Invariant set generated by a nonreal number is everywhere dense

We will show that this construction is impossible if a straightedge and compass are the only tools which we may use. We precede the proof with a couple of definitions and facts which will be needed. A complex number is algebraic if it is a zero of a polynomial with integer coefficients. Numbers which are not algebraic are called transcendental. The set of all algebraic numbers is a field. This implies, in particular, that for any algebraic number a and any natural number k the number ak iS also algebraic. Recall also that each constructible number, i.e., a number which can be constructed using a straightedge and compass only, is algebraic. A crucial result for us is the following theorem due to Lindemann. If x 7s 0 is an algebraic number, then ex is a transcendental number. We refer to [2, 3] for unexplained terminology and details.

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

Algebraic Numbers with Integral Power Traces

Let $\{ {x_i } \}$ be any sequence approximating an algebraic number $\alpha $ of degree r, and let $x_{i + 1} = \varphi (x_i ,x_{i - 1} , \cdots ,x_{i - d + 1} )$, for some rational function $\varphi $ with integral coefficients. Let M denote the number of multiplications or divisions needed to compute $\varphi $ and let $\bar M$ denote the number of multiplications or divisions, except by constants, needed to compute $\varphi $. Define the multiplicative efficiency measure of $\{ {x_i } \}$ as $E = {{(\log _2 p)} / M}$ or as $\bar E = {{(\log _2 p)} / {\bar M}}$, where p is the order of convergence of $\{ {x_i } \}$. Kung [1] showed that $\bar E \leqq 1$ or equivalently, $\bar M \geqq \log _2 p$. In this paper we show that (i) $\bar M \geqq \log _2 [r(\lceil p \rceil - 1) + 1] - 1$; (ii) if $E = 1$ then $\alpha $ is a rational number; (iii) if $\bar E = 1$ then $\alpha $ is a rational or quadratic irrational number. This settles the question of when the multiplicative efficiency E or $\bar E$ achieves its optimal value of unity. Also, as a consequence of result (i), we show that the maximal efficiency $\bar E$ achievable by algebraic numbers of degree r drops at least as $O[(\log r)^{ - 1} ]$, provided that we only consider sequences $\{ {x_i } \}$ of bounded order of convergence.

Every algebraic number except zero and unity (e), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in 7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

Every algebraic number except zero and unity (e°), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, § succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in §7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the non-denumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagonalization argument, which did not appear until 1891. The third proof is of the existence of real transcendental (i.e., non-algebraic) numbers. It also appeared in Cantor’s 1874 paper, as a corollary to the non-denumerability of the reals. What Cantor ingeniously showed is that the algebraic numbers are denumerable, so every open interval must contain at least one transcendental number.

Let Q denote the rational number field and α be an algebraic number of degree s. Then the algebraic number field F s = Q(α) is the field given by the polynomials in α of degree < s with rational coefficients.

Given k ≥ 2 let α1, ..., αk be transcendental numbers such that α1, ..., αk–1 are algebraically independent over Q and αk $\in$ Q(α1, ..., αk–1), but αk, ≠ (aαi + c)/b for some i $\in$ {1, ..., k – 1} and some a, b $\in$ N, c $\in$ Z satisfying gcd(a,b) = 1. We prove that then there exists a nonnegative integer q such that the set of so-called Dirichlet exponents log(n + αj, where n runs through the set of all nonnegative integers for j = 1, ..., k – 1 and n = q, q + 1, q + 2, ... for j = k, is linearly independent over Q. As an application we obtain a joint universality theorem for corresponding Hurwitz zeta functions ζ(s, α1), ..., ζ(s, αk) in the strip {s $\in$ C: 1/2 < $\Re$(s) < 1}. In our approach we follow a recent result of Mishou who analyzed the case k = 2.

In this paper, a method for image encryption is proposed based on a novel approach using chaos and transcendental numbers for generating the substitution box and the ciphering keys. The Chaotic Pi Ciphering (CPC) is a robust symmetric cryptosystem of the substitution-permutation-network type, that ciphers images in ten rounds. The objective of CPC is to avoid some general weakness points in cryptosystems, such as the vulnerability to certain types of attacks or the encryption with data loss. The number of rounds can be increased up to 14, with an even further increment of the ciphering security. An 8×8 substitution box is constructed using chaos, and its value of the Differential Power Analysis is lower than the corresponding to the Advanced Encryption Standard (AES). It makes CPC more resistant to differential and linear attacks than AES. CPC has approximately 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">512</sup> keys, presenting a high resistance to brute-force attacks. Chaos is produced by a nonlinear differential equation, and the Walsh function is applied to measure the nonlinearity of the box. The transcendental number pi and a positive integer are employed to generate the schedule keys, and the key length is equal to the image size. The randomness of the encrypted images is evaluated with four tests: entropy, correlation, Discrete Fourier Transform, and a goodness-of-fit test using the chi square distribution χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . CPC was tested using well-known images from the related literature, and its performance was compared to the corresponding to AES. The results show a good entropy and a high randomness, indicating a high-quality ciphering.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

We begin with some preliminaries on algebraic numbers. Let α be an algebraic number. Then we observe that α satisfies the polynomial of minimal degree such that it has relatively prime integer coefficients and the leading coefficient positive. This is the minimal polynomial of α. The degree of this polynomial is called the degree of α. The maximum of the absolute values of the coefficients of this polynomial is called the height of α. We write ν = ν(α) for the least positive integer such that να is an algebraic integer i.e., να satisfies a monic polynomial with integer coefficients. The integer ν exists and we say that it is the denominator of α.

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures u ( s ) , u ( r ) u(s),u(r) with r r being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if s = r s = r or s = 1 − r s = 1 - r . We also show that for each positive integer n &gt; 2 n{\text { &gt; 2}} , there are algebraic integers and fractionals of degree n n in the unit interval that are binomial numbers.

We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [ J. Comb. Theory Ser. B , 101 (2011), 315–377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mspace width="0.166667em"/> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≈</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> <mml:mn>40958</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> <mml:mn>41501</mml:mn> </mml:mrow> </mml:math> is an algebraic number of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>10</mml:mn> </mml:math> .