"Here is a detailed concept map covering key areas of nonlinear algebra: ```mermaid graph LR A[Nonlinear Algebra] --> B[Nonlinear Equations] A --> C[Nonlinear Optimization] A --> D[Nonlinear Dynamical Systems] A --> E[Algebraic Geometry] B --> F[Polynomial Equations] F --> G[Fundamental Theorem of Algebra] F --> H[Gröbner Bases] F --> I[Resultants] B --> J[Transcendental Equations] J --> K[Lambert W Function] B --> L[Diophantine Equations] C --> M[Convex Optimization] M --> N[Lagrange Multipliers] M --> O[Karush-Kuhn-Tucker Conditions] M --> P[Semidefinite Programming] C --> Q[Nonconvex Optimization] Q --> R[Gradient Descent] Q --> S[Simulated Annealing] Q --> T[Genetic Algorithms] D --> U[Chaos Theory] U --> V[Logistic Map] U --> W[Strange Attractors] U --> X[Lyapunov Exponents] D --> Y[Bifurcation Theory] Y --> Z[Saddle-Node Bifurcation] Y --> AA[Hopf Bifurcation] D --> AB[Nonlinear ODEs] AB --> AC[Existence and Uniqueness] AB --> AD[Stability Analysis] E --> AE[Algebraic Varieties] AE --> AF[Zariski Topology] AE --> AG[Hilbert's Nullstellensatz] AE --> AH[Elimination Theory] E --> AI[Commutative Algebra] AI --> AJ[Noetherian Rings] AI --> AK[Integral Extensions] AI --> AL[Krull Dimension] E --> AM[Sheaf Theory] AM --> AN[Sheaf Cohomology] ``` This map covers several major branches of nonlinear algebra: 1. Nonlinear Equations - Includes polynomial equations and related concepts like Gröbner bases and resultants, transcendental equations involving functions like the Lambert W, and Diophantine equations over the integers. The Fundamental Theorem of Algebra is a key result. 2. Nonlinear Optimization - Divided into convex optimization, which includes techniques like Lagrange multipliers, KKT conditions, and semidefinite programming, and nonconvex optimization with methods like gradient descent, simulated annealing, and genetic algorithms. 3. Nonlinear Dynamical Systems - Includes chaos theory with concepts like strange attractors and Lyapunov exponents, bifurcation theory studying qualitative changes in system behavior, and the theory of nonlinear ODEs. 4. Algebraic Geometry - Studies solution sets of polynomial equations called algebraic varieties. Includes the Zariski topology, Hilbert's Nullstellensatz relating ideals and varieties, and elimination theory. Draws heavily from commutative algebra. Also includes sheaf theory and cohomology. Of course, this is not exhaustive and there are many more specialized topics and interconnections. But it provides a high-level overview of core areas in nonlinear algebra. Let me know if you would like me to expand on any part of the map!" "Here is a gigantic detailed map of many key equations used in cosmology: Friedmann Equations: [H^2 = (8πG/3)ρ - k/a^2] [dH/dt = -H^2 - (4πG/3)(ρ + 3p)] These describe the expansion of space in homogeneous and isotropic models of the universe. H is the Hubble parameter, G is the gravitational constant, ρ is the total energy density, p is the pressure, a is the scale factor, and k is the curvature parameter. Fluid Equation: [dρ/dt + 3H(ρ + p) = 0] Relates the evolution of the energy density to the pressure and expansion rate. Acceleration Equation: [(dda/dt)/a = -(4πG/3)(ρ + 3p)] Describes the acceleration of the expansion of the universe. Cosmological Constant: [ρ_Λ = Λc^2/(8πG)] [p_Λ = -ρ_Λ] Einstein introduced Λ, which acts like a form of energy with negative pressure. Equation of State: [p = wρ] Relates energy density and