Largest minimal inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where 1 \le i \textless{} j \le n, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Carath\'eodory numbers for certain abstract convexity structures on the $(n-1)$-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever $n \ge 4$

Largest minimally inversion-complete and pair-complete sets of permutations

We solve two related extremal problems in the theory of permutations. A set Q of permutations of the integers 1 to n is inversion-complete (resp., pair-complete) if for every inversion (j; i), where 1 j), where i 6= j), there exists a permutation in Q where j is before i. It is minimally inversion-complete if in addition no proper subset of Q is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion- complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (n1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide when ever n>=4

A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

Strain intermittency in shape-memory alloys

We study experimentally the intermittent progress of the mechanically induced martensitic transformation in a Cu-Al-Be single crystal through a full-field measurement technique: the grid method. We utilize an in- house, specially designed gravity-based device, wherein a system controlled by water pumps applies a perfectly monotonic uniaxial load through very small force increments. The sample exhibits hysteretic superelastic behavior during the forward and reverse cubic-monoclinic transformation, produced by the evolution of the strain field of the phase microstructures. The in-plane linear strain components are measured on the sample surface during the loading cycle, and we characterize the strain intermittency in a number of ways, showing the emergence of power-law behavior for the strain avalanching over almost six decades of magnitude. We also describe the nonstationarity and the asymmetry observed in the forward versus reverse transformation. The present experimental approach, which allows for the monitoring of the reversible martensitic transformation both locally and globally in the crystal, proves useful and enhances our capabilities in the analysis and possible control of transition-related phenomena in shape-memory alloys.Comment: Four supplementary video

Some specific features and consequences of the thermal response of rubber under cyclic mechanical loading

International audienceThe present paper deals with the specificities of the thermal response of rubber under cyclic mechanical loading at constant ambient temperature. This question is important, since the stabilized thermal response is used in fatigue life criteria, especially for the fast evaluation of fatigue life. For this purpose, entropic coupling in a thermo-hyperelastic framework is first used to predict the variation in the heat source produced or absorbed by the material during cyclic loading. The heat diffusion equation is then used to deduce temperature variations under adiabatic and non-adiabatic conditions. The influence of several parameters on the stabilized thermal response is studied: signal shape, frequency, minimum and maximum stretch levels, multiaxiality of the mechanical state. The results show that, in the steady-state regime, the mean value between the maximum and minimum temperature variations over a mechanical cycle is different from zero. This is due to the specific variation in the heat source, which depends on both the stretch rate and the stretch level. This result has numerous consequences, in particular for fatigue. Indeed, the stabilized mean value between the maximum and minimum temperature variations during fatigue tests does not reflect only fatigue damage, since the entropic coupling also leads to a value different from zero. This is a major difference with respect to materials exhibiting only isentropic coupling, such as metallic materials

Création de structures à précontrainte adaptative à base d'alliages à mémoire de forme

International audienceL'étude porte sur l'utilisation de fils en alliage à mémoire de forme (AMF) pour créer des systèmes mécaniques actifs. Il s'agit en pratique de contrôler l'état de contrainte ou de déformation d'une structure par activation thermique de l'effet mémoire de fils AMF connectés à la structure. Un point clé consiste à n'utiliser que des fils AMF classiques, i.e. à effet mémoire simple-sens, pour créer une activation à double-sens de la structure. Trois dispositifs sont développés et étudiés. Dans un premier temps, le cas simple de deux fils AMF placés en série est considéré. Le chauffage des fils est assuré par effet Joule. Ce dispositif permet d'analyser comment deux fils à effet mémoire simple-sens peuvent interagir pour permettre le pilotage double-sens d'un déplacement. Le cas d'une poutre console équipée de fils AMF est ensuite considéré. Il est montré qu'il est possible de contrôler en temps réel la flèche de la poutre par un chauffage sélectif des fils. Enfin, le cas d'une poutrelle sur deux appuis et soumise à une charge mobile est étudié. Une limitation en temps réel de la valeur de la flèche a été possible en utilisant un unique fil AMF précontraint. Cette étude montre que des AMF peuvent être utilisés pour contrôler l'état de déformation d'une structure ou d'adapter un état de précontrainte en fonction d'une variation de charge

FIGENIX: Intelligent automation of genomic annotation: expertise integration in a new software platform

BACKGROUND: Two of the main objectives of the genomic and post-genomic era are to structurally and functionally annotate genomes which consists of detecting genes' position and structure, and inferring their function (as well as of other features of genomes). Structural and functional annotation both require the complex chaining of numerous different software, algorithms and methods under the supervision of a biologist. The automation of these pipelines is necessary to manage huge amounts of data released by sequencing projects. Several pipelines already automate some of these complex chaining but still necessitate an important contribution of biologists for supervising and controlling the results at various steps. RESULTS: Here we propose an innovative automated platform, FIGENIX, which includes an expert system capable to substitute to human expertise at several key steps. FIGENIX currently automates complex pipelines of structural and functional annotation under the supervision of the expert system (which allows for example to make key decisions, check intermediate results or refine the dataset). The quality of the results produced by FIGENIX is comparable to those obtained by expert biologists with a drastic gain in terms of time costs and avoidance of errors due to the human manipulation of data. CONCLUSION: The core engine and expert system of the FIGENIX platform currently handle complex annotation processes of broad interest for the genomic community. They could be easily adapted to new, or more specialized pipelines, such as for example the annotation of miRNAs, the classification of complex multigenic families, annotation of regulatory elements and other genomic features of interest