For a degree-$m$ extension $L/K$, substitute a $K$-basis of $L$ into a homogeneous degree-$d$ polynomial in $N>d$ variables over $L$, then take its field norm. The resulting degree-$md$ polynomial has $mN>md$ variables over $K$. A nonzero zero over a C1 field gives a nonzero zero of the original form because the norm of a field element vanishes only at zero. Separability is unnecessary.

A central division algebra of degree $d>1$ has an anisotropic reduced norm of degree $d$ in $d^2>d$ variables. That contradicts the C1 field property. Thus only degree one is possible, and every central simple algebra is split.

In the area normalization $d^2$, the buoyancy-gradient mixing-length closure transports dense-fluid buoyancy downward with magnitude $\mathcal{F}=d^{2}K{G}_z=d^4(G_z{)}^{3/2}$. Its convergence in an upward coordinate is ${\mathcal{F}}_z$. Transport has units $L^4{T}^{-3}$ and convergence units $L^3{T}^{-3}$. Constant positive $G_z$ has nonzero transport and zero convergence; the two quantities cannot be identified.

An upward volume flux $Q$ and a downward turbulent buoyancy transport $d^4(G_z{)}^{3/2}$ balance in a steady reactor. Matching $G=G_z=0$ below $-H$ gives $QG=d^4(G_z{)}^{3/2}$. Integrating yields the cubic profile above the front. If downward source buoyancy flux is $B_s$, then $H=3d^{8/3}{B}_s^{1/3}/Q$. The corresponding eddy diffusivity is $K=Q(z+H)/(3d^2)$. All these prefactors use the same area and mixing-length normalization; an actual cross-sectional area $S$ replaces the denominator in $K$ by $3S$.

With fixed structural parameters and $K=v/a$, the Bühlmann credibility estimate has form $({\sum }_{i=1}^n{x}_i+K{m}_0)/(n+K)$. The next observation updates it by the displayed rule. Consequently the estimate decreases exactly when the new observation is less than the previous credibility estimate, with strict equality handled separately.

When a rarefaction wave attaches to a saturation shock, its terminal characteristic speed must match the Rankine-Hugoniot condition speed. This gives a tangent from the initial state $(s_0,F(s_0))$ to the fractional-flow curve at $s_s$. It selects the upstream shock saturation; the actual dimensional shock speed still includes total flux divided by porosity.

Eliminating phase pressures with $p_c=p_n-p_w$ gives $\varphi s_t+QF(s{)}_x=(D_c{s}_x{)}_x$. For decreasing capillary pressure, $D_c\ge 0$. This nonlinear diffusion broadens a saturation shock; it can degenerate where a phase mobility vanishes. Endpoint conservation retains the limiting Rankine-Hugoniot condition.

The ordinary time integral of standard Brownian motion is a centered Gaussian process with continuously differentiable paths and derivative $B_t$. For $0\le s\le t$, its covariance is $\mathbb{E}(I_s{I}_t)=s^2(3t-s)/6$, obtained by integrating the Brownian covariance $\min (u,v)$. Consequently $I_t$ has normal distribution $N(0,t^3/3)$. This time integral is distinct from the Itô integral with respect to Brownian motion.

At integer times, $B_n/\sqrt{n}$ has the same standard normal distribution for every $n$. The event that its limit superior exceeds a fixed finite level is a tail event of the independent unit-time Brownian increments, since changing any finite initial segment contributes only a term tending to zero. The Kolmogorov zero-one law applies. Its probability is positive because each fixed-time exceedance has the same positive probability, giving a positive probability of infinitely many exceedances by the decreasing union-of-tail-events argument. Thus its probability is one. Intersecting over integer levels proves the stated divergence, without the lower iterated-logarithm bound.

An increment of a Brownian motion between deterministic times $s<t$ is $W_t-W_s$. It has normal distribution $N(0,t-s)$ and is independent of the filtration up to time $s$. Increments on disjoint time intervals are independent. Their covariance is the length of overlap of the corresponding intervals; this also follows from the Brownian covariance kernel.

For the completed natural Brownian filtration, every event in the germ sigma-field of Brownian motion has probability zero or one. Indeed, an event measurable at every positive time is independent of all increments after each such time. Letting that time decrease to zero and using path continuity makes the event independent of the entire Brownian path sigma-field. Since it is itself measurable in that sigma-field, its probability equals its square.

