Unit-circle-preserving mappings

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible. The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles. 1.1. Relation between mutual distances of 4 points in a plane. Let O, A, B, C be the four given points.

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Mappings which preserve regular dodecahedrons

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/58479 ◽

2006 ◽

Vol 2006 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Byungbae Kim

Keyword(s):

Linear Isometry ◽

One To One

We prove that if a one-to-one mappingf:ℝ3→ℝ3preserves regular dodecahedrons, thenfis a linear isometry up to translation.

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On Pure Quasi-Quantum Quadratic Operators of 𝕄2(ℂ) II

Open Systems & Information Dynamics ◽

10.1142/s1230161215500249 ◽

2015 ◽

Vol 22 (04) ◽

pp. 1550024 ◽

Cited By ~ 1

Author(s):

Farrukh Mukhamedov

Keyword(s):

Unit Circle ◽

Early Paper ◽

Unit Sphere ◽

Quadratic Operator ◽

Completely Positive ◽

Potential Interest ◽

Unit Circles ◽

Weaker Conditions

In this paper we study quasi quantum quadratic operators (QQO) acting on the algebra of [Formula: see text] matrices [Formula: see text]. We consider two kinds of quasi QQO the corresponding quadratic operator maps from the unit circle into the sphere and from the sphere into the unit circle, respectively. In our early paper we have defined a q-purity of quasi QQO. This notion is equivalent to the invariance of the unit sphere in [Formula: see text]. But to check this condition, in general, is tricky. Therefore, it would be better to find weaker conditions to check the q-purity. One of the main results of this paper is to provide a criterion of q-purity of quasi QQO in terms of the unit circles. Moreover, we are able to classify all possible kinds of quadratic operators which can produce q-pure quasi QQO. We think that such result will allow one to check whether a given mapping is a pure channel or not. This finding suggests us to study such a class of nonpositive mappings. Correspondingly, the complement of this class will be of potential interest for physicist since this set contains all completely positive mappings.

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A Support Program

Language Speech and Hearing Services in Schools ◽

10.1044/0161-1461.2502.112 ◽

1994 ◽

Vol 25 (2) ◽

pp. 112-114 ◽

Cited By ~ 3

Author(s):

Henna Grunblatt ◽

Lisa Daar

Keyword(s):

Hearing Aids ◽

School Counselor ◽

Support Program ◽

Educational Audiology ◽

Part Program ◽

One To One

A program for providing information to children who are deaf about their deafness and addressing common concerns about deafness is detailed. Developed by a school audiologist and the school counselor, this two-part program is geared for children from 3 years to 15 years of age. The first part is an educational audiology program consisting of varied informational classes conducted by the audiologist. Five topics are addressed in this part of the program, including basic audiology, hearing aids, FM systems, audiograms, and student concerns. The second part of the program consists of individualized counseling. This involves both one-to-one counseling sessions between a student and the school counselor, as well as conjoint sessions conducted—with the student’s permission—by both the audiologist and the school counselor.

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