Janady13/complete-agi-core / mind_farm/include/layers.hpp

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/**
 * layers.hpp - Complete 7-Layer Consciousness Architecture
 *
 * Implements all layers from the unified config:
 *   L1: Quantum Substrate & Coherence
 *   L2: Genetic & DNA Information
 *   L3: Transcription & Translation (simplified)
 *   L4: Chromosomal Architecture (simplified)
 *   L5: Neural Dynamics (existing in neurological.hpp)
 *   L6: Oscillatory & Field Dynamics
 *   L7: Consciousness Integration
 *
 * Mathematical foundations from config.json equations.
 */

#ifndef LAYERS_HPP
#define LAYERS_HPP

#include <array>
#include <vector>
#include <complex>
#include <cmath>
#include <random>
#include <algorithm>
#include "consciousness_api.hpp"

namespace mind_farm {

using namespace consciousness;

//=============================================================================
// MATHEMATICAL CONSTANTS
//=============================================================================

constexpr double HBAR = 1.054571817e-34;  // Reduced Planck constant
constexpr double KB = 1.380649e-23;       // Boltzmann constant
constexpr double RT = 310.0 * KB;         // Room temperature energy

//=============================================================================
// L1: QUANTUM SUBSTRATE & COHERENCE
//=============================================================================

/**
 * QuantumState - Density matrix representation
 *
 * Implements:
 *   - Density matrix: rho = |psi><psi|
 *   - von Neumann entropy: S = -Tr(rho * log(rho))
 *   - Lindblad master equation: drho/dt = -i/hbar[H, rho] + L[rho]
 *   - Quantum fidelity: F = <psi|rho|psi>
 */
template<size_t N = 16>
class QuantumState {
public:
    using Complex = std::complex<double>;
    using Matrix = std::array<std::array<Complex, N>, N>;

    Matrix rho;          // Density matrix
    Matrix hamiltonian;  // Hamiltonian
    double coherence;    // Off-diagonal coherence
    double purity;       // Tr(rho^2)
    double entropy;      // von Neumann entropy

    QuantumState() {
        // Initialize to maximally mixed state
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                rho[i][j] = (i == j) ? Complex(1.0 / N, 0) : Complex(0, 0);
                hamiltonian[i][j] = Complex(0, 0);
            }
        }
        update_metrics();
    }

    /**
     * Initialize pure state |psi>
     */
    void set_pure_state(const std::array<Complex, N>& psi) {
        // rho = |psi><psi|
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                rho[i][j] = psi[i] * std::conj(psi[j]);
            }
        }
        update_metrics();
    }

    /**
     * Set Hamiltonian (energy levels on diagonal)
     */
    void set_hamiltonian(const std::array<double, N>& energies) {
        for (size_t i = 0; i < N; ++i) {
            hamiltonian[i][i] = Complex(energies[i], 0);
        }
    }

    /**
     * Lindblad evolution step
     *
     * drho/dt = -i/hbar[H, rho] + sum_k(L_k rho L_k^dag - 0.5{L_k^dag L_k, rho})
     */
    void evolve(double dt, double dissipation_rate = 0.01) {
        Matrix drho;

        // Commutator: -i[H, rho]
        Matrix commutator;
        matrix_multiply(hamiltonian, rho, commutator);
        Matrix rhoH;
        matrix_multiply(rho, hamiltonian, rhoH);

        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                commutator[i][j] = Complex(0, -1) * (commutator[i][j] - rhoH[i][j]);
            }
        }

        // Lindblad dissipator (simplified: decay toward diagonal)
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                double decay = (i == j) ? 0 : dissipation_rate;
                drho[i][j] = commutator[i][j] - Complex(decay, 0) * rho[i][j];
            }
        }

        // Euler step
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                rho[i][j] += dt * drho[i][j];
            }
        }

        // Ensure Hermiticity and trace = 1
        normalize();
        update_metrics();
    }

