Names: letter first, no specials, < 32 chars, case-sensitive. A(row,col) rows split by ; · A(1,:) row · A(1,1:2:5) cols 1,3,5 · .* ./ .^ element-wise, * matrix (inner dims agree) · zeros ones eye rand · for i=1:n … end, if … elseif … else … end, odd test fix(x/2)~=x/2 · index starts at 1 → a(n+1).
Course matrix A = [12 10 13 15 16; 22 45 65 1 0; 22 33 41 23 45; 21 30 12 6 2; 1 0 0 1 7]: sum(A) = [78 118 131 46 70], sum(sum(A)) = 443, diag = [12 45 41 6 7], trace 111. [3,5].*[4,8] = [12,40].
[0 2 10 20 3 15] with +1 if > 10 else −1 → [−1 1 9 21 2 16]. Plot: t=0:0.01:2*pi (0:1:2π gives only 7 points).
uint8 0–255 (uint16 65,535); toy 4×4 histogram [5 4 3 4]. medfilt2(X,[3 3]) odd window → middle of 9: {0,20,0,127,112,100,128,135,0} → 100; {0,5,8,60,99,99,109,125,155} → 99; ECG 1-D [1 3]. 3-D: V(:,:,k), isosurface(…,15), daspect([1 1 .4]), plot3, trisurf.
Coefficients highest power first: x³+4x²+9x+16 → [1 4 9 16]. roots ↔ poly; conv = multiply ([1 2 3 4]⋆[1 4 9 16] = [1 6 20 50 75 84 64]); deconv; polyder → [3 8 9]; polyint → [0.25 1.333 4.5 16 0].
$$\text{polyfit: }\min_p\sum_i(y_i-P(x_i))^2\ \Rightarrow\ (V^TV)p=V^Ty\quad(\text{41 pts}\to p\approx[0.965,\,0.140,\,4.969]);\qquad\text{spline passes exactly through the data}$$Solution exists iff rank(A) = rank([A b]) = r; unique if r = n (det ≠ 0); infinite if r < n (pinv, rref); A x = 0 non-trivial iff rank < n. Over-determined consistent → exact; inconsistent → least squares. Slide system A = [1 2 3; 4 5 6; 7 8 0], y = [366; 804; 351] → x = [25; 22; 99] (y/A is a bug). [3 −4; 6 −8] singular. Circuit A = [1 −1 1; −1 1 −1; 4 2 0; 0 2 5], b = [0;0;8;9] → i = [1, 2, 1] A. Chemical CO₂ + H₂O → O₂ + C₆H₁₂O₆: null space t·[6 6 6 1]. dot = projection, cross = moment.
Series: periodic only ("main limitation"); transform: periodic and aperiodic ("main advantage"). syms t; fourier(exp(-t^2)) = √π e^{−w²/4}; fourier(exp(-abs(t))) = 2/(1+w²).
"Backslash is Gaussian elimination (LU); the inverse is only for theory." "A solution exists when b lies in the column space: rank(A) = rank([A b])." "Fitting minimises squared error and need not touch the points; interpolation must." "A square wave has only odd harmonics falling as 1/k, so its edges need infinite bandwidth." "Sampling replicates the spectrum every fs; keep the copies apart with fs ≥ 2fm, then low-pass to recover."
MATLAB indices start at 1 (Hist(i+1), a(n+1)). uint8 saturates at 255: convert to double. medfilt2 needs an odd window. y/A ≠ A\y. det ≈ 0 means numerically singular. Variable names are case-sensitive (items vs Items bug). Fourier series only for periodic signals.
f = i·r (illumination × reflectance); bits = M·N·k (1024² × 8 = 8,388,608). N4 / ND / N8; m-connectivity removes multiple paths. D4 = |Δx|+|Δy|, D8 = max, De = √: (1,2)→(4,6) = 7, 4, 5. Two-pass labelling with equivalence table. Point ops: negative (L−1)−r, log c·log(1+r), power s = c·rᵞ (γ<1 brightens), slicing.
