Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality

A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations

AbstractWe introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions

Computational Approaches to Simulation and Analysis of Large Conformational Transitions in Proteins

abstract: In a typical living cell, millions to billions of proteins—nanomachines that fluctuate and cycle among many conformational states—convert available free energy into mechanochemical work. A fundamental goal of biophysics is to ascertain how 3D protein structures encode specific functions, such as catalyzing chemical reactions or transporting nutrients into a cell. Protein dynamics span femtosecond timescales (i.e., covalent bond oscillations) to large conformational transition timescales in, and beyond, the millisecond regime (e.g., glucose transport across a phospholipid bilayer). Actual transition events are fast but rare, occurring orders of magnitude faster than typical metastable equilibrium waiting times. Equilibrium molecular dynamics (EqMD) can capture atomistic detail and solute-solvent interactions, but even microseconds of sampling attainable nowadays still falls orders of magnitude short of transition timescales, especially for large systems, rendering observations of such "rare events" difficult or effectively impossible. Advanced path-sampling methods exploit reduced physical models or biasing to produce plausible transitions while balancing accuracy and efficiency, but quantifying their accuracy relative to other numerical and experimental data has been challenging. Indeed, new horizons in elucidating protein function necessitate that present methodologies be revised to more seamlessly and quantitatively integrate a spectrum of methods, both numerical and experimental. In this dissertation, experimental and computational methods are put into perspective using the enzyme adenylate kinase (AdK) as an illustrative example. We introduce Path Similarity Analysis (PSA)—an integrative computational framework developed to quantify transition path similarity. PSA not only reliably distinguished AdK transitions by the originating method, but also traced pathway differences between two methods back to charge-charge interactions (neglected by the stereochemical model, but not the all-atom force field) in several conserved salt bridges. Cryo-electron microscopy maps of the transporter Bor1p are directly incorporated into EqMD simulations using MD flexible fitting to produce viable structural models and infer a plausible transport mechanism. Conforming to the theme of integration, a short compendium of an exploratory project—developing a hybrid atomistic-continuum method—is presented, including initial results and a novel fluctuating hydrodynamics model and corresponding numerical code.Dissertation/ThesisDoctoral Dissertation Physics 201

Fine-grained complexity and algorithm engineering of geometric similarity measures

Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür

The Lean mathematical library

This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here

Advancing performance-based design and assessment of exposed column base plates and welded column splices in steel moment resisting frames

Exposed column base plate (ECBP) and welded column splice (WCS) connections are critical load-carrying structural connections and are commonly used in steel moment-resisting frames (SMRFs). However, they have received relatively lower research attention than welded beam-to-column (WBC) connections, leaving several relevant aspects of their performance and their effects on the overall seismic performance of SMRFs not well investigated. For instance, although the current design approach for ECBP connections is relatively well-established from a mechanistic standpoint, the reliability of such designed connections (i.e., the structural performance of ECBPs at the design level) is not as well understood. Therefore, some prospective refinements to the current approach may be developed to ensure acceptable and consistent failure probabilities across the various components of the ECBP connections. In the context of WCS connections constructed before the 1994 Northridge earthquake (i.e., pre-Northridge WCSs), their potential fracture due to earthquake shaking has been recently revealed in some research studies. However, these studies did not take advantage of recent advancements in performance-based earthquake engineering (PBEE), and made several simplifying assumptions for practical purposes. Some refinements and research tools within the PBEE framework may be required to more accurately estimate the fracture demand and capacity distributions, and the associated fragility and risk of pre-Northridge WCSs. This doctoral dissertation attempts to address these mentioned issues in a rigorous manner. Specifically, this dissertation presents the following research studies: 1. Detailed reliability analysis of ECBPs designed as per the current design method and two modified approaches (improved from the current one) for a set of 59 design scenarios subjected to combinations of gravity, wind, and seismic loads. This also includes the Monte Carlo sampling to characterize the uncertainty sources in the load, material properties, component geometry, and demand/capacity models for various components within the connection. 2. Refined probabilistic fracture fragility assessment of pre-Northridge WCSs, accounting for the seismic demand and fracture capacity uncertainties. Optimal ground-motion intensity measures, the effect of vertical ground accelerations, and the WCS capacity uncertainties are included to improve the fracture fragility estimation. 3. Expanded fracture fragility and risk assessment of pre-Northridge WCSs in near-fault regions to address the effect of pulse-like ground motions on the distribution/increase of WCS seismic demands. Near-source probabilistic seismic hazard analysis is conducted to facilitate the fracture risk assessment. The findings of the first study can contribute to the better scientific knowledge of reliability-based design and assessment of ECBP connections in SMRFs, whereas the last two studies can help better understand the fracture risk of WCS connections in SMRFs, and inform the planning of retrofitting strategies