For a thin illuminated surface $z=z_s(R)$ and a central point source, the radial light ray has slope $z_s/R$ whereas the surface tangent has slope $z_s^{\prime }$. Their angle difference is the grazing angle. The incident flux is $L_{\ast }\alpha /(4\pi R^2)$ to leading order. This requires a positive grazing angle and an unobstructed ray; a negative formal flux instead signals failure of direct illumination, not negative heating.

With $b_0=\beta b_{\mathrm{in}}-\alpha b_1$ and $b_2=\alpha b_{\mathrm{in}}+\beta b_1$, real ${\alpha }^2+{\beta }^2=1$ and $|\alpha |<1$, eliminate the internal fields to obtain $\dot{a}=-{\gamma }_{\mathrm{eff}}a/2-c{b}_{\mathrm{in}}$, where $c=\beta \sqrt{\gamma }/(1+\alpha )$ and $c^2={\gamma }_{\mathrm{eff}}$. The internal output transfer is $b_1/b_{\mathrm{in}}=\beta G/(1+\alpha G)$; the external output transfer is $b_2/b_{\mathrm{in}}=(\alpha +G)/(1+\alpha G)=(s-{\gamma }_{\mathrm{eff}}/2)/(s+{\gamma }_{\mathrm{eff}}/2)$. The loop is asymptotically stable in this nondegenerate range. At $\alpha =1$ the cavity mode is undamped and decoupled; at $\alpha =-1$ the instantaneous algebraic feedback connection is singular.

The Young lattice is a partially ordered set whose elements are integer partitions, including the empty partition. Each partition of an integer is identified with its diagram, and the order is inclusion of Young diagrams. A cover adds exactly one box. A saturated path from the empty diagram to $\lambda$ is equivalent to a standard Young tableau of shape $\lambda$.

Fix a prime $p$. Replace each part $p^{a}m$, with $p\nmid m$, by $p^a$ copies of $m$. This is a size-preserving bijection from regular partitions to partitions of an integer whose parts are not divisible by $p$. To invert it, write each multiplicity of a part $m$ not divisible by $p$ in base $p$ and use its digit in position $a$ as the multiplicity of $p^{a}m$. Every resulting multiplicity is below $p$. The bijection explains why the number of simple symmetric-group modules equals the number of p-regular elements' conjugacy classes.

Pad partitions with trailing zeros. In the descending-dictionary convention used for symmetric-function reverse lexicographic ordering, the partition with the larger first differing part precedes the other. This is the reverse of increasing dictionary order on integer partitions, and is a total order. It is distinct from comparing the last differing part in a monomial reverse-lexicographic order.

A colouring assigns to each member of a set $X$ a colour in another set $C$. Its nonempty fibres form a partition of a set. The colour set may be infinite; a finite colouring is the special case with finitely many colours. A homogeneous set for a colouring of finite subsets has all its subsets of the specified size assigned one colour.

Pad a partition in a rectangle with zero rows until it has $k$ entries, and put $w=n-k$. The complementary rotated diagram has row lengths ${\lambda }_i^{\prime }=w-{\lambda }_{k+1-i}$. This is again a partition in a rectangle, and its size is $kw-|\lambda |$. Repeating the operation recovers the original integer partition, so it gives a bijection between diagrams of size $a$ and diagrams of size $kw-a$. Consequently the coefficients of the Gaussian polynomial read the same in either direction. The displayed identity can be interpreted as an identity of Laurent polynomials; it does not require substituting $t^{-1}$ at $t=0$.

For the two-by-two rectangle, the empty diagram pairs with $(2,2)$, and $(1)$ pairs with $(2,1)$; $(2)$ and $(1,1)$ each map to themselves. This establishes symmetry, while unimodality of Gaussian binomial coefficients is a stronger statement about the growth and decrease of successive coefficients.

Indistinguishable particles have the same intrinsic particle properties, so exchanging their particle labels does not identify a different observable arrangement. Two electrons are an example; an electron and a proton belong to different species. Labels used as slots in a wavefunction are mathematical coordinates, not permanent identity tags.

