scholarly journals Adding Layers to Bumped-Body Polyforms with Minimum Perimeter Preserves Minimum Perimeter

10.37236/1032 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Winston C. Yang
Keyword(s):  
Two Dimensions ◽  
Grid Cells ◽  
Minimum Area ◽  
Hexagonal Grid ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Additional Assumptions

In two dimensions, a polyform is a finite set of edge-connected cells on a square, triangular, or hexagonal grid. A layer is the set of grid cells that are vertex-adjacent to the polyform and not part of the polyform. A bumped-body polyform has two parts: a body and a bump. Adding a layer to a bumped-body polyform with minimum perimeter constructs a bumped-body polyform with min perimeter; the triangle case requires additional assumptions. A similar result holds for 3D polyominos with minimum area.

Comments on convex hull of a finite set of points in two dimensions

1979 ◽  
Vol 9 (3) ◽  
pp. 141-142 ◽  
Author(s):  
Ferenc Dévai ◽  
Tibor Szendrényi
Keyword(s):  
Convex Hull ◽  
Two Dimensions ◽  
Finite Set ◽  
Set Of Points

MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

1995 ◽  
Vol 05 (03) ◽  
pp. 243-256 ◽  
Author(s):  
DAVID RAPPAPORT
Keyword(s):  
Convex Hull ◽  
General Problem ◽  
Line Segment ◽  
Simple Polygon ◽  
Fixed Number ◽  
Line Segments ◽  
Geometric Objects ◽  
Finite Set ◽  
Minimum Perimeter ◽  
Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

Minimum-Area ellipse containing a finite set of points. II

1998 ◽  
Vol 50 (8) ◽  
pp. 1253-1261 ◽  
Author(s):  
B. V. Rublev ◽  
Yu. I. Petunin
Keyword(s):  
Minimum Area ◽  
Finite Set ◽  
Set Of Points

scholarly journals A Product Version of Dynamic Linear Time Temporal Logic

1997 ◽  
Vol 4 (9) ◽  
Author(s):  
Jesper G. Henriksen ◽  
P. S. Thiagarajan
Keyword(s):  
Temporal Logic ◽  
Decision Procedure ◽  
Linear Time ◽  
Dynamic Logic ◽  
Two Dimensions ◽  
Sequential Programs ◽  
Propositional Dynamic Logic ◽  
The Core ◽  
Finite Set ◽  
Linear Time Temporal Logic

We present here a linear time temporal logic which simultaneously extends LTL, the propositional temporal logic of linear time, along two dimensions. Firstly, the until operator is strengthened by indexing it with the regular programs of propositional dynamic logic (PDL). Secondly, the core formulas of the logic are decorated with names of sequential agents drawn from fixed finite set. The resulting logic has a natural semantics in terms of the runs of a distributed program consisting of a finite set of sequential programs that communicate by performing common actions together. We show that our logic, denoted DLTL, admits an exponential time decision procedure. We also show that DLTL is expressively equivalent to the so called regular product languages.

scholarly journals Adaptive optimal transport

2019 ◽  
Vol 8 (4) ◽  
pp. 789-816 ◽  
Author(s):  
Montacer Essid ◽  
Debra F Laefer ◽  
Esteban G Tabak
Keyword(s):  
Optimal Transport ◽  
A Priori ◽  
Two Dimensions ◽  
Local Optimality ◽  
Discrete Point ◽  
Variational Characterization ◽  
Leibler Divergence ◽  
Local Problems ◽  
Finite Set ◽  
Push Forward

AbstractAn adaptive, adversarial methodology is developed for the optimal transport problem between two distributions $\mu $ and $\nu $, known only through a finite set of independent samples $(x_i)_{i=1..n}$ and $(y_j)_{j=1..m}$. The methodology automatically creates features that adapt to the data, thus avoiding reliance on a priori knowledge of the distributions underlying the data. Specifically, instead of a discrete point-by-point assignment, the new procedure seeks an optimal map $T(x)$ defined for all $x$, minimizing the Kullback–Leibler divergence between $(T(x_i))$ and the target $(y_j)$. The relative entropy is given a sample-based, variational characterization, thereby creating an adversarial setting: as one player seeks to push forward one distribution to the other, the second player develops features that focus on those areas where the two distributions fail to match. The procedure solves local problems that seek the optimal transfer between consecutive, intermediate distributions between $\mu $ and $\nu $. As a result, maps of arbitrary complexity can be built by composing the simple maps used for each local problem. Displaced interpolation is used to guarantee global from local optimality. The procedure is illustrated through synthetic examples in one and two dimensions.

Efficient Pathway Identification from Geospatial Trajectories

2021 ◽  
Author(s):  
Carola Trahms ◽  
Patricia Handmann ◽  
Willi Rath ◽  
Matthias Renz ◽  
Martin Visbeck
Keyword(s):  
State Of The Art ◽  
Particle Density ◽  
Transition Strength ◽  
Grid Cells ◽  
Hexagonal Grid ◽  
Small Areas ◽  
Physics Community ◽  
The Earth ◽  
Novel Method ◽  
Temperature Depth

<p>In the earth-physics community Lagrangian trajectories are used within multiple contexts – analyzing the spreading of pollutants in the air or studying the connectivity between two ocean regions of interest. Huge amounts of data are generated reporting the geo position and other variables e.g. temperature, depth or salinity for particles spreading in the ocean. As state-of-the-art, these experiments are analyzed and visualized by binning the particle positions to pre-defined rectangular boxes. For each box a particle density is computed which then yields a probability map to visualize major pathways. Identifying the main pathways directly still remains a challenge when huge amounts of particles and variables are involved.</p><p>We propose a novel method that focuses on linking the net fluctuation of particles between adaptable hexagonal grid cells. For very small areas the rectangular boxing does not imply big differences in area or shape, though when gridding larger areas it introduces rather large distortions. Using hexagons instead provides multiple advantages, such as constant distances between the centers of neighboring boxes or more possibilities of movement due to 6 edges instead of 4 with a lower number of neighbors at the same time (6 instead of 9). The net fluctuation can be viewed as transition strength between the cells.Through this network perspective, the density of the transition strength can be visualized clearly. The main pathways are the transitions with the highest net fluctuation. Thus, simple statistical filtering can be used to reveal the main pathways. The combination of network analysis and adaptable hexagonal grid cells yields a surprisingly time and resource efficient way to identify main pathways.</p>

Minimum-Area ellipse containing a finite set of points. I

1998 ◽  
Vol 50 (7) ◽  
pp. 1115-1124 ◽  
Author(s):  
B. V. Rublev ◽  
Yu. I. Petunin
Keyword(s):  
Minimum Area ◽  
Finite Set ◽  
Set Of Points

The tangled web of agency

2018 ◽  
Vol 41 ◽  
Author(s):  
Alain Pe-Curto ◽  
Julien A. Deonna ◽  
David Sander
Keyword(s):  
Two Dimensions ◽  
Bad Company

AbstractWe characterize Doris's anti-reflectivist, collaborativist, valuational theory along two dimensions. The first dimension is socialentanglement, according to which cognition, agency, and selves are socially embedded. The second dimension isdisentanglement, the valuational element of the theory that licenses the anchoring of agency and responsibility in distinct actors. We then present an issue for the account: theproblem of bad company.

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