15 research outputs found

### On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction $q$ of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph $G \in \mathcal{G}(n,c/n)$ the threshold value of $c$ for the appearance of a linear-sized rigid component as a function of $q$, generalizing results of Kasiviswanathan et al. We show that this appearance of a giant component undergoes a continuous transition for $q \leq 1/2$ and a discontinuous transition for $q > 1/2$. In our proofs, we introduce a generalized notion of orientability interpolating between 1- and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2+1-core and 3+2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan et al. about the size of the 3+2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.Comment: 32 pages, 1 figur

### ON RIGIDITY, ORIENTABILITY AND CORES OF RANDOM GRAPHS WITH SLIDERS

International audienceSuppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G â G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [8]. We show that this appearance of a giant component undergoes a continuous transition for q â€ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between the 2-core and the 3-core, and of extended cores interpolating between the 2 + 1-core and the 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [8] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

### Detection of Motor Cerebral Activity After Median Nerve Stimulation During General Anesthesia (STIM-MOTANA): Protocol for a Prospective Interventional Study

International audienceBackground Accidental awareness during general anesthesia (AAGA) is defined as an unexpected awareness of the patient during general anesthesia. This phenomenon occurs in 1%-2% of high-risk practice patients and can cause physical suffering and psychological after-effects, called posttraumatic stress disorder. In fact, no monitoring techniques are satisfactory enough to effectively prevent AAGA; therefore, new alternatives are needed. Because the first reflex for a patient during an AAGA is to move, but cannot do so because of the neuromuscular blockers, we believe that it is possible to design a brain-computer interface (BCI) based on the detection of movement intention to warn the anesthetist. To do this, we propose to describe and detect the changes in terms of motor cortex oscillations during general anesthesia with propofol, while a median nerve stimulation is performed. We believe that our results could enable the design of a BCI based on median nerve stimulation, which could prevent AAGA. Objective To our knowledge, no published studies have investigated the detection of electroencephalographic (EEG) patterns in relation to peripheral nerve stimulation over the sensorimotor cortex during general anesthesia. The main objective of this study is to describe the changes in terms of event-related desynchronization and event-related synchronization modulations, in the EEG signal over the motor cortex during general anesthesia with propofol while a median nerve stimulation is performed. Methods STIM-MOTANA is an interventional and prospective study conducted with patients scheduled for surgery under general anesthesia, involving EEG measurements and median nerve stimulation at two different times: (1) when the patient is awake before surgery (2) and under general anesthesia. A total of 30 patients will receive surgery under complete intravenous anesthesia with a target-controlled infusion pump of propofol. Results The changes in event-related desynchronization and event-related synchronization during median nerve stimulation according to the various propofol concentrations for 30 patients will be analyzed. In addition, we will apply 4 different offline machine learning algorithms to detect the median nerve stimulation at the cerebral level. Recruitment began in December 2022. Data collection is expected to conclude in June 2024. Conclusions STIM-MOTANA will be the first protocol to investigate median nerve stimulation cerebral motor effect during general anesthesia for the detection of intraoperative awareness. Based on strong practical and theoretical scientific reasoning from our previous studies, our innovative median nerve stimulationâbased BCI would provide a way to detect intraoperative awareness during general anesthesia. Trial Registration Clinicaltrials.gov NCT05272202; https://clinicaltrials.gov/ct2/show/NCT05272202 International Registered Report Identifier (IRRID) PRR1-10.2196/4387

Optical heterodyne analog radio-over-fiber (A-RoF) links provide an efficient solution for future millimeter wave (mm-wave) wireless systems. The phase noise of the photo-generated mm-wave carrier limits the performance of such links, especially, for the transmission of low subcarrier baud rate multi-carrier signals. In this work, we present three different techniques for the compensation of the laser frequency offset (FO) and phase noise (PN) in an optical heterodyne A-RoF system. The first approach advocates the use of an analog mm-wave receiver; the second approach uses standard digital signal processing (DSP) algorithms, while in the third approach, the use of a photonic integrated mode locked laser (MLL) with reduced DSP is advocated. The compensation of the FO and PN with these three approaches is demonstrated by successfully transmitting a 1.95 MHz subcarrier spaced orthogonal frequency division multiplexing (OFDM) signal over a 25 km 61 GHz mm-wave optical heterodyne A-RoF link. The advantages and limitations of these approaches are discussed in detail and with regard to recent 5G recommendations, highlighting their potential for deployment in next generation wireless systems

### ON RIGIDITY, ORIENTABILITY AND CORES OF RANDOM GRAPHS WITH SLIDERS

International audienceSuppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G â G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [8]. We show that this appearance of a giant component undergoes a continuous transition for q â€ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between the 2-core and the 3-core, and of extended cores interpolating between the 2 + 1-core and the 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [8] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

### A Streaming Algorithm for Graph Clustering

International audienceWe introduce a novel algorithm to perform graph clustering in the edge streaming setting. In this model, the graph is presented as a sequence of edges that can be processed strictly once. Our streaming algorithm has an extremely low memory footprint as it stores only three integers per node and does not keep any edge in memory. We provide a theoretical justification of the design of the algorithm based on the modularity function, which is a usual metric to evaluate the quality of a graph partition. We perform experiments on massive real-life graphs ranging from one million to more than one billion edges and we show that this new algorithm runs more than ten times faster than existing algorithms and leads to similar or better detection scores on the largest graphs

### A Streaming Algorithm for Graph Clustering

International audienceWe introduce a novel algorithm to perform graph clustering in the edge streaming setting. In this model, the graph is presented as a sequence of edges that can be processed strictly once. Our streaming algorithm has an extremely low memory footprint as it stores only three integers per node and does not keep any edge in memory. We provide a theoretical justification of the design of the algorithm based on the modularity function, which is a usual metric to evaluate the quality of a graph partition. We perform experiments on massive real-life graphs ranging from one million to more than one billion edges and we show that this new algorithm runs more than ten times faster than existing algorithms and leads to similar or better detection scores on the largest graphs

### Analog link performance of mode-locked laser diodes in the 60 GHz range

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