A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

An effective opening area may absorb both the discharge coefficient and the factor of two in an orifice law. If $A_e=\sqrt{2}{C}_d{A}_{\mathrm{geom}}$, its discharge is $q=A_e\sqrt{\delta p/\rho }$. Two equal openings carrying the same flux share the total pressure drop, so $q^2=A_e^2\Delta p/(2\rho )$. They do not share equal drops when an internal volume source makes their fluxes unequal. State the effective-area convention before comparing numerical factors in natural ventilation models.

For a distributed heat source, a well-mixed interior with temperature excess $\theta$ gains temperature-volume flux $\mathcal{H}$ and loses $|q|\theta$ by ventilation in either direction. With signed upward throughflow, $q|q|=K_A(G_T\theta \pm W)$, where $G_T=g{\alpha }_T{H}_b$ and $W$ is wind pressure divided by mass density. The plus sign is assisting wind. Steady temperatures satisfy ${\theta }^2(G_T\theta +W)={\mathcal{H}}^2/K_A$ for assistance, or ${\theta }^2|G_T\theta -W|={\mathcal{H}}^2/K_A$ for opposition. Direction restrictions must be applied to roots.

Every nonempty subset of the natural numbers has a least element in their usual order. Applying this principle to positive integer linear combinations proves Bézout's identity. This principle for the usual natural-number order is distinct from the general well-ordering theorem asserting that every set can be equipped with some well-order.

For a fixed sufficiently regular image edge set, the Mumford–Shah functional is a strictly convex quadratic energy on the Sobolev space of its complement. A first variation gives $u-\alpha \Delta u=g$ within each region and separate one-sided Neumann boundary conditions on free edges. A jump need not vanish across an edge. The weak solution is unique by the Lax-Milgram theorem; this does not prove uniqueness when the edge set is also optimized.

On the unit square, let $g=0$ and take $u_0={\chi }_{\{x_1>0.4\}}$, $u_1={\chi }_{\{x_1>0.6\}}$. Both have one unit-length jump and zero ordinary gradient. Their mean has two unit-length jumps. For squared-error coefficient one, $[E(u_0)+E(u_1)]/2=0.5+\beta$ while $E((u_0+u_1)/2)=0.45+2\beta$, violating convexity when $\beta >0.05$. The Mumford–Shah functional can therefore be nonconvex even though the fixed-edge reconstruction subproblem is strictly convex.

Removing the jump-length cost from the Mumford–Shah functional lets finer piecewise-constant partitions reduce squared fidelity without charging for their proliferating image edges. For continuous data on a compact image domain, fine-cell averages converge in squared error while the ordinary within-cell gradient is zero. A finite interface penalty makes that geometry costly.

The Mumford–Shah functional can be approximated by an auxiliary edge field $0\le v\le 1$ and energyThe field is near one inside regions and near zero across an image edge, weakening smoothing there. The profile $v(s)=1-e^{-|s|/(2\epsilon )}$ has unit edge energy per unit length in this normalization. The 1990 construction is a Gamma-convergence approximation; its global-minimizer limit does not guarantee global optimality of an alternating numerical solve. Each fixed-field subproblem is quadratic, but the joint energy is nonconvex.

For $\rho >0$, these coordinates cover a wedge of Minkowski spacetime and give $d{s}^2=-{\kappa }^2{\rho }^{2}d{t}^2+d{\rho }^2$ in two dimensions. A constant-$\rho$ observer has proper acceleration $1/\rho$ and the Unruh effect temperature $1/(2\pi \rho )$ in natural units. The apparent degeneration at $\rho =0$ is a coordinate horizon.

Let $\mathfrak{m}$ be a maximal ideal of a ring $A$, $k=A/\mathfrak{m}$, and $I\subseteq {\mathfrak{m}}^d$. Since $\mathfrak{m}I\subseteq {\mathfrak{m}}^{d+1}$, elements of $I$ with linearly independent classes in ${\mathfrak{m}}^d/{\mathfrak{m}}^{d+1}$ have independent classes in $I/\mathfrak{m}I$. Any generating set must span this latter vector space. Thus $r$ independent degree-$d$ initial forms give a lower bound of $r$ generators, without assuming that height equals the generator count.

Methanogenesis is biological production of methane by anaerobic microorganisms. Waterlogged soil and lake sediments can favour it. Warming or thaw changes substrate availability, but production is distinct from atmospheric emission because oxidation and transport intervene.