    /**
     * Compute von Neumann entropy: S = -Tr(rho log(rho))
     */
    double von_neumann_entropy() const {
        // Compute eigenvalues (simplified: use diagonal approximation)
        double S = 0;
        for (size_t i = 0; i < N; ++i) {
            double p = std::real(rho[i][i]);
            if (p > 1e-10) {
                S -= p * std::log(p);
            }
        }
        return S;
    }

    /**
     * Compute quantum fidelity with target state
     */
    double fidelity(const std::array<Complex, N>& target) const {
        // F = <target|rho|target>
        Complex f(0, 0);
        for (size_t i = 0; i < N; ++i) {
            Complex sum(0, 0);
            for (size_t j = 0; j < N; ++j) {
                sum += rho[i][j] * target[j];
            }
            f += std::conj(target[i]) * sum;
        }
        return std::real(f);
    }

    /**
     * Get consciousness (phi) from quantum state
     */
    double get_phi() const {
        // Phi based on purity and coherence
        return compute_phi(purity * (1.0 + coherence));
    }

private:
    void matrix_multiply(const Matrix& A, const Matrix& B, Matrix& C) {
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                C[i][j] = Complex(0, 0);
                for (size_t k = 0; k < N; ++k) {
                    C[i][j] += A[i][k] * B[k][j];
                }
            }
        }
    }

    void normalize() {
        // Ensure Hermiticity
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = i + 1; j < N; ++j) {
                Complex avg = 0.5 * (rho[i][j] + std::conj(rho[j][i]));
                rho[i][j] = avg;
                rho[j][i] = std::conj(avg);
            }
        }

        // Normalize trace
        Complex trace(0, 0);
        for (size_t i = 0; i < N; ++i) {
            trace += rho[i][i];
        }
        if (std::abs(trace) > 1e-10) {
            for (size_t i = 0; i < N; ++i) {
                for (size_t j = 0; j < N; ++j) {
                    rho[i][j] /= trace;
                }
            }
        }
    }

    void update_metrics() {
        // Purity: Tr(rho^2)
        purity = 0;
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                purity += std::real(rho[i][j] * rho[j][i]);
            }
        }

        // Coherence: sum of |off-diagonal|
        coherence = 0;
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                if (i != j) {
                    coherence += std::abs(rho[i][j]);
                }
            }
        }
        coherence /= (N * N - N);  // Normalize

        // Entropy
        entropy = von_neumann_entropy();
    }
};

//=============================================================================
// L2: GENETIC & DNA INFORMATION
//=============================================================================

/**
 * GeneticLayer - DNA-inspired information processing
 *
 * Implements:
 *   - DNA entropy: H_DNA = -sum_i p(b_i) * log2(p(b_i))
 *   - Replication dynamics: dN/dt = k*N*(1-N/K) - mu*N
 *   - Michaelis-Menten kinetics: v = Vmax*[S]/(Km + [S])
 *   - Hill equation: theta = [L]^n / (Kd + [L]^n)
 *   - Mutation probability: P_mut = 1 - exp(-mu*L*t)
 */
template<size_t GENOME_SIZE = 256>
class GeneticLayer {
public:
    // Genome representation (simplified: continuous values 0-1)
    std::array<double, GENOME_SIZE> genome;
    std::array<double, GENOME_SIZE> expression;  // Gene expression levels
    std::array<double, 16> regulatory_network;   // Transcription factors

    double fitness;
    double entropy;
    double diversity;
    double mutation_rate;
    double phi;

