Exact Sequences for the Homology of the Matching Complex

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex $M_n$, which is the simplicial complex of matchings in the complete graph $K_n$. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of $M_n$. First, we demonstrate that there is nonvanishing 3-torsion in $H_d(M_n;Z)$ whenever \nu_n \le d \le (n-6}/2, where $\nu_n= \lceil (n-4)/3 \rceil$. By results due to Bouc and to Shareshian and Wachs, $H_{\nu_n}(M_n;Z)$ is a nontrivial elementary 3-group for almost all $n$ and the bottom nonvanishing homology group of $M_n$ for all $n \neq 2$. Second, we prove that $H_d(M_n;Z)$ is a nontrivial 3-group whenever $\nu_n \le d \le (2n-9)/5$. Third, for each $k \ge 0$, we show that there is a polynomial $f_k(r)$ of degree 3k such that the dimension of $H_{k-1+r}(M_{2k+1+3r};Z_3)$, viewed as a vector space over $Z_3$, is at most $f_k(r)$ for all $r \ge k+2$.Comment: 31 page

Double or nothing: Deconstructing cultural heritage

This paper draws on the deconstruction(ist) toolbox and specifically on the textual unweaving tactics of supplementarity, exemplarity, and parergonality, with a view to critically assessing institutional (UNESCO’s) and ordinary tourists’ claims to authenticity as regards artifacts and sites of ‘cultural heritage’. Through the ‘destru[k]tion’ of claims to ‘originality’ and ‘myths of origin’, that function as preservatives for canning such artifacts and sites, the cultural arche-writing that forces signifiers to piously bow before a limited string of ‘transcendental signifieds’ is brought to full view. The stench of the aeons is thus forced to evaporate through a post-transcendentalist opening towards originary myths’ original doubles

On moduli of rings and quadrilaterals: algorithms and experiments

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by K\"uhnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new $hp$-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the $hp$-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

FracKfinder: A MATLAB Toolbox for Computing 3‐D Hydraulic Conductivity Tensors for Fractured Porous Media

Fractures in porous media have been documented extensively. However, they are often omitted from groundwater flow and mass transport models due to a lack of data on fracture hydraulic properties and the computational burden of simulating fractures explicitly in large model domains. We present a MATLAB toolbox, FracKfinder, that automates HydroGeoSphere (HGS), a variably-saturated, control volume finite-element model, to simulate an ensemble of discrete fracture network (DFN) flow experiments on a single cubic model mesh containing a stochastically-generated fracture network. Because DFN simulations in HGS can simulate flow in both a porous media and a fracture domain, this toolbox computes tensors for both the matrix and fractures of a porous medium. Each model in the ensemble represents a different orientation of the hydraulic gradient, thus minimizing the likelihood that a single hydraulic gradient orientation will dominate the tensor computation. Linear regression on matrices containing the computed 3-D hydraulic conductivity (K) values from each rotation of the hydraulic gradient is used to compute the K tensors. This approach shows that the hydraulic behavior of fracture networks can be simulated where fracture hydraulic data are limited. Simulation of a bromide tracer experiment using K tensors computed with FracKfinder in HydroGeoSphere demonstrates good agreement with a previous large-column, laboratory study. The toolbox provides a potential pathway to upscale groundwater flow and mass transport processes in fractured media to larger scales