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Normalized hypergraph extremal numbers decrease under vertex sampling and converge to Turán density. At a fixed sample size where the extremal density is close to that limit, excess density forces a positive fraction of samples to contain $H$. Counting incidences of copies and samples proves the displayed lower bound. Shrinking $\delta$ also handles the finitely many small orders when the required count is rounded down.

For the function field $K$ of a curve over a finite full constant field $k$, extend constants to $\overline{k}$. The resulting curve function field is $C_1$ by the Tsen theorem, so every central simple algebra splits there. The finite collection of coefficients of a splitting isomorphism descends to some finite constant extension $k^{\prime }/k$. Regularity makes $K{k}^{\prime }/K$ cyclic, giving a finite cyclic splitting field.

To draw $N(\mu ,1)$ restricted to a positive value, let $a=\Phi (-\mu )$ and return $\mu +{\Phi }^{-1}(a+(1-a)U)$ for uniform $U\in (0,1)$. For restriction to a negative value use $\mu +{\Phi }^{-1}(aU)$. These are inverse transform sampling constructions. Repeatedly drawing an untruncated normal until its sign agrees is also exact, but inefficient for rare signs.

For the triangular current with capillary retention, the initial trailing face is already the maximum for $0<x<L$. Further upslope, the maximum occurs when the moving crest passes. HenceThe final trapped phase has saturation $s$ below this envelope. Its volume is $\varphi s\int M_{\infty }dx=\varphi a{L}^2$, exactly the initial mobile volume. Maximum invaded thickness is distinct from residual saturation: $s$ multiplies occupied pore volume rather than shrinking the geometrical envelope.

The Maurer-Cartan form of the translation-dilation group of the plane has these coefficients in its dilation–translation basis, with $a=1,2$. Left multiplication rescales each coordinate differential and $\rho$ equally, proving they are left-invariant differential forms. Their dual left-invariant vector fields are $\rho {\partial }_{\rho }$ and $\rho {\partial }_{x^a}$. The Maurer-Cartan equation is $d{\lambda }^0=0$, $d{\lambda }^a=-{\lambda }^0\wedge {\lambda }^a$. The tensor ${\sum }_j{\lambda }^j\otimes {\lambda }^j$ is a positive definite left-invariant metric.

For the local flow of a smooth compactly supported vector field and any scalar BV space function $w$, the change of variables formula and polar decomposition of a vector measure giveThe two cofactor matrices are $I\pm tB+O(t^2)$ uniformly. On $|{\sigma }_w|=1$, the linear terms in their Euclidean norm expansions cancel. Integration bounds the remainder by $C{t}^2|Dw|(\Omega )$. The identity includes all components of the decomposition of a BV derivative; it does not require smoothness of $w$ or its jump interfaces.

For exponential distribution claims of mean $\mu$ and compound Poisson distribution count parameter $\lambda$, the total party variance under excess of loss reinsurance is $g(M)=2\lambda {\mu }^2(1-(M/\mu )e^{-M/\mu })$. Its derivative is $2\lambda e^{-M/\mu }(M-\mu )$, so $M=\mu$ is the unique global minimum, with value $2\lambda {\mu }^2(1-e^{-1})$.

For càdlàg processes on a finite horizon, compact containment together with uniformly small increments over pairs of bounded stopping times whose separation tends to zero is a sufficient criterion for tightness in the Skorokhod J1 topology. The increment condition is in probability, not merely at deterministic times. If in addition the largest jump tends to zero in probability, every limiting law is supported on continuous paths.

For a film with fixed lower temperature $T_0$ and upper condition $-\kappa T_z=\alpha (T_s-T_0+\Delta T)$, negligible horizontal thermal diffusion and thermal advection reduce the heat equation to $T_{zz}=0$. Thus $T=T_0-\alpha \Delta Tz/(\kappa +\alpha h)$. The local approximation requires $(h/L{)}^2\ll 1$ and $U{h}^2/(\kappa L)\ll 1$, together with slow time variation relative to $h^2/\kappa$. For $\alpha h/\kappa \ll 1$, $T_s\simeq T_0-\Delta T\alpha h/\kappa$. The coefficient $\kappa$ is thermal diffusivity, and the boundary coefficient $\alpha$ is normalized accordingly.

