Search results for "REAL NUMBER"
showing 10 items of 31 documents
Banach spaces of general Dirichlet series
2018
Abstract We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let { λ n } be a strictly increasing sequence of positive real numbers such that lim n → ∞ λ n = ∞ . We denote by H ∞ ( λ n ) the complex normed space of all Dirichlet series D ( s ) = ∑ n b n λ n − s , which are convergent and bounded on the half plane [ Re s > 0 ] , endowed with the norm ‖ D ‖ ∞ = sup Re s > 0 | D ( s ) | . If (⁎) there exists q > 0 such that inf n ( λ n + 1 q − λ n q ) > 0 , then H ∞ ( λ n ) is a Banach space. Further, if there exists a strictly increasing sequ…
Periodicity, morphisms, and matrices
2003
In 1965, Fine and Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn = gn for 0 ≤ n > h + k - gcd(h,k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h,k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn ≤ gn, instead of fn = gn. We also obtain generalizations to more than two periods.We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if h : Σ* → Σ* is a morphism with |Σ|= n, and w is a word with |w| < |h(w)| < |h2(w)| < ... < |hk(w)|, then k ≤ n.
Varieties with at most quadratic growth
2010
Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V); n = 1; 2, … and here we study varieties of polynomial growth. Recently, for any real number a, 3 < a < 4, a variety V was constructed satisfying C1n^a < cn(V) < C2n^a; for some constants C1;C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) < Cn^a; with 0 < a < 2, for some constant C. We prove that if 0 < a < 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 < a < 2, then lim logn cn(V) = 1 o…
Rapid construction of algebraic axioms from samples
1991
Abstract An axiom is called reliable if it is confirmed in several places in a given sample of algebra. A very effective algorithm for enumerating such axioms is described.
Variations on a Theorem of Fine & Wilf
2001
In 1965, Fine & Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of periods h and k respectively, and fn = gn for 0 ≤ n ≤ h+k-gcd(h, k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h, k) is best possible. In this paper we consider some variations on this theorem. In particular, we study the case where fn ≤ gn instead of fn = gn. We also obtain a generalization to more than two periods.
Convergence Analysis of Distributed Set-Valued Information Systems
2016
This paper focuses on the convergence of information in distributed systems of agents communicating over a network. The information on which the convergence is sought is not rep- resented by real numbers, as often in the literature, rather by sets. The dynamics of the evolution of information across the net- work is accordingly described by set-valued iterative maps. While the study of convergence of set-valued iterative maps is highly complex in general, this paper focuses on Boolean maps, which are comprised of arbitrary combinations of unions, intersections, and complements of sets. For these important class of systems, we provide tools to study both global and local convergence. A distr…
Saddle index properties, singular topology, and its relation to thermodynamic singularities for aϕ4mean-field model
2004
We investigate the potential energy surface of a ${\ensuremath{\phi}}^{4}$ model with infinite range interactions. All stationary points can be uniquely characterized by three real numbers ${\ensuremath{\alpha}}_{+},{\ensuremath{\alpha}}_{0},{\ensuremath{\alpha}}_{\ensuremath{-}}$ with ${\ensuremath{\alpha}}_{+}+{\ensuremath{\alpha}}_{0}+{\ensuremath{\alpha}}_{\ensuremath{-}}=1$, provided that the interaction strength $\ensuremath{\mu}$ is smaller than a critical value. The saddle index ${n}_{s}$ is equal to ${\ensuremath{\alpha}}_{0}$ and its distribution function has a maximum at ${n}_{s}^{\mathrm{max}}=1∕3$. The density $p(e)$ of stationary points with energy per particle $e$, as well as…
Robustness and Randomness
2008
The study of robustness problems for computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floating-point instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discusses approaches based on exact arithmetic, interval arithmetic and probabilistic methods. The paper investigates the possibility to use randomness at certain levels of reasoning to make geometric constructions more robust.
On lazy representations and Sturmian graphs
2011
In this paper we establish a strong relationship between the set of lazy representations and the set of paths in a Sturmian graph associated with a real number α. We prove that for any non-negative integer i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ostrowski numeration system associated with α. Moreover, we provide several properties of the representations of the natural integers in this numeration system.
Ultrametric Vs. Quantum Query Algorithms
2014
Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. This paper introduces the notion of ultrametric query algorithms and shows an example of advantages of ultrametric query algorithms over deterministic, probabilistic and quantum query algorithms.