Delving Deeper: A Specific Construction of a Conic From An Ellipse

2004 ◽  
Vol 98 (2) ◽  
pp. 98-103
Author(s):  
J. Todd Lee
Keyword(s):  
Mathematics Teacher ◽  
Major Axis ◽  
Geometric Construction ◽  
The Other ◽  
Conic Section ◽  
Center Point ◽  
Variable Point ◽  
Specific Construction ◽  
General Setup

Consider the interesting geometric construction given by Manuel Santos-Trigo in the “Technology Tips” in the January 2004 Mathematics Teacher (Santos-Trigo 2004). He starts with an ellipse, its center point O, a variable point R on the line along the major axis of the ellipse, and variable points S and S', which are points on the ellipse that are reflections with respect to the major axis. Figure 1 shows the general setup and various placements of S. Santos-Trigo constructs a pair of lines, one through R and S and the other through O and S'. He describes a set of discovery exercises involving the locus of points generated by the intersection of these lines as the point S roams around the ellipse. The empirical conclusion of these exercises is that the locus is a conic section, the nature of which is determined in a fairly simple way by the location of R relative to O and the major vertices of the ellipse.

scholarly journals 22. The relation between orbits and physical characteristics of meteors

1968 ◽  
Vol 33 ◽  
pp. 217-235 ◽  
Author(s):  
Ľ. Kresák
Keyword(s):  
Major Axis ◽  
Progressive Increase ◽  
Physical Characteristics ◽  
The Other ◽  
Asteroid Belt ◽  
Angle Of Incidence ◽  
Semi Major Axis ◽  
Short Period ◽  
Broad Variety ◽  
Bona Fide

The relation of physical characteristics of meteors to their orbital elements is investigated using Harvard Super-Schmidt data. A set of characteristic indices is defined, allowing for the effects of geocentric velocity, angle of incidence, magnitude and mass, wherever a correction appears appropriate according to the correlations found by Jacchia et al. (1967). The medians for representative meteor samples are plotted in the semi-major axis/eccentricity diagram and the distribution of each parameter is derived. Although the differences are moderate compared to the measuring errors, six regions of different nature can be distinguished.The existence of two families of asteroidal meteors is indicated, one of them brought to crossing with the Earth's orbit by drag effects and the other by collision effects in the main asteroid belt. These meteors are characterized by low and uniform beginning heights, high fragmentation, low ablation, low deceleration, and bright wakes. A direct counterpart to this is represented by meteors moving in short-period orbits of higher eccentricity and shorter perihelion distance, which bear resemblance to the long-period and retrograde cometary meteors. Meteors with perihelion distances of less than 0·15 AU tend to resemble the bona fide asteroidal meteors by a progressive increase of fragmentation and decrease of reduced beginning heights and decelerations as the perihelion approaches the Sun. This is attributed to the selective destruction effects of solar radiation.With the exception of the Draconids, the mean characteristics of meteor showers agree well with those of sporadic meteors moving in similar orbits. It is suggested that the Draconid stream includes a broad variety of meteoric material and that the two peculiar Super-Schmidt meteors on record represent only the less resistive, short-lived component which has already been eliminated from the other showers.

Identifying Sets of Four and Five Positions That Generate Distinctive Center-Point Curves

2009 ◽  
Author(s):  
David H. Myszka ◽  
Andrew P. Murray
Keyword(s):  
Closed Form ◽  
The Other ◽  
Specific Design ◽  
Opposite Pole ◽  
Center Point ◽  
Point Curve ◽  
Position Synthesis ◽  
Two Sides ◽  
Cubic Function

The fixed pivots of a planar 4R linkage that can achieve four design positions are constrained to a center-point curve. The curve is a circular cubic function and plots can take one of five different forms. The center-point curve can be generated with a compatibility linkage obtained from an opposite pole quadrilateral of the four design positions. This paper presents a method to identify design positions that generate distinctive shapes of the center-point curves. The form of the center-point curve is dependent on whether the shape of the opposite pole quadrilateral is an open or closed form of a rhombus, kite, parallelogram, or when the sum of two sides equals the other two. Interesting cases of three and five position synthesis are also explored. Four and five position cases are generated that have center points at infinity allowing a PR dyad with line of slide in any direction to achieve the design positions. Further, a center-point curve for five specific design positions is revealed.