Positive steady solutions of the balanced thermocapillary thin-film equation with zero volume flux satisfy $H_{XX}+\ln H=0$ after choosing the thickness scale. Their first integral has potential $V(H)=H(\ln H-1)$, with minimum $V(1)=-1$ and limiting value $V(0^{+})=0$. At $E=-1$ the film is uniform; for $-1<E<0$ it is positive and periodic. At $E=0$ a limiting drop has peak $e$, half-width $\sqrt{\pi e}$, zero limiting contact angle and divergent curvature at zero thickness. Small periodic profiles have period $2\pi$ and approach the neutral, rather than fastest-growing, normal mode of the thermocapillary film instability.

For a nonnegative terminal wealth floor whose state-price density cost is strictly below the available budget, pointwise maximization of $U(y)-\lambda {\zeta }_{T}y$ over $y\ge \xi$ gives the displayed payoff. The positive multiplier exhausts the state-price budget constraint. This requires the usual utility and integrability hypotheses ensuring that the multiplier exists and the payoff is replicable.

A free acoustic membrane wave is a zero of the outgoing dispersion relation $D$. At this zero, the impedance seen from one side equals the negative of that side's outgoing normal acoustic impedance. Thus the continued by analytic continuation reflection coefficient has a pole. A real guided mode is evanescent normal to the sheet; a complex root can describe a leaky mode. The pole describes nontrivial homogeneous motion with no incoming wave, rather than unbounded passive reflection at an ordinary real incidence angle.

For a prime $p$ and $n\ge 1$, the unique residue $y$ modulo $p^n$ satisfying $y\equiv x(\mathrm{mod}p)$ and $y^p\equiv y(\mathrm{mod}{p}^n)$ is the displayed residue. Existence follows from Fermat's little theorem and prime-power contraction under pth powers. Uniqueness follows by raising any other such residue repeatedly to the $p$th power and applying the same contraction to its congruence with $x$. This is the finite-modulus version of a Teichmuller representative, including the zero class.

A small transverse Taylor swimming sheet in a Brinkman medium obeys $({\nabla }^2-A^2){\nabla }^2\psi =0$ with $A=\alpha /k$. Its no-slip material velocity is imposed on the moving sheet. The matrix's preferred frame and the mean force-free condition must be included when finding the second-order swimming speed.

The top-degree differential form ${\omega }^n/n!$ on a $2n$-dimensional symplectic manifold. Every canonical transformation preserves this form. For $n>1$, preservation of this volume alone does not imply preservation of the symplectic form.

In row-block order $(W^{\prime },U,W)$ with dual bases, a unit-diagonal symplectic flag stabilizer satisfies $E=J_U{D}^T$ and $F-F^T=D{J}_U{D}^T$. The matrix $D$ is arbitrary. The latter right side is alternating, so each off-diagonal pair of entries of $F$ gives one free choice and each diagonal entry is free. ConsequentlyReversed dual-basis order inserts a reversal matrix but does not change the count. The argument holds in characteristic two without division by two.

An unbroken gauge symmetry forbids an explicit local Proca mass for its gauge vector. A Higgs mechanism can generate that mass through a charged vacuum expectation value, but the stability of the resulting vector scale then depends on the scalar sector that fixes the expectation. Gauge invariance alone does not protect a fundamental scalar mass.

A fermion mass that violates an otherwise restored chiral symmetry is multiplicatively rather than additively renormalized in perturbation theory: corrections vanish with the symmetry-breaking mass or coupling. A chiral gauge representation can forbid a bare mass altogether until a Higgs mechanism and Yukawa coupling make a gauge-invariant term possible. A scalar-partner bilinear may respect all those ordinary symmetries and therefore needs separate supersymmetry protection.

Over a field of characteristic zero, degree-$n$ symmetric tensors are spanned by pure powers $a^{\otimes n}$. Inclusion-exclusion expands ${\sum }_{J\subseteq [n]}(-1{)}^{n-|J|}({\sum }_{j\in J}{a}_j{)}^{\otimes n}$ into the sum of all permutation orderings of $a_1\otimes \cdots \otimes a_n$. Every surviving term uses each of the $n$ labels once. This converts an invariant-tensor spanning problem into pure tensor powers.

The continuous gain map takes a probability simplex to itself. The Brouwer fixed-point theorem supplies a fixed point, whose positive gains would contradict the zero strategy-weighted average payoff excess. Thus all gains vanish, giving a strategy that is a best response to itself. In a symmetric bimatrix game, this produces a symmetric equilibrium.