$$p(g)=\frac{h(g)}{RC},\quad s_k=\mathrm{round}\big((L-1)\textstyle\sum_{j\le k}p_j\big):\ h=[5,4,0,0,2,1,3,0,4,1],\ L=10\ \to\ s=2,4,4,4,5,5,7,7,9,9\ (\text{"almost, not completely, flat"})$$Freq 8,7,2,6,9,4 (N 36): m_G 2.3611; σ²_B = 1.593, 2.564, 2.629, 2.142, 0.870 → k* = 2, η ≈ 0.84. Limitation: histogram only, no spatial info. Iterative T = ½(m₁+m₂); Niblack T = m + k·s. Split/merge (std < 5): 4 quadrants pass, R1∪R3 and R2∪R4 merge → 2 regions. K-means 7 points: C₁(1,1), C₂(5,7) → it.1 {1,2,3}/{4–7} → it.2 point 3 moves → (1.25,1.5), (3.9,5.1) → it.3 stable.
Opening = erode→dilate (salt, thin joints); closing = dilate→erode (pepper, holes). 3×3 block: erode by 3×3 ones → centre only; by vertical line → middle row; dilate by 3×3 ones → full 5×5; by cross → plus with zero corners. Boundary β = A − (A⊖B); fill X_k = (X_{k−1}⊕B) ∩ Aᶜ; coins: binarise → erode → label → count.
Unsharp g = f + k(f − f_LP). Homomorphic: ln f = ln i + ln r, H = (γ_H−γ_L)[1−e^{−cD²/D₀²}]+γ_L (0.25, 2, 1, 80). Notch pairs ±(u_k,v_k) kill stripes. Sampling: fs > 2·f_max else aliasing. Mask weights sum 1 → DC passes (low-pass); sum 0 → edge detector; [0 −1 0; −1 5 −1; 0 −1 0] emphasises edges.
Box 3×3 of 106,104,99,95,100,108,98,90,85 → 98.33; median = 5th of the 9 sorted values (slide: 0,0,1,1,1,2,2,2,4 → 1). Borders: discard (512→510), zero-pad (false edges), replicate. Second derivative: thin double edges, fine detail. Canny: Gaussian 5×5/159 (→ 41) → Sobel G_x −191, G_y −181, |G| 263, θ 134° → NMS (263 vs 7, 255 → kept) → hysteresis T_H 200 / T_L 50 (weak kept if 8-connected to strong).
Contraharmonic Q>0 pepper, Q<0 salt; median for salt-and-pepper. Local filter μ 75.14, σ² 6259.5, v² 400: 186→178.9, 95→93.7, 36→38.5. Diffusion 186 with N255 S157 E212 W208, K 30, λ 1/7: c = .005, .392, .472, .584 → 188.0 (edge kept).
Frame differencing = boundaries + ghosts; temporal median or Stauffer–Grimson GMM background. E-step example x=1, N(0,1)/N(4,1): γ₁ = 0.982. Tracking = detection + prediction; Markov on state and observation. Kalman: linear-Gaussian, mean + covariance, one object. Particle filter: weighted samples, clutter, multimodal. Too strong dynamics → ignores data; too strong observation → repeated detection; drift.
"Otsu maximises between-class variance from the histogram alone." "Erosion keeps a pixel only where the structuring element fits; dilation wherever it hits; they are duals." "Filtering in space is multiplication in frequency, but DFT convolution is circular, so we pad." "Canny: smooth, gradient, thin by non-maximum suppression, link by hysteresis." "EM alternates responsibilities and parameter updates; a Kalman filter predicts then corrects."
Ideal low-pass rings (sinc kernel). Zero padding of borders creates false edges. Otsu fails on unimodal or unevenly lit images. Second derivative amplifies noise: smooth first. 8-connectivity merges diagonal blobs. Mask sums: 1 passes DC, 0 rejects it.
$e=SP-PV$. P only → steady-state error (0 output at $e=0$); I removes it; D damps, amplifies noise. Tune $K_p$, then $K_d$, then $K_i$ (0.0001). $K_p=125/3500=0.0357$. Example (2.0,0.5,0.1), $\Delta t$ 0.1, $e=-0.48$, $e_{prev}=-0.30$, $\sum=-0.12$: $-0.96-0.06-0.18=\mathbf{-1.20}$.
