g = cost so far from the start. h = estimated cost to the goal (here Manhattan distance). w = 0 gives Dijkstra, w = 1 gives A*.

h never overestimates the true remaining cost. Manhattan distance is admissible on a 4-connected grid with unit steps, so A* returns a shortest path.

With w > 1 the search trusts h more, expands fewer cells, and finds a path at most w times the optimal cost (bounded suboptimal).

Each cell is expanded at most once; with a binary heap the work is O(E log V). A good heuristic does not change the worst case, but it shrinks the explored area a lot in practice.

Each node has at most m children and m − 1 keys. Here m = 4. Every node except the root has at least ⌈m/2⌉ − 1 keys (here 1).

An overflowing node (m keys) splits into two halves; the median key moves up into the parent. If the root splits, a new root appears: the tree grows from the top.

With 200 keys per page, a billion rows need about 4 page reads.

This is how a relational database finds a customer by ID without scanning the table. Index design (which columns get a B-tree) is a core database-administration decision.

Two scalar equations, two unknowns (θ₃, θ₄). Geometrically: B is where the circle of radius b around A meets the circle of radius c around O₄. Two intersections = two assembly branches (open and crossed).

s = shortest, l = longest link. If true, at least one link turns fully. Which one depends on where the shortest link sits: as the crank it gives a crank-rocker, as the ground a drag link, as the coupler a double rocker.

The angle between coupler and rocker. Near 0° or 180° the coupler pushes almost along the rocker and force transmission is poor; designers keep μ roughly between 40° and 140°.

Points on the coupler trace closed curves of surprising shapes. The Hoekens proportions (2 : 1 : 2.5 : 2.5, point at 5) give a near-straight segment, used in walking machines.

Solved here numerically with 4th-order Runge–Kutta (step 2 ms).

ζ < 1 underdamped (rings), ζ = 1 critically damped (fastest return, no overshoot), ζ > 1 overdamped.

Peaks near r = 1. With light damping the peak is about 1/(2ζ).

Below resonance the mass follows the force; at r = 1 it lags 90°; far above it moves opposite to the force.

D = annual demand, S = cost per order, H = holding cost per unit per year. At EOQ, yearly ordering cost equals yearly holding cost.

Expected demand during the lead time plus safety stock. z sets the cycle service level: the chance a replenishment cycle ends without a stockout.

With daily review, inventory drops past s before you notice, so the effective safety stock is smaller. The formula above assumes continuous review.

Fill rate counts units served from stock; cycle service counts cycles without any shortage. A 95% cycle service usually means a fill rate above 99%.

Lee, Padmanabhan & Whang (1997): demand-signal processing (forecasting from orders, shown here), order batching, rationing and shortage gaming, and price fluctuations.

L = lead time + one review period, N = weeks in the moving average. Longer lead times or shorter memory amplify more.

Without shared data the factors multiply; with shared data the lead times add. Sharing reduces the effect but does not remove it.

Point-of-sale data sharing, EDI, vendor-managed inventory (VMI), collaborative planning (CPFR), and shorter lead times. These are information-system decisions with direct cost effects.