If idempotents $e_1,\dots ,e_N$ satisfy $e_i{e}_j=0$ for $i>j$, then the left ideals $A{e}_i$ have an internal direct sum. Multiply a relation $\sum z_i=0$ by $e_1$ on the right to obtain $z_1=0$, and continue in order. Pairwise orthogonality in both directions is unnecessary. If the sum has dimension $\dim A$, it is the whole algebra.

Frost heave is displacement caused by segregated ice growth supplied by liquid flow, rather than solely by the expansion of water on freezing. A premelting film allows liquid access beneath the ice; thermomolecular pressure draws supply against viscous resistance and load.

Thermomolecular pressure is the force per area transmitted through a thin premelting liquid film between a solid and a substrate. Let $d$ be film thickness. For a repulsive, nonretarded dispersion interaction, define a positive interaction amplitude $A$ byThis sign convention assigns positive $p_T$ to repulsion; a signed Hamaker constant in another convention must be translated before using this formula. The strong thickness dependence permits a thin film to support appreciable load while remaining hydraulically connected to a liquid supply.

In the equal-phase-density approximation, the Clapeyron pressure relation for equal-density phases gives $p_s-p_l\simeq \rho L(T_m-T_i)/T_m$. A molecular film force can balance this phase-pressure difference and an applied load. With $\rho \simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, $L\simeq 3.34\times 10^5\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$ and $T_m\simeq 273\mathrm{K}$, an undercooling of $0.1\mathrm{K}$ corresponds to a pressure scale about $0.12\mathrm{MPa}$. This is an equilibrium driving or limiting scale, not a guaranteed growth rate; liquid delivery through Darcy flow can limit frost heave.^[ref]

A normalized positive linear functional on $C(\Omega )$ for Cantor space gives a finitely additive set function on the algebra of clopen sets. If a countable disjoint union of such sets is itself clopen, compactness gives a finite subcover; all other terms are empty. Finite additivity therefore gives the required countable additivity, making $\ell$ a premeasure. The Caratheodory extension theorem produces a unique Borel probability measure.

For $C(K)=\mathbb{E}(S-K{)}_{+}$ and $\mathbb{E}S<\infty$, split the static call integral at a fixed positive strike. The first moment bounds the integrand near zero; polynomial call decay controls the integral at infinity for $0<\epsilon <\delta$. The endpoint can fail, as shown by a Pareto distribution with survival exponent $1+\delta$.

For $K>0$ and $s>K$, minimize $(s/K{)}^{1+\epsilon }/(s/K-1)$. Its minimum occurs at $s/K=(1+\epsilon )/\epsilon$, giving the sharp constant. Taking expected values yields a uniform bound on $K^{\epsilon }C(K)$ from a finite moment of order $1+\epsilon$.

On a bounded connected Lipschitz domain, let $u\ge 0$ belong to the BV space, and let $T$ be a positivity-preserving operator with $T1\ne 0$. Writing $u=c1+(u-c1)$, $c=u_{\Omega }\ge 0$, the triangle inequality and Poincaré inequality for total variation give $c\Vert T1{\Vert }_1\le \Vert Tu{\Vert }_1+\Vert T\Vert \Vert u-c{\Vert }_1\le \Vert Tu{\Vert }_1+C_{\Omega }\Vert T\Vert \mathrm{TV}(u)$. Thus forward-image and variation bounds control the missing constant mode, and hence the full $BV$ norm.

On ${\ell }^2$, the bounded self-adjoint operator $(Lv{)}_j=v_j/j$ is strictly positive. Yet $f_j=1/j$ lies in ${\ell }^2$ and $Lu=f$ would require the non-square-summable vector $u_j=1$. A positive uniform lower bound, rather than strict positivity alone, guarantees a bounded inverse for a bounded self-adjoint operator.

For a finite mixture with positive weights $p_i$ and rates ${\lambda }_i>0$, the survivor function is $S(t)={\sum }_i{p}_i{e}^{-{\lambda }_{i}t}$ and the hazard function is the surviving-population mean of the component rates. Write $w_i(t)=p_i{e}^{-{\lambda }_{i}t}/S(t)$. Differentiation gives $w_i^{\prime }=w_i(h-{\lambda }_i)$ and therefore $h^{\prime }=h^2-{\sum }_i{w}_i{\lambda }_i^2=-\mathrm{Var}(\Lambda \mid T>t)$. The hazard is decreasing, strictly so when distinct rates have positive weights. For finitely many rates it tends to the smallest rate with positive weight. This is survival selection even though each individual component has a constant hazard.

