Plane Maps with Denominator. Part II: Noninvertible Maps with Simple Focal Points

2003 ◽  
Vol 13 (08) ◽  
pp. 2253-2277 ◽  
Author(s):  
Gian-Italo Bischi ◽  
Laura Gardini ◽  
Christian Mira
Keyword(s):  
Focal Point ◽  
Single Point ◽  
Focal Points ◽  
Previous Part ◽  
Noninvertible Maps ◽  
Set Of Points

This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or "focalized") into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous Part I appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points.

scholarly journals PLANE MAPS WITH DENOMINATOR. PART III: NONSIMPLE FOCAL POINTS AND RELATED BIFURCATIONS

2005 ◽  
Vol 15 (02) ◽  
pp. 451-496 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
LAURA GARDINI ◽  
CHRISTIAN MIRA
Keyword(s):  
Focal Point ◽  
Dynamic Properties ◽  
Focal Points ◽  
One Dimensional ◽  
Global Dynamic ◽  
Critical Sets ◽  
One To One ◽  
The One ◽  
Noninvertible Maps ◽  
Qualitative Changes

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as the particular structures of the basins and the global bifurcations related to the presence of such singularities, this third paper is devoted to the analysis of nonsimple focal points, and the bifurcations associated with them. We prove the existence of a one-to-one relation between the points of a prefocal curve and arcs through the focal point having all the same tangent but different curvatures. In the case of nonsimple focal points, such a relation replaces the one-to-one correspondence between the slopes of arcs through a focal point and the points along the associated prefocal curve that have been proved and extensively discussed in the previous papers. Moreover, when dealing with noninvertible maps, other kinds of relations can be obtained in the presence of nonsimple focal points or prefocal curves, and some of them are associated with qualitative changes of the critical sets, i.e. with the structure of the Riemann foliation of the plane.

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

2009 ◽  
Vol 19 (02) ◽  
pp. 545-555 ◽  
Author(s):  
F. TRAMONTANA ◽  
L. GARDINI ◽  
D. FOURNIER-PRUNARET ◽  
P. CHARGE
Keyword(s):  
Focal Point ◽  
Singular Line ◽  
Two Dimensional ◽  
Focal Points ◽  
Singular Set ◽  
One Dimensional ◽  
Attracting Set ◽  
Straight Line ◽  
Special Character ◽  
Stable And Unstable Sets

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

Wave Height Measurements Behind Submerged Lens-Shaped Structures

2011 ◽  
Author(s):  
Andrew L. Bloxom ◽  
Karl D. von Ellenrieder ◽  
Matthew R. Anderson ◽  
Ryan S. Mieras ◽  
William S. Weidle
Keyword(s):  
Wave Height ◽  
Focal Point ◽  
Focal Length ◽  
Experimental Tests ◽  
Geometrical Optics ◽  
Width Ratio ◽  
Focal Points ◽  
Wave Amplification ◽  
Strip Theory ◽  
Longitudinal Position

The ability of submerged lens-shaped structures to focus linear surface waves in deep water is explored through a series of experimental tests in a wave making basin. Three lenses were designed using a combination of linear strip theory and a surface wave analogy to geometrical optics. Two of these lenses were designed to focus waves of a single wavelength of 0.482 m (18.97 in.), one with a focal length to lens width ratio (f-number) of 2.0 and the other with an f-number of 0.5. The third lens was designed to function as a compound lens that could focus a range of wavelengths of between 0.39 m (15.37 in.) and 0.694 m (27.32 in.) at an f-number of 2.0. Using resistance wave height gauges, the sensitivity of wave height at the focus to variations in wavelength from between 0.39 m (15.37 in.) to 0.61 m (24.01 in.) was experimentally measured for all three lenses; the sensitivity of wave height at the focus to variations of lens depths of submergence spanning the range of between 0.75 to 1.25 times the design submergence depth was also explored for the two simple lenses. It was found that the linear strip theory and geometrical optics approach predicted the wave amplification to within ten percent at the design wavelengths and depths, but that the longitudinal position of the experimentally observed focal lengths differed substantially from that expected, by as much as a factor of 2.2 for an f-number of 0.5. Additionally, while the theory predicted a single focal point for each lens, multiple focal points were found to exist behind the compound lens.

