Give $N$ the subspace filtration and $M/N$ the quotient filtration. If the degree-$i$ class of $m\in F_{i}M$ maps to zero, then $m=n+m^{\prime }$ with $n\in N$ and $m^{\prime }\in F_{i-1}M$. Thus $n\in N\cap F_{i}M$ and the class comes from $\mathrm{gr}N$. Inclusion is injective in every degree and the quotient map is surjective, proving the displayed short exact sequence of associated graded modules. Exactness can fail for arbitrarily chosen incompatible filtrations.

An Asian option whose averaging uses the geometric mean. In the constant-coefficient Black-Scholes model, the logarithm of the continuous geometric average is Gaussian with mean $\log S_0+(r-{\sigma }^2/2)T/2$ and variance ${\sigma }^{2}T/3$, because $\mathrm{Var}({\int }_0^T{W}_{t}dt)=T^3/3$. Its forward mean is $S_0{e}^{rT/2-{\sigma }^{2}T/12}$. The discounted lognormal call formula gives its price. It is strictly cheaper than the terminal European call when $\sigma ,K,T>0$ and $r\ge -{\sigma }^2/6$, since both its forward mean and log-variance are no larger, with strictly smaller variance. No unconditional comparison holds for arbitrary negative interest rates: for $r<-{\sigma }^2/6$, sufficiently small positive strikes reverse the inequality.

If a traded payoff matrix $A$ has initial price vector $q=A^{T}p$ with $p\ge 0$, there cannot be a negative-cost portfolio with nonnegative terminal payoff. Nevertheless a nonzero nonnegative attainable payoff supported only in zero-price states can cost zero and give an arbitrage. When every state has positive physical probability, excluding this possibility requires strictly positive Arrow state prices, not merely a nonnegative supporting vector.

Write a presheaf on $0\to 1$ as an arrow $r:B\to A$. Evaluation $\Gamma (r)=B$ has the displayed chain, where $\Pi (r)=A$, $\Delta (S)=(S\stackrel{1}{\to }S)$, $\nabla (S)=(S\to 1)$ and $\Lambda (S)=(\varnothing \to S)$. The Hom-set bijections follow by solving the commuting-square equation. The functor $\nabla$ has no right adjoint because it fails to preserve the initial object: $\nabla (\varnothing )=(\varnothing \to 1)$ is not $(\varnothing \to \varnothing )$.

The normalized information matrix for a homoscedastic linear regression model is the average outer product of its regressor vector. It is positive-definite precisely when the supported regressors span the parameter space. The full Fisher information matrix under normal errors is $NM(\xi )/{\sigma }^2$.

Use the north-pole stereographic projection and the cross-ratio convention $[a,b;c,d]=(a-c)(b-d)/[(a-d)(b-c)]$. Antipodes have coordinates $-1/\bar{u}$. Rotate the first point to the south pole, whose coordinate is zero: its antipode becomes infinity, while a point at spherical distance $d$ has projected modulus $\tan (d/2)$. Möbius invariance of the cross-ratio then gives the displayed identity. Antipodal endpoints require the extended-value interpretation.

For three Lie-algebra-valued functions of one variable, the Nahm equations areThey arise by taking a solution of the ASDYM equations with orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge dt$, imposing independence of $x^1,x^2,x^3$, and setting $A_t=0$ by a local Yang-Mills gauge transformation. Then $F_{ab}=[A_a,A_b]$, $F_{at}=-{\dot{A}}_a$, and anti-self-duality of gauge curvature gives the displayed equations. The opposite orientation reverses their sign.

An isolated axisymmetric magnetic field cannot be sustained by ordinary resistive dynamo action with scalar positive magnetic diffusivity. Its poloidal magnetic flux function satisfies a homogeneous scalar advection-diffusion equation with no toroidal-to-poloidal source. The maximum principle excludes regeneration of that flux; after it decays the toroidal field has no lasting source either. This constrains the symmetry of the field. An axisymmetric flow may still generate a nonaxisymmetric field, and a turbulent mean-field electromotive force is not covered by the unmodified scalar-induction proof.

