scholarly journals Complexity of linear relaxations in integer programming

2021 ◽  
Author(s):  
Gennadiy Averkov ◽  
Matthias Schymura
Keyword(s):  
Residue Class ◽  
Integer Point ◽  
Three Dimensional ◽  
Quantifier Elimination ◽  
Geometry Of Numbers ◽  
Integer Points ◽  
Open Questions ◽  
Lattice Width ◽  
Rational Polyhedra ◽  
Linear Relaxations

AbstractFor a setXof integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with Xis called the relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$rc(X). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of Xwithout using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding$${{\,\mathrm{rc}\,}}(X)$$rc(X)and its variant$${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rcQ(X), restricting the descriptions of Xto rational polyhedra. As our main results we show that$${{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$rc(X)=rcQ(X)when: (a)Xis at most four-dimensional, (b)Xrepresents every residue class in$$(\mathbb {Z}/2\mathbb {Z})^d$$(Z/2Z)d, (c) the convex hull of Xcontains an interior integer point, or (d) the lattice-width of Xis above a certain threshold. Additionally,$${{\,\mathrm{rc}\,}}(X)$$rc(X)can be algorithmically computed when Xis at most three-dimensional, orXsatisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on$${{\,\mathrm{rc}\,}}(X)$$rc(X)in terms of the dimension of X.

scholarly journals Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

4OR ◽  
2020 ◽  
Author(s):  
Michele Conforti ◽  
Marianna De Santis ◽  
Marco Di Summa ◽  
Francesco Rinaldi
Keyword(s):  
Integer Point ◽  
Cutting Plane ◽  
Integer Program ◽  
Compact Set ◽  
Cutting Plane Method ◽  
Integer Points ◽  
Gomory Cuts ◽  
The Family ◽  
Split Cuts ◽  
Number Of Iterations

AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

Quick phases control ocular torsion during smooth pursuit

2011 ◽  
Vol 106 (5) ◽  
pp. 2151-2166 ◽  
Author(s):  
Bernhard J. M. Hess ◽  
Jakob S. Thomassen
Keyword(s):  
Eye Movements ◽  
Visual Field ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Oculomotor Control ◽  
Visually Guided ◽  
Open Questions ◽  
Spatial Frequencies ◽  
Eye Rotation ◽  
Smooth Tracking

One of the open questions in oculomotor control of visually guided eye movements is whether it is possible to smoothly track a target along a curvilinear path across the visual field without changing the torsional stance of the eye. We show in an experimental study of three-dimensional eye movements in subhuman primates ( Macaca mulatta) that although the pursuit system is able to smoothly change the orbital orientation of the eye's rotation axis, the smooth ocular motion was interrupted every few hundred milliseconds by a small quick phase with amplitude <1.5° while the animal tracked a target along a circle or ellipse. Specifically, during circular pursuit of targets moving at different angular eccentricities (5°, 10°, and 15°) relative to straight ahead at spatial frequencies of 0.067 and 0.1 Hz, the torsional amplitude of the intervening quick phases was typically around 1° or smaller and changed direction for clockwise vs. counterclockwise tracking. Reverse computations of the eye rotation based on the recorded angular eye velocity showed that the quick phases facilitate the overall control of ocular orientation in the roll plane, thereby minimizing torsional disturbances of the visual field. On the basis of a detailed kinematic analysis, we suggest that quick phases during curvilinear smooth tracking serve to minimize deviations from Donders' law, which are inevitable due to the spherical configuration space of smooth eye movements.

scholarly journals 3D survey for the archaeological study and virtual reconstruction of the “Sanctuary of Isis” in the ancient Lilybaeum (Italy)

2020 ◽  
Vol 11 (22) ◽  
pp. 1 ◽  
Author(s):  
Leonarda Fazio ◽  
Mauro Lo Brutto
Keyword(s):  
Three Dimensional ◽  
Archaeological Site ◽  
3D Models ◽  
Archaeological Remains ◽  
Archaeological Heritage ◽  
Digital Reconstruction ◽  
3D Environment ◽  
3D Data ◽  
Open Questions ◽  
Virtual Reconstruction

<p class="VARKeywords">In recent years, the use of three-dimensional (3D) models in cultural and archaeological heritage for documentation and dissemination purposes has increased. New geomatics technologies have significantly reduced the time spent on fieldwork surveys and data processing. The archaeological remains can be documented and reconstructed in a digital 3D environment thanks to the new 3D survey technologies. Furthermore, the products generated by modern surveying technologies can be reconstructed in a virtual environment on effective archaeological bases and hypotheses coming from a detailed 3D data analysis. However, the choice of technologies that should be used to get the best results for different archaeological remains and how to use 3D models to improve knowledge and dissemination to a wider audience are open questions.</p><p class="VARKeywords">This paper deals with the use of terrestrial laser scanners and photogrammetric surveys for the virtual reconstruction of an archaeological site. In particular, the work describes the study for the 3D documentation and virtual reconstruction of the “Sanctuary of Isis” in <em>Lilybaeum,</em> the ancient city of Marsala (southern Italy). The "Sanctuary of Isis" is the only Roman sacred building known in this archaeological area. Based on the survey data, it has been possible to recreate the original volumes of the ancient building and rebuild the two best-preserved floors –a geometric mosaic and an <em>opus spicatum</em>– for a first digital reconstruction of the archaeological complex in a 3D environment.</p>

scholarly journals The positive integral points on the elliptic curve 𝒚𝟐=𝟕𝒑𝒙(𝒙𝟐+𝟖)

