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For a connected capacitated minimum-cost flow network, a spanning tree supplies basic flow variables. Every non-tree edge is fixed at its lower or upper bound. Flow balance determines the tree flows; these must also obey capacity constraints. An entering edge creates one cycle, and adjusting flow on that cycle preserves balance until a bound is hit and an edge leaves the tree.

A subnet is obtained from a net by an order-preserving cofinal map from another directed index set. It preserves any limit of the original net, and eventual properties still hold along it. Every net in a compact set has a convergent subnet. This supplies compactness arguments when sequential compactness is unavailable.

A small mass is technically protected when its vanishing restores a symmetry that restricts the corresponding quantum corrections. Gauge protection of a vector mass, chiral protection of a fermion mass, and supersymmetry protection of scalar masses use different symmetry constraints. This is a radiative-stability statement, not a proof that all small numerical parameters require such protection.

A finite nondegenerate bimatrix game has a finite odd number of Nash equilibria. After shifting to positive payoffs, fully labelled vertex pairs in the bounded best-response polytopes correspond to equilibria plus one artificial zero pair. Dropping a fixed label gives a finite path graph whose endpoints are exactly these pairs. The number of endpoints is even, leaving an odd number of genuine equilibria. This is the endpoint-parity argument underlying the Lemke-Howson algorithm.

The product of the two players' gains over the disagreement point is maximized over the individually rational feasible set in the Nash bargaining solution. Positive affine changes of utility multiply this product by a positive constant and leave the selected payoff correspondence unchanged after transforming coordinates.

Normalize a positive Nash product maximizer and the disagreement point to $(1,1)$ and zero. The first-order product inequality along every feasible segment places the transformed set in $z_1+z_2\le 2$. Compactness allows a symmetric containing triangle with finite lower coordinate bounds. Bargaining symmetry and Pareto efficiency choose $(1,1)$ in that triangle; bargaining independence of irrelevant alternatives transfers this choice to the original normalized set. Positive affine invariance in bargaining then proves the full characterization.

With Fourier convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$, the Transverse-field Ising model has $H=J{\sum }_k{\Psi }_k^{†}[(g-\cos k){\sigma }_z-\sin k{\sigma }_x]{\Psi }_k$. Expanding the spinor retains the constant $-JgN$ and pairs $k$ with $-k$. The physical quasiparticle energy is twice the positive matrix eigenvalue times $J$, because both momentum partners occur in the Nambu sum.

In a bosonic Nambu spinor expression, the second diagonal entry commonly contains $a_{-k}{a}_{-k}^{†}$. The canonical commutation relations add one constant per mode when this is normal ordered. For an antiferromagnetic chain coefficient $JS$, rewriting the quadratic Hamiltonian as a full Nambu sum therefore changes the exterior constant from $-NJ{S}^2$ to $-NJS(S+1)$. Omitting the shift corrupts the ground-state energy even when the dispersion is unchanged.

In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

For an affine variety with coordinate ring $A$ acted on by a finite group $G$, the affine quotient is $\mathrm{Spec}{A}^G$. Elements of $A$ satisfy monic orbit polynomials over $A^G$, so the quotient map is a finite morphism. Over an algebraically closed field, invariant functions separate finite orbits; in characteristic not dividing $|G|$, interpolate and average a function separating two orbits. The quotient fibres are consequently the orbits.

The Engquist-Osher method is the finite volume method using the Engquist-Osher flux. Under $(k/d)\sup |f\prime |\le 1$ on the invariant state interval, its update is monotone. It preserves the state range, is contractive for summable perturbations in $L^1$, and decreases discrete total variation.

Let $T$ be a monotone conservative scalar update on a periodic grid. Since $T(U\vee V)\ge TU,TV$, one has $(TU-TV{)}_{+}\le T(U\vee V)-TV$. Conservation makes the sum of the right side equal to $\sum (U-V{)}_{+}$. Reverse $U,V$ and add to obtain $\Vert TU-TV{\Vert }_1\le \Vert U-V{\Vert }_1$. On an infinite grid the same proof holds for summable differences with a Lipschitz flux on the state range.