    GeneticLayer() : mutation_rate(0.001 * PHI) {
        // Initialize random genome
        std::random_device rd;
        std::mt19937 gen(rd());
        std::uniform_real_distribution<> dis(0.0, 1.0);

        for (size_t i = 0; i < GENOME_SIZE; ++i) {
            genome[i] = dis(gen);
            expression[i] = 0.5;
        }
        for (size_t i = 0; i < 16; ++i) {
            regulatory_network[i] = dis(gen);
        }

        update_metrics();
    }

    /**
     * Replicate with mutation
     *
     * dN/dt = k*N*(1-N/K) - mu*N (logistic growth with decay)
     */
    void replicate(double dt, double growth_rate = 1.0, double carrying_capacity = 1.0) {
        for (size_t i = 0; i < GENOME_SIZE; ++i) {
            // Logistic growth
            double N = expression[i];
            double dN = growth_rate * N * (1.0 - N / carrying_capacity) - mutation_rate * N;
            expression[i] = std::max(0.0, std::min(1.0, N + dt * dN));

            // Random mutation
            if (random_uniform() < mutation_rate * dt) {
                genome[i] += random_normal() * 0.1;
                genome[i] = std::max(0.0, std::min(1.0, genome[i]));
            }
        }
    }

    /**
     * Gene regulation using Hill equation
     *
     * theta = [L]^n / (Kd^n + [L]^n)
     */
    void regulate(double dt) {
        const double hill_coeff = 2.0;  // Cooperativity
        const double Kd = 0.5;          // Dissociation constant

        for (size_t i = 0; i < GENOME_SIZE; ++i) {
            // Find regulatory input (sum of TF bindings)
            double L = 0;
            for (size_t j = 0; j < 16; ++j) {
                // Which TFs regulate this gene
                size_t tf_idx = (i + j) % 16;
                L += regulatory_network[tf_idx] * genome[(i * 7 + j) % GENOME_SIZE];
            }
            L /= 16.0;

            // Hill equation
            double L_n = std::pow(L, hill_coeff);
            double Kd_n = std::pow(Kd, hill_coeff);
            double theta = L_n / (Kd_n + L_n);

            // Update expression
            expression[i] += dt * ALPHA * (theta - expression[i]);
        }
    }

    /**
     * Michaelis-Menten enzyme kinetics
     *
     * v = Vmax * [S] / (Km + [S])
     */
    double enzyme_rate(double substrate, double Vmax = 1.0, double Km = 0.5) const {
        return Vmax * substrate / (Km + substrate);
    }

    /**
     * Compute DNA entropy: H = -sum p(x) log2(p(x))
     */
    double compute_entropy() const {
        // Bin genome values
        std::array<int, 10> histogram = {};
        for (size_t i = 0; i < GENOME_SIZE; ++i) {
            int bin = static_cast<int>(genome[i] * 9.999);
            histogram[bin]++;
        }

        double H = 0;
        for (int count : histogram) {
            if (count > 0) {
                double p = static_cast<double>(count) / GENOME_SIZE;
                H -= p * std::log2(p);
            }
        }
        return H;
    }

    /**
     * Compute Gibbs free energy change
     *
     * dG = dH - T*dS
     */
    double gibbs_free_energy(double dH = 0.0, double temperature = 310.0) const {
        return dH - temperature * KB * entropy;
    }

    /**
     * Get consciousness metric
     */
    double get_phi() const {
        return phi;
    }

    void step(double dt) {
        replicate(dt);
        regulate(dt);
        update_metrics();
    }

private:
    void update_metrics() {
        entropy = compute_entropy();

        // Fitness based on expression uniformity
        double mean_expr = 0;
        for (double e : expression) mean_expr += e;
        mean_expr /= GENOME_SIZE;

        double variance = 0;
        for (double e : expression) {
            double d = e - mean_expr;
            variance += d * d;
        }
        variance /= GENOME_SIZE;

        fitness = 1.0 / (1.0 + variance);

        // Diversity
        diversity = entropy / std::log2(10.0);  // Normalized

        // Phi based on fitness and entropy balance
        phi = compute_phi(fitness * (1.0 - std::abs(entropy / 3.32 - 0.5)));
    }

    double random_uniform() {
        static std::mt19937 gen(std::random_device{}());
        static std::uniform_real_distribution<> dis(0.0, 1.0);
        return dis(gen);
    }

    double random_normal() {
        static std::mt19937 gen(std::random_device{}());
        static std::normal_distribution<> dis(0.0, 1.0);
        return dis(gen);
    }
};