Normal form: vertical lines finite. (3,3),(4,3),(5,3) → $A(3,90°)=3$. RANSAC $P$ 0.99, $p$ 0.5: $k$=2→17, 3→35, 4→72. Reject lane parabola $|a|\ge0.003$.
(700,400), $Z$ 10, $f$ 800, $c$ (640,360) → (0.75, 0.5, 10). Stereo $d=80$, $f_x$ 795, $B$ 0.2 → $Z=1.9875$ m. One image: 2 eq, 3 unknowns. Quality = reprojection error.
Output $S\times S\times(5B+C)$: 7×7×30. Loss: $\sqrt w,\sqrt h$; $\lambda_{noobj}=0.5$; NMS IoU > 0.5. MIO: in lane if $x_L(y)\le x\le x_R(y)$, $x(y)=(y-b)/m$; MIO = argmax $y_{bottom}$. Three-car example: Car 3 (x 500) out of lane → Car 2 (290 > 260). FCW: tracks (confirm [2 3], delete 5), closest in lane; $d=1.2v+v^2/(2\cdot0.4\cdot9.8)$ → 24.8 m at 10 m/s.
Derivation: $N(\mu_p,p)\times N(z,r)$ → $\frac1{\sigma^2}=\frac1p+\frac1r$, $\mu=\frac{r\mu_p+pz}{p+r}$; set $K=\frac{p}{p+r}$. $K\to1$ trust sensor, $K\to0$ trust prediction; $0 Fusion: $z_f=\frac{\sum z_i/r_i}{\sum 1/r_i},\ r_f=\frac1{\sum1/r_i}$; (0.9,1.1),(1,4) → 0.94, 0.8; prior 10.1 → $K=0.927$, $x=0.87$, $p=0.74$. $r_i$ never changes during filtering.
"Bang-bang chooses a direction; PID chooses how much." "Hough votes in $(\rho,\theta)$ because slope is infinite for vertical lines." "RANSAC keeps the model most points agree with; least squares is pulled by outliers." "The Kalman filter is a recursive Bayesian estimator: predict widens, update shrinks." "MIO comes from confirmed tracks because detections flicker."
Pull-up button pressed = LOW. Never delay(), use millis(). Pooling has no parameters; PID I-term needs anti-windup in practice. Behaviour cloning is regression (ELU + regression layer), not softmax. Gazebo = world, RViz = belief.
NRMSE punishes large errors quadratically (selection criterion); MAE is what an operator reads; capacity normalisation avoids dividing by night-time zeros. Bootstrap: resample MAPEs m = 10,000 times → 95% CI of the mean; width grows with horizon.
LSTM adds a cell state with forget/input/output gates (more parameters, similar accuracy here). Sliding window of past samples (1, 2, 3, 6, 24 h, HISIMI+x) → 21 outputs: 1–6 h ahead at 15-min steps (multi-target regression). FFNN sees the flattened window; recurrent models see the sequence.
Phase 1 feature/window combinations + extensive grid HL ∈ {2,3}, HU powers of 2, mb ∈ {32,64} (≥ 240 runs/model). Phase 2 guided grid search: expand any hyper-parameter sitting on the grid edge until accuracy saturates. Phase 3 mb ∈ {16,128,256}. Fixed: MSE, Adam 1e-4, early stopping patience 20. Clustering: Euclidean distance between inverter series → 84×84 matrix → K-means K = 10 (elbow, CH, gap) → tune one representative per cluster, assign its top-2 settings to the rest.
$$d(a,b)=\sqrt{\textstyle\sum_t(a_t-b_t)^2},\qquad J=\sum_{k}\sum_{a\in C_k}\|a-\mu_k\|^2$$Macro model: one network on the plant total. Aggregated inverter-level: 84 networks (one per inverter), summed, plus a loss-correction FFNN for the difference between Σ inverters and the plant meter. Hypothesis: local effects (wind, dust, shading) are visible per inverter and lost in the total. Weather types (clear, overcast, variable) are compared separately.
GRU ≥ LSTM > FFNN at all horizons; inverter-level beats macro by a small margin (8.02 vs 8.12 NRMSE), mainly on variable-cloud days and short horizons; gains vanish on clear/overcast days and at 6 h. Cluster-based tuning matches exhaustive tuning. Error grows roughly linearly with horizon; CIs widen with horizon.
