scholarly journals How often do we reject a superior value? (Extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Kamilla Oliver ◽  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Natural Numbers ◽  
List Structure ◽  
Skip List ◽  
International Audience

International audience Words $a_1 a_2 \ldots a_n$ with independent letters $a_k$ taken from the set of natural numbers, and a weight (probability) attached via the geometric distribution $pq^{i-1}(p+q=1)$ are considered. A consecutive record (motivated by the analysis of a skip list structure) can only advance from $k$ to $k+1$, thus ignoring perhaps some larger (=superior) values. We investigate the number of these rejected superior values. Further, we study the probability that there is a single consecutive maximum and show that (apart from fluctuations) it tends to a constant. On considère des mots $a_1a_2 \ldots a_n$ formés de lettres à valeurs entières, tirées de façon indépendante avec une distribution géométrique $pq^{i-1}(p+q=1)$. Un record $k+1$ est dit consécutif si la lettre précédente est $k$. La notion est motivée par des considérations algorithmiques. Les autres records sont rejetés. Nous étudions le nombre de records rejetés. Nous étudions aussi la probabilité qu'il y ait un seul maximum consécutif, et montrons qu'elle converge vers une constante, à certaines fluctuations près.

scholarly journals On congruences for certain sums of E. Lehmer's type

2014 ◽  
Vol Volume 38 ◽  
Author(s):  
Shigeru Kanemitsu ◽  
Takako Kuzumaki ◽  
Jerzy Urbanowicz
Keyword(s):  
Natural Number ◽  
Natural Numbers ◽  
Principal Character ◽  
International Audience

International audience Let n > 1 be an odd natural number and let r (1 < r < n) be a natural number relatively prime to n. Denote by χn the principal character modulo n. In Section 3 we prove some new congruences for the sums T r,k (n) = n r ] i=1 (χn(i) i k) (mod n s+1) for s ∈ {0, 1, 2}, for all divisors r of 24 and for some natural numbers k.We obtain 82 new congruences for T r,k (n), which generalize those obtained in [Ler05], [Leh38] and [Sun08] if n = p is an odd prime. Section 4 is an appendix by the second and third named authors. It contains some new congruences for the sums Ur(n) = n

scholarly journals Right-cancellability of a family of operations on binary trees

1998 ◽  
Vol Vol. 2 ◽  
Author(s):  
Philippe Duchon
Keyword(s):  
Binary Trees ◽  
Natural Numbers ◽  
The Family ◽  
International Audience

International audience We prove some new results on a family of operations on binary trees, some of which are similar to addition, multiplication and exponentiation for natural numbers. The main result is that each operation in the family is right-cancellable.

scholarly journals Position of the maximum in a sequence with geometric distribution

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Margaret Archibald
Keyword(s):  
Geometric Distribution ◽  
Asymptotic Results ◽  
Mellin Transforms ◽  
International Audience

International audience As a sequel to [arch04], the position of the maximum in a geometrically distributed sample is investigated. Samples of length n are considered, where the maximum is required to be in the first d positions. The probability that the maximum occurs in the first $d$ positions is sought for $d$ dependent on n (as opposed to d fixed in [arch04]). Two scenarios are discussed. The first is when $d=αn$ for $0 < α ≤ 1$, where Mellin transforms are used to obtain the asymptotic results. The second is when $1 ≤ d = o(n)$.

scholarly journals The unreasonable ubiquitousness of quasi-polynomials

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Kevin Woods
Keyword(s):  
General Class ◽  
Linear Inequalities ◽  
Boolean Operations ◽  
Natural Numbers ◽  
Ehrhart Theory ◽  
Presburger Arithmetic ◽  
Polynomial Structure ◽  
Special Cases ◽  
International Audience

International audience A function $g$, with domain the natural numbers, is a quasi-polynomial if there exists a period $m$ and polynomials $p_0,p_1,\ldots,p_m-1$ such that $g(t)=p_i(t)$ for $t\equiv i\bmod m$. Quasi-polynomials classically – and ``reasonably'' – appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form $a_1x_1+⋯+a_dx_d≤ b(t)$. Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the $a_i$ are also allowed to vary with $t$. We discuss these ``unreasonable'' results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets $S_t$ that are defined with quantifiers $(\forall , ∃)$, boolean operations (and, or, not), and statements of the form $a_1(t)x_1+⋯+a_d(t)x_d ≤ b(t)$, where $a_i(t)$ and $b(t)$ are polynomials in $t$. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures.

Accurate Location in Batch Dynamic Provable Data Possession

2014 ◽  
Vol 519-520 ◽  
pp. 9-12
Author(s):  
Ming Chu Li ◽  
Liang Zhang ◽  
Cheng Guo
Keyword(s):  
Performance Evaluation ◽  
List Structure ◽  
Trade Off ◽  
Storage Model ◽  
Provable Data Possession ◽  
Storage Cost ◽  
Proposed Model ◽  
Skip List ◽  
And Performance ◽  
And Storage

The paper focused on the construction of an efficient DPDP for public audit. We improved the existing proof of storage model by manipulating authenticated skip list structure for authentication. We further explored embedded MHT structure helping our scheme to accurate locate the incorrect part in batch auditing. Extensive security and performance evaluation showed the proposed model is highly efficient and a nice trade-off between robust construction and storage cost.