A New Imaging Method for Arterial Tubes Based on Vibration Measurement

2003 ◽  
Author(s):  
Xiaoming Zhang ◽  
Mostafa Fatemi ◽  
James F. Greenleaf
Keyword(s):  
Focal Point ◽  
Single Point ◽  
Vibration Measurement ◽  
Ultrasound Transducer ◽  
Imaging Method ◽  
Large Artery ◽  
Large Radius ◽  
Modal Shape ◽  
Measured Velocity ◽  
Gel Phantom

A new method for imaging and detecting modal shapes of vessels is introduced. Theory is developed that predicts the measured velocity is proportional to the value of the mode shape at the focal point of the ultrasound beam. Experimental a cylindrical gel phantom of large radius. This model simulates approximately a large artery and the surrounding body. The fundamental frequency was measured 83 Hz for the tube-phantom system. At this frequency the ultrasound transducer was scanned across the vessel plane with velocity measurement at one single point on the vessel and on the phantom by laser. The images obtained show clearly the interior tube and the modal shape of the tube.

Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards

2003 ◽  
Vol 93 (5) ◽  
pp. 1703-1729 ◽  
Author(s):  
Christopher R Knittel ◽  
Victor Stango
Keyword(s):  
Credit Card ◽  
Focal Point ◽  
Credit Cards ◽  
State Level ◽  
Tacit Collusion ◽  
Focal Points ◽  
National Integration ◽  
Credit Card Market ◽  
Using Data ◽  
Perverse Effect

We test whether a nonbinding price ceiling may serve as a focal point for tacit collusion, using data from the credit card market during the 1980’s. Our empirical model can distinguish instances when firms match a binding ceiling from instances when firms tacitly collude at a nonbinding ceiling. The results suggest that tacit collusion at nonbinding state-level ceilings was prevalent during the early 1980’s, but that national integration of the market reduced the sustainability of tacit collusion by the end of the decade. The results highlight a perverse effect of price regulation.

scholarly journals On Some Properties of Focal Points

2009 ◽  
Vol 2009 ◽  
pp. 1-11
Author(s):  
M. R. Ferchichi ◽  
I. Djellit
Keyword(s):  
Fixed Point ◽  
Focal Point ◽  
Two Dimensional ◽  
Focal Points ◽  
Dynamical Properties

We consider some dynamical properties of two-dimensional maps having an inverse with vanishing denominator. We put in evidence a link between a fixed point of a map with fractional inverse and a focal point of this inverse.

scholarly journals Development of a Regional Focal Point for animal genetic resources: the European example

2002 ◽  
Vol 32 ◽  
pp. 13-17
Author(s):  
D. Planchenault ◽  
L. Ollivier
Keyword(s):  
Annual Meeting ◽  
Focal Point ◽  
Steering Committee ◽  
Global Strategy ◽  
First Year ◽  
Focal Points ◽  
Animal Genetic Resources ◽  
Animal Diversity ◽  
Work Programme ◽  
Ressources Génétiques

SummaryThis paper describes the process that lead to the creation of the European Regional Focal Point (ERFP). The action was suggested by the FAO Global Strategy (1995) aimed to assist countries to stop animal diversity erosion by helping them with a better use and preservation of their livestock resources.In 1997, France accepted the responsibility of developing an ERFP though its Bureau des Ressources Génétiques (BRG). During the first year, the ERFP held meaningful discussions with the different European countries with the objective of finding a general agreement for an organisational structure as well as a medium-term work programme. The following step was settled during the Annual Meeting of the European Association for Animal Production (EAAP) in Warsaw where it was agreed that the new body had to have a light structure and respect national sovereignty regarding the AnGR.In February 2000, following a difficult internal debate involving the establishment of a basic strategy and further steps, an enquiry was launched among the National Coordinators in order to have an overall picture and to evaluate the usefulness of the proposed organisation to be established. In 2000 during the 6th Workshop of the European NCs, the ERFP was created.The new structure is based ona) an Annual Meeting of National Focal Points;b) a Steering Committee; andc) a Secretariat to be elected among the National Focal Points to serve for a limited period.

scholarly journals Characterizing Regenerative Aspects of Living Root Bridges

2020 ◽  
Vol 12 (8) ◽  
pp. 3267
Author(s):  
Wilfrid Middleton ◽  
Amin Habibi ◽  
Sanjeev Shankar ◽  
Ferdinand Ludwig
Keyword(s):  
Focal Point ◽  
Economic Systems ◽  
Design Tools ◽  
Focal Points ◽  
User Perspective ◽  
Global Tourism ◽  
Living Root ◽  
Living Environments ◽  
Regenerative Design ◽  
Whole Systems

