Almost everywhere one-to-one functions and an n-cube decomposition

In this note, we investigate the regularity of Cantor’s one-to-one mapping between the irrational numbers of the unit interval and the irrational numbers of the unit square. In particular, we explore the fractal nature of this map by showing that its Hölder regularity lies between 0.35 and 0.72 almost everywhere (with respect to the Lebesgue measure).

Download Full-text

Existence of Continuous Functions That Are One-to-One Almost Everywhere

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-23688 ◽

2016 ◽

Vol 118 (2) ◽

pp. 269 ◽

Cited By ~ 1

Author(s):

Alexander J. Izzo

Keyword(s):

Metric Space ◽

Borel Measure ◽

Measure Zero ◽

Continuous Functions ◽

Almost Everywhere ◽

Regular Borel Measure ◽

One To One

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

Download Full-text

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

International Journal of Number Theory ◽

10.1142/s179304211850149x ◽

2018 ◽

Vol 14 (09) ◽

pp. 2473-2516 ◽

Cited By ~ 1

Author(s):

Thomas Garrity ◽

Peter Mcdonald

Keyword(s):

Continued Fractions ◽

Singular Function ◽

Question Mark ◽

Basic Properties ◽

Naturally Occurring ◽

Almost Everywhere ◽

Multidimensional Continued Fractions ◽

One To One ◽

Almost All ◽

Minkowski Question Mark Function

The Minkowski question mark function [Formula: see text] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus [Formula: see text] is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

Download Full-text

The Pisot conjecture for -substitutions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.44 ◽

2016 ◽

Vol 38 (2) ◽

pp. 444-472 ◽

Cited By ~ 2

Author(s):

MARCY BARGE

Keyword(s):

Dynamical System ◽

Discrete Spectrum ◽

Minimal Polynomial ◽

Pisot Number ◽

Companion Matrix ◽

Pisot Numbers ◽

Almost Everywhere ◽

One To One

We prove the Pisot conjecture for$\unicode[STIX]{x1D6FD}$-substitutions: if$\unicode[STIX]{x1D6FD}$is a Pisot number, then the tiling dynamical system$(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$associated with the$\unicode[STIX]{x1D6FD}$-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.

Download Full-text

Microstructural Characterization of Au-Ge-Ni based ohmic contacts to GaAs/AlGaAs modfet device

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100126433 ◽

1987 ◽

Vol 45 ◽

pp. 326-327

Author(s):

A.K. Rai ◽

A.K. Petford-Long ◽

A. Ezis ◽

D.W. Langer

Keyword(s):

Contact Resistance ◽

Ohmic Contact ◽

Ohmic Contacts ◽

Microstructural Characterization ◽

Almost Everywhere ◽

Spacer Layer ◽

Good Contact ◽

The Relationship ◽

Lower Contact

Considerable amount of work has been done in studying the relationship between the contact resistance and the microstructure of the Au-Ge-Ni based ohmic contacts to n-GaAs. It has been found that the lower contact resistivity is due to the presence of Ge rich and Au free regions (good contact area) in contact with GaAs. Thus in order to obtain an ohmic contact with lower contact resistance one should obtain a uniformly alloyed region of good contact areas almost everywhere. This can possibly be accomplished by utilizing various alloying schemes. In this work microstructural characterization, employing TEM techniques, of the sequentially deposited Au-Ge-Ni based ohmic contact to the MODFET device is presented.The substrate used in the present work consists of 1 μm thick buffer layer of GaAs grown on a semi-insulating GaAs substrate followed by a 25 Å spacer layer of undoped AlGaAs.

Download Full-text

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.

Download Full-text