scholarly journals Record statistics in integer compositions

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Arnold Knopfmacher ◽  
Toufik Mansour
Keyword(s):  
Generating Functions ◽  
Natural Numbers ◽  
Mean Values ◽  
Fixed Subset ◽  
Record Statistics ◽  
Weak Records ◽  
International Audience ◽  
Positive Integers

International audience A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.

scholarly journals On the Frobenius’ Problem of three numbers

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Francesc Aguiló ◽  
Alícia Miralles
Keyword(s):  
Upper Bound ◽  
Frobenius Number ◽  
Explicit Computation ◽  
Natural Numbers ◽  
Frobenius Problem ◽  
Geometrical Proof ◽  
International Audience ◽  
Alpha Beta

International audience Given $k$ natural numbers $\{a_1, \ldots ,a_k\} \subset \mathbb{N}$ with $1 \leq a_1 < a_2 < \ldots < a_k$ and $\mathrm{gcd} (a_1, \ldots ,a_k)=1$, let be $R(a_1, \ldots ,a_k) = \{ \lambda_1 a_1+ \cdots + \lambda_k a_k | \space \lambda_i \in \mathbb{N}, i=1 \div k\}$ and $\overline{R}(a_1, \ldots ,a_k) = \mathbb{N} \backslash R (a_1, \ldots ,a_k)$. It is easy to see that $| \overline{R}(a_1, \ldots ,a_k)| < \infty$. The $\textit{Frobenius Problem}$ related to the set $\{a_1, \ldots ,a_k\}$ consists on the computation of $f(a_1, \ldots ,a_k)=\max \overline{R} (a_1, \ldots ,a_k)$, also called the $\textit{Frobenius number}$, and the cardinal $| \overline{R}(a_1, \ldots ,a_k)|$. The solution of the Frobenius Problem is the explicit computation of the set $\overline{R} (a_1,\ldots ,a_k)$. In some cases it is known a sharp upper bound for the Frobenius number. When $k=3$ this bound is known to be $$F(N)=\max\limits_{\substack{0 \lt a \lt b \lt N \\ \mathrm{gcd}(a,b,N)=1}} f(a,b,N)= \begin{cases} 2(\lfloor N/2 \rfloor -1)^2-1 & \textrm{if } N \equiv 0 (\mod 2),\\ 2 \lfloor N/2 \rfloor (\lfloor N/2 \rfloor -1) -1 & \textrm{if } N \equiv 1 (\mod 2).\\ \end{cases}$$ This bound is given in [Dixmier1990]. In this work we give a geometrical proof of this bound which allows us to give the solution of the Frobenius problem for all the sets $\{\alpha ,\beta ,N\}$ such that $f(\alpha ,\beta ,N)=F(N)$.

scholarly journals A combinatorial and probabilistic study of initial and end heights of descents in samples of geometrically distributed random variables and in permutations

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Guy Louchard ◽  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Analysis Of Algorithms ◽  
Markov Chain Model ◽  
Random Variables ◽  
Discrete Distributions ◽  
Chain Model ◽  
Initial Height ◽  
Probabilistic Study ◽  
International Audience ◽  
Expected Values

Analysis of Algorithms International audience In words, generated by independent geometrically distributed random variables, we study the lth descent, which is, roughly speaking, the lth occurrence of a neighbouring pair ab with a>b. The value a is called the initial height, and b the end height. We study these two random variables (and some similar ones) by combinatorial and probabilistic tools. We find in all instances a generating function Ψ(v,u), where the coefficient of vjui refers to the jth descent (ascent), and i to the initial (end) height. From this, various conclusions can be drawn, in particular expected values. In the probabilistic part, a Markov chain model is used, which allows to get explicit expressions for the heights of the second descent. In principle, one could go further, but the complexity of the results forbids it. This is extended to permutations of a large number of elements. Methods from q-analysis are used to simplify the expressions. This is the reason that we confine ourselves to the geometric distribution only. For general discrete distributions, no such tools are available.

scholarly journals d-records in geometrically distributed random variables

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Geometric Distribution ◽  
Random Variables ◽  
Random Permutations ◽  
Weak Model ◽  
Asymptotic Formulae ◽  
International Audience

International audience We study d-records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d-record occurs, when one computes the d-largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the "strict model," but a "weak model" is also briefly investigated. We also discuss the limit q → 1 (q the parameter of the geometric distribution), which leads to the model of random permutations.

scholarly journals On generalised Carmichael numbers.

1999 ◽  
Vol Volume 22 ◽  
Author(s):  
L Halbeisen ◽  
N Hungerbühler
Keyword(s):  
Natural Numbers ◽  
Carmichael Numbers ◽  
Special Cases ◽  
Prime Factors ◽  
International Audience

International audience For arbitrary integers $k\in\mathbb Z$, we investigate the set $C_k$ of the generalised Carmichael number, i.e. the natural numbers $n< \max\{1, 1-k\}$ such that the equation $a^{n+k}\equiv a \mod n$ holds for all $a\in\mathbb N$. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that $C_1$ is infinite and that $C_k$ is infinite, whenever $1-k>1$ is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers $n$ which satisfy the equation $a^n\equiv a \mod n$ only for $a=2$, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares.

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