Recreational Mathematics

1950 ◽  
Vol 43 (6) ◽  
pp. 290-291
Author(s):  
Aaron Bakst
Keyword(s):  
Mathematics Teacher ◽  
Classroom Teacher ◽  
The Other ◽  
Mathematical Instruction ◽  
Other Hand ◽  
Points Of View ◽  
Mathematical Operations ◽  
Interest And Motivation ◽  
Mathematical Literature

This is the beginning of a new department in The Mathematics Teacher. This department has a purpose. Its aim is to assist the classroom teacher in putting color and life in everyday teaching. There are many ways and means how this might be achieved. Generally, recreations are supposed to introduce elements of interest and motivation. On the other hand, recreations, as they have been known in the mathematical literature for centuries, have been centered around the puzzling and the play with mathematical operations. This may be interesting, but only for a while. Soon the interest in such things may wear off. This acts as a warning that we should not become too enthusiastic over such types of recreations. If we teach mathematics from such recreational points of view only, we may obscure the more important aims of the mathematical instruction.

1a. Qualified Quantitative Geography

10.1068/d292 ◽  
2001 ◽  
Vol 19 (5) ◽  
pp. 555-572 ◽  
Author(s):  
Marcus A Doel
Keyword(s):  
The Other ◽  
Opening Scene ◽  
Qualitative Approaches ◽  
Ethical Status ◽  
Other Hand ◽  
The One ◽  
Ethical Response ◽  
General Setup ◽  
The Way

The rapprochement of humanism and structuralism on the one hand, and quantitative and qualitative approaches on the other hand, has not addressed an implacable difficulty which continues to haunt both spatial science and ‘critical’ human geographies. That difficulty concerns the ontological and ethical status of numbers, and their relationship to concepts, events, and sensations. The paper engages with this difficulty through a combination of theoretical and literary writings, most notably Woody Allen's film Deconstructing Harry, Samuel Beckett's play Not I, and Derrida's work of Dissemination. Insofar as ‘one’ lacks consistency—by disavowing difference, alterity, and innumerable numbers—its deployment is invariably unbecoming, repressive, and ill-mannered. The ethical response is to divine ‘another way of working with numbers’, as Derrida once intimated; to prevent some ones from taking hold. The outcome is a form of poststructuralist geography that takes flight from all kinds of pointillism. After an opening scene that lays out the general setup of quantification and its qualification, the first section of the paper employs the notion of a soft ontology in order to prepare the way for ‘another way of working with numbers’ that is occasioned by a sensitivity towards the ontological buzzing and solicitation that accompanies processes of subjectification, objectification, identification, and enumeration. The paper concludes with an affirmation of a ‘disturbing geography’ that leaves everything in perpetual suspense.

scholarly journals Vanishing or non-vanishing rainbow? Reduction formulas of electric dipole moment

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Motoko Fujiwara ◽  
Junji Hisano ◽  
Takashi Toma
Keyword(s):  
Electric Dipole ◽  
Dipole Moments ◽  
Form Factors ◽  
The Other ◽  
Vertex Correction ◽  
Electric Dipole Moments ◽  
Self Energy ◽  
The Standard Model ◽  
General Setup ◽  
Exact Cancellation

Abstract In this paper, we derive a simplified formula of electric dipole moments (EDMs) of a fermion. In the Standard Model, it is well-known that non-trivial cancellations between some rainbow-type diagrams induced by W boson exchanges occur in the calculation of the neutron EDM at the two-loop level due to the gauge symmetry. The fermion self-energy and the vertex correction are related through the Ward-Takahashi identity, and this relation causes the exact cancellation of the EDM. We derive EDM formulas for a more general setup by introducing the form factors for the fermion self-energy and the vertex correction so that the derived formulas can be applicable to a larger class of models. We conclude that the non-zero EDM contributions are induced from rainbow-type diagrams with the chirality flipping effects for internal fermions. We also discuss the other possible generalization of the EDM calculation which is applicable to the other classes of models.

Intuitive Approach to a−m≡1am

1967 ◽  
Vol 60 (7) ◽  
pp. 737
Author(s):  
Pete Murray
Keyword(s):  
Mathematics Teacher ◽  
The Other ◽  
The Relationship

In the article “Intuitive Approach to X0 ≡ 1” by Curtiss Mallory (The Mathematics Teacher, January 1967, page 41), two approaches to the definition X0≡ 1 were suggested, one depending on the relationship am · an = am + n, the other depending on the pattern that occurs as one writes a sequence of descending powers of some (nonzero) base.

