About the p-paperfolding words

AbstractLet p be an integer greater than or equal to 2. The aim of this paper is to study the language associated to a p-paperfolding sequence. It is known that the number of factors of length n of a 2-paperfolding sequence (i.e. its complexity function) is P(n) = 4n for n ⩾ 7. It is also known that the language of all the factors of all 2-paperfolding sequences is not context-free and that its generating function is transcendental.We show that the complexity function of a p-paperfolding sequence is either strictly subaffine or ultimately linear. The first case never happens if p = 2 or 3. In the second case, the complexity function is either P(n) = 2n or P(n) = 4n for n large enough. We give a simple necessary and sufficient condition for the number of special factors to be p-automatic. We finally show that, for any given p, the language of all factors of all p-paperfolding sequences is not context-free, and that the associated generating series is not algebraic

Spatial cluster detection using the number of connected components of a graph

The aim of this work is to detect spatial clusters. We link Erdös graph and Poisson point process. We give the probability distribution function (pdf) of the number of connected component for an Erdös graph and obtain the pdf of the number of cluster for a Poisson process. Using this result, we directly obtain a test for complete spatial randomness and also obtain the clusters that violates the CSR hypothesis. Border effects are computed. We illustrate our results on a tropical forest example

Homomorphic sign evaluation using functional bootstrapping with a RNS representation of integers

In the context of fully-homomorphic-encryption, we consider the representation of large integers by their decomposition over a product of rings (through the Chinese Remainder Theorem) and introduce a new algorithm for the determination of the sign solely through the knowledge of ring-components. We then prove that our algorithm delivers a correct result with a very high probability

Fully Homomorphic Encryption on large integers

At the core of fully homomorphic encryption lies a procedure to refresh the ciphertexts whose noise component has grown too big. The efficiency of the so-called bootstrap is of paramount importance as it is usually regarded as the main bottleneck towards a real-life deployment of fully homomorphic crypto-systems. In two of the fastest implementations so far, the space of messages is limited to binary integers. If the message space is extended to the discretized torus $T_{p_i}$ or equivalently to $Z_{p_i}$ with values of $p_i$ large as compared to the dimension of the quotient ring in which the operations are realised, the bootstrap delivers incorrect results with far too high probability. As a consequence, the use of a residue numeral system to address large integers modulo $p=p_1 \times \ldots \times p_\kappa$ would be of limited interest in practical situations without the following remedy: rather than increasing the polynomial degree and thus the computational cost, we introduce here a novel and simple technique (hereafter referred to as ``collapsing ) which, by grouping the components of the mask, attenuates both rounding errors and computational costs, and greatly helps to sharpen the correctness of the bootstrap. We then rigorously estimate the probability of success as well as the output error and determine practical parameters to reach a given correctness threshold

Diffuse laser illumination for Maxwellian view Doppler holography of the retina

We describe the advantages of diffuse illumination in laser holography for ophthalmology. The presence of a diffusing element introduces an angular diversity of the optical radiation and reduces its spatial coherence, which spreads out the energy distribution of the illumination beam in the focal plane of the eyepiece. The field of view of digitally computed retinal images can easily be increased as the eyepiece can be moved closer to the cornea to obtain a Maxwellian view of the retina without compromising ocular safety. Compliance with American and European safety standards for ophthalmic devices is more easily obtained by preventing the presence of a laser hot spot observed in front of the cornea in the absence of a scattering element. Diffuse laser illumination does not introduce any adverse effects on digitally computed laser Doppler images.Comment: 9 page

a Hierarchical Database Manager

This paper describes a new algorithm dealing with databases. This algorithm allows to fully manage a database, but their most natural field of applications is the datawarehouse (OLAP). It lies on a de-normalized representation of the database. The data is stored in thesauruses and radix trees (a hierarchical representation of bitmaps) which have interesting properties.ou