2020 ◽  
Vol 165 ◽  
pp. 03046
Author(s):  
Du Xiancun ◽  
Jianhong Zhao ◽  
Lixing Yang
Keyword(s):  
Number Theory ◽  
Elliptic Curve ◽  
Integer Point ◽  
Integral Point ◽  
Analytic Number Theory ◽  
Elementary Number Theory ◽  
Integral Points ◽  
Integer Points ◽  
Elementary Methods ◽  
Special Case

The integral point of elliptic curve is a very important problem in both elementary number theory and analytic number theory. In recent years, scholars have paid great attention to solving the problem of positive integer points on elliptic curve 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), where 𝑘,𝑎,𝑏,𝑐 are integers. As a special case of 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), when 𝑎 = 1,𝑏 = 0,𝑐 = 22𝑡−1, it turns into 𝑦2 = 𝑘𝑥(𝑥2+22𝑡−1), which is a very important case. However ,at present, there are only a few conclusions on it, and the conclusions mainly concentrated on the case of 𝑡 = 1,2,3,4. The case of 𝑡 = 1, main conclusions reference [1] to [7]. The case of 𝑡 = 2, main conclusions reference [8]. The case of 𝑡 = 3, main conclusions reference [9] to [11]. The case of 𝑡 = 4, main conclusions reference [12] and [13]. Up to now, there is no relevant result on the case of 𝑘 = 7𝑝 when 𝑡 = 2, here the elliptic curve is 𝑦2 = 7𝑝(𝑥2 + 8), this paper mainly discusses the positive integral points of it. And we obtained the conclusion of the positive integral points on the elliptic curve 𝑦2 = 7𝑝(𝑥2 + 8). By using congruence, Legendre symbol and other elementary methods, it is proved that the elliptic curve in the title has at most one integer point when 𝑝 ≡ 5,7(𝑚𝑜𝑑8).

scholarly journals The vortex instability pathway in stratified turbulence

2013 ◽  
Vol 716 ◽  
pp. 1-4 ◽  
Author(s):  
Michael L. Waite
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Small Scale ◽  
Nonlinear Interactions ◽  
Stratified Turbulence ◽  
Transfer Energy ◽  
Columnar Vortex ◽  
Open Questions ◽  
Small Scales ◽  
Flow Transitions

AbstractThe parameter regime of strong stable density stratification and weak rotation is an important one in geophysical fluid dynamics. These conditions exist at intermediate length scales in the atmosphere and ocean (mesoscale and sub-mesoscale, respectively), and turbulence here links large-scale quasi-geostrophic motions with small-scale dissipation. While major advances in the theory of stratified turbulence have been made over the last few decades, many open questions remain, particularly about the nature of the energy cascade. Recent numerical experiments and analysis by Augier, Chomaz & Billant (J. Fluid Mech., vol. 713, 2012, pp. 86–108) present a remarkably vivid illustration of the nonlinear interactions that drive such turbulence. They consider a columnar vortex dipole, which naturally three-dimensionalizes under the influence of strong stratification. Kelvin–Helmholtz instabilities subsequently transfer energy directly to small scales, where the flow transitions into three-dimensional turbulence. This direct link between large and small scales is quite distinct from the usual picture of a turbulent cascade, in which nonlinear interactions are local in scale. But how important is this mechanism in the atmosphere and ocean?

The Effect of Product Representation in Visual Conjoint Analysis

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
Brian Sylcott ◽  
Seth Orsborn ◽  
Jonathan Cagan
Keyword(s):  
Conjoint Analysis ◽  
Choice Model ◽  
Three Dimensional ◽  
Model Performance ◽  
Consumer Preference ◽  
Product Representation ◽  
Consumer Responses ◽  
Open Questions ◽  
3D Printed ◽  
Preference Models