In a population mixture with initial component probabilities $p_i$, conditioning on survival to time $t$ changes them to the displayed probabilities by Bayes' theorem. Components with larger survivor functions receive greater relative weight among survivors. The population hazard function is the weighted mean of the component hazards with these time-dependent weights. This selection can change the population hazard even when all component hazards are constant, as shown by the hazard derivative for a mixture of exponential distributions.

For the permutation action on $n$ coordinates with $n\ge 2$, every alternating-group invariant is a symmetric polynomial plus the Vandermonde product $\Delta$ times a symmetric polynomial. Thus the invariant ring is $\mathbb{C}[e_1,\dots ,e_n,Z]/(Z^2-D(e_1,\dots ,e_n))$, where $D$ is the discriminant polynomial. The two summands are independent over the symmetric polynomial ring.

A finite group permutes the associate classes of irreducible polynomial factors of an invariant. The product of one representative from each orbit transforms by a scalar character of the group. If that character is trivial, the product is invariant. Its transitive factor orbit makes it prime in the invariant ring: an invariant polynomial divisible by one orbit factor is divisible by all of them.

If a finite group has no nontrivial homomorphism to ${\mathbb{C}}^{\times }$, its polynomial invariant ring is a unique factorization domain. Factor an invariant in the ambient polynomial ring and collect its irreducible factors into orbit products. Each orbit product transforms by a linear character, hence is invariant. It is prime in the invariant ring, and the invariant factorization is a product of these primes.

The scalar sign action of a cyclic group of order two on ${\mathbb{C}}^2$ has invariant ring $\mathbb{C}[X^2,XY,Y^2]\cong \mathbb{C}[a,b,c]/(ac-b^2)$. It is not factorial: $a,b,c$ are pairwise nonassociate irreducibles, since every nonconstant invariant has degree at least two, while $ac=b^2$ gives two distinct factorizations.

Let $\zeta =e^{\pi i/n}$ and let the binary dihedral group act on ${\mathbb{C}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{-1}y)$ and $(x,y)\mapsto (-y,x)$. For $n\ge 2$, the invariantspresent the invariant ring as $\mathbb{C}[U,V,W]/(W^2-V{U}^2+4V^{n+1})$. First take the cyclic invariants $A=x^{2n},B=y^{2n},C=xy$ with $AB=C^{2n}$. The residual involution interchanges $A,B$ and negates $C$, so invariant normal forms are polynomials in $A+B,C^2$ plus $C(A-B)$ times such polynomials. For $n=3$ the group has order twelve and the equation is $W^2-V{U}^2+4V^4=0$.

For three distinct times, the columns $1,t,t^2$ are linearly independent by the Vandermonde determinant. Thus a quadratic Poisson regression for the log mean can reproduce any three positive time-group means. A grouped goodness-of-fit test of that mean form has zero residual degrees of freedom. Replicated observations still allow checks of the Poisson distribution and overdispersion within each time group; saturation of the mean does not establish the response distribution.

A Poisson distribution of parameter $a$ conditioned to lie in $\{0,\dots ,C\}$. It is the occupancy law of a unit-resource loss network. Its mean is the carried load of an Erlang loss resource, $a[1-E(C,a)]$. Differentiating the mean with respect to $\log a$ gives its variance, which is positive for $a>0$ and $C\ge 1$; this proves strict monotonicity of the carried load.

For $u\ge 0$ and a positivity-preserving operator, this is the negative log-likelihood for independent Poisson observations with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge 0$, its scalar derivatives are $1-g/(1+s)$ and $g/(1+s{)}^2$ on $s=Tu\ge 0$. It is convex, and strictly convex in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the Jensen inequality gives $D_g(u)\ge \Vert Tu{\Vert }_1-\Vert g{\Vert }_{\infty }\log (1+\Vert Tu{\Vert }_1)$. The shift makes $\log (1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.

For a particle initially having semimajor axis one and encountering a circular planet at radius one, the incoming stellar-frame speed is one. A conservative encounter at planet-relative speed $u$ changes the stellar-frame velocity by at most $2u$. Since stellar escape requires speed $\sqrt{2}$, a necessary bound is $u>(\sqrt{2}-1)/2$. Retaining the outgoing relative-velocity geometry sharpens the necessary bound to $u>\sqrt{2}-1$. Neither inequality by itself guarantees a realizable deflection at a specified impact parameter.

Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.