For equal phase thermal properties, a roof undercooling $\Delta T$ and liquid superheat $\Theta$, the planar Stefan problem has interface $a=2\lambda \sqrt{\kappa t}$. The solid error function profile and liquid complementary-error-function profile give $S\lambda =e^{-{\lambda }^2}[1/\mathrm{erf}\lambda -(\Theta /\Delta T)/\mathrm{erfc}\lambda ]/\sqrt{\pi }$. Here $S=L/(c_p\Delta T)$ is the latent-to-sensible heat ratio. Liquid heat supplied to the interface reduces crust growth.

For an object of the nested partial unary operation category ${\mathcal{C}}_n$, adjoining the next operation freely gives one new value at each defined fixed point of the previous operation. Each new value starts an infinite free chain under the first unary operation; the other old operations are undefined on those new points. The new top operation consequently has no fixed points, so subsequent free extensions add no points. This gives a left adjoint to forgetting the last operation; extensions of maps along the chains are forced by the first target operation.

For a linear spin-wave approximation, the ground state is empty of the diagonal quasiparticles but contains the original spin bosons. If $a_k=u_k{\alpha }_k-v_k{\alpha }_{-k}^{†}$, their occupation number is $v_k^2$. The staggered magnetization is reduced from $S$ to $S-N^{-1}{\sum }_k{v}_k^2$. For the one-dimensional nearest-neighbor Heisenberg antiferromagnet, $v_k^2=(|\sin k{|}^{-1}-1)/2$ and the integral diverges logarithmically. This invalidates finite ordered magnetization within the approximation; it is not itself a determination of the exact spectral gap.

The implication introduction rule discharges a temporary assumption $A$ from a derivation of $B$, producing $A\to B$. Under the Curry-Howard correspondence, a body $t:B$ under $a:A$ gives the lambda abstraction $\lambda a^A.t:A\to B$. Unused assumptions may also be discharged.

From $A\to B$ and $A$, implication elimination derives $B$. Under the Curry-Howard correspondence, this is application: from lambda terms $f:A\to B$ and $a:A$ form $fa:B$.

The rule with premises $C,D$ and conclusion $C$ retains the first derivation while allowing an unused second assumption. Under the Curry-Howard correspondence, the output term is the first premise term, with the enlarged context. This allows a lambda abstraction whose bound variable does not occur in its body.

The Lemke-Howson algorithm follows an almost completely labeled path in the product of two best-response polytopes of a bimatrix game. Drop one label at the artificial zero pair, then alternately pivot away the duplicated label until the dropped label returns. A nonzero completely labeled endpoint, after normalization, gives a Nash equilibrium. The two players' vectors must be kept separate even in a symmetric game.

For a symmetric bimatrix game with payoff matrices $P,P^{\mathsf{T}}$, any nonzero $x$ satisfying these equations gives the symmetric Nash equilibrium $(x/{\sum }_i{x}_i,x/{\sum }_i{x}_i)$. Every positive coordinate attains the same maximal payoff. This one-vector construction differs from a two-vector Lemke-Howson algorithm path, which can return asymmetric equilibria.

A symmetric equilibrium of a symmetric game has every player use the same strategy or the same distribution of strategies. Symmetry between players does not require identical behavior across contests with different prizes.

A stagnant mushy-layer model has no bulk advection, equal phase mass densities, pure solid, and local equilibrium on the liquidus. With liquid fraction $\varphi$, specific enthalpy $E=c_{p}T+L\varphi$ and bulk solute concentration $B=\varphi C$, neglecting solute diffusion inside the mushy layer gives $T_t+(L/c_p){\varphi }_t=\kappa T_{zz}$ and $(\varphi C{)}_t=0$. The latter fixes composition at each stationary material level; it does not automatically set every level to the initial liquid concentration when solute can diffuse ahead of the growing mush.

A multiplicative subset is saturated if every factor of a product in the subset also lies in it. We include $1$ in a multiplicative subset. A saturated subset containing $0$ is the whole ring. The units form a saturated subset in a commutative ring.

An exponentially weighted consumption habit is $h_t=e^{-\lambda t}{h}_0+{\int }_0^t\lambda e^{-\lambda (t-s)}{c}_{s}ds$. Differentiation gives $d{h}_t=\lambda (c_t-h_t)dt$. The state remains positive from positive initial habit and nonnegative consumption.

When habit evolves by $dh=\lambda (c-h)dt$, increasing consumption both costs wealth and increases future habit. Consequently the consumption coefficient in the Hamilton-Jacobi-Bellman equation is $V_w-\lambda V_h$. A finite interior optimum requires this effective shadow price to be positive.

After scaling wealth by habit and applying the wealth-variable Legendre dual, the effective consumption shadow price is proportional to $\mathcal{D}=z-\lambda Rz{J}^{\prime }+\lambda (R-1)J$. Its power in the utility conjugate keeps the dual equation nonlinear. When $\lambda =0$, habit is fixed and the dual equation is a linear Euler differential equation with effective relative risk aversion $R+\alpha$.

For finite sets of positive real numbers $A,B$ with $|B|\ge 2$, $E(A,B)\le 4\lceil \log |B|\rceil |A+A||B+B|$, with natural or binary logarithm. Partition ray occupancies of $A\times B$ into $\lceil \log |B|\rceil$ classes of multiplicative width $e$, or width $2$ for the binary convention. Injectivity of sums on two distinct rays and disjoint open planar sectors bound each class's sum of squared occupancies by $4|A+A||B+B|$. A singleton class contributes at most $|A||B|$. Combining this with the energy lower bound gives a sum-product inequality.

Concentration of the prime omega function near $\log \log N$, together with $\omega (ab)=\omega (a)+\omega (b)-\omega (\gcd (a,b))$ and the bounded mean of $\omega (\gcd (a,b))$, gives $M(N)\ll N^2/\log \log N$. This elementary argument uses only distinct prime factors.

A bilinear map is a map $B:V\times W\to U$ between vector spaces that is linear in each argument when the other is fixed. The nonlinear temperature advection can be written as the diagonal of a bilinear map, $\mathcal{N}(\varphi ,\chi )=-\mathbf{u}[\varphi ]\cdot \nabla \chi$, when velocity depends linearly on temperature. Unlike a bilinear form, its output need not be a scalar.

A causal square-integrable linear time series can be written $X_t=\mu +{\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$. Square summability of the coefficients ensures mean-square convergence for white noise. An innovation representation additionally identifies its driving noise as the innovation process.

For coefficients ${\theta }_0=1,{\theta }_1,\dots ,{\theta }_q$, the long-run variance of a stationary process sampled every second time is ${\sigma }^2[({\sum }_{j\text{ even}}{\theta }_j{)}^2+({\sum }_{j\text{ odd}}{\theta }_j{)}^2]$. It equals ${\sigma }^2[\Theta (1{)}^2+\Theta (-1{)}^2]/2$.

On the unit round sphere $S^n$, a geodesic of speed $L$ on $[0,1]$ has normal Jacobi equation $y^{\prime \prime }+L^{2}y=0$. Its conjugate parameters are $j\pi /L$, each with multiplicity $n-1$. If $L/\pi$ is not an integer, its index is $(n-1)\lfloor L/\pi \rfloor$. For $L=m\pi >0$, its index is $(m-1)(n-1)$ and its nullity is $n-1$.