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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.

In a balanced category with pullbacks in a category, pull a monomorphism back across the bottom map of a lifting square whose left map is an epimorphism. The resulting monic projection is epic because its composite with the lifted top map is epic. Balancedness makes that projection invertible, producing the unique lift. Conversely, if all monomorphisms are strong monomorphisms, applying the lifting property of a bimorphism to its own square gives its inverse, so the category is balanced. Pullback stability of epimorphisms is not needed.

Under true density $q$, the advantage of truthful reporting is the Kullback-Leibler divergence $D_{\mathrm{KL}}(q\Vert p)$. Its nonnegativity and equality condition prove strict propriety when the expected log scores are well defined.

A strictly increasing function from a closed real interval onto the interval between its endpoint values is continuous. Given a desired output tolerance, choose nearby output values on either side and use their preimages to trap nearby inputs. At the endpoints only one side is needed. Equivalently, a discontinuity of a monotone function creates a gap in its range, which is incompatible with surjectivity onto an interval.

With one A and one B observation per stratum and no ties, the stratified log-rank statistic receives a nonzero contribution only when the earlier observation is a failure while both remain at risk. It is $+1/2$ if A fails first and $-1/2$ if B fails first. After the first observation, only one member remains and every further observed-minus-expected contribution is zero. Under an exchangeable null with independent censoring, conditional on being informative the two signs have equal probability, with mean zero and variance $1/4$. Thus standardization is the normal approximation to a sign test on informative pairs. Failure-time distances do not affect this statistic.

For a finite stopping time $T$, use the shifted filtration ${\mathcal{G}}_s={\mathcal{F}}_{T+s}$. An martingale with a bounded $L^2$ norm shifts to the martingale $M_{T+s}-M_T$; a continuous bounded adapted integrand shifts to a predictable integrand. Their quadratic variation and integration intervals shift by subtracting the value at $T$. Both integrals have identical left-endpoint sums, and the Itô isometry passes the equality to the limit. The starting jump at $T$ is excluded.

Two scalar SDEs driven by the same Brownian motion, with the same globally Lipschitz diffusion coefficient and ordered globally Lipschitz drifts, preserve a strict initial order at every finite time. If $Z=Y-X$ starts positive, apply a reciprocal barrier to the first hit of zero. A Gronwall bound on $(Z+ϵ{)}^{-1}$ makes the hitting probability at most $ϵe^{(L+K^2)t}/(Z_0+ϵ)$, which tends to zero. Common noise and the Lipschitz bound on the diffusion difference are essential to this argument.

In Laplace method for the Gamma function, scaling $t=x(1+s/\sqrt{x})$ gives a standard Gaussian integral. The first even correction polynomial is $s^2-7s^4/12+s^6/18$. Its expectation under a unit-variance centered Gaussian is $1-7\cdot 3/12+15/18=1/12$. The amplitude expansion contributes to this coefficient as well as the phase expansion; expanding only the latter gives the wrong Stirling formula correction.

On the unit sphere, use $\zeta =(x+iy)/(1-z)$ away from the north pole and $\eta =(x-iy)/(1+z)$ away from the south pole. Their overlap relation $\eta =1/\zeta$ is a holomorphic map, giving a holomorphic atlas for the Riemann sphere. The opposite sign of $iy$ in the second manifold chart ensures a holomorphic rather than antiholomorphic transition.

For a Steiner chain between concentric circles, tangencies of neighbouring circles are midpoints of their centres and lie on the circle of radius $\sqrt{R\rho }$. Under a Möbius transformation, their locus is a generalized circle, which may be a line. An ordinary-circle conclusion requires the pole of the inverse transformation to avoid the original locus.

If a simple Steiner chain closes after $n\ge 3$ circles, rotating the starting circle between concentric boundaries gives another closed $n$-circle chain. Applying a Möbius transformation proves the corresponding statement for arbitrary boundary circles. Construction must stay in the same intervening family and proceed without reversing the chosen direction. Merely choosing arbitrary circles tangent to the preceding circle permits backtracking and does not assert closure.

For any nonzero limit ordinal $\delta$, a club set is an unbounded subset closed under limit points below $\delta$. A subset is stationary if it meets every such club set. Statements concerning the completeness of the club filter or Fodor lemma require regular uncountable height; they should not be transferred to arbitrary singular or countable limit ordinals without checking their hypotheses.

A subset $X\subseteq \lambda$ is non-reflecting in the indicated sense if it is not stationary below any $\delta <\lambda$ of uncountable cofinality. The subset itself need not be stationary. Nonreflection is inherited by subsets. With this definition every subset of ${\omega }_1$ is non-reflecting, since all smaller ordinals have countable cofinality.

A local martingale deflator only forces deflated asset prices to be local martingales. True expectation pricing requires additional martingale and integrability properties. In a Brownian one-factor market, continuity of the coefficients ensures a pathwise finite market-price-of-risk square integral, but not expectation-one of its stochastic exponential. The reciprocal three-dimensional Bessel strict local martingale gives a counterexample to promoting this local conclusion automatically.