scholarly journals On the algebraic numbers computable by some generalized Ehrenfest urns

2012 ◽  
Vol Vol. 14 no. 2 ◽  
Author(s):  
Marie Albenque ◽  
Lucas Gerin
Keyword(s):  
Distributed Computing ◽  
Algebraic Number ◽  
Fixed Number ◽  
Algebraic Numbers ◽  
Population Protocols ◽  
Black And White ◽  
International Audience

International audience This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be ''computed'' this way.

scholarly journals Remarks on the approximation to an algebraic number by algebraic numbers.

1986 ◽  
Vol 33 (1) ◽  
pp. 83-93 ◽  
Author(s):  
E. Bombieri ◽  
J. Mueller
Keyword(s):  
Algebraic Number ◽  
Algebraic Numbers

scholarly journals Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Vichian Laohakosol ◽  
Suton Tadee
Keyword(s):  
Growth Factor ◽  
Real Number ◽  
Algebraic Number ◽  
Positive Real Number ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Quantitative Version

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of f similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers.

scholarly journals Top Coefficients of the Denumerant

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Velleda Baldoni ◽  
Nicole Berline ◽  
Brandon Dutra ◽  
Matthias Köppe ◽  
Michele Vergne ◽  
...  
Keyword(s):  
Nonnegative Integer ◽  
Fixed Number ◽  
Lattice Points ◽  
Combinatorial Number Theory ◽  
International Audience ◽  
Integer Solutions ◽  
Combinatorial Function ◽  
Rational Generating Function ◽  
Rational Generating Functions ◽  
The Right

International audience For a given sequence $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\alpha)(t)$ that counts the nonnegative integer solutions of the equation $\alpha_1x_1+\alpha_2 x_2+ \ldots+ \alpha_Nx_N+ \alpha_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\alpha)(t)$ is a quasipolynomial function of $t$ of degree $N$. In combinatorial number theory this function is known as the $\textit{denumerant}$. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\alpha)(t)$ as step polynomials of $t$. Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\alpha)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a $\texttt{MAPLE}$ implementation will be posted separately. Considérons une liste $\alpha = [\alpha_1,\alpha_2,\ldots , \alpha_N, \alpha_{N+1}]$ de $N+1$ entiers positifs. Le dénumérant $E(\alpha)(t)$ est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation $\sum^{N+1}_{i=1}x_i\alpha_i=t$, où $t$ varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de $t$, de degré $N$. Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé $k$ (mais $N$ n’est pas fixé, les $k+1$ plus hauts coefficients du quasi-polynôme $E(\alpha)(t)$ en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de $E(\alpha)(t)$. Les $k+1$ plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale à $k$.

scholarly journals The work of K. Ramachandra in algebraic number theory

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
M. Ram Murty
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Cyclotomic Field ◽  
Algebraic Number Theory ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
International Audience ◽  
Abelian Fields ◽  
Fundamental Units

International audience We give a brief survey of three papers of K. Ramachandra in algebraic number theory. The first paper is based on his thesis and appeared in the Annals of Mathematics and titled, ``Some Applications of Kronecker's Limit Formula.'' The second paper determines a system of fundamental units for the cyclotomic field and is titled, ``On the units of cyclotomic fields.'' This appeared in Acta Arithmetica. The third deals with relative class numbers and is titled, ``The class number of relative abelian fields.'' This appeared in Crelle's Journal.

scholarly journals The New Gulliver: Russian Mountains and Soviet Fantasmagoria

Slovo ◽  
2019 ◽  
Vol To the East of Pixar :... ◽  
Author(s):  
Pascal Vimenet
Keyword(s):  
Cultural Context ◽  
The Political ◽  
Roller Coaster ◽  
Feature Film ◽  
Avant Garde ◽  
Black And White ◽  
Mise En Scene ◽  
International Audience ◽  
The Subject ◽  
Homo Sovieticus