Instant Insanity: That Ubiquitous Baffler

1972 ◽  
Vol 65 (2) ◽  
pp. 131-135
Author(s):  
Dewey C. Duncan
Keyword(s):  
Mathematics Teacher ◽  
The Other ◽  
Letter To The Editor

This discussion is occasioned by two notes in the October 1969 issue of the Mathematics Teacher, one a letter to the editor from Barbara Almli on page 446, and the other Philip Peak's review on page 477 in his “Have You Read …?” section of T. A. Brown's “A Note on ‘Instant Insanity’” (Mathematics Magazine, September 1968, pp. 167-69). Each of these items describes the puzzling challenge known as “Instant Insanity.”

Call for Manuscripts: Focus Issues on the Role of Proof throughout the Mathematics Curriculum

1996 ◽  
Vol 89 (6) ◽  
pp. 507
Keyword(s):  
Secondary School ◽  
Mathematics Teacher ◽  
Mathematics Curriculum ◽  
School Mathematics ◽  
The Other ◽  
Secondary School Mathematics

Proof has traditionally been the touchstone of mathematics—that which distinguishes it from the other sciences. The role and nature of proof in a Standards-based curriculum merit reexamination in an era of reform. In 1998 a fall issue of the Mathematics Teacher will focus on the theme of proof in all aspects of secondary school mathematics. The Editorial Panel is seeking manuscripts for this issue.

Call for Manuscripts: Focus Issue on the Role of Proof throughout the Mathematics Curriculum

1997 ◽  
Vol 90 (1) ◽  
pp. 19
Keyword(s):  
Secondary School ◽  
Mathematics Teacher ◽  
Mathematics Curriculum ◽  
School Mathematics ◽  
The Other ◽  
Secondary School Mathematics

Proof has traditionally been the touchstone of mathematics—that which distinguishes it from the other sciences. The role and nature of proof in a Standards-based curriculum merit reexamination in an era of reform. In 1998 a fall issue of the Mathematics Teacher will focus on the theme of proof in all aspects of secondary school mathematics. The Editorial Panel is seeking manuscripts for this issue.

scholarly journals Accumulated and recent migratory at traction in Mexico and Veracruz. Analysis with special reference to the state of Veracruz

2018 ◽  
pp. 134-153
Author(s):  
Violeta Alejandra Bastián Lima ◽  
José Dionicio Vázquez Vázquez
Keyword(s):  
Special Reference ◽  
Internal Migration ◽  
National Level ◽  
The State ◽  
The Other ◽  
Future Research ◽  
Cultural Dynamics ◽  
Urban Centers ◽  
Other Hand ◽  
Specific Construction

La atracción migratoria (reciente y acumulada) es una construcción específica de la migración interna, la cual contiene dinámicas socioeconómicas y culturales, entre otras, desde los centros rurales o urbanos, con perspectivas territoriales y regionales. En México, el indicador de la atracción migratoria reciente muestra mejores condiciones y dinámicas de la economía desde hace siete años. Por su parte, Veracruz presenta desde hace una década un importante crecimiento demográfico y de inmigración interna, además de una significativa migración hacia entidades del norte. En este contexto, el objetivo del artículo es construir indicadores de atracción migratoria y reciente a nivel nacional y para el estado de Veracruz, que sirvan como base para futuras investigaciones que profundicen en las causas y factores de la migración interna, y expliquen las dinámicas entre un territorio y otro. The migratory attraction (recent and accumulated) is a specific construction of internal migration, which contains socio - economic and cultural dynamics, among others; from rural or urban centers, with territorial and regional perspectives. In Mexico, the indicator of recent migration attraction shows the best conditions and dynamics of the economy since seven years ago. On the other hand, Veracruz has had an important demographic and internal immigration growth since a decade, as well as an important migration towards northern entities. In this context, the objective of the article is to build indicators of mig ratory and recent attraction at the national level and for the state of Veracruz, which serve as a basis for future research that delves into the causes and factors of internal migration and explain the dynamics between one territory and another.

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