walling off” procedures and procedures for “wall crossings”; a review of proprietary training when the firm is in possession of material, non‐public information; employee training and education, and documentation.Originality/valueReviews the key elements of an investment firm's insider trading policies.

Filtrations on G 1 T -Modules

Let G be an almost simple algebraic group defined over Fp for some prime p. Denote by G1 the first Frobenius kernel in G and let T be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of G1T. We have such filtrations on the baby Verma modules Zλ, where λ is a character of T. They are obtained via a certain deformation of the natural homomorphism from Zλ into its contravariant dual Zλτ. Using the same deformation we construct for each projective G1T-module Q a filtration of the vector space . We then prove that this filtration may also be described in terms of the above-mentioned homomorphism Z(λ) → Z(λ)τ and this leads us to a sum formula for our filtrations. When Q is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on Fλ(Q) explicitly for all λ. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation betweenthe dimensions of the layers in the filtrations of Fλ(Q) for an arbitrary indecomposable projective Q and the coefficients in the corresponding Kazhdan–Lusztig polynomials. We conclude the paper by proving that the above results in the G1T theory have some analogues in the representation theory of G (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.

Quasi-periodic structures in the large-scale galaxy distribution and three-dimensional Voronoi tessellation

It has been suggested that the void and wall structure associated with the large-scale galaxy distribution might be qualitatively, or perhaps even physically, modeled by a Voronoi tessellation, and that such structure might account for the surprisingly regular, sharp peaks in the galaxy redshift distributions obtained from 'pencil beam' surveys. Taking cell wall crossings by random line segments to correspond to such redshift peaks, an exact expression is derived for the distribution of spacings of these intersections in a three-dimensional Voronoi tessellation. This result verifies that the spacings are nonrandom and quasi-periodic, qualitatively resembling the observed pattern, even though the cell wall structure is generated from randomly placed seeds. Finally, moments of the spacing distribution are used to show that apparently periodic samples, similar to those recently reported, represent only one to two sigma fluctuations in a Voronoi tesselation.

Statistical Mechanics of Phase Transitions with a Hierarchy of Structures

AbstractRescaling (renormalization-group) [1] and restructuring [2] transformations are used in the statistical mechanics of systems with different complexities at different length scales, starting with the microscopic scale. These transformations are effected by a partial trace of the partition function, yielding effective coupling constants that incorporate the free energy contribution of the structures and length scales that are summed out of the problem. An example of this approach is a quantitative treatment [3] of the phase diagram of krypton adsorbed onto graphite. At the smallest length scales, adatom adsorption (first and second layer) and vibrations at adsorption sites are considered. At the next scale, sublattice formation and vacancy occurrence on the hexagonal graphite surface are considered. At a larger scale, heavy and superheavy domain wall formation, wall crossings and annihilations (dislocations) [4] are considered. This problem is treated by a sequence of 5 different transformations. Starting with the microscopic potentials, phase diagrams are obtained in temperature versus pressure or coverage, featuring disordered, commensurate and incommensurate solid phases, in good agreement with experiments [5]. The mechanism for reentrant phase transitions is divulged. Such solutions can be considered approximate treatments of physical systems or, alternatively, exact solutions of “hierarchical models” [6]. This “realizability” on hierarchical lattices insures the robustness of the approximation.