International audience Новый Гулливер (1935), первый черно-белый советскиймультипликационно-игровой фильм, режиссера Александра Птушко, не былдо сих пор предметом специального исследования. Нижеследующее являетсяего контекстуальным, биографическим, равно как и кинематографическим анализом. Творческий путь Птушко (1900-1973) дается в новом освещении, особенно 1920-30 годы, на которые приходится становление его кино.Осуществив постановку Это случилось на стадионе (1928) и Властелинбыта (1932), Птушко принял руководство секцией мультипликационногофильма на студии Совкино, ставшей Москинокомбинат, затем Мосфильм.Подробно рассматривается политико-культурный контекст, в период созданияего игрового фильма, между 1927 и 1935 годами, который сводит воедино всёмногообразие советского авангарда, но и является свидетелем торжествадогмы соцреализма. Фильм Птушко оказался восприимчив к теориям бурлескаоснователей Фабрики эксцентрического актера (ФЭКС). Превращениеперсонажей фильма Птушко в кукол-марионеток, символическое обыгрываниемасштабов могло зародиться в недрах этого движения. Но как сумел «НовыйГулливер», этот «фильм-сказка» обмануть политический контекст ипротивостоять сиренам соцреализма или отклониться от известныхшаблонных кодов? Птушко избирает философскую повесть, усугубленнуюполитическим памфлетом, что дает ему законное основание привязать чудесноек политическому. Смелая экранизация Свифта, московский дорогостоящийкинофильм, с его полуторатысячей шарнирных кукол, сотнями пластилиновыхфигурок, их сочетание с игровым кино, и два десятка массовых сцен претендуетна соперничество с «Кинг Конгом». Своим остроумием и развлекательностьюфильм Птушко вызывал восхищение Чарли Чаплина. Данное исследованиепрепарирует аллегорию – держащий каркас фильма, и стремитсяидентифицировать в кинопостановке всё то, в чем отголоском дают о себе знатьсоветские времена. И когда оказывается, что гимн лиллипутов, это воспеваниеотвратительного монархизма, жутковатым образом предвосхищает ОдуСталину 1939 года, то совершенно внятно становится, что Новый Гулливерне столько выставляет на показ всепобедительность homo sovieticus, сколькоделает из истории опасный аттракцион падений и взлетов и внедряет в неёсубверсивное понятие ухронии. The New Gulliver (1935), the first black and white Soviet animatedfeature film directed by Alexander Ptusko, has not been the subject of a specific studyso far. The following is contextual, biographical, as well as filmic analysis. It offers anew light on the route of Ptusko (1900-1973), especially on the 1920-1930 whichsees emerge his cinema. After having realized It Arrived at the Stadium (1928)and The Master of Everyday Life (1932), Ptusko took the lead of the section of theanimated film of the Sovkino studios become Moskinokombinat, then Mosfilm. Thepolitico-cultural context of the emergence of his feature film, 1927 and 1935, whichbrings together all the diversity of the Soviet avant-garde but sees the triumph of thedogma of socialist realism, is examined at length. Ptusko’s film seems receptive to theburlesque theories of the founders of the FEKS. The puppetisation of the characters inPtusko’s film, the symbolic games of scale, may have originated in this movement. Buthow could The New Gulliver, this « film-tale », have imposed itself in the politicalcontext and resist the realistic socialist sirens or divert certain codes? Ptusko choosesa philosophical tale doubled by a political pamphlet that authorizes him to link themarvelous to the political. An audacious adaptation of Swift, a Muscovite spectacular,The New Gulliver aims to compete, with its 1 500 puppets articulated, hundreds ofplastic figurines, their combination to the real shot and its 20 sets, with King Kong.Ptusko’s film provokes Chaplin’s admiration for his facetiousness. The study dissects theallegory that structures the film and seeks to identify in the staging of everything thatechoes the Soviet present. And when it appears that the Lilliputian hymn, the songof the hated kingship, anticipates in an unsettling way The Ode to Stalin of 1939, itbecomes certain that The New Gulliver, more than to demonstrate the invincibility ofthe homo sovieticus, transforms the history in a roller coaster and makes penetrate theexplosive concept of uchronia. Le Nouveau Gulliver (1935), premier long métrage d’animationsoviétique noir et blanc, réalisé par Aleksandr Ptouchko, n’a pas fait l’objetd’études spécifiques jusqu’ici. Celle qui suit est contextuelle, biographique, toutautant qu’analyse filmique. Elle propose un nouvel éclairage sur l’itinéraire dePtouchko (1900-1973), particulièrement sur les années 1920-1930 qui voitémerger son cinéma. Après avoir réalisé C’est arrivé au stade (1928) et Le Maîtredu quotidien (1932), Ptouchko prend la tête de la section du film animé des studiosSovkino devenus Moskinokombinat, puis Mosfilm. Le contexte politico-cultureld’émergence de son long métrage, entre 1927 et 1935, qui met en présence toutela diversité de l’avant-garde soviétique mais voit triompher le dogme du réalismesocialiste, est longuement examiné. Le film de Ptouchko semble réceptif aux théoriesburlesques des fondateurs de la FEKS. La marionnettisation des personnages dufilm de Ptouchko, les jeux symboliques d’échelle ont peut-être pris leur sourcedans ce mouvement. Mais comment Le Nouveau Gulliver, ce « ciné-conte », a-t-ilpu s’imposer dans le contexte politique et résister aux sirènes réalistes socialistesou en détourner certains codes ? Ptouchko choisit un conte philosophiquedoublé d’un pamphlet politique qui l’autorise à lier le merveilleux au politique.Audacieuse adaptation de Swift, superproduction moscovite, Le Nouveau Gulliverambitionne de rivaliser, avec ses 1 500 marionnettes articulées, ses centaines defigurines en plastiline, leur combinaison à la prise de vue réelle et ses vingt décors,avec King-Kong. Le film de Ptouchko provoque l’admiration de Chaplin, quien apprécie l’esprit facétieux. L’étude dissèque l’allégorie qui structure le film ets’attache à repérer dans la mise en scène tout ce qui fait écho au présent soviétique.Et quand il apparaît que l’hymne lilliputien, chant de la royauté honnie, anticipede manière troublante L’Ode à Staline de 1939, il devient certain que Le NouveauGulliver, plus que de démontrer l’invincibilité de l’homo sovieticus, transformel’histoire en montagnes russes et y fait pénétrer le concept explosif de l’uchronie.

