scholarly journals Distance Product Cubics

KoG ◽  
2020 ◽  
pp. 29-40
Author(s):  
Boris Odehnal
Keyword(s):  
Geometric Properties ◽  
The Real ◽  
Cubic Curve ◽  
Triangle Center ◽  
Triangle Centers ◽  
Trilinear Coordinates

The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of cubics that contains only one rational and non-degenerate cubic curve which is known as the Bataille acnodal cubic determined by the product of the actual trilinear coordinates of the centroid of the base triangle. Each triangle center defines a distance product cubic. It turns out that only a small number of triangle centers share their distance product cubic with other centers. All distance product cubics share the real points of inflection which lie on the line at infinity. The cubics' dual curves, their Hessians, and especially those distance product cubics that are defined by particular triangle centers shall be studied.

Triangle Center Technology

2020 ◽  
Vol 113 (3) ◽  
pp. 237-243
Author(s):  
Anne Quinn
Keyword(s):  
Dynamic Geometry ◽  
Dynamic Geometry Software ◽  
Web Based ◽  
Triangle Center ◽  
Triangle Centers

The paper discusses technology that can help students master four triangle centers -- circumcenter, incenter, orthocenter, and centroid. The technologies are a collection of web-based apps and dynamic geometry software. Through use of these technologies, multiple examples can be considered, which can lead students to generalizations about triangle centers.

Periodic orbits of difference equations

1995 ◽  
Vol 125 (4) ◽  
pp. 657-674 ◽  
Author(s):  
A. F. Beardon ◽  
S. R. Bullett ◽  
P. J. Rippon
Keyword(s):  
Dynamical System ◽  
Difference Equation ◽  
Periodic Orbits ◽  
Difference Equations ◽  
Real Difference ◽  
Geometric Properties ◽  
Circle Maps ◽  
The Real ◽  
Rotation Numbers

The real difference equationan+2− (λ|an+1| + μan+1) +an= 0 may be interpreted as a dynamical system Φ:(an,an+1) ↦ (an+1,an+2) acting in the plane. The set ΛPof points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP(for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.

scholarly journals Starlikeness, convexity and landau type theorem of the real kernel α-harmonic mappings

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2629-2644
Author(s):  
Bo-Yong Long ◽  
Qi-Han Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Type Theorem ◽  
Second Order ◽  
Harmonic Mappings ◽  
Partial Differential ◽  
Geometric Properties ◽  
The Real ◽  
Landau Type Theorem ◽  
Univalence Criteria

In [26], Olofsson introduced a kind of second order homogeneous partial differential equation. We call the solution of this equation real kernel ?-harmonic mappings. In this paper, we study some geometric properties of this real kernel ?-harmonic mappings. We give univalence criteria and sufficient coefficient conditions for real kernel ?-harmonic mappings that are fully starlike or fully convex of order ?, ? ? [0, 1). Furthermore, we establish a Landau type theorem for real kernel ?-harmonic mappings.

Teacher to Teacher: Cardboard Tubes Bring Geometry from Home

1998 ◽  
Vol 4 (2) ◽  
pp. 120-122
Author(s):  
Nakonia Hayes
Keyword(s):  
Real World ◽  
Geometric Properties ◽  
The Real ◽  
The Relationship ◽  
Paper Towels

No matter what their ages, students who see mathematics used in the real world-a world they really do see and use every day-begin to appreciate the relationship between mathematics and their lives. Spending a lot of money to see that spark of connection in students' eyes is unnecessary. even with today's glitzy, worldly attractions. For example, the cardboard tube found inside a roll of tissue or paper towels can help students examine and extend their understanding of geometric properties of parallelograms.

Affine cubic functions

1980 ◽  
Vol 87 (1) ◽  
pp. 1-14 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Complex Case ◽  
Real Case ◽  
Dynamic Approach ◽  
Cubic Curves ◽  
The Real ◽  
Level Curves ◽  
Cubic Curve

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.

