Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms

Les traces écrites en sciences à l'école maternelle et au cours préparatoire de l'école primaire en France: intérêts et limites pour la didactique des sciences

http://www.inrp.fr/biennale/8biennale/contrib/longue/185.pdfSusciter la production d'écrits en sciences à l'école maternelle et au cours préparatoire de l'école primaire renvoie à deux séries d'objectifs : permettre aux élèves de découvrir le monde et se familiariser avec les langages écrits. Cette production s'inscrit dans une approche pédagogique respectueuse de la progressivité du développement pour des sujets non encore lisant ou écrivant de façon autonome. Le support technique sur lequel sont recueillies les traces est le cahier de sciences.L'étude de notre objet est référée à un double cadre théorique : Didactique des sciences et didactique du français (INRP PROG Brigaudiot et al., 2000). Le passage d'une lecture ingénue à une lecture critique du monde passe par une démarche structurée en trois étapes : questionnement, investigation et institutionnalisation des savoirs. L'élève conceptualise le monde en produisant des représentations à l'aide de divers registres sémiotiques et symboles, en dessinant, en dictant des textes de plus en plus longs à l'adulte. Permettre aux jeunes enfants d'accéder progressivement à la compréhension et à la production d'écrits suppose de donner au cahier de sciences, le statut d'instrument de traitement de l'information et de communication. L'utilisation de formes d'écrit privilégiées s'impose.La structuration des savoirs, leur mémorisation et leur évocation par les élèves dépendent d'un ensemble de choix didactiques et pédagogiques mis en lumière par nos recherches parmi lesquelles figurent les degrés d'iconicité et de personnalisation, l'utilisation conjointe des langages analogiques, mathématiques et textuels, la structuration globale du cahier personnel.L'hypothèse qui guide nos travaux est que la production écrite avec des élèves de 3 à 6 ans, apprentis lecteurs et scripteurs, induit une conceptualisation en sciences en dépassant l'activité manipulatoire pour elle-même. Les traces constituent un objet pertinent de la didactique des sciences. De ce cadre théorique, leur analyse vise à faire ressortir leurs intérêts didactiques à côté de leurs limites

Computing minimal interpolation bases

International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor computing shifted minimal bases of relations between $m$ vectors of size$\sigma$ it uses $O~( m^{\omega-1} (\sigma + |s|) )$ field operations, where$\omega$ is the exponent of matrix multiplication, and $|s|$ is the sum of theentries of the input shift $s$, with $\min(s) = 0$. This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation

A new binary floating-point division algorithm and its software implementation on the ST231 processor

This paper deals with the design and implementation of low latency software for binary floating-point division with correct rounding to nearest. The approach we present here targets a VLIW integer processor of the ST200 family, and is based on fast and accurate programs for evaluating some particular bivariate polynomials. We start by giving approximation and evaluation error conditions that are sufficient to ensure correct rounding. Then we describe the heuristics used to generate such evaluation programs, as well as those used to automatically validate their accuracy. Finally, we propose, for the binary32 format, a complete C implementation of the resulting division algorithm. With the ST200 compiler and compared to previous implementations, the speed-up observed with our approach is by a factor of almost 1.8

Systematic mapping of regions of human cardiac troponin I involved in binding to cardiac troponin C: N- and C-terminal low affinity contributing regions

AbstractThe Spot method of multiple peptide synthesis was used to map in a systematic manner regions of the human cardiac troponin I sequence (hcTnI) involved in interactions with its physiological partner, troponin C (cTnC). Ninety-six 20-mer peptides describing the entire hcTnI sequence were chemically assembled; their reactivity with [125I]cTnC, in the presence of 3 mM Ca2+, enabled the assignment of six sites of interaction (residues 19–32, 45–54, 129–138, 145–164, 161–178 and 191–210). For several sites, a good correlation with literature data was obtained, thus validating this methodological approach. Synthetic peptides, each containing in their sequence an interaction site, were prepared. As assessed by BIACORE, all of them exhibited an affinity for cTnC in the range of 10−6–10−7 M, except for hcTnI [39–58] which showed a nanomolar affinity. This peptide was also able to block the interaction between hcTnI and cTnC. We therefore postulate that despite the existence of multiple cTnC interaction sites on the hcTnI molecule, only that region of hcTnI allows a stabilization of the complex. Residues 19–32 from the N-terminal cardio-specific extension of hcTnI were also found to be involved in interaction with cTnC; residues 19–32 may correspond to the minimal sequence of the extension which could switch between the N- and C-terminal TnC domains, depending on its phosphorylation state. Finally, two Ca2+-dependent cTnC binding domains within the C-terminal part of hcTnI (residues 164–178 and 191–210) were also mapped. The latter site may be linked with the cardiac dysfunction observed in stunned myocardium