For ordinary identical particles in three spatial dimensions, the symmetrization postulate takes the total quantum state to be symmetric for bosons or antisymmetric for fermions. In the antisymmetric case, exchanging all one-particle variables changes the sign of the wavefunction while leaving detection probabilities unchanged. The symmetry includes spin and other internal coordinates, not just position.

Spatial modes can nevertheless be addressed separately. Measuring the spin in one localized mode differs from measuring an imagined permanently labelled electron. Conditional spin measurement of identical fermions makes that distinction explicit. In two spatial dimensions, anyons allow more general exchange behavior; the boson/fermion alternative is not an unrestricted statement about every dimension.

A linear extension is a total ordering of the elements of a partially ordered set that preserves all its comparisons. For a finite set it can be recorded as a list. If two adjacent elements in that list are incomparable, their interchange is again a linear extension.

A filter is a nonempty upward-closed subset $G$ of a partial order such that every two members have a common lower bound in $G$. With standard notation for forcing, lower means stronger, so these are exactly the directed filters used to define a generic filter. The projection of a product-generic filter uses these properties before genericity of its factor filters is proved.

If $\mathrm{Ch}(P)$ is the inclusion-ordered collection of every chain in a partial order, including the empty chain, it has no order-preserving injection into $P$. Supposing $j$ exists, recursively set $p_{\alpha }=j(\{p_{\beta }:\beta <\alpha \})$ for $\alpha <h(P)$. By induction the earlier values form a chain; its proper initial chains are distinct, so monotonicity and injectivity make $p_{\alpha }$ strictly greater than every earlier value. This would inject the Hartogs ordinal into $P$.

A strict partial order is an irreflexive transitive relation. In a classical set, it corresponds to a partial order through reflexification, but recovering the strict relation from the weak one requires inequality. In an arbitrary topos, this reverse operation need not be geometric or define an inverse interpretation.

The product order on a Cartesian product of partially ordered sets compares every coordinate separately. Points that improve one coordinate and worsen another are incomparable. It is generally a partial order, not a total order.

A family of order-completeness notions used in semantics: in particular, a directed-complete partial order has every directed join, while a pointed one additionally has a least element. The precise completeness condition should be specified, since this terminology is not synonymous with complete lattice.

A chain-complete partially ordered set has a supremum for every chain, with the empty-chain condition requiring a least element if empty chains are included. This implies the weaker upper-bound hypothesis of Zorn lemma, which requires only that each chain have some upper bound, not necessarily a least one. Some treatments use “chain-complete” for that weaker condition, so the convention should be specified. In the inclusion order on lattice ideals avoiding a specified element, the union of a nonempty chain is an ideal and is its supremum. This union observation is the step needed to apply Zorn in prime-ideal separation arguments.

A lower bound of a subset $S$ of a partially ordered set is an element $\ell$ no greater than every member of $S$. For minimization, a number below every feasible objective value is therefore a lower bound on the optimum; a relaxation often provides such a certificate.

// Target: mathematical-optimization.bigb

Every finite partially ordered set can be partitioned into as many chains in a partial order as its largest antichain. Reachability in an acyclic directed graph is a partial order, so the theorem supplies chains which can be expanded into directed paths, possibly overlapping at intervening vertices.

Every partial order has a total order extension. Apply Zorn's lemma to extensions ordered by inclusion: the union of a chain is a partial order. If a maximal extension leaves two elements incomparable, adjoining one comparison and taking its transitive closure of a relation creates no cycle, contradicting maximality. Quotienting a preorder by its indifference relation and lifting a total order gives a complete weak-order extension preserving its strict part.

A real-valued function on a partially ordered set is increasing when it preserves its order. Indicators of increasing events are examples. In spin or edge configuration spaces the order is coordinatewise, so increasing observables grow when a zero coordinate is replaced by one. Positive association of random variables concerns pairs of such observables, and stochastic domination compares their expectations.

An upper bound of a subset is an element above every member, and a lower bound is below every member. They need not lie in the subset or be unique. Infimum and supremum are the greatest lower and least upper bounds when those exist.