scholarly journals Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 360
Author(s):  
Douglas Lenseth ◽  
Boris Goldfarb
Keyword(s):  
Vector Fields ◽  
Persistent Homology ◽  
Gradient Field ◽  
Basic Question ◽  
Wrong Answer ◽  
Discrete Gradient ◽  
Gradient Fields ◽  
The Given ◽  
Well Posed ◽  
Local Nature

We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grey-scale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use, it is natural to want to distribute the computation over patches Pi, apply the chosen algorithm to compute the fields Vi associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy-to-arrange covering patterns.

scholarly journals Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

2021 ◽  
Author(s):  
Claudia Landi ◽  
Sara Scaramuccia
Keyword(s):  
Morse Theory ◽  
Vector Fields ◽  
Persistent Homology ◽  
Natural Generalization ◽  
Heterogeneous Data ◽  
Gradient Vector ◽  
Discrete Gradient ◽  
The One ◽  
Definition Of ◽  
Geometric Content

AbstractThe combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

scholarly journals The Constant Changes in US Strategy in Afghanistan: Achievement or Failure

2017 ◽  
Vol 10 (2) ◽  
pp. 41
Author(s):  
Behnam Sahranavard ◽  
Ali Asghar Kazemi
Keyword(s):  
United States ◽  
Decision Making ◽  
September 11 ◽  
International Terrorism ◽  
Basic Question ◽  
Collection Method ◽  
External Conditions ◽  
Post September 11 ◽  
The Given

The nations take various strategies in exposure to different developments and phenomena and impact on foreign and internal policies of countries in international scene proportional to their internal and external conditions and rivals and at international arena. What US implemented after September 11 Event and targeted accusation finger toward Taliban and Al-Qaeda in Afghanistan is deemed as a type of strategy that has occurred in created nostalgic climate together with hasty decision making and negligence to domestic issues in Afghan Community while their output was to take different and even paradoxical strategies in this crisis-stricken region since 1980s. In this article that has been written in order to analyze US Post September- 11 Strategies in Afghanistan this basic question will be answered that how changes in US macro policies influenced in orientation of diplomacy of this country and why this country has adapted different policies in occupation of Afghanistan. Afterwards, it is deduced according to the given findings from librarian data collection method that the constant changes in US strategy in Afghanistan were due to overlooking of domestic issues and historic, ethnic, cultural, political, and ideological complexities of this country that has resulted in degradation of US position in world scene and its failure in suppression of Taliban.This article has been excerpted from my PhD treatise under title of ‘The role of United States in the regional crisis (e.g. Afghan and Iraqi crises) and the rise of revolutionary and radicalism on the emergence of international terrorism’.

scholarly journals The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

2006 ◽  
Vol 9 ◽  
pp. 287-329 ◽  
Author(s):  
Katsiaryna Krupchyk ◽  
Jukka Tuomela
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Formal Theory ◽  
Elliptic Boundary Value Problems ◽  
Constructive Method ◽  
Elliptic Boundary Value ◽  
The Given ◽  
Well Posed

AbstractElliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

scholarly journals Discrete gradient fields on infinite complexes

2011 ◽  
Vol 30 (3) ◽  
pp. 623-639 ◽  
Author(s):  
Rafael Ayala ◽  
◽  
Jose Antonio Vilches ◽  
Gregor Jerše ◽  
Neža Mramor Kosta ◽  
...  
Keyword(s):  
Discrete Gradient ◽  
Gradient Fields

scholarly journals The Frenet Vector Fields and the Curvatures of the Natural Lift Curve

2012 ◽  
Vol 2 ◽  
pp. 38-43 ◽  
Author(s):  
Evren Ergün ◽  
Mustafa Bilici ◽  
Mustafa Çalişkan
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Similar Calculation ◽  
Darboux Vector ◽  
Lift Curve ◽  
Curvature And Torsion ◽  
The Given ◽  
Made In

In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in 3/1 . Also, a similar calculation is made in 3/1 considering timelike or spacelike Darboux vector field.

scholarly journals Correlation analysis of the interaction between oceanic heat fluxes and the geopotential gradient fields in the middle troposphere during meridional and zonal processes

2019 ◽  
Vol 59 (4) ◽  
pp. 521-528
Author(s):  
O. A. Razorenova ◽  
P. A. Shabanov
Keyword(s):  
Correlation Analysis ◽  
Correlation Coefficients ◽  
Heat Fluxes ◽  
Gradient Field ◽  
Middle Troposphere ◽  
The North ◽  
Gradient Fields ◽  
Linear Correlation Analysis ◽  
European Sector

Investigation of the interaction between oceanic heat fluxes and formation of the geopotential gradient fields in the middle troposphere during meridional and zonal processes has been carried out by applying linear correlation analysis. Analysis of the spatial distribution of the correlation coefficients has demonstrated that the structure of the interaction of heat fluxes in the North Atlantic with the geopotential gradient field in the middle troposphere over the Atlantic-European sector differs in periods with the predominance of meridional and zonal circulation forms, which indicates the significant role of the ocean heat flow in the formation and development of circulation regimes in the atmosphere.