When most designers set out to develop a new product, they solicit feedback from potential consumers. These data are incorporated into the design process in an effort to more effectively meet customer requirements. Often these data are used to construct a model of consumer preference capable of evaluating candidate designs. Although the mechanics of these models have been extensively studied, there are still some open questions, particularly with respect to models of aesthetic preference. When constructing preference models, simplistic product representations are often favored over high fidelity product models in order to save time and expense. This work investigates how choice of product representation can affect model performance in visual conjoint analysis. Preference models for a single product, a table knife, are derived using three different representation schemes: simple sketches, solid models, and three dimensional (3D)-printed models. Each of these representations is used in a separate conjoint analysis survey. The results from this study show that the choice model based on 3D-printed photopolymer prototypes underperformed. Additionally, consumer responses were inconsistent and potentially contradictory between different representations. Consequently, when using conjoint analysis for product innovation, obtaining a true understanding of consumer preference requires selecting representations based on how accurately they convey the product details in question.

scholarly journals Multi-D magnetohydrodynamic modelling of pulsar wind nebulae: recent progress and open questions

2016 ◽  
Vol 82 (6) ◽  
Author(s):  
B. Olmi ◽  
L. Del Zanna ◽  
E. Amato ◽  
N. Bucciantini ◽  
A. Mignone
Keyword(s):  
Gamma Ray ◽  
Crab Nebula ◽  
Three Dimensional ◽  
Pulsar Wind Nebulae ◽  
Dissipation Process ◽  
Tearing Instability ◽  
Open Questions ◽  
Pulsar Wind ◽  
The Crab Nebula

In the last decade, the relativistic magnetohydrodynamic (MHD) modelling of pulsar wind nebulae, and of the Crab nebula in particular, has been highly successful, with many of the observed dynamical and emission properties reproduced down to the finest detail. Here, we critically discuss the results of some of the most recent studies: namely the investigation of the origin of the radio emitting particles and the quest for the acceleration sites of particles of different energies along the termination shock, by using wisp motions as a diagnostic tool; the study of the magnetic dissipation process in high magnetization nebulae by means of new long-term three-dimensional simulations of the pulsar wind nebula evolution; the investigation of the relativistic tearing instability in thinning current sheets, leading to fast reconnection events that might be at the origin of the Crab nebula gamma-ray flares.

scholarly journals Correlative tomography of an exceptionally preserved Jurassic ammonite implies hyponome-propelled swimming

Geology ◽  
2021 ◽  
Author(s):  
Lesley Cherns ◽  
Alan R.T. Spencer ◽  
Imran A. Rahman ◽  
Russell J. Garwood ◽  
Christopher Reedman ◽  
...  
Keyword(s):  
Soft Tissue ◽  
Functional Morphology ◽  
Middle Jurassic ◽  
Three Dimensional ◽  
The Body ◽  
Tissue Preservation ◽  
Soft Body ◽  
X Ray ◽  
Open Questions ◽  
First Time

The extreme rarity of soft-tissue preservation in ammonoids has meant there are open questions regarding fundamental aspects of their biology. We report an exceptionally preserved Middle Jurassic ammonite with unrivaled information on soft-body organization interpreted through correlative neutron and X-ray tomography. Three-dimensional imaging of muscles and organs of the body mass for the first time in this iconic fossil group provides key insights into functional morphology. We show that paired dorsal muscles withdrew the body into the shell, rather than acting with the funnel controlling propulsion as in Nautilus. This suggests a mobile, retractable body as a defense strategy and necessitates a distinct swimming mechanism of hyponome propulsion, a trait that we infer evolved early in the ammonoid-coleoid lineage.

scholarly journals Facilitating Accessibility: A Study on Innovative Didactic Materials to Generate Emotional Interactions with Pictorial Art

2021 ◽  
Author(s):  
Carmen Carpio de los Pinos ◽  
Arturo Galán González
Keyword(s):  
Three Dimensional ◽  
Inclusive Design ◽  
Test Model ◽  
3D Printer ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Audio Recording ◽  
Audio Description ◽  
Open Questions ◽  
Points Of View

This research has been undertaken to establish criteria for the construction of didactic materials to be experienced through touch (using a three-dimensional model) and hearing (through the provision of an audio description of the chosen painting) to provide learning and emotions. Eleven experts examined the didactic tools in which the scene of the painting had been depicted, through the use of white plastic figures modeled using a 3D printer. The models had been positioned to accurately correspond with the reference painting, with an explanatory narration supplied as an audio recording. Each of the experts involved were asked the same open questions in interviews that were audio-recorded and later transcribed. This feedback was analyzed and eleven concerns for consideration were determined: 1. How the figures felt to touch 2. Modeling and placement of the figure, 3. Position of the character, 4. Size, 5. The accurate 3D depiction of the 2D image, 6. Perspectives or visual points of view of the scene, 7. Enough representation of the painting in the model, 8. Distribution of visual components within the scene, 9. Perceptual appraisals, 10. Size of the model, 11. Touch of the whole model. The results indicated that the size of the model and the figurines was appropriate for their function. The figurines felt pleasant to handle and adequately described the postures and placement. Suggestions for further improvements were including more figurines in the model and adding color (omitted in the test model) to belong an inclusive design.

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