For a primitive root of unity $\omega =e^{2\pi i/n}$ and labels $r,s$ modulo $n$, the displayed $n^2$ maximally entangled states form an orthonormal basis. Inner products equal ${\delta }_{s{s}^{\prime }}{n}^{-1}{\sum }_j{\omega }^{j(r^{\prime }-r)}={\delta }_{s{s}^{\prime }}{\delta }_{r{r}^{\prime }}$, by the geometric series. A nonprimitive root repeats phase labels and does not give a complete orthonormal basis.

Spatially separated simultaneous events in one inertial frame are generally not simultaneous in a relatively moving frame. The Lorentz transformation explicitly mixes position into time. This is essential when comparing distances between moving light pulses: one must compare their positions at the same time in the measuring frame.

At $v=0.6c$, the rod's simultaneous laboratory endpoint measurements in length contraction have $(ct,x)=(0,0)$ and $(0,8\mathrm{m})$. They become $(c{t}^{\prime },x^{\prime })=(0,0)$ and $(-6\mathrm{m},10\mathrm{m})$. In contrast, simultaneous rod-frame measurements at $t^{\prime }=0$ use $(ct,x)=(0,0)$ and $(7.5\mathrm{m},12.5\mathrm{m})$. The latter front-end event occurs later in the laboratory, after the front has moved. The sloping $t^{\prime }=0$ line in the figure is a set of simultaneous spacetime events, not a signal's trajectory. ^[ref].

Expand the Einstein tensor about the Minkowski metric. The quadratic term in the metric perturbation defines $t_{\mu \nu }=-G_{\mu \nu }^{(2)}[h,h]/(8\pi )$, a symmetric tensor relative to the background. For a first-order vacuum solution, the expanded contracted Bianchi identity gives ${\partial }^{\mu }{t}_{\mu \nu }=0$. Equivalently, a second-order correction $j$ obeys $G^{(1)}[j]=8\pi t$, whose flat divergence vanishes identically. This conservation is an on-shell statement. The local $t$ depends on perturbative gauge and the background split; conservation alone does not make it a gauge-independent local gravitational stress-energy tensor.

For a source surface parameterized by $y(p,q,\tau )$, the retarded time equation is $t=\tau +|x-y(p,q,\tau )|/c_0$. Integrating a Dirac delta distribution enforcing that equation produces the reciprocal derivative $|1-M_r{|}^{-1}$ at each simple root. For subsonic motion there is a unique retarded root. Multiple roots require summation; a zero derivative is outside the simple-root rule. This factor is the moving-source counterpart of a delta change-of-variable Jacobian.

The acoustic far field is the radiation region of an acoustic wave. For wavenumber $k_0=\omega /c_0$, $k_{0}R\gg 1$ places an observer beyond the reactive near field. For a source of extent $\ell$, a range-independent acoustic directivity pattern generally also requires $R\gg \ell$ and $k_0{\ell }^2/R\ll 1$. The last condition controls the quadratic phase error in the Fraunhofer approximation: distance expands as $|R\mathbf{n}-\mathbf{y}|=R-\mathbf{n}\cdot \mathbf{y}+O({\ell }^2/R)$.^[ref]

Outgoing acoustic density perturbations spread as $R^{-1}$ in three dimensions and $R^{-1/2}$ in two. These observer-distance conditions differ from the acoustic compact-source approximation $k_0\ell \ll 1$, which constrains the source itself.

The original mathematical optimization problem relative to a chosen Lagrangian dual problem. In convex perturbation duality, its objective is $\phi (x)=f(x,0)$ and its value is $p(0)$. Strong duality concerns optimal values; primal attainment additionally requires an actual minimizing point.

For $\gamma >0$ and $c\ne 0$, minimize $c^{T}u+\gamma z^2$ subject to $\Vert u{\Vert }^2+z=R$. Adding a positive Lagrange multiplier times the constraint gives a positive-definite quadratic optimization Lagrangian, whose minimizer is $u=-c/(2\lambda )$ and $z=-\lambda /(2\gamma )$. Feasibility is the displayed scalar equation. Its left side strictly decreases from positive infinity to negative infinity on $\lambda >0$, so it has a unique root for every real $R$. The Lagrangian sufficiency theorem certifies a unique global optimum on the possibly nonconvex equality surface.

A knot invariant assigns the same value to isotopic knots. Examples include the knot group, Alexander polynomial of a knot, Seifert genus, knot signature and Jones polynomial. Different knot invariants can retain different information: isomorphic knot groups do not alone imply isotopic knots.

The incoming two-particle distribution at a collision is approximated by the product of one-particle distributions. This closes the binary collision term into the quadratic nonlinear Boltzmann collision operator. It does not assert exact independence after collisions: outgoing pairs can be correlated, and time reversal generally converts those correlations into incoming correlations. A rigorous kinetic limit needs propagation and convergence estimates rather than a bare independence assertion.