Continuously defective crystals with a prescribed dislocation density

2014 ◽  
Vol 11 (09) ◽  
pp. 1460034 ◽  
Author(s):  
Marek Z. Elżanowski ◽  
Serge Preston
Keyword(s):  
Dislocation Density ◽  
Lie Algebra ◽  
Vector Fields ◽  
Density Tensor ◽  
Kinematic Theory ◽  
Dislocation Density Tensor ◽  
The Given ◽  
Elastic Crystals ◽  
Defective Crystals

We analyze some aspects of the kinematic theory of non-uniformly defective elastic crystals. Concentrating on the problem of identifying continuous defective lattices possessing the given defectiveness, as defined by the dislocation density tensor, we investigate the relation between the dislocation density tensor and the Lie algebra of vector fields associated with a defective lattice.

VISUALIZATION OF CHAOTIC DYNAMICAL SYSTEMS BASED ON MANDELBROT SET METHODOLOGY

Fractals ◽  
2008 ◽  
Vol 16 (01) ◽  
pp. 89-97 ◽  
Author(s):  
JIN CHENG ◽  
JIANRONG TAN ◽  
CHUNBIAO GAN
Keyword(s):  
Parameter Space ◽  
Chaotic Systems ◽  
Mandelbrot Set ◽  
Control Parameters ◽  
Limited Information ◽  
Visualization Technique ◽  
Lyapunov Spectra ◽  
The Given ◽  
Local Nature

Although there are lots of methods to analyze chaotic systems, they tend to be of local nature and reveal only limited information about a certain group of control parameters. This paper explores a visualization technique on the basis of Mandelbrot set (M-set) methodology to give an overall view of a chaotic system's dynamical performance in the parameter space. Firstly, the Lyapunov spectra with regard to different points in the parameter space are calculated, according to which the color of these points is defined. Then points in the given parameter space are mapped to the computer screen and a colorful image named Lyapunov distribution map (LDM) is generated, which conveys a wealth of information about a system's dynamic behavior across variations in its control parameters. The results of visualizing two typical chaotic systems proved the feasibility and validity of this technique. The study suggests that desired dynamical performance of a system can be conveniently achieved by selecting a point from some color region in its LDM and adjusting the control parameters according to the point value. Furthermore, there is no need for any prior knowledge of the system under evaluation and the complicated mathematical analysis before the formulation of appropriate rules for a new system can also be avoided since uniform rules are employed in the classification of parameter points for any system and the LDM is generated via the same algorithm.

Thermal state in the north Viking Graben (North Sea) determined from oil exploration well data

Geophysics ◽  
1992 ◽  
Vol 57 (1) ◽  
pp. 69-88 ◽  
Author(s):  
F. Brigaud ◽  
G. Vasseur ◽  
G. Caillet
Keyword(s):  
Thermal Conductivity ◽  
North Sea ◽  
Thermal Gradient ◽  
Geometric Mean ◽  
Gradient Field ◽  
Thermal Gradients ◽  
Oil Exploration ◽  
The North ◽  
Basin Scale ◽  
Gradient Fields

We can deduce thermal conductivities and thermal gradients from extensively available oil exploration data: geophysical well logs, cores, cuttings, formation thicknesses and temperatures. Thermal conductivity is predicted at three significant scales. First, it is computed at the scale of well‐log electrofacies (thicknesses from 1 to 10 m) using a geometric mean model calibrated on laboratory measurements made on the main sedimentary rocks—the electrofacies conductivity is calculated as a function of the mineralogy, the porosity and the saturating fluids. Second, it is estimated at formation scale at each well site (thicknesses from 100 to a few thousand meters) using a serial model that accounts for the anisotropy due to sediment stacking and for temperature effects. Finally, for each formation (thicknesses on the order of 1 km), the average conductivity field is mapped at basin scale (extent on the order of 100 km) using a geostatistical treatment accounting for lateral facies and/or porosity changes. For thermal gradient field reconstruction, the systematic errors associated with the drilling history are removed from temperatures (bottom‐hole temperatures) using various techniques depending on data quality. The formation thermal gradient fields are then estimated using a stochastic inversion for temperatures and thicknesses, considering lateral correlations between thermal gradients at well sites. The technique is applied to the Norwegian Viking Graben, a multistage rift basin in the North Sea, where previous studies indicate large lateral and vertical variations in thermal conductivity and thermal gradient fields.

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