scholarly journals Local Reasoning for Global Graph Properties

2020 ◽  
pp. 308-335
Author(s):  
Siddharth Krishna ◽  
Alexander J. Summers ◽  
Thomas Wies
Keyword(s):  
Small Region ◽  
Algebraic Equations ◽  
Proof Technique ◽  
Modular Reasoning ◽  
Graph Properties ◽  
Local Modification ◽  
Non Local ◽  
Local Reasoning ◽  
Path Lengths ◽  
Mathematical Foundations

AbstractSeparation logics are widely used for verifying programs that manipulate complex heap-based data structures. These logics build on so-called separation algebras, which allow expressing properties of heap regions such that modifications to a region do not invalidate properties stated about the remainder of the heap. This concept is key to enabling modular reasoning and also extends to concurrency. While heaps are naturally related to mathematical graphs, many ubiquitous graph properties are non-local in character, such as reachability between nodes, path lengths, acyclicity and other structural invariants, as well as data invariants which combine with these notions. Reasoning modularly about such graph properties remains notoriously difficult, since a local modification can have side-effects on a global property that cannot be easily confined to a small region.In this paper, we address the question: What separation algebra can be used to avoid proof arguments reverting back to tedious global reasoning in such cases? To this end, we consider a general class of global graph properties expressed as fixpoints of algebraic equations over graphs. We present mathematical foundations for reasoning about this class of properties, imposing minimal requirements on the underlying theory that allow us to define a suitable separation algebra. Building on this theory, we develop a general proof technique for modular reasoning about global graph properties expressed over program heaps, in a way which can be directly integrated with existing separation logics. To demonstrate our approach, we present local proofs for two challenging examples: a priority inheritance protocol and the non-blocking concurrent Harris list.

scholarly journals A non-local scalar conservation law describing navigation processes

2020 ◽  
Vol 17 (04) ◽  
pp. 809-841
Author(s):  
Paulo Amorim ◽  
Florent Berthelin ◽  
Thierry Goudon
Keyword(s):  
Conservation Law ◽  
Stability Result ◽  
Planck Equation ◽  
Small Region ◽  
Effective Velocity ◽  
Fokker Planck Equation ◽  
Scalar Conservation Law ◽  
Non Local ◽  
Scalar Conservation ◽  
Fokker Planck

We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.

scholarly journals Graph Properties of the Adult Drosophila Central Brain

2020 ◽  
Author(s):  
Louis K. Scheffer
Keyword(s):  
Experimental Error ◽  
Electrical Response ◽  
Brain Connectome ◽  
Input And Output ◽  
Graph Properties ◽  
Central Brain ◽  
Path Lengths ◽  
Efficient Packing ◽  
Adult Drosophila ◽  
Computational Structures

AbstractThe recent Drosophila central brain connectome offers the possibility of analyzing the graph properties of the fly brain. Crucially, this connectome is dense, meaning all nodes and links are represented, within the limits of experimental error. We consider the connectome as a directed graph with weighted edges. This enables us to look at a number of graph properties, compare them to human designed logic systems, and speculate on how this may affect function. We look at input and output distributions, randomness of wiring, differences between compartments, path lengths, proximity of strong connections, known computational structures, electrical response as a function of compartment structure, and evidence for efficient packing.

scholarly journals Analysis of Waterman’s Method in the Case of Layered Scatterers

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Victor Farafonov ◽  
Vladimir Il’in ◽  
Vladimir Ustimov ◽  
Evgeny Volkov
Keyword(s):  
Light Scattering ◽  
Infinite System ◽  
Inverse Matrix ◽  
Numerical Calculations ◽  
Algebraic Equations ◽  
Numerical Computations ◽  
Scattering Problems ◽  
Strong Similarity ◽  
Linear Algebraic Equations ◽  
Mathematical Foundations

The method suggested by Waterman has been widely used in the last years to solve various light scattering problems. We analyze the mathematical foundations of this method when it is applied to layered nonspherical (axisymmetric) particles in the electrostatic case. We formulate the conditions under which Waterman’s method is applicable, that is, when it gives an infinite system of linear algebraic equations relative to the unknown coefficients of the field expansions which is solvable (i.e., the inverse matrix exists) and solutions of the truncated systems used in calculations converge to the solution of the infinite system. The conditions obtained are shown to agree with results of numerical computations. Keeping in mind the strong similarity of the electrostatic and light scattering cases and the agreement of our conclusions with the numerical calculations available for homogeneous and layered scatterers, we suggest that our results are valid for light scattering as well.