"Operators need intra-day solar forecasts. Almost all models forecast the plant total. On a 75 MW plant with 84 inverters, local wind, dust and shading differ across the site, so the authors forecast every inverter and sum them. Four years of data, equal-effort three-phase tuning, FFNN/LSTM/GRU, 21 outputs over 1–6 h. GRU is best; inverter-level forecasts are slightly better (8.02 vs 8.12 % NRMSE), mostly on variable days. Clustering inverters with K-means lets you tune ten instead of 84 at no loss. The gain is real but small, and its cost and significance are not quantified."
σ = density (is the point inside the robot?), α = visibility (does the camera see it?). Gradient to a point ∝ pixel error × α, so hidden points are not trained. Plain NeRF volume rendering failed (no depth ordering from one view).
IK by gradient descent through the differentiable model, warm-started from the previous pose (1,000-point 3-D spiral). Planning: whole-body FFKSM flags configurations whose occupied points hit an obstacle; end-effector FFKSM gives the heuristic; RRT grows the tree. Damage: bent link → re-babble, re-train, re-plan; mirror self-recognition by comparing predicted and observed silhouettes.
A self-model is a learned simulator of the robot's own morphology and kinematics. Instead of CAD and forward-kinematics equations, a neural field maps (3-D query point, joint angles) → occupancy and visibility; the first two joints are handled analytically as rotations (virtual coordinates), the last two by the network. Self-supervised: the only labels are the robot's own silhouettes.
Morphology predicted in simulation and reality for two arms; end-effector isolated; spiral tracked by gradient-descent IK; collision-free RRT paths executed; after a bent link the re-learned model restores task performance. Baselines (random search RS, nearest-neighbour NN) are 2.5–7× worse in silhouette MSE.
"Robots usually need a hand-built simulator. Here a 4-DOF arm builds its own by watching itself with one camera. It babbles, segments its silhouette, and trains a small neural field that says for any 3-D point and joint angles whether the point is occupied and visible. Rendering sums density × visibility along each pixel ray and is compared with the silhouette by MSE. With 12,000 images the model predicts the robot's shape in simulation and reality, isolates the end effector, tracks a spiral by gradient-descent inverse kinematics, plans collision-free RRT paths, and recovers from a bent link. The weak point is that everything is judged by 2-D silhouettes from one view."
α learning rate, γ discount, R reward; A public random matrix, x secret, e small noise, q modulus. The noise e turns an easy linear system (Gaussian elimination) into a hard lattice problem: that is the LWE assumption.
| state | block | counter-attack | learned action |
|---|---|---|---|
| no attack | – | – | monitor |
| mild attack | 10 | 5 | block |
| severe attack | 5 | 15 | counter-attack |
States {no, mild, severe}, actions {block, counter}: a 3×2 table, so the "learned" policy is essentially the hand-set rewards. Response effectiveness score ranks actions.
Signature-based defences miss zero-day and disguised attacks, so detect surprise: train on normal traffic, flag large prediction errors. Integrity by keyed hashing (an HMAC in spirit). Adaptive response by tabular reinforcement learning. Confidentiality that survives quantum computers via lattices (LWE): keys are matrices, ciphertexts expand, arithmetic mod q is heavier than AES, hence precomputed keys and offloading to gateways.
Qualitative only: the LSTM flags injected anomalies in a simulated network, tampered packets fail hash verification, the Q-table converges to block-for-mild and counter-for-severe, and LWE encryption is described as deployable with precomputation. No accuracy, false-positive rate, latency or energy figures.
"IoT devices are numerous and weak, and signature defences miss new attacks. The paper combines four parts: an LSTM trained on normal traffic flags prediction errors above the 95th percentile; a salted SHA-256 detects tampering; Q-learning over three states learns to block mild and counter severe attacks; and LWE lattice encryption gives post-quantum confidentiality. Heavy parts run on gateways, light hashing on devices. The design is sensible, but the evaluation is qualitative on 150,000 simulated events, the hashing is wrongly called homomorphic, and no accuracy, latency or key-size numbers are given."