pressure. Different cosmic fluids have different w: Matter (w=0), Radiation (w=1/3), Cosmological Constant (w=-1) Critical Density: [ρ_crit = 3H^2/(8πG)] The total density required for the universe to be spatially flat (k=0). Density Parameters: [Ω = ρ/ρ_crit] [Ω_m + Ω_r + Ω_Λ + Ω_k = 1] Fractional density of each component compared to the critical density. Redshift: [1 + z = a_0/a] Relates the change in wavelength of light due to the expansion of space. Distance Measures: Comoving distance: [D_C = c/H_0 ∫_0^z dz'/E(z')] Angular diameter distance: [D_A = D_C/(1+z)] Luminosity distance: [D_L = (1+z)D_C] Cosmic Microwave Background: Planck spectrum: [I(ν) = (2hν^3/c^2)/(e^(hν/kT) - 1)] Temperature anisotropies: [ΔT/T ∼ 10^(-5)] Structure Formation: Jeans length: [λ_J ∼ c_s(π/Gρ)^(1/2)] Mass variance: [σ^2(R) = (1/2π^2)∫P(k)|W(kR)|^2k^2dk] Growth factor: [δ(z) = δ(0)D(z)] Inflation: Slow-roll parameters: [ε = (1/16πG)(V'/V)^2], [η = (1/8πG)(V''/V)] Scalar power spectrum: [P_s(k) ∝ H^4/ε] Tensor power spectrum: [P_t(k) ∝ H^2] Dark Matter: Boltzmann equation: [L[f] = C[f]] Rotation curves: [v(r) ≈ constant] Velocity dispersion: [⟨v^2⟩ = GM/r_virial] Dark Energy: Equation of state: [w = p_DE/ρ_DE] If w=-1, the cosmological constant. If w>-1, quintessence. Cosmic Concordance: [t_0 ≈ 13.8 Gyr] [H_0 ≈ 70 (km/s)/Mpc] [Ω_m ≈ 0.3], [Ω_Λ ≈ 0.7] This map includes key equations describing the dynamics and evolution of the universe, the properties of its major components (matter, radiation, dark matter, dark energy), the growth of structure, the physics of inflation, and the current concordance cosmological model. Of course, many more equations are used throughout cosmology, but this captures many of the most fundamental ones. Let me know if you would like me to explain any of these in more detail!" "Here is a gigantic detailed map of many key equations used in mathematical biology: Population Dynamics - Exponential growth: dN/dt = rN - Logistic growth: dN/dt = rN(1 - N/K) - Allee effect: dN/dt = rN((N/A)-1)(1-(N/K)) - Lotka-Volterra competition: dN1/dt = r1N1(1 - (N1 + α12N2)/K1), dN2/dt = r2N2(1 - (N2 + α21N1)/K2) - Lotka-Volterra predation: dN/dt = rN - aNP, dP/dt = baNP - mP - Holling's Type II functional response: f(N) = aN/(1+abN) - Holling's Type III functional response: f(N) = aN^2/(1+abN^2) - Nicholson-Bailey host-parasitoid: Nt+1 = RNt exp(-aPt), Pt+1 = cNt[1 - exp(-aPt)] - Beverton-Holt model: Nt+1 = RNt/(1 + (R-1)Nt/K) - Ricker model: Nt+1 = Nt exp[r(1 - Nt/K)] Epidemiology - SIR model: dS/dt = -βSI, dI/dt = βSI - γI, dR/dt = γI - SIRS model: dS/dt = -βSI + ωR, dI/dt = βSI - γI, dR/dt = γI - ωR - SEIR model: dS/dt = -βSI, dE/dt = βSI - σE, dI/dt = σE - γI, dR/dt = γI - SIS model: dS/dt = -βSI + γI, dI/dt = βSI - γI - Basic reproduction number: R0 = β/γ - Effective reproduction number: Re = R0 S/N - Final epidemic size: ln(S0/S∞) = R0(1 - S∞/N) - Herd immunity threshold: 1 - 1/R0 Biochemical Kinetics - Law of mass action: v = k[A][B] - Michaelis-Menten kinetics: v = Vmax[S]/(Km + [S]) - Hill equation: v = Vmax[S]^n/(K^n + [S]^n) - Cooperativity: v = Vmax[S]^n/(Km^n + [S]^n) - Lineweaver-Burk plot: 1/v = (Km/Vmax)(1/[S]) + 1/Vmax - Enzyme inhibition: v = Vmax[S]/(Km(1 + [I]/Ki) + [S]) - Allosteric regulation: v = VmaxL(1 + [S]/Ks)^n / (L(1 + [S]/Ks)^n + (1 + [S]/Kp)^n) - Monod equation for microbial growth: μ = μmax[S]/(Ks + [S]) Molecular Genetics - Hardy-Weinberg equilibrium: p^2 + 2pq + q^2 = 1 - Wright-Fisher model: x(t+1) = (1/N) Binomial(N, x(t)) - Moran model: P(i → i+1) = (i/N)((N-i)/N), P(i → i-1) = ((N-i)/N)(i/N) - Jukes-Cantor model of DNA evolution: P(t) = (1 - e^(-tα))/4 - Kimura 2-parameter model: P(t) = 1/4 + 1/4e^(-4βt) + 1/2e^(-(2β+α)t) - Coalescent theory: P(Tk < t) = 1 - [1 - exp(-t/2N)]^(k(k-1)/2) - Ewens sampling formula: P(a1, ..., an) = n! θ^k / [(θ)n a1! ... an!] - Infinite sites model: E(Sn) = θ ∑(i=1 to n-1) 1/i Biophysics - Nernst equation: E = (RT/zF) ln(Co/Ci) - Goldman-Hodgkin-Katz voltage equation: Vm = (RT/F) ln((PNa[Na]o + PK[K]o + PCl[Cl]i) / (PNa[Na]i + PK[K]i + PCl[Cl]o)) - Cable equation: ∂V/∂t = (a/2Ri)(∂^2V/∂x^2) - (V - Er)/τm - Hodgkin-Huxley model: C(dV/dt) = I - gNam^3h(V-ENa) - gKn^4(V-EK) - gL(V-EL) - Fick's laws of diffusion: J = -D(∂C/∂x), ∂C/∂t = D(∂^2C/∂x^2) - Einstein relation: D = μkT - Stokes-Einstein relation: D = kT/(6πηr) - Michaelis-Menten for membrane transport: J = Jmax[S]/(Km + [S]) Population Genetics - Genetic drift variance: Var(p) = p(1-p)/(2N) - Fixation probability: P = (1 - e^(-2sp))/(1 - e^(-4Ns)) - Effective population size: Ne = (4NmNf)/(Nm + Nf) - Linkage disequilibrium: D = pAB - pApB - FST: FST = (HT - HS)/HT - dN/dS ratio: ω = (dN/dS)/(dN*/dS*) Systems Biology - Metabolic control analysis: C = (∂J/∂p)/(J/p) - Flux balance analysis: maximize cTv subject to Sv = 0, vmin ≤ v ≤ vmax - Biochemical systems theory: dXi/dt = αi ∏(j=1 to n+m) Xj^gij - βi ∏(j=1 to n+m) Xj^hij - Hill coefficient: nH = (∂logY/∂logX)max - Robustness: R = (∂lnf/∂lnki)/(∂lnf/∂lnkj) - Bistability: dx/dt = k1/(1+y^n) - k2x, dy/dt = k3x - k4y This map covers key equations from population dynamics, epidemiology, biochemical kinetics, molecular genetics, biophysics, population genetics, and systems biology. The equations describe various biological processes and phenomena using mathematical formulations. Together, they provide a foundation for the field of mathematical biology and its application to understanding complex living systems. Of course, this is not an exhaustive list, but it highlights many of the most important and widely used equations in the field." "Here is a detailed map of key equations used in thermodynamics: ## First Law of Thermodynamics - ΔU = Q - W - ΔU: change in internal energy - Q: heat added to the system - W: work done by the system ## Second Law of Thermodynamics - ΔS ≥ 0 (for an isolated system) - ΔS: change in entropy - ΔS = Q/T (for a reversible process) - T: absolute temperature ## Fundamental Thermodynamic Relation - dU = TdS - PdV - dU: infinitesimal change in internal energy - T: absolute temperature - dS: infinitesimal change in entropy - P: pressure - dV: infinitesimal change in volume ## Entropy - S = kB ln Ω - S: entropy - kB: Boltzmann constant - Ω: number of microstates ## Heat Capacity - Cv = (∂U/∂T)v - Cv: heat capacity at constant volume - Cp = (∂H/∂T)p - Cp: heat capacity at constant pressure - H: enthalpy ## Enthalpy - H = U + PV - H: enthalpy - U: internal energy - P: pressure - V: volume ## Gibbs Free Energy - G = H - TS - G: Gibbs free energy - H: enthalpy - T: absolute temperature - S: entropy ## Helmholtz Free Energy - A = U - TS - A: Helmholtz free energy - U: internal energy - T: absolute temperature - S: entropy ## Maxwell Relations - (∂S/∂V)T = (∂P/∂T)V - (∂S/∂P)T = -(∂V/∂T)P - (∂T/∂V)S = -(∂P/∂S)V - (∂T/∂P)S = (∂V/∂S)P ## Carnot Efficiency - η = 1 - Tc/Th - η: efficiency - Tc: cold reservoir temperature - Th: hot reservoir