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.

In a nondegenerate symmetric bimatrix game, player interchange pairs every nonsymmetric Nash equilibrium with a distinct swapped equilibrium. The equilibrium oddness theorem gives an odd total; the remaining fixed points, which are the symmetric equilibria, therefore also number an odd amount.

For a mixture distribution of exponential distributions, introduce one convolution state $A_j(u)={\int }_0^u\phi (z)e^{-{\beta }_j(u-z)}dz$ for each rate ${\beta }_j$. It satisfies $A_j^{\prime }=\phi -{\beta }_j{A}_j$. Together with the survival integro-differential equation these form a constant-coefficient first-order system; applying the differential operators $D+{\beta }_j$ eliminates the convolution states. For equal mixing weights at rates one and one half, with $r=\lambda /c$, the result is ${\phi }^{\prime \prime \prime }+(3/2-r){\phi }^{\prime \prime }+(1/2-3r/4){\phi }^{\prime }=0$. The original integral equation supplies initial conditions lost in elimination.

For one canonical chiral field at $W^{\prime }(v)=0$, the quadratic scalar potential is $|W^{\prime \prime }(v){|}^2|\delta \phi {|}^2$. Writing $\delta \phi =(a+ib)/\sqrt{2}$ gives two canonically normalized real scalars with squared mass $|W^{\prime \prime }(v){|}^2$. The chiral-superfield fermion mass matrix has the same physical mass. A complex bilinear coefficient or its algebraic square is not itself a signed physical squared mass.

In a sheaf topos on a topological space, subterminal sheaves correspond to open sets $U$. Their sections over $V$ are a singleton if $V\subseteq U$ and empty otherwise. An open cover gives a jointly epimorphic family of these subobjects. The open cover criterion for a local sheaf topos follows by applying a colimit-preserving global-sections functor to that family.

If the real error depends only on the second index and ${\alpha }_n/n\to 0$, the normalized sequence has a limit in $[-\infty ,\infty )$. For fixed $k$, write $n=qk+r$ with $1\le r\le k$ and iterate to obtain $x_n\le x_r+q(x_k+{\alpha }_k)$. It follows that $\mathrm{limsup}{x}_n/n\le (x_k+{\alpha }_k)/k$ for every $k$; a subsequence attaining the lower limit proves convergence. The limit is ${\inf }_k(x_k+{\alpha }_k)/k$, even when the errors can be negative.

A triad is the set of triples supported on one prescribed pair cell for each of the three pairs of vertex classes. Relative triple density is measured among those supported triples. In strong hypergraph regularity, the pair cells must themselves be sufficiently regular and have controlled density before a small relative three-dimensional box norm can give a tetrahedron counting lemma.

A morphism $m:A\to B$ is strict when every $g:C\to B$ satisfying $hm=km\Rightarrow hg=kg$ for all parallel arrows $h,k$ out of $B$ factors uniquely through $m$. Every regular monomorphism is strict by its equalizer property. A strict monomorphism is strong: in a lifting square with an epimorphism, cancellation of the epimorphism makes the bottom arrow admissible, and its unique factorization gives the diagonal.