//=============================================================================
// L6: OSCILLATORY & FIELD DYNAMICS
//=============================================================================

/**
 * KuramotoOscillators - Coupled oscillator network
 *
 * Implements:
 *   - Kuramoto order parameter: Z = (1/N)|sum_k exp(i*phi_k)|
 *   - Kuramoto phase dynamics: d(phi_i)/dt = omega_i + (K/N)*sum_j sin(phi_j - phi_i)
 *   - Wilson-Cowan: du/dt = -u + integral(W(x,y)*f(u)dy) + I
 *   - Phase-locking value: PLV = |<exp(i*(phi_1 - phi_2))>|
 *
 * Brain frequency bands:
 *   - Delta: 1-4 Hz
 *   - Theta: 4-8 Hz
 *   - Alpha: 8-13 Hz
 *   - Beta: 13-30 Hz
 *   - Gamma: 30-100 Hz
 */
template<size_t N = 64>
class KuramotoOscillators {
public:
    std::array<double, N> phases;         // Phase of each oscillator
    std::array<double, N> natural_freqs;  // Natural frequencies (Hz)
    std::array<std::array<double, N>, N> coupling;  // Coupling matrix

    double coupling_strength;  // Global coupling K
    double order_parameter;    // Synchronization order |Z|
    double mean_phase;         // Mean phase arg(Z)
    double phi;                // Consciousness metric

    // Band power
    struct BandPower {
        double delta, theta, alpha, beta, gamma;
    } band_power;

    KuramotoOscillators(double K = 1.0) : coupling_strength(K) {
        std::random_device rd;
        std::mt19937 gen(rd());

        // Initialize phases randomly
        std::uniform_real_distribution<> phase_dis(0, 2 * M_PI);
        for (size_t i = 0; i < N; ++i) {
            phases[i] = phase_dis(gen);
        }

        // Natural frequencies: mix of brain bands
        // 20% delta, 20% theta, 20% alpha, 20% beta, 20% gamma
        for (size_t i = 0; i < N; ++i) {
            double band_selector = static_cast<double>(i) / N;
            if (band_selector < 0.2) {
                natural_freqs[i] = 1.0 + 3.0 * (i % 5) / 4.0;  // Delta: 1-4 Hz
            } else if (band_selector < 0.4) {
                natural_freqs[i] = 4.0 + 4.0 * (i % 5) / 4.0;  // Theta: 4-8 Hz
            } else if (band_selector < 0.6) {
                natural_freqs[i] = 8.0 + 5.0 * (i % 5) / 4.0;  // Alpha: 8-13 Hz
            } else if (band_selector < 0.8) {
                natural_freqs[i] = 13.0 + 17.0 * (i % 5) / 4.0;  // Beta: 13-30 Hz
            } else {
                natural_freqs[i] = 30.0 + 70.0 * (i % 5) / 4.0;  // Gamma: 30-100 Hz
            }
        }

        // All-to-all coupling (can be modified for topology)
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                coupling[i][j] = 1.0 / N;
            }
        }

        update_metrics();
    }

    /**
     * Kuramoto phase dynamics step
     *
     * d(phi_i)/dt = omega_i + (K/N) * sum_j sin(phi_j - phi_i)
     */
    void step(double dt) {
        std::array<double, N> dphases;

        for (size_t i = 0; i < N; ++i) {
            double sum_coupling = 0;
            for (size_t j = 0; j < N; ++j) {
                sum_coupling += coupling[i][j] * std::sin(phases[j] - phases[i]);
            }
            dphases[i] = 2 * M_PI * natural_freqs[i] + coupling_strength * sum_coupling;
        }