For a bounded conducting sphere with uniform magnetic diffusivity and an insulating exterior, an incompressible flow tangent to concentric spheres cannot sustain dynamo action. Neither the velocity nor the magnetic field needs to be axisymmetric. The radial magnetic scalar has a homogeneous advection-diffusion equation, so the poloidal field decays. Once it is absent, the toroidal magnetic potential also has a homogeneous scalar advection-diffusion equation. This is a theorem about spherical flow geometry, not about a general cylindrical flow with an axial component.

In the standard two-dimensional setting, with planar velocity and magnetic field independent of the third coordinate and homogeneous energy-closed boundary conditions, the in-plane magnetic potential is an advected diffusing scalar. Its quadratic integral decreases, preventing regeneration of a dynamo field. A third field component independent of that coordinate also obeys a passive advection-diffusion equation. A flow depending on two coordinates but possessing a third velocity component is not covered by this strictly planar formulation.

For a solenoidal three-component velocity depending on two periodic coordinates and a zero-mean magnetic field with the same dependence, the in-plane flux potential satisfies a homogeneous advection-diffusion equation. The Poincaré inequality gives $⟨A^2⟩\le ⟨A_0^2⟩e^{-2\eta k^{2}t}$. Integrating the vertical stretching term by parts bounds it by $Q\sqrt{⟨A^2⟩⟨|\nabla B_z{|}^2⟩}$, where $Q=\Vert \nabla u_z{\Vert }_{\infty }$. Maximizing against resistive dissipation and integrating gives the displayed transient bound. Splitting off half the dissipation instead gives $E^{\prime }+\eta k^{2}E\le (Q^2/\eta )⟨A_0^2⟩e^{-2\eta k^{2}t}$, proving vertical energy decays. This permits transient generation by a nonzero vertical velocity while excluding sustained fields in this zero-mean, z-independent periodic class.

For an ordered sequence of nested linear regression models $M_0\subset M_1\subset \cdots$, the sequential sum of squares for the $j$th added term is $\mathrm{RSS}(M_{j-1})-\mathrm{RSS}(M_j)$. These are also called type-I sums of squares. They depend on term order when the design is not orthogonal. Dividing by the number of newly added coefficients and the full model's residual mean square gives the sequential F-test. A covariate added first is tested before adjustment for later terms, whereas a treatment added after the covariate is tested with that covariate already included.

An additive two-factor Gaussian model separates row and column effects. With one observation in each of $rc$ cells, constraints fixing one row and column effect leave $r+c-1$ mean parameters and $(r-1)(c-1)$ residual degrees of freedom. Balanced least squares gives fitted values equal to row mean plus column mean minus grand mean. With no cell replication, unrestricted interactions are confounded with residual error; the additive interpretation requires an assumption that those interactions are absent or negligible.

Multivariate analysis of variance compares mean vectors of several groups while accounting for correlations between their response variables. In the one-way normal model, independent observations $Y_{ri}\sim N_p({\mu }_r,\Sigma )$ share a positive-definite covariance matrix. Put $E={\sum }_{r,i}(Y_{ri}-{\bar{Y}}_r)(Y_{ri}-{\bar{Y}}_r{)}^T$ and $H={\sum }_r{n}_r({\bar{Y}}_r-\bar{Y})({\bar{Y}}_r-\bar{Y}{)}^T$. Under equal means, these matrices have independent Wishart distributions with degrees of freedom $N-g$ and $g-1$, respectively. This follows by separating the Gaussian data into orthogonal within-group and between-group projections. The Wilks lambda statistic tests equality of the means using both matrices, rather than independently testing each coordinate.

For $x^{\prime \prime }+x=-2\epsilon y{x}^{\prime }$ and $y^{\prime }=\epsilon g(x)$, the method of multiple scales gives the displayed amplitude-phase equations, together with constant leading phase. For $g(x)=x$, $Y$ is constant and $R$ changes exponentially. For $g(x)=\log |x|$ and positive amplitude, the integrable phase average is $\log (R/2)$. Thus $L=\log (R/2)$ obeys $L_{TT}+L=0$ with $L_T=-Y$. The identically zero oscillator is inadmissible for logarithmic feedback, whose force is undefined there.