A translation-invariant conservative monotone update decreases discrete total variation. If $S$ is the one-cell shift, $TS=ST$, and L1 contraction of a monotone conservative scheme applied to $U,SU$ gives ${\sum }_j|(TU{)}_{j+1}-(TU{)}_j|\le {\sum }_j|U_{j+1}-U_j|$. This prevents creation of new total variation, though it does not make every discontinuity sharply resolved.

For $V=C_p(\Phi /M_{\mathrm{Pl}}{)}^p$ and the leading endpoint estimate ${ϵ}_V=1$, the slow-roll approximation gives ${\Phi }_N^2/M_{\mathrm{Pl}}^2=2pN+p^2/2$ and ${ϵ}_V=p/(4N+p)$. Thus ${\mathcal{P}}_{\mathcal{R}}=C_p(2pN+p^2/2{)}^{p/2}(4N+p)/(24{\pi }^2{M}_{\mathrm{Pl}}^{4}p)$ and the scalar spectral index obeys $n_s-1=-2(p+2)/(4N+p)$. The coefficient sets the amplitude, whereas the monomial power and exit e-fold count set the leading tilt. Changing the power without specifying the coefficient leaves the overall normalization undetermined.

For any commutative ring $A$ and a monic polynomial $F$ of positive degree $D$, monic polynomial division over a ring gives unique representatives of degree less than $D$. Thus the classes of $1,T,\dots ,T^{D-1}$ form a basis of a module over $A$. This is a finite free algebra, hence flat, and the map $A\to A[T]/(F)$ is injective. The monic hypothesis makes leading-degree arguments valid even when $A$ has zero divisors.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

If $X\ge 0$ and $\mathbb{E}{e}^{tX}<\infty$ for some finite $t\ge 0$, the estimator ${\hat{m}}_n(s)=n^{-1}{\sum }_i{e}^{s{X}_i}$ satisfies ${\sup }_{s\in [0,t]}|{\hat{m}}_n(s)-\mathbb{E}{e}^{sX}|\to 0$ almost surely. The dominated convergence theorem makes the population moment-generating function uniformly continuous on this interval. Parameter subintervals give function brackets whose endpoints are ordered exponentials and whose widths are the corresponding increments of the population mean. Apply the uniform strong law from finite L1 bracketing. Continuity in the parameter makes the supremum measurable.

Suppose a positive function $M$ satisfies the displayed fourth-power identity and $\log M(h)=h^2/2+o(h^2)$ at zero. Iteration gives $\log M(t)=4^m\log M(t/2^m)$, which tends to $t^2/2$. A related first-power symmetry step is: if $\psi (t)=\psi (t/2{)}^2$ and $\psi (h)=1+o(h^2)$, then $\log \psi (t)=2^m\log \psi (t/2^m)\to 0$. These two different dyadic exponents must not be interchanged in the Gaussian characterization by independent sum and difference.

Two disjoint ordinary circle boundaries can be taken to distinct concentric circles by a Möbius transformation. After their centres are put at $0,d$ with radii $a,b$, the map $(z-s)/(z-t)$ works when $st=a^2$ and $s+t=(d^2+a^2-b^2)/d$. Disjointness makes the quadratic have distinct real roots. Already concentric circles require no transformation.

The special linear group $S{L}_2(\mathbb{R})$ acting by Möbius transformations has exactly three group orbits on the Riemann sphere: the extended real line and the two open half-planes. The identity $\mathrm{Im}((az+b)/(cz+d))=\mathrm{Im}z/|cz+d{|}^2$ preserves these sets. Upper triangular determinant-one matrices send $i$ to any prescribed point of the complex upper half-plane, and similarly send $-i$ through the lower half-plane. The stabilizer subgroups of $i$ and $-i$ are both $SO(2)$.