Faster Algorithms for Multivariate Interpolation with Multiplicities and Simultaneous Polynomial Approximations

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for Reed-Solomon list-decoding with re-encoding, our approach has complexity $\mathcal{O}\tilde{~}(\ell^{\omega-1}m^2(n-k))$, where $\ell,m,n,k$ are the list size, the multiplicity, the number of sample points and the dimension of the code, and $\omega$ is the exponent of linear algebra; this accelerates the previously fastest known algorithm by a factor of $\ell / m$.Comment: Version 2: Generalized our results about Problem 1 to distinct multiplicities. Added Section 4 which details several applications of our results to the decoding of Reed-Solomon codes (list-decoding with re-encoding technique, Wu algorithm, and soft-decoding). Reorganized the sections, added references and corrected typo

Creatine synthesis and transport during rat embryogenesis: Spatiotemporal expression of AGAT, GAMT and CT1

BACKGROUND: Creatine (Cr) is synthesized by a two-step mechanism involving arginine:glycine amidinotransferase (AGAT) and guanidinoacetate methyltransferase (GAMT), and is taken up by cells through a specific Cr transporter, CT1. Recently, genetic defects of this pathway have been described, that lead to Cr deficiency, neurological symptoms in early infancy and severe neurodevelopmental delay. To investigate the involvement of Cr synthesis and uptake pathways during embryonic development, we determined the spatiotemporal expression of AGAT, GAMT and CT1 during the rat embryogenesis, at the mRNA and protein level. RESULTS: We show that AGAT and GAMT are expressed in hepatic primordium as soon as 12.5 days, then progressively acquire their adult pattern of expression, with high levels of AGAT in kidney and pancreas, and high levels of GAMT in liver and pancreas. AGAT and CT1 are prominent in CNS, skeletal muscles and intestine, where they appear earlier than GAMT. High levels of CT1 are found in epithelia. CONCLUSION: Our results suggest that de novo synthesis of Cr by AGAT and GAMT, as well as cellular Cr uptake by CT1, are essential during embryonic development. This work provides new clues on how creatine can be provided to developing tissues, and suggests that Cr deficiencies might induce irreversible damages already in utero, particularly on the nervous system

Specific antibody responses against membrane proteins of erythrocytes infected by Plasmodium falciparum of individuals briefly exposed to malaria

International audienceBACKGROUND: Plasmodium falciparum infections could lead to severe malaria, principally in non-immune individuals as children and travellers from countries exempted of malaria. Severe malaria is often associated with the sequestration of P. falciparum-infected erythrocytes in deep micro-vascular beds via interactions between host endothelial receptors and parasite ligands expressed on the surface of the infected erythrocyte. Although, serological responses from individuals living in endemic areas against proteins expressed at surface of the infected erythrocyte have been largely studied, seldom data are available about the specific targets of antibody response from travellers. METHODS: In order to characterize antigens recognized by traveller sera, a comparison of IgG immune response against membrane protein extracts from uninfected and P. falciparum-infected red blood cells (iRBC), using immunoblots, was performed between non exposed individuals (n = 31) and briefly exposed individuals (BEI) (n = 38) to malaria transmission. RESULTS: Immune profile analysis indicated that eight protein bands from iRBC were significantly detected more frequently in the BEI group. Some of these antigenic proteins were identified by an original immuno-proteomic approach. CONCLUSION: Collectively, these data may be useful to characterize the singular serological immune response against a primary malaria infection in individuals briefly exposed to transmission

PEPOP: Computational design of immunogenic peptides

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