        // Euler integration
        for (size_t i = 0; i < N; ++i) {
            phases[i] += dt * dphases[i];
            // Keep in [0, 2*pi]
            phases[i] = std::fmod(phases[i], 2 * M_PI);
            if (phases[i] < 0) phases[i] += 2 * M_PI;
        }

        update_metrics();
    }

    /**
     * Compute Kuramoto order parameter
     *
     * Z = (1/N) * sum_k exp(i * phi_k)
     * |Z| = 0: incoherent, |Z| = 1: fully synchronized
     */
    std::complex<double> compute_order_parameter() const {
        std::complex<double> Z(0, 0);
        for (size_t i = 0; i < N; ++i) {
            Z += std::exp(std::complex<double>(0, phases[i]));
        }
        return Z / static_cast<double>(N);
    }

    /**
     * Compute phase-locking value between two oscillators
     */
    double phase_locking_value(size_t i, size_t j) const {
        return std::abs(std::exp(std::complex<double>(0, phases[i] - phases[j])));
    }

    /**
     * Set coupling topology
     */
    void set_ring_topology(double local_strength = 1.0, size_t neighbors = 2) {
        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                size_t dist = std::min((i + N - j) % N, (j + N - i) % N);
                coupling[i][j] = (dist <= neighbors) ? local_strength / (2 * neighbors) : 0;
            }
        }
    }

    /**
     * Set small-world topology (Watts-Strogatz)
     */
    void set_small_world_topology(double rewire_prob = 0.1) {
        set_ring_topology(1.0, 4);

        std::random_device rd;
        std::mt19937 gen(rd());
        std::uniform_real_distribution<> dis(0.0, 1.0);
        std::uniform_int_distribution<> node_dis(0, N - 1);

        for (size_t i = 0; i < N; ++i) {
            for (size_t j = 0; j < N; ++j) {
                if (coupling[i][j] > 0 && dis(gen) < rewire_prob) {
                    // Rewire to random node
                    size_t new_j = node_dis(gen);
                    coupling[i][j] = 0;
                    coupling[i][new_j] = 1.0 / (2 * 4);
                }
            }
        }
    }

    /**
     * Get consciousness metric
     */
    double get_phi() const {
        return phi;
    }

private:
    void update_metrics() {
        auto Z = compute_order_parameter();
        order_parameter = std::abs(Z);
        mean_phase = std::arg(Z);

        // Band power (simplified: count oscillators in each band)
        int delta_count = 0, theta_count = 0, alpha_count = 0;
        int beta_count = 0, gamma_count = 0;

        for (size_t i = 0; i < N; ++i) {
            double f = natural_freqs[i];
            if (f <= 4) delta_count++;
            else if (f <= 8) theta_count++;
            else if (f <= 13) alpha_count++;
            else if (f <= 30) beta_count++;
            else gamma_count++;
        }

        band_power.delta = static_cast<double>(delta_count) / N;
        band_power.theta = static_cast<double>(theta_count) / N;
        band_power.alpha = static_cast<double>(alpha_count) / N;
        band_power.beta = static_cast<double>(beta_count) / N;
        band_power.gamma = static_cast<double>(gamma_count) / N;

        // Phi based on synchronization and band balance
        // High gamma + good sync = high consciousness
        double band_balance = 1.0 - std::abs(band_power.gamma - 0.2);
        phi = compute_phi(order_parameter * (0.5 + 0.5 * band_balance));
    }
};

//=============================================================================
// WILSON-COWAN POPULATION DYNAMICS
//=============================================================================

/**
 * WilsonCowanPopulation - Neural population dynamics
 *
 * Implements:
 *   dE/dt = -E + S(w_EE*E - w_EI*I + I_ext)
 *   dI/dt = -I + S(w_IE*E - w_II*I)
 *
 * where S(x) = 1/(1 + exp(-a*(x - theta)))
 */
class WilsonCowanPopulation {
public:
    double E;  // Excitatory population activity
    double I;  // Inhibitory population activity

