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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]

A flow is an antisymmetric function on directed edges whose divergence of a flow vanishes away from designated sources and sinks.

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 $f$.

The Löwenheim--Skolem theorems control the cardinalities of models of a first-order theory without changing its first-order truths.

A set of first-order sentences has a model if and only if every finite subset has a model.

Every sentence derivable from a first-order theory is true in every model of that theory.

This is soundness of the first-order deduction system relative to its structure semantics.

A structure supplies a nonempty domain and an interpretation of every constant, operation, and relation symbol in its language.

This is a structure for a first-order language, with finitary interpretations of its nonlogical symbols.

A first-order theory is a set of sentences in one first-order language. Depending on convention, the term may instead mean its deductive closure.

This is a theory formulated in first-order logic.

A first-order sentence is a formula with no free variables, so its truth in a structure is independent of a variable assignment.

A sentence has no free variables; a first-order sentence is its first-order instance.