With the environment initially in $|0⟩$, take $U$ to fix $|00⟩$ and $|11⟩$, and rotate the span of $|10⟩,|01⟩$ by the displayed action and $U|01⟩=\sqrt{1-p}|01⟩-\sqrt{p}|10⟩$. This is a unitary operator. Its environment matrix elements $⟨0{|}_{E}U|0{⟩}_E$ and $⟨1{|}_{E}U|0{⟩}_E$ are the Kraus operators of the amplitude damping channel. Tracing out the environment multiplies the excited population by $1-p$, transfers the remainder to the ground state, and multiplies off-diagonal entries by $\sqrt{1-p}$.

For risk-neutral stock dynamics $dS=(r-\delta )Sdt+\sigma SdW$, with $r>0$ and $\sigma >0$, let ${\beta }_{-}<0$ solve ${\sigma }^2\beta (\beta -1)/2+(r-\delta )\beta -r=0$. A fixed downward trigger $L\in (0,X)$ gives continuation value $(X-L)(S/L{)}^{{\beta }_{-}}$ for $S>L$. Optimizing over the trigger, or imposing smooth fit, gives the displayed boundary. The optimal value equals $X-S$ below $S^{\ast }$ and the continuation expression above. A verification argument uses its payoff majorization and nonpositive discounted generator. At $r=0$ and $\delta \ge 0$, the perpetual value is $X$ for positive spot, approached by triggers tending to zero rather than attained at a positive finite trigger.

For equal ion and neutral mass densities, let $\mu ={\mu }_{in}={\mu }_{ni}$, $a=k_{\parallel }{B}_0/\sqrt{4\pi {\rho }_i}$ and use Fourier time dependence $e^{-i\omega t}$. Solenoidal displacements obey ${\ddot{\xi }}_i=-a^2{\xi }_i-\mu ({\dot{\xi }}_i-{\dot{\xi }}_n)$ and ${\ddot{\xi }}_n=-\mu ({\dot{\xi }}_n-{\dot{\xi }}_i)$. For nonzero frequency, the neutral equation gives ${\xi }_i=(1-i\omega /\mu ){\xi }_n$. Substitution in the ion equation givesEquivalently, the growth rate $s=-i\omega$ obeys $s^3+2\mu s^2+a^{2}s+\mu a^2=0$. A time-independent neutral displacement with zero velocity is a relabeling of a uniform fluid, not an additional wave root of the velocity/magnetic-field system.

For a regular category that is a small category, a regular functor to the Category of sets that reflects isomorphisms exists exactly when the category has almost total support for a regular category. Necessity follows because a nonempty image forces the object's support to be terminal, while an empty image makes all projections $A\times X\to A$ invertible and makes all equalizers of maps out of $A$ invertible. This proves that $A$ is strict initial. For sufficiency, use the capitalization theorem to map conservatively and regularly into a capital regular category, then take global sections of a capital regular category. Well-supported objects retain points; proper strict initial objects retain empty point sets. A bijection between nonempty point sets makes the image subobject entire and makes the kernel-pair diagonal entire, by well-pointedness. Thus the image arrow is both a regular epimorphism and a monomorphism, and is invertible. The capitalization theorem is a substantive existence input; this argument does not infer arbitrary large-category representability from small-category hypotheses.

Suppose $X_n\ge 0$ is integrable and adapted, and $Y_n\ge 0$ is finite and adapted. Use the stopped predictable budget for adapted increments with an integer budget $a$. The process $X_{n\wedge {\tau }_a}+a-U_{n\wedge {\tau }_a}$ is an integrable nonnegative supermartingale, because its budget decrement cancels the localized drift. It has a finite almost-sure limit. On the event that the total drift is smaller than $a$, the stopping time is infinite, so $X_n$ itself converges. Taking the countable union over integer budgets proves finite almost sure convergence. No expectation of the total drift is needed.