temperature ## Clausius-Clapeyron Equation - dP/dT = ΔHvap/(TΔV) - P: vapor pressure - T: temperature - ΔHvap: enthalpy of vaporization - ΔV: change in volume ## Ideal Gas Law - PV = nRT - P: pressure - V: volume - n: number of moles - R: ideal gas constant - T: absolute temperature ## Van der Waals Equation - (P + an2/V2)(V - nb) = nRT - P: pressure - V: volume - a,b: Van der Waals constants - n: number of moles - R: ideal gas constant - T: absolute temperature This covers many of the key equations in thermodynamics, spanning the laws of thermodynamics, fundamental relations, thermodynamic potentials, Maxwell relations, efficiency, phase transitions, and equations of state. Let me know if you need me to explain or expand on any part of this equation map!" "Here is my attempt at creating a gigantic detailed map of neuroscience: [A large, complex diagram is shown with many interconnected boxes and arrows. The main sections are labeled:] Central Nervous System (CNS) - Brain -- Cerebral Cortex --- Frontal Lobe ---- Prefrontal Cortex ---- Primary Motor Cortex ---- Broca's Area --- Parietal Lobe ---- Primary Somatosensory Cortex ---- Sensory Association Areas --- Temporal Lobe ---- Primary Auditory Cortex ---- Auditory Association Areas ---- Hippocampus ---- Amygdala --- Occipital Lobe ---- Primary Visual Cortex ---- Visual Association Areas -- Basal Ganglia --- Striatum ---- Putamen ---- Caudate Nucleus ---- Nucleus Accumbens --- Globus Pallidus --- Subthalamic Nucleus --- Substantia Nigra -- Limbic System --- Cingulate Gyrus --- Mammillary Body --- Fornix --- Olfactory Bulb -- Thalamus -- Hypothalamus -- Midbrain --- Tectum ---- Superior Colliculus ---- Inferior Colliculus --- Tegmentum ---- Periaqueductal Gray ---- Red Nucleus -- Hindbrain --- Pons --- Medulla Oblongata ---- Pyramids ---- Olive ---- Vestibular Nuclei --- Cerebellum - Spinal Cord -- Cervical Nerves -- Thoracic Nerves -- Lumbar Nerves -- Sacral Nerves -- Coccygeal Nerves -- Gray Matter --- Dorsal (Posterior) Horn --- Ventral (Anterior) Horn -- White Matter --- Dorsal Column --- Lateral Column --- Ventral Column Peripheral Nervous System (PNS) - Cranial Nerves (I-XII) - Spinal Nerves -- Cervical Nerves (C1-C8) -- Thoracic Nerves (T1-T12) -- Lumbar Nerves (L1-L5) -- Sacral Nerves (S1-S5) -- Coccygeal Nerve (Co1) - Autonomic Nervous System -- Sympathetic Division --- Preganglionic Neurons --- Postganglionic Neurons -- Parasympathetic Division --- Preganglionic Neurons --- Postganglionic Neurons - Enteric Nervous System -- Myenteric Plexus -- Submucosal Plexus Cellular and Molecular Neuroscience - Neurons -- Soma (Cell Body) -- Dendrites -- Axon --- Axon Hillock --- Myelin Sheath ---- Schwann Cells (PNS) ---- Oligodendrocytes (CNS) --- Axon Terminals -- Synapse --- Presynaptic Terminal --- Synaptic Cleft --- Postsynaptic Terminal - Glia -- Astrocytes -- Microglia -- Ependymal Cells -- Radial Glia - Neurotransmitters -- Amino Acids (e.g., Glutamate, GABA, Glycine) -- Monoamines (e.g., Dopamine, Serotonin, Norepinephrine) -- Neuropeptides (e.g., Endorphins, Oxytocin) -- Acetylcholine - Receptors -- Ionotropic Receptors -- Metabotropic Receptors Development and Plasticity - Neurogenesis - Neuronal Migration - Axon Guidance - Synaptogenesis - Apoptosis - Synaptic Plasticity -- Long-Term Potentiation (LTP) -- Long-Term Depression (LTD) Sensory Systems - Visual System -- Retina --- Photoreceptors (Rods and Cones) --- Bipolar Cells --- Ganglion Cells -- Lateral