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Exercices Corrigés - Polynomes d'endomorphismes Exercices Corrigés - Polynomes d'endomorphismes Exercice 1 : Vrai ou Faux Soit \( E \) un \( K \)-espace vectoriel de dimension finie, et \( u \in \mathcal{L}(E) \). Dire si les assertions suivantes sont vraies ou fausses : 1. Si \( F \) et \( G \) sont deux sous-espaces supplémentaires de \( E \) stables par \( u \), alors \( u \) est diagonalisable si et seulement si les deux endomorphismes induits \( u_F \) et \( u_G \) sont diagonalisables. 2. Deux matrices semblables ont le même polynôme caractéristique et le même polynôme minimal. 3. Si le polynôme caractéristique d'une matrice est égal à son polynôme minimal, alors la matrice est diagonalisable. Correction de l'exercice 1 1. Vrai ...

How Motor Neurons Find Their Paths (FR/EN) Motor Neuron Pathfinding — Axonal Guidance English Français On this page · Overview · 1) Genetic identity · 2) Exit & boundary cap · 3) Segmentation · 4) Axial muscles · 5) Limb guidance · 6) Branching · Key principles Overview During neuroembryogenesis, newly born motor neurons (MNs) must project axons to precise peripheral targets to build functional circuits. They do this via a stepwise process called axonal pathfinding , using the growth cone’s ability to sense attractive and repulsive cues in the environment. Core reference: Bonanomi & Pfaff, Cold Spring Harbor Perspectives in Biology (2010) . 1)...

⚙️ Mécanique du Point – Cours et Travaux Dirigés (S1) SMP-SMC-SMA-SMI-MIP-MI Le module Mécanique du Point fait partie intégrante du programme du Semestre 3 des filières Mathématiques et Physique (SMP/SMA) . Il s’agit d’une matière fondamentale qui permet de comprendre les lois régissant le mouvement des corps ponctuels soumis à des forces. Ce cours établit un pont essentiel entre les mathématiques (vecteurs, dérivées, intégrales) et la physique (mouvement, énergie, dynamique). Au travers de six chapitres, l’étudiant découvre la cinématique , la dynamique et l’ énergétique du point matériel, tout en apprenant à résoudre des problèmes concrets de mécanique classique. 📘 Objectifs du Module Comprendre la modélisation du mouvement d’un point matériel dans un espace donné. Appliquer les lois fondamentales de Newton et les principes énergétiques. Maîtriser les changements de référentiels et les transformations de vitesses et accélérations. Résoudre des ex...

🧬 Le Séquençage de l’ADN – Comprendre le Langage de la Vie Le séquençage de l’ADN est l’une des découvertes les plus puissantes de la biologie moderne. Il permet aux scientifiques de lire les instructions génétiques qui définissent chaque être vivant — des bactéries aux humains. Depuis l’achèvement du Projet Génome Humain en 2003, les technologies de séquençage ont considérablement évolué, devenant plus rapides, moins coûteuses et plus précises. Aujourd’hui, le séquençage de l’ADN se trouve au cœur de la médecine, de l’évolution, de l’agriculture et même des sciences médico-légales. 🔍 1. Qu’est-ce que le Séquençage de l’ADN ? Le séquençage de l’ADN est le processus qui permet de déterminer l’ordre exact des nucléotides dans une molécule d’ADN — c’est-à-dire la séquence des quatre bases : Adénine (A) , Thymine (T) , Cytosine (C) et Guanine (G) . Lire cette séquence permet aux chercheurs de comprendre quels gènes sont présents, comment ils fonctionnent e...

🧬 DNA Sequencing – Understanding the Language of Life DNA sequencing is one of the most powerful discoveries in modern biology. It allows scientists to read the genetic instructions that define every living organism — from bacteria to humans. Since the completion of the Human Genome Project in 2003, sequencing technologies have evolved dramatically, becoming faster, cheaper, and more accurate. Today, DNA sequencing lies at the heart of medicine, evolution, agriculture, and even forensic science. 🔍 1. What Is DNA Sequencing? DNA sequencing is the process of determining the exact order of nucleotides in a DNA molecule — that is, the sequence of the four bases: Adenine (A) , Thymine (T) , Cytosine (C) , and Guanine (G) . Reading them allows researchers to understand which genes are present, how they work, and how they vary between individuals or species. 🖼️ [Insert here: Diagram showing DNA double helix with labeled bases (A, T, C, G).] ⚙️ 2. The Main Steps...