scholarly journals On real algebraic numbers in which the derivative of their minimal polynomial is small

2021 ◽  
Vol 57 (2) ◽  
pp. 135-147
Author(s):  
D. V. Koleda
Keyword(s):  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Numbers ◽  
Absolute Value ◽  
Positive Degree ◽  
Integer Polynomials ◽  
The Absolute

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

scholarly journals ON SUBFIELDS OF A FIELD GENERATED BY TWO CONJUGATE ALGEBRAIC NUMBERS

2004 ◽  
Vol 47 (1) ◽  
pp. 119-123 ◽  
Author(s):  
Paulius Drungilas ◽  
Artūras Dubickas
Keyword(s):  
Algebraic Number ◽  
Monic Polynomial ◽  
Subject Classification ◽  
Real Field ◽  
Algebraic Numbers ◽  
Root Of Unity ◽  
Conjugate Algebraic Numbers

AbstractLet $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals The Size of One-Way Cellular Automata

2010 ◽  
Vol DMTCS Proceedings vol. AL,... (Proceedings) ◽  
Author(s):  
Martin Kutrib ◽  
Jonas Lefèvre ◽  
Andreas Malcher
Keyword(s):  
Cellular Automata ◽  
Real Time ◽  
Finite Automata ◽  
Fixed Number ◽  
Point Of View ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Order Of Magnitude ◽  
International Audience ◽  
Number Of Cells

International audience We investigate the descriptional complexity of basic operations on real-time one-way cellular automata with an unbounded as well well as a fixed number of cells. The size of the automata is measured by their number of states. Most of the bounds shown are tight in the order of magnitude, that is, the sizes resulting from the effective constructions given are optimal with respect to worst case complexity. Conversely, these bounds also show the maximal savings of size that can be achieved when a given minimal real-time OCA is decomposed into smaller ones with respect to a given operation. From this point of view the natural problem of whether a decomposition can algorithmically be solved is studied. It turns out that all decomposition problems considered are algorithmically unsolvable. Therefore, a very restricted cellular model is studied in the second part of the paper, namely, real-time one-way cellular automata with a fixed number of cells. These devices are known to capture the regular languages and, thus, all the problems being undecidable for general one-way cellular automata become decidable. It is shown that these decision problems are $\textsf{NLOGSPACE}$-complete and thus share the attractive computational complexity of deterministic finite automata. Furthermore, the state complexity of basic operations for these devices is studied and upper and lower bounds are given.

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