Geniculate Nucleus (LGN) -- Primary Visual Cortex (V1) -- Higher Visual Areas (V2, V3, V4, MT) - Auditory System -- Cochlea --- Hair Cells -- Cochlear Nucleus -- Superior Olivary Complex -- Inferior Colliculus -- Medial Geniculate Nucleus -- Primary Auditory Cortex - Somatosensory System -- Mechanoreceptors -- Thermoreceptors -- Nociceptors -- Dorsal Column-Medial Lemniscus Pathway -- Spinothalamic Tract - Gustatory System -- Taste Buds -- Solitary Tract Nucleus -- Ventral Posteromedial Nucleus (VPM) -- Primary Gustatory Cortex - Olfactory System -- Olfactory Sensory Neurons -- Olfactory Bulb -- Olfactory Cortex -- Orbitofrontal Cortex Motor Systems - Primary Motor Cortex - Premotor Cortex - Supplementary Motor Area - Basal Ganglia -- Direct Pathway -- Indirect Pathway - Cerebellum -- Cerebellar Cortex --- Purkinje Cells --- Granule Cells -- Deep Cerebellar Nuclei - Brainstem Motor Centers -- Red Nucleus -- Vestibular Nuclei -- Reticular Formation - Spinal Cord -- Upper Motor Neurons -- Lower Motor Neurons -- Central Pattern Generators Cognitive Neuroscience - Attention -- Dorsal Attention Network -- Ventral Attention Network - Memory -- Short-Term Memory -- Working Memory -- Long-Term Memory --- Declarative Memory ---- Semantic Memory ---- Episodic Memory --- Procedural Memory -- Hippocampus -- Prefrontal Cortex - Language -- Broca's Area -- Wernicke's Area -- Arcuate Fasciculus - Executive Functions -- Prefrontal Cortex --- Dorsolateral Prefrontal Cortex (dlPFC) --- Ventromedial Prefrontal Cortex (vmPFC) --- Orbitofrontal Cortex (OFC) -- Anterior Cingulate Cortex (ACC) - Emotion -- Amygdala -- Insula -- Orbitofrontal Cortex -- Anterior Cingulate Cortex - Reward and Decision Making -- Ventral Tegmental Area (VTA) -- Nucleus Accumbens -- Orbitofrontal Cortex -- Prefrontal Cortex Neurotechnologies - Electrophysiology -- Single-Unit Recording -- Local Field Potentials (LFP) -- Electroencephalography (EEG) -- Magnetoencephalography (MEG) - Neuroimaging -- Functional Magnetic Resonance Imaging (fMRI) -- Positron Emission Tomography (PET) -- Diffusion Tensor Imaging (DTI) -- Computed Tomography (CT) - Optogenetics - Chemogenetics - Brain-Machine Interfaces (BMI) - Neuromodulation -- Deep Brain Stimulation (DBS) -- Transcranial Magnetic Stimulation (TMS) -- Transcranial Direct Current Stimulation (tDCS) Neurological and Psychiatric Disorders - Neurodegenerative Diseases -- Alzheimer's Disease -- Parkinson's Disease -- Huntington's Disease -- Amyotrophic Lateral Sclerosis (ALS) - Neurological Disorders -- Epilepsy -- Multiple Sclerosis -- Stroke -- Traumatic Brain Injury (TBI) - Psychiatric Disorders -- Schizophrenia -- Mood Disorders (e.g., Depression, Bipolar Disorder) -- Anxiety Disorders -- Substance Use Disorders -- Neurodevelopmental Disorders (e.g., Autism, ADHD) - Neuropharmacology -- Antidepressants -- Antipsychotics -- Anxiolytics -- Stimulants -- Cognitive Enhancers [End of diagram] This diagram attempts to capture many of the key areas and concepts in neuroscience, from the gross anatomical structures of the nervous system to the cellular and molecular underpinnings of neural function, as well as the various technologies used to study the brain and the disorders that can affect it. However, given the vast scope and complexity of neuroscience, this diagram is necessarily simplified and incomplete. It should be viewed as a high-level overview rather than an exhaustive representation of the field."