scholarly journals Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

2021 ◽  
Vol 5 (3) ◽  
pp. 61
Author(s):  
Stanislav Harizanov ◽  
Nikola Kosturski ◽  
Ivan Lirkov ◽  
Svetozar Margenov ◽  
Yavor Vutov
Keyword(s):  
Computational Complexity ◽  
Rational Approximation ◽  
Elliptic Equations ◽  
Spectral Space ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Linear Algebraic Equations ◽  
Solution Methods ◽  
Non Local ◽  
General Geometry

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

scholarly journals Aneris: A Mechanised Logic for Modular Reasoning about Distributed Systems

2020 ◽  
pp. 336-365 ◽  
Author(s):  
Morten Krogh-Jespersen ◽  
Amin Timany ◽  
Marit Edna Ohlenbusch ◽  
Simon Oddershede Gregersen ◽  
Lars Birkedal
Keyword(s):  
Distributed Systems ◽  
Higher Order ◽  
Separation Logic ◽  
Two Phase ◽  
Modular Reasoning ◽  
Multiple Servers ◽  
Local Reasoning ◽  
Load Balancer ◽  
Concurrent And Distributed Systems ◽  
Great Complexity

AbstractBuilding network-connected programs and distributed systems is a powerful way to provide scalability and availability in a digital, always-connected era. However, with great power comes great complexity. Reasoning about distributed systems is well-known to be difficult. In this paper we present , a novel framework based on separation logic supporting modular, node-local reasoning about concurrent and distributed systems. The logic is higher-order, concurrent, with higher-order store and network sockets, and is fully mechanized in the Coq proof assistant. We use our framework to verify an implementation of a load balancer that uses multi-threading to distribute load amongst multiple servers and an implementation of the two-phase-commit protocol with a replicated logging service as a client. The two examples certify that is well-suited for both horizontal and vertical modular reasoning.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

An improved magnetic prism design for a transmission electron microscope energy filter

1978 ◽  
Vol 36 (1) ◽  
pp. 40-41 ◽  
Author(s):  
John W. Andrew ◽  
F.P. Ottensmeyer ◽  
E. Martell
Keyword(s):  
Energy Resolution ◽  
Symmetry Plane ◽  
Fringe Field ◽  
Focal Distance ◽  
Electron Trajectories ◽  
Focusing Effect ◽  
Transmission Electron ◽  
Path Lengths ◽  
Electron Microscopes ◽  
Focusing Properties

Energy selecting electron microscopes of the Castaing-Henry prism-mirror-prism design suffer from a loss of image and energy resolution with increasing field of view. These effects can be qualitatively understood by examining the focusing properties of the prism shown in Fig. 1. A cone of electrons emerges from the entrance lens crossover A and impinges on the planar face of the prism. The task of the prism is to focus these electrons to a point B at a focal distance f2 from the side of the prism. Electrons traveling in the plane of the diagram (i.e., the symmetry plane of the prism) are focused toward point B due to the different path lengths of different electron trajectories in the triangularly shaped magnetic field. This is referred to as horizontal focusing; the better this focusing effect the better the energy resolution of the spectrometer. Electrons in a plane perpendicular to the diagram and containing the central ray of the incident cone are focused toward B by the curved fringe field of the prism.

Chromatic aberration and low-voltage SEM

1987 ◽  
Vol 45 ◽  
pp. 546-547
Author(s):  
Zhifeng Shao ◽  
A.V. Crewe
Keyword(s):  
Electron Energy ◽  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Primary Electron ◽  
Probe Size ◽  
Non Local ◽  
Optical Treatment ◽  
Electron Microscopes ◽  
Scanning Electron Microscopes

For scanning electron microscopes, it is plausible that by lowering the primary electron energy, one can decrease the volume of interaction and improve resolution. As shown by Crewe /1/, at V0 =5kV a 10Å resolution (including non-local effects) is possible. To achieve this, we would need a probe size about 5Å. However, at low voltages, the chromatic aberration becomes the major concern even for field emission sources. In this case, δV/V = 0.1 V/5kV = 2x10-5. As a rough estimate, it has been shown that /2/ the chromatic aberration δC should be less than ⅓ of δ0 the probe size determined by diffraction and spherical aberration in order to neglect its effect. But this did not take into account the distribution of electron energy. We will show that by using a wave optical treatment, the tolerance on the chromatic aberration is much larger than we expected.

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