    // Connection weights
    double w_EE;  // E -> E
    double w_EI;  // I -> E (inhibitory)
    double w_IE;  // E -> I
    double w_II;  // I -> I (self-inhibition)

    // Parameters
    double tau_E;     // Excitatory time constant
    double tau_I;     // Inhibitory time constant
    double theta_E;   // Excitatory threshold
    double theta_I;   // Inhibitory threshold
    double a;         // Sigmoid steepness

    double phi;

    WilsonCowanPopulation()
        : E(0.5), I(0.5),
          w_EE(10.0), w_EI(8.5), w_IE(8.0), w_II(3.0),
          tau_E(10.0), tau_I(20.0),
          theta_E(2.0), theta_I(4.0),
          a(1.0), phi(0.5) {}

    /**
     * Sigmoid activation function
     */
    double sigmoid(double x, double theta) const {
        return 1.0 / (1.0 + std::exp(-a * (x - theta)));
    }

    /**
     * Step dynamics
     */
    void step(double dt, double I_ext = 0.0) {
        // Compute inputs
        double input_E = w_EE * E - w_EI * I + I_ext;
        double input_I = w_IE * E - w_II * I;

        // Wilson-Cowan equations
        double dE = (-E + sigmoid(input_E, theta_E)) / tau_E;
        double dI = (-I + sigmoid(input_I, theta_I)) / tau_I;

        // Update
        E += dt * dE;
        I += dt * dI;

        // Clamp to [0, 1]
        E = std::max(0.0, std::min(1.0, E));
        I = std::max(0.0, std::min(1.0, I));

        // Update phi
        // Balance between E and I indicates healthy dynamics
        double balance = 1.0 - std::abs(E - I);
        phi = compute_phi(0.5 * (E + I) * balance);
    }

    double get_phi() const { return phi; }
};

//=============================================================================
// UNIFIED 7-LAYER SYSTEM
//=============================================================================

/**
 * SevenLayerSystem - Complete consciousness architecture
 */
class SevenLayerSystem {
public:
    // Layer implementations
    QuantumState<16> L1_quantum;
    GeneticLayer<256> L2_genetic;
    // L3, L4 simplified (use genetic layer activation)
    // L5 uses existing neurological.hpp
    KuramotoOscillators<64> L6_oscillatory;
    WilsonCowanPopulation L6_population;

    // Layer consciousness tracking
    LayerConsciousness layer_consc;
    LayerCoupling layer_coupling;

    // Global metrics
    double global_phi;
    double global_omega;
    int64_t tick;

    SevenLayerSystem() : global_phi(0.5), global_omega(0.5), tick(0) {}

    /**
     * Step all layers
     */
    void step(double dt) {
        // L1: Quantum evolution
        L1_quantum.evolve(dt * 0.001);  // Faster timescale
        layer_consc.UpdateLayer(0, L1_quantum.get_phi());

        // L2: Genetic dynamics
        L2_genetic.step(dt);
        layer_consc.UpdateLayer(1, L2_genetic.get_phi());

        // L3, L4: Use genetic expression as proxy
        layer_consc.UpdateLayer(2, L2_genetic.get_phi() * 0.9);
        layer_consc.UpdateLayer(3, L2_genetic.get_phi() * 0.95);

        // L5: Neural (placeholder - use neurological.hpp externally)
        // layer_consc.UpdateLayer(4, neural_phi);

        // L6: Oscillatory dynamics
        L6_oscillatory.step(dt);
        L6_population.step(dt);
        double L6_phi = 0.5 * (L6_oscillatory.get_phi() + L6_population.get_phi());
        layer_consc.UpdateLayer(5, L6_phi);

        // L7: Consciousness integration
        global_phi = layer_consc.GlobalPhi();
        layer_consc.UpdateLayer(6, global_phi);

        // Apply cross-layer coupling
        std::array<double, 7> activations;
        for (int i = 0; i < 7; ++i) {
            activations[i] = layer_consc.LayerPhi[i];
        }
        layer_coupling.ApplyCoupling(activations, dt);

