Hierarchy of general invariants for bivariate LPDOs

We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of general invariants are studied and some examples are presented. We also show that classical Laplace invariants correspond to some particular cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200

Gradual sub-lattice reduction and a new complexity for factoring polynomials

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes

Homeostasis of mitochondrial Ca²⁺ stores is critical for signal amplification in Drosophila melanogaster olfactory sensory neurons

SIMPLE SUMMARY: The evolution of flight imposed new challenges on insects when locating and identifying food sources, mates, or enemies. As an adaptation, flying insects developed a remarkably sensitive olfactory system to detect faint odor traces. This ability is linked to the olfactory receptor class of odorant receptors, which are found in insect olfactory sensory neurons. In a subgroup of these neurons, sensitivity can be further enhanced through a process called sensitization. Extracellular calcium ions, calmodulin, and protein kinase C are known to be key factors in this process. While manipulation of mitochondrial calcium im- and export has been shown to influence odor responses in general, the connection of intracellular calcium stores to sensitization has so far been only speculative. Using two pharmacological approaches, we disrupted mitochondrial calcium management in order to explore its importance to sensitization. Overall, our findings reveal that mitochondrial calcium stores are important players in the complex intracellular signaling pathways required for sensitization. ABSTRACT: Insects detect volatile chemosignals with olfactory sensory neurons (OSNs) that express olfactory receptors. Among them, the most sensitive receptors are the odorant receptors (ORs), which form cation channels passing Ca(2+). OSNs expressing different groups of ORs show varying optimal odor concentration ranges according to environmental needs. Certain types of OSNs, usually attuned to high odor concentrations, allow for the detection of even low signals through the process of sensitization. By increasing the sensitivity of OSNs upon repetitive subthreshold odor stimulation, Drosophila melanogaster can detect even faint and turbulent odor traces during flight. While the influx of extracellular Ca(2+) has been previously shown to be a cue for sensitization, our study investigates the importance of intracellular Ca(2+) management. Using an open antenna preparation that allows observation and pharmacological manipulation of OSNs, we performed Ca(2+) imaging to determine the role of Ca(2+) storage in mitochondria. By disturbing the mitochondrial resting potential and induction of the mitochondrial permeability transition pore (mPTP), we show that effective storage of Ca(2+) in the mitochondria is vital for sensitization to occur, and release of Ca(2+) from the mitochondria to the cytoplasm promptly abolishes sensitization. Our study shows the importance of cellular Ca(2+) management for sensitization in an effort to better understand the underlying mechanics of OSN modulation

Targeting insect olfaction in vivo and in vitro using functional imaging

Insects decode volatile chemical signals from its surrounding environment with the help of its olfactory system, in a fast and reliable manner for its survival. In order to accomplish this task, odorant receptors (ORs) expressed in olfactory sensory neurons (OSNs) in the fly’s antenna process such odor information. In order to study such a sophisticated process, we require access to the sensory neurons to perform functional imaging. In this article, we present different preparations to monitor odor information processing in Drosophila melanogaster OSNs using functional imaging of their Ca(2+) dynamics. First, we established an in vivo preparation to image specific OSN population expressing the fluorescent Ca(2+) reporter GCaMP3 during OR activation with airborne odors. Next, we developed a method to extract and to embed OSNs in a silica hydrogel with OR activation by dissolved odors. The odor response dynamics under these different conditions was qualitatively similar which indicates that the reduction of complexity did not affect the concentration dependence of odor responses at OSN level

Fast construction of irreducible polynomials over finite fields

International audienceWe present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+o(1)} \times (\log q)^{5+o(1)}$ elementary operations. The $o(1)$ in $d^{1+o(1)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o(1)$ in $(\log q)^{5+o(1)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. In particular, the complexity is quasi-linear in the degree $d$

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201