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A forcing order is $\kappa$-closed when every decreasing sequence of conditions of length below $\kappa$ has a lower bound. Countably closed forcing adds no new countable sequences of ground-model elements and in particular adds no new real numbers.

A forcing order has the $\kappa$-chain condition when every antichain has cardinality below $\kappa$. It preserves cardinals and cofinalities at least $\kappa$.

The order $\mathrm{Fn}(X,Y,\kappa )$ consists of partial functions $p:X\rightharpoonup Y$ with $|\mathrm{dom}p|<\kappa$, ordered by reverse inclusion so that larger functions are stronger conditions.

An eventually different condition is a finite sequence together with finitely many ground-model functions that all newly appended coordinates must avoid. The generic union eventually differs from every ground-model function.

Cohen forcing consists of finite partial approximations to a new subset or function, ordered by reverse inclusion. Meeting the ground-model dense sets makes the union a total generic object.

A countable transitive model is a countable set $M$ such that membership on $M$ is the actual membership relation, $M$ is transitive, and $(M,\in )$ satisfies the specified axioms of set theory.

A countable transitive model is in particular a standard model (set theory) with both countability and transitivity.

A filter $G\subseteq \mathbb{P}$ is generic over a model $M$ when it meets every dense subset of $\mathbb{P}$ belonging to $M$.

A subset $D$ of a forcing order $P$ is dense if every $p\in P$ has a stronger condition $q\le p$ in $D$. This order-density condition is the one used by a generic filter; it is distinct from density in an arbitrarily specified topology.

Two forcing conditions are compatible when they have a common stronger extension. In standard notation this means that some $r$ satisfies $r\le p,q$; in Jerusalem notation for forcing it means that some $r$ satisfies $p,q\le r$.

In Jerusalem notation, $p\le q$ says that $q$ is stronger than $p$: the larger condition carries more information.

In standard forcing notation, $q\le p$ says that $q$ is stronger than $p$: the smaller condition carries more information.

Writing a Fokker-Planck equation as ${\partial }_{t}P=-{\partial }_{x}J$ defines its probability current. For ${\partial }_{t}P={\partial }_x(gP)+\frac{1}{2}{\partial }_x^2(hP)$,

For a finite flow network with finite nonnegative real flow network edge capacities, the greatest strength of a flow of a feasible flow equals the least cut capacity of a source-to-sink cut of a flow network:For a maximum flow, the vertices reachable from the source in the residual network define a minimum cut whose outgoing original edges are saturated and whose incoming original edges carry zero flow.^[ref]

Why is a cut of a flow network an upper bound? Sum flow conservation on its source side $S$. Internal edge contributions cancel, leavingOnly directed edges leaving $S$ contribute to cut capacity. Directed edges entering $S$ are subtracted in net flow, so adding their capacities to the upper bound would lose the intended bottleneck. This inequality is weak duality; the theorem says the bound is attained, an instance of strong duality. Its beauty is that a routing plan and a separating boundary independently witness each other's optimality.

In the running flow network, the two internal vertices give four possible source sides. For $S=\{s\}$, the cut capacity is $c_{sa}+c_{sb}=3+2=5$. For $S=\{s,a\}$ it is $c_{sb}+c_{ab}+c_{at}=2+1+2=5$. For $S=\{s,b\}$ it is $c_{sa}+c_{bt}=3+3=6$; the incoming edge $a\to b$ contributes nothing to that cut capacity. Finally, $S=\{s,a,b\}$ gives $c_{at}+c_{bt}=2+3=5$. Thus the minimum cut has cut capacity $5$, matching the explicitly computed maximum flow.

Ford and Fulkerson's 1956 paper opens with a railway-capacity question attributed there to T. Harris and proves a minimal-cut theorem as Theorem 1. Its original formulation uses graph paths carrying nonnegative amounts; the residual reachability certificate for maximum flow below gives the finite directed flow network proof used here.^[ref]

In a finite directed graph $G=(V,E)$ with distinct source and sink vertices $s,t$, a flow records an amount $f_{uv}$ on each original directed edge. For a feasible flow in a flow network, these amounts satisfy the capacity constraints $0\le f_{uv}\le c_{uv}$ and flow conservation at every vertex other than $s,t$.^[ref]

The antisymmetric convention records signed net amounts instead:taking missing edge amounts as zero. The paired signs describe one net transfer, rather than two independent transfers.^[ref]

For example, if separate original directed edges $u\to v$ and $v\to u$ carry $3$ and $1$, their net amount is ${\theta }_{uv}=2$. This signed record loses the separate amounts: $f_{uv}=2$, $f_{vu}=0$ would give the same net amount. When checking capacity constraints on actual edges, keep the original amounts or specify the signed bounds carefully; when summing flow conservation, either record gives the same divergence of a flow.

For the running flow network, choose the original edge amountsEvery capacity constraint holds. At $a$, incoming flow $3$ equals outgoing flow $1+2$; at $b$, incoming flow $2+1$ equals outgoing flow $3$. This feasible flow has flow value $5$, computed at $s$ or $t$.

The total of all five edge amounts is $11$, but the flow value is $5$. A unit passing through several edges appears several times in that total. Concretely, route $2$ units along each of the directed paths $s\to a\to t$ and $s\to b\to t$, and $1$ along $s\to a\to b\to t$. Their delivered amounts sum to $2+2+1=5$, while their edge usages sum to $2\cdot 2+2\cdot 2+1\cdot 3=11$. The cut of a flow network $S=\{s\}$ certifies that this delivery rate is already a maximum flow.

Flow conservation is useful because it makes a local ledger predict a global delivery rate: intermediate vertices only redistribute the resource. An additional network circulation can change the individual edge amounts without changing the flow value. Counting net delivery rather than total edge usage prevents such recirculation from masquerading as increased output.

The Floquet multipliers solveBounded Bloch waves occur when $|\mathrm{tr}M|\le 2$, with $\mu =e^{\pm iKa}$. Band edges satisfy $\mathrm{tr}M=\pm 2$; outside the bands the multipliers are real reciprocal numbers and one solution grows exponentially.

Cone isometries preserve its angular identification; those fixing a point are the identity or an axial-plane reflection.

A developing map locally unfolds a flat surface into the Euclidean plane.

The physical tension is $f=k_{B}T\tau$. At small extension,At fixed tension the extension is $L=na\tanh (fa/(k_{B}T))$ and decreases as temperature rises.

For continuous $f:[0,1]\to [0,1]$, the function $f(x)-x$ is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.

If $h:A\to B$ is a homeomorphism and $A$ has the fixed-point property, then a continuous $f:B\to B$ induces $h^{-1}fh:A\to A$. A fixed point of the conjugate map is carried by $h$ to a fixed point of $f$.