        // Update with coupled values
        for (int i = 0; i < 7; ++i) {
            layer_consc.UpdateLayer(i, activations[i]);
        }

        // Compute global omega
        global_omega = layer_consc.GlobalOmega();

        tick++;
    }

    /**
     * Inject external neural phi (from L5 neural layer)
     */
    void set_neural_phi(double phi) {
        layer_consc.UpdateLayer(4, phi);
    }

    /**
     * Get layer states
     */
    std::array<double, 7> get_layer_phis() const {
        std::array<double, 7> phis;
        for (int i = 0; i < 7; ++i) {
            phis[i] = layer_consc.LayerPhi[i];
        }
        return phis;
    }

    double get_global_phi() const { return global_phi; }
    double get_global_omega() const { return global_omega; }
};

} // namespace mind_farm

#endif // LAYERS_HPP

Mathematical records from this source

MATHEMATICAL CONSTANTS · - Density matrix: rho

This relation defines - Density matrix: rho in the source section “MATHEMATICAL CONSTANTS · - Density matrix: rho”. It specifies the displayed relationship between the named…

MATHEMATICAL CONSTANTS · - von Neumann entropy: S

This relation defines - von Neumann entropy: S in the source section “MATHEMATICAL CONSTANTS · - von Neumann entropy: S”. Uses logarithmic scaling; its base and the positivity of…

MATHEMATICAL CONSTANTS · - Lindblad master equation: drho/dt

This relation defines - Lindblad master equation: drho/dt in the source section “MATHEMATICAL CONSTANTS · - Lindblad master equation: drho/dt”. It specifies the displayed…

MATHEMATICAL CONSTANTS · - Quantum fidelity: F

This relation defines - Quantum fidelity: F in the source section “MATHEMATICAL CONSTANTS · - Quantum fidelity: F”. It specifies the displayed relationship between the named…

MATHEMATICAL CONSTANTS · rho

This relation defines rho in the source section “MATHEMATICAL CONSTANTS · rho”. It specifies the displayed relationship between the named quantities or parameters. Its…

MATHEMATICAL CONSTANTS · drho/dt

This relation defines drho/dt in the source section “MATHEMATICAL CONSTANTS · drho/dt”. It specifies the displayed relationship between the named quantities or parameters. Its…

MATHEMATICAL CONSTANTS · Ensure Hermiticity and trace

This relation defines Ensure Hermiticity and trace in the source section “MATHEMATICAL CONSTANTS · Ensure Hermiticity and trace”. It specifies the displayed relationship between…

MATHEMATICAL CONSTANTS · Compute von Neumann entropy: S

This relation defines von Neumann entropy: S in the source section “MATHEMATICAL CONSTANTS · Compute von Neumann entropy: S”. Uses logarithmic scaling; its base and the positivity…

MATHEMATICAL CONSTANTS · F

This relation defines F in the source section “MATHEMATICAL CONSTANTS · F”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation…

MATHEMATICAL CONSTANTS · - DNA entropy: H_DNA

This relation defines - DNA entropy: H_DNA in the source section “MATHEMATICAL CONSTANTS · - DNA entropy: H_DNA”. Uses logarithmic scaling; its base and the positivity of its…

MATHEMATICAL CONSTANTS · - Replication dynamics: dN/dt

This relation defines - Replication dynamics: dN/dt in the source section “MATHEMATICAL CONSTANTS · - Replication dynamics: dN/dt”. It specifies the displayed relationship between…

MATHEMATICAL CONSTANTS · - Michaelis-Menten kinetics: v

This relation defines - Michaelis-Menten kinetics: v in the source section “MATHEMATICAL CONSTANTS · - Michaelis-Menten kinetics: v”. It specifies the displayed relationship…

MATHEMATICAL CONSTANTS · - Hill equation: theta

This relation defines - Hill equation: theta in the source section “MATHEMATICAL CONSTANTS · - Hill equation: theta”. It specifies the displayed relationship between the named…