📘 Cours et Travaux Dirigés de Mécanique du Solide – S3 La Mécanique du Solide est une discipline fondamentale du parcours Mathématiques et Physique (S3) . Elle explore les principes du mouvement et de l’équilibre des corps rigides, reliant les bases de la mécanique classique aux applications modernes en ingénierie, en physique et en modélisation mathématique. Ce module s’adresse aux étudiants des filières Sciences Mathématiques (SMP, SMI) et Sciences de la Matière Physique . 🎯 Objectifs du module Comprendre les principes fondamentaux de la mécanique classique appliqués aux corps solides. Modéliser les mouvements de translation et de rotation à l’aide des outils vectoriels. Maîtriser les notions de torseurs, de cinématique et de dynamique du solide. Appliquer les théorèmes généraux du mouvement et de l’équilibre mécanique. 📚 Contenu du cours Le cours complet de Mécanique du Solide (basé sur le document de Mohammed Loulidi – Université Mohammed ...

Gene Therapy Slows Huntington’s Disease: What Just Happened — and Why It Matters Breakthrough Medicine Gene Therapy Slows Huntington’s Disease for the First Time A plain‑English guide to what HD is, how the new therapy works, what the trial actually showed, and what happens next. Updated: Sept 29, 2025 TL;DR (Short Version) What happened? A one‑time gene therapy (AAV-delivered microRNA targeting HTT ) appears to slow Huntington’s disease in early‑stage clinical studies. How big is the effect? Public updates reported substantially less clinical progression (≈ three‑quarters less decline) in a high‑dose group over ~3 years versus matched controls. Why is this historic? Until now, no treatment convincingly changed the course of HD — only symptoms could be managed. Caveats: Small study; surgical brain infusion; more peer‑reviewed data and confirmatory trials are needed. ...

Gene Therapy for Huntington’s Disease | Breakthrough in Neurodegenerative Research Gene Therapy for Huntington’s Disease: A Landmark Breakthrough Huntington’s disease has long stood as one of the most devastating neurodegenerative disorders . Now, in a major medical breakthrough , a new gene therapy has been shown to slow the progression of Huntington’s disease by nearly 75% over three years in early clinical trials. This finding marks a turning point in the search for effective treatments. Understanding Huntington’s Disease Huntington’s disease (HD) is an inherited, progressive brain disorder caused by a mutation in the HTT gene . This mutation leads to abnormal repetition of a DNA sequence known as a CAG trinucleotide repeat . The result is a toxic form of the huntingtin protein, which damages neurons in specific brain regions. Symptoms Uncontrolled movements (chorea) Cognitive decline and memory loss Mood disturbances (depression,...

Biological Qubits & Quantum Sensing in Cells | Quantum Biology Breakthrough Biological Qubits & Quantum Sensing in Cells A quantum biology breakthrough is taking shape at the University of Chicago, where researchers engineered fluorescent proteins into biological qubits . This discovery could transform how we study living cells , allowing nanoscale quantum sensing of magnetic and electrical signals in real time. What Are Biological Qubits? Qubits (quantum bits) are the foundation of quantum computing and quantum sensing. Unlike classical bits, they exist in a superposition of states, providing unparalleled sensitivity. By reprogramming fluorescent proteins , scientists created stable quantum sensors inside cells, bridging quantum mechanics with biology. Why Quantum Sensing in Living Cells? Quantum sensors promise unmatched precision in detecting weak signals. Inside cells, this means researchers can: Measure nanoscale magn...

Will AI Replace the Microscope? The Rise of Digital Pathology 🔬 Will AI Replace the Microscope? The Rise of Digital Pathology Pathologists have long been the quiet heroes of medicine, peering into microscopes to identify cancers, infections, and rare diseases. But human expertise has its limits — fatigue, subjectivity, and a global shortage of trained pathologists are pressing concerns. Artificial intelligence (AI) now promises to transform this field, ushering in the era of digital pathology where algorithms scan biopsy slides with speed and precision once thought impossible. ⚠️ The Limitations of Human Pathologists Diagnosing cancer is not always straightforward. Two pathologists examining the same biopsy may not always agree, particularly in borderline cases. Fatigue, workload, and even differences in training can introduce subjectivity into diagnoses. With cancer cases rising globally, and a...

How AI is Helping Discover New Antibiotics Faster than Ever 🧬 How AI is Helping Discover New Antibiotics Faster than Ever Antibiotic resistance is one of the most urgent health crises of the 21st century. The World Health Organization (WHO) has repeatedly warned that if current trends continue, common infections and minor injuries could once again become deadly. At the same time, the pipeline for new antibiotics has nearly dried up, as traditional discovery methods are slow, expensive, and often unsuccessful. Enter artificial intelligence (AI) — a new hope for finding life-saving drugs at unprecedented speed. 🌍 The Growing Crisis of Antibiotic Resistance Antibiotics transformed medicine in the 20th century, turning once-lethal infections like pneumonia and tuberculosis into treatable conditions. However, bacteria evolve rapidly. Overuse and misuse of antibiotics in healthcare and agriculture have ac...