Living root bridges (LRBs) are functional load-bearing structures grown from Ficus elastica by rural Khasi and Jaintia communities in Meghalaya (India). Formed without contemporary engineering design tools, they are a unique example of vernacular living architecture. The main objective of this study is to investigate to what extent LRBs can be seen as an example of regenerative design. The term "regenerative" describes processes that renew the resources necessary for their function. Whole systems thinking underpins regenerative design, in which the integration of human and non-human systems improves resilience. We adapted the living environments in natural, social, and economic systems (LENSES) framework (living environments in natural, social, and economic systems) to reflect the holistic, integrated systems present in LRBs. The regenerative / sustainable / degenerative scale provided by LENSES Rubrics is applied to 27 focal points in nine flow groups. Twenty-two of these points come from LENSES directly, while five were created by the authors, as advised by the LENSES framework. Our results show 10 focal points in which LRBs are unambiguously regenerative. One focal point is unambiguously sustainable, while 16 are ambiguous, showing regenerative, sustainable, and degenerative aspects. User perspective determines how some focal points are evaluated. The contrast between a local, indigenous perspective and a global, tourism-focused perspective is demonstrated by the results.

Shape Measurement of Solder Bumps by Shape-from-Focus Using Varifocal Mirror

2006 ◽  
Vol 18 (6) ◽  
pp. 722-727
Author(s):  
Jun Mitsudo ◽  
◽  
Akira Ishii
Keyword(s):  
Copper Alloy ◽  
Solder Bump ◽  
Optical Axis ◽  
Focal Point ◽  
Shape Measurement ◽  
Depth Range ◽  
Focal Points ◽  
Depth Data ◽  
Solder Bumps ◽  
Shape From Focus

We measured solder bumps on an LSI package board presented for inspection based on the shape from focus. We used a copper-alloy mirror deformed by a piezoelectric actuator as a varifocal mirror to build a motionless yet fast focusing mechanism. The varifocal mirror was at the image focal point of the image-taking lens so that lateral magnification was constant during focusing and orthographic projection was established. A focused plane was shifted along the optical axis with a precision of 1.4μm in a depth range of 1.5mm by driving the varifocal mirror. A magnification of 1.97 was maintained during focusing. Evaluating the curvature of field and removing its effect from the depth data reduced errors. The shape of 208 solder bumps 260-μm high spaced 500μm on the board was measured. The 10mm×10mm board was segmented into partly overlapping 3×4 sections. We captured 101 images in each section with a high-resolution camera at different focal points at 15-μm intervals. The shape of almost the entire upper-hemisphere of a solder bump could be measured. Error in measuring bump heights was less than 12μm.

Hobbes's Confusing “Clarity”—The Case of “Liberty”

1960 ◽  
Vol 54 (2) ◽  
pp. 428-436 ◽  
Author(s):  
J. Roland Pennock
Keyword(s):  
Social Contract ◽  
Focal Point ◽  
Psychological Explanation ◽  
Political Obligation ◽  
Focal Points ◽  
Central Concept ◽  
Natural Right ◽  
The Social ◽  
Analytical System ◽  
Acceptable Theory

The Leviathan has been described as “original, persuasive, solid, coherent.” General commentaries on Hobbes usually single out his logic for special praise; more detailed critiques generally unearth a mass of confusions and inconsistencies. Confusions and inconsistencies there certainly are; more, I believe, than one would expect to find in the work of a man of such undeniable logical powers. Speculation upon the psychological explanation of this fact is intriguing, but no part of the purpose of the present article. It is part of my purpose, however, to contend that Hobbes's passion for clarity and certainty may have played a part in leading him to adopt perverse definitions, to which even he did not consistently adhere and which constituted a major source of confusion. Conversely, I disagree with those who say his analytical system is sound and only his empirical assumptions about human nature are open to serious criticism.More specifically, one may profitably inquire whether there is some central concept that serves as a focal point for many of these difficulties. For example, it is often suggested, with merit, that Hobbes's perversion, or inversion, of the traditional meaning of “jus naturale” plays such a role. Without making any exclusive claim or denying the insights that may be gained by concentrating attention upon other focal points, my hypothesis is that understanding of Hobbes may be deepened by an examination of his use of the word “liberty.” I shall deal first with his definitions of the term, and then in turn with his applications of it to natural right and natural law, to sovereignty by acquisition, and finally to the social contract. I shall argue that his method, as illustrated by his definitions, leads him occasionally into confusion or inconsistency, and more frequently tends to confuse the reader and so to enable Hobbes to make an unsound conclusion appear sound, by means of specious reasoning. In particular, I shall contend that Hobbes's treatment of liberty (1) leads him into self-contradiction regarding the extent of natural liberty, (2) enables him to argue persuasively but speciously in support of the obligation to obey a sovereign who has attained his position by violence, and (3) prevents him from developing an acceptable theory of political obligation.

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