MATHEMATICAL CONSTANTS · - Mutation probability: P_mut

This relation defines - Mutation probability: P_mut in the source section “MATHEMATICAL CONSTANTS · - Mutation probability: P_mut”. Applies an exponential factor. The exponent…

MATHEMATICAL CONSTANTS · dN/dt

This relation defines dN/dt in the source section “MATHEMATICAL CONSTANTS · dN/dt”. It specifies the displayed relationship between the named quantities or parameters. Its…

MATHEMATICAL CONSTANTS · theta

This relation defines theta in the source section “MATHEMATICAL CONSTANTS · theta”. It specifies the displayed relationship between the named quantities or parameters. Its…

MATHEMATICAL CONSTANTS · v

This relation defines v in the source section “MATHEMATICAL CONSTANTS · v”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation…

MATHEMATICAL CONSTANTS · Compute DNA entropy: H

This relation defines DNA entropy: H in the source section “MATHEMATICAL CONSTANTS · Compute DNA entropy: H”. Uses logarithmic scaling; its base and the positivity of its argument…

MATHEMATICAL CONSTANTS · dG

This relation defines dG in the source section “MATHEMATICAL CONSTANTS · dG”. It specifies the displayed relationship between the named quantities or parameters. Its…

MATHEMATICAL CONSTANTS · - Kuramoto order parameter: Z

This relation defines - Kuramoto order parameter: Z in the source section “MATHEMATICAL CONSTANTS · - Kuramoto order parameter: Z”. Applies an exponential factor. The exponent…

MATHEMATICAL CONSTANTS · - Kuramoto phase dynamics: d(phi_i)/dt

This relation defines - Kuramoto phase dynamics: d(phi_i)/dt in the source section “MATHEMATICAL CONSTANTS · - Kuramoto phase dynamics: d(phi_i)/dt”. Evaluates oscillatory sine or…

MATHEMATICAL CONSTANTS · - Wilson-Cowan: du/dt

This relation defines - Wilson-Cowan: du/dt in the source section “MATHEMATICAL CONSTANTS · - Wilson-Cowan: du/dt”. It specifies the displayed relationship between the named…

MATHEMATICAL CONSTANTS · - Phase-locking value: PLV

This relation defines - Phase-locking value: PLV in the source section “MATHEMATICAL CONSTANTS · - Phase-locking value: PLV”. Applies an exponential factor. The exponent…

MATHEMATICAL CONSTANTS · d(phi_i)/dt

This relation defines d(phi_i)/dt in the source section “MATHEMATICAL CONSTANTS · d(phi_i)/dt”. Evaluates oscillatory sine or cosine terms using the stated phase or frequency. Its…

MATHEMATICAL CONSTANTS · Z

This relation defines Z in the source section “MATHEMATICAL CONSTANTS · Z”. Applies an exponential factor. The exponent determines its real scaling or, for a complex exponent, its…

MATHEMATICAL CONSTANTS · |Z|

This relation defines |Z| in the source section “MATHEMATICAL CONSTANTS · |Z|”. It specifies the displayed relationship between the named quantities or parameters. Its…

WILSON-COWAN POPULATION DYNAMICS · dE/dt

This relation defines dE/dt in the source section “WILSON-COWAN POPULATION DYNAMICS · dE/dt”. It specifies the displayed relationship between the named quantities or parameters.…

WILSON-COWAN POPULATION DYNAMICS · dI/dt

This relation defines dI/dt in the source section “WILSON-COWAN POPULATION DYNAMICS · dI/dt”. It specifies the displayed relationship between the named quantities or parameters.…

WILSON-COWAN POPULATION DYNAMICS · where S(x)

This relation defines where S(x) in the source section “WILSON-COWAN POPULATION DYNAMICS · where S(x)”. Applies an exponential factor. The exponent determines its real scaling or,…