A Wiener sausage is the region swept out by a ball of fixed radius along a Brownian motion path. A deterministic continuous displacement $f(s)$ gives the variant ${\bigcup }_{s\le t}\mathcal{B}(B_s+f(s),r)$. On each bounded time interval it has finite expected volume: the region lies inside a ball whose radius is the path's maximum displacement plus $r$, and the Brownian maximum has Gaussian tail bounds and finite moments of every order.

A Brownian motion starting outside a fixed affine subspace of codimension at least two almost surely never hits it. Choose a two-dimensional orthogonal projection annihilating the subspace's direction and with nonzero projected initial displacement. The projected process is planar Brownian motion starting away from zero, and a hit of the subspace would hit zero in that projection. The polar point for planar Brownian motion result excludes this. In three dimensions this shows avoidance of every fixed line when started outside it.

Porous-matrix drag depends on fluid velocity relative to the matrix, so it is not invariant under adding a uniform fluid velocity while leaving the matrix fixed. In a swimmer frame with stationary laboratory matrix, the drag is proportional to $\mathbf{u}-U{\mathbf{e}}_x$. A constant shift can alternatively be absorbed into a linear pressure term, whose physical interpretation must then be retained.

For $h_t=\kappa (h{h}_x{)}_x$ with a drained boundary $h(0,t)=0$ and a finite current vanishing at its nose, integration by parts gives $\dot{D}=-\kappa [h^2{]}_0^{\infty }/2=0$. The outlet can have finite volume flux because its contribution to the first-moment flux is multiplied by $x=0$. A positive prescribed boundary height instead gives $\dot{D}=\kappa h(0,t{)}^2/2$, so drainage is essential to this invariant.

For the unit-modulus hyperbolic scattering block with elementary mass $m$, a pair bound at rapidity difference $iu$ has mass $m_2=2m\cos (u/2)$. Bootstrap fusion with another elementary particle gives a further direct pole at $3iu/2$. It is inside the physical rapidity strip with positive imaginary residue only for $0<u<2\pi /3$. In that range the new bound state has rest-frame constituent rapidities $iu,0,-iu$, so its energy is $m(1+2\cos u)=m\sin (3u/2)/\sin (u/2)$. Charges add at the bound-state vertex. At $u=2\pi /3$ the apparent pole cancels; beyond that point it lies outside the physical strip. The formula is conditional on this particular scattering block and its direct-pole interpretation, not a universal three-body binding law.

Canonical diagonalization of $A_k{b}_k^{†}{b}_k$ plus an anomalous pair term gives the vacuum shift $(E_k-A_k)/2$ per momentum. Thus $H=C+\frac{1}{2}{\sum }_k(E_k-A_k)+{\sum }_k{E}_k{\alpha }_k^{†}{\alpha }_k$. The subtraction of $A_k/2$ accompanies the quasiparticle zero-point term; omitting it changes the ground-state energy.

For a finite physical supermultiplet at fixed positive energy, fermion parity anticommutes with each odd supercharge. Trace cyclicity makes $\mathrm{Tr}[(-1{)}^F\{Q,Q^{†}\}]=0$. Summing the Super-Poincaré algebra diagonal entries replaces the anticommutator by $4E$, giving equal bosonic and fermionic state counts. A zero-energy vacuum is an exception to the division by $E$, so it need not have a partner. The argument requires an invariant fixed-momentum representation, which is the step lost under explicit supersymmetry breaking.

A dilute condensed Bose system whose interaction is sufficiently weak for a small-depletion expansion. Its low-energy excitations are described by a Bogoliubov transformation rather than free particles. The homogeneous quadratic theory is controlled only while noncondensate occupation remains small compared with the condensate and its excitation energies are stable.

A Bose-Einstein condensate is a state with macroscopic occupation of a one-particle mode. A phase-selected mean-field description can use $⟨\psi ⟩\ne 0$, while a fixed-number finite-volume state need not have a nonzero field expectation. Bose-Einstein condensation and superfluidity are related but distinct notions; long-wavelength phase fluctuations can invalidate a uniform condensate saddle in low dimensions.