Scanning electron microscopy of human iliac bone by a simple preparation method

1990 ◽  
Vol 48 (3) ◽  
pp. 226-227
Author(s):  
Toshihiko Takita ◽  
Tomonori Naguro ◽  
Toshio Kameie ◽  
Akihiro Iino ◽  
Kichizo Yamamoto
Keyword(s):  
Electron Microscopy ◽  
Scanning Electron Microscopy ◽  
Preparation Method ◽  
Specimen Preparation ◽  
Real Surface ◽  
Public Attention ◽  
Preparation Methods ◽  
The Real ◽  
Simple Preparation ◽  
Scanning Electron

Recently with the increase in advanced age population, the osteoporosis becomes the object of public attention in the field of orthopedics. The surface topography of the bone by scanning electron microscopy (SEM) is one of the most useful means to study the bone metabolism, that is considered to make clear the mechanism of the osteoporosis. Until today many specimen preparation methods for SEM have been reported. They are roughly classified into two; the anorganic preparation and the simple preparation. The former is suitable for observing mineralization, but has the demerit that the real surface of the bone can not be observed and, moreover, the samples prepared by this method are extremely fragile especially in the case of osteoporosis. On the other hand, the latter has the merit that the real information of the bone surface can be obtained, though it is difficult to recognize the functional situation of the bone.

Personal Audio System: Hearing Symptoms, Habits, and Sound Pressure Levels Measured in Real Ear and a Manikin

2020 ◽  
Vol 63 (6) ◽  
pp. 2016-2026
Author(s):  
Tamara R. Almeida ◽  
Clayton H. Rocha ◽  
Camila M. Rabelo ◽  
Raquel F. Gomes ◽  
Ivone F. Neves-Lobo ◽  
...  
Keyword(s):  
Sound Pressure ◽  
Daily Basis ◽  
Cross Sectional Study ◽  
Normal Hearing ◽  
Chi Square ◽  
Cross Sectional ◽  
The Real ◽  
Significant Difference ◽  
Sound Pressure Levels ◽  
Audio System

Purpose The aims of this study were to characterize hearing symptoms, habits, and sound pressure levels (SPLs) of personal audio system (PAS) used by young adults; estimate the risk of developing hearing loss and assess whether instructions given to users led to behavioral changes; and propose recommendations for PAS users. Method A cross-sectional study was performed in 50 subjects with normal hearing. Procedures included questionnaire and measurement of PAS SPLs (real ear and manikin) through the users' own headphones and devices while they listened to four songs. After 1 year, 30 subjects answered questions about their usage habits. For the statistical analysis, one-way analysis of variance, Tukey's post hoc test, Lin and Spearman coefficients, the chi-square test, and logistic regression were used. Results Most subjects listened to music every day, usually in noisy environments. Sixty percent of the subjects reported hearing symptoms after using a PAS. Substantial variability in the equivalent music listening level (Leq) was noted ( M = 84.7 dBA; min = 65.1 dBA, max = 97.5 dBA). A significant difference was found only in the 4-kHz band when comparing the real-ear and manikin techniques. Based on the Leq, 38% of the individuals exceeded the maximum daily time allowance. Comparison of the subjects according to the maximum allowed daily exposure time revealed a higher number of hearing complaints from people with greater exposure. After 1 year, 43% of the subjects reduced their usage time, and 70% reduced the volume. A volume not exceeding 80% was recommended, and at this volume, the maximum usage time should be 160 min. Conclusions The habit of listening to music at high intensities on a daily basis seems to cause hearing symptoms, even in individuals with normal hearing. The real-ear and manikin techniques produced similar results. Providing instructions on this topic combined with measuring PAS SPLs may be an appropriate strategy for raising the awareness of people who are at risk. Supplemental Material https://doi.org/10.23641/asha.12431435

Current Issues: Evidence-Based Practice for the Real World: Thoughts on Clinical Education in Communication Disorders

2010 ◽  
Vol 20 (3) ◽  
pp. 100-105 ◽  
Author(s):  
Anne K. Bothe
Keyword(s):  
Clinical Education ◽  
Real World ◽  
Evidence Based Practice ◽  
Communication Disorders ◽  
University Faculty ◽  
Evidence Based ◽  
Speech Language Pathology ◽  
Basic Principles ◽  
The Real ◽  
Language Pathology

This article presents some streamlined and intentionally oversimplified ideas about educating future communication disorders professionals to use some of the most basic principles of evidence-based practice. Working from a popular five-step approach, modifications are suggested that may make the ideas more accessible, and therefore more useful, for university faculty, other supervisors, and future professionals in speech-language pathology, audiology, and related fields.

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