scholarly journals Universality probability of a prefix-free machine

2012 ◽  
Vol 370 (1971) ◽  
pp. 3488-3511 ◽  
Author(s):  
George Barmpalias ◽  
David L. Dowe
Keyword(s):  
Turing Degree ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Halting Problem ◽  
The Real ◽  
The Third ◽  
Free Machine

We study the notion of universality probability of a universal prefix-free machine, as introduced by C. S. Wallace. We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers that are universality probabilities of universal prefix-free machines.

scholarly journals Undecidable, Unrecognizable, and Quantum Computing

2020 ◽  
Vol 2 (3) ◽  
pp. 337-342
Author(s):  
Michael Siomau
Keyword(s):  
Quantum Computing ◽  
Turing Machine ◽  
Quantum Computer ◽  
Practical Realization ◽  
Real Numbers ◽  
Theoretical Limit ◽  
Halting Problem ◽  
The Real ◽  
Computing Model ◽  
Successful Design

Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.

COMPUTING STRENGTH OF STRUCTURES RELATED TO THE FIELD OF REAL NUMBERS

2017 ◽  
Vol 82 (1) ◽  
pp. 137-150 ◽  
Author(s):  
GREGORY IGUSA ◽  
JULIA F. KNIGHT ◽  
NOAH DAVID SCHWEBER
Keyword(s):  
Analytic Function ◽  
Real Numbers ◽  
Computing Power ◽  
The Real ◽  
The Third ◽  
Ordered Field ◽  
Image Position

AbstractIn [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure ${\cal R}$ℚ consisting of the real numbers with just the ordering and constants naming the rationals, all o-minimal expansions of ${\cal R}$ℚ are equivalent to ${\cal R}$. Using this, we show that for any analytic function f, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$. (This is so even if $\left( {{\cal R},f} \right)$ is not o-minimal.)

On the quartic surface

1944 ◽  
Vol 40 (2) ◽  
pp. 121-145 ◽  
Author(s):  
B. Segre
Keyword(s):  
Geometric Characterization ◽  
Discontinuous Group ◽  
Quartic Surface ◽  
The Real ◽  
Birational Transformations ◽  
Projective Transformations ◽  
The Third ◽  
Real Domain ◽  
Different Types

Summary1. The projective transformations of F into itself … 1212. The flecnodal curve, and the lines of F … 1223. A geometric characterization of F, and the six different types of F in the real domain … 1234. The τ-points and τ-planes, and a notation for the lines of F … 1245. The incidence conditions for the lines of F … 1256. The tetrads of the first kind … 1267. The tetrads of the second kind … 1268. The pairs of lines of F … 1279. The tetrads of the third kind … 12910. The 16-tangent quadrics of F … 13011. The conics of the first kind … 13112. The conics of the second kind … 13213. No other irreducible conics lie on F … 13314. The tangent planes of F of multiplicity greater than 3 … 13615. On twisted curves, especially cubics and quartics, lying on F … 13816. The T-transformations … 13917. Construction of an infinite discontinuous group of birational transformations of F into itself … 14218. Deduction of an infinity of unicursal curves lying on F … 143

scholarly journals On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
J. M. Sepulcre
Keyword(s):  
Important Class ◽  
Exponential Polynomial ◽  
Critical Interval ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Polynomials ◽  
The Real ◽  
Linearly Independent ◽  
Pointwise Characterization

We provide the proof of a practical pointwise characterization of the setRPdefined by the closure set of the real projections of the zeros of an exponential polynomialP(z)=∑j=1ncjewjzwith real frequencieswjlinearly independent over the rationals. As a consequence, we give a complete description of the setRPand prove its invariance with respect to the moduli of thecj′s, which allows us to determine exactly the gaps ofRPand the extremes of the critical interval ofP(z)by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

scholarly journals On the characterization of Jensen m-convex polynomials

2017 ◽  
Vol 3 (2) ◽  
pp. 140-148
Author(s):  
Teodoro Lara ◽  
Nelson Merentes ◽  
Roy Quintero ◽  
Edgar Rosales
Keyword(s):  
Convex Functions ◽  
Polynomial Functions ◽  
Real Polynomial ◽  
Real Numbers ◽  
The Real ◽  
Class Of Functions

AbstractThe main objective of this research is to characterize all the real polynomial functions of degree less than the fourth which are Jensen m-convex on the set of non-negative real numbers. In the first section, it is established for that class of functions what conditions must satisfy a particular polynomial in order to be starshaped on the same set. Finally, both kinds of results are combined in order to find examples of either Jensen m-convex functions which are not starshaped or viceversa.

scholarly journals Coloring the $n$-Smooth Numbers with $n$ Colors

10.37236/8492 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Andrés Eduardo Caicedo ◽  
Thomas A. C. Chartier ◽  
Péter Pál Pach
Keyword(s):  
High School ◽  
Abelian Group ◽  
General Version ◽  
Real Numbers ◽  
Smooth Numbers ◽  
The Third ◽  
School Journal ◽  
Natural Density ◽  
Positive Integers

For which values of $n$ is it possible to color the positive integers using precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all receive different colors? The third-named author posed the question around 2008-2009. Particular cases appeared in the Hungarian high school journal KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by D. Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings. Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ that is compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is  a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$. There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\leqslant 5$ and all multiplicative colorings for $n\leqslant 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then it admits infinitely many and in fact the set of such $q$ has positive natural density in the set of all primes. We also show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.

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