The two-dimensional Lie algebra of infinitesimal dilations and translations of a line has a basis $h,e$ with $[h,e]=e$. Its derived algebra is the abelian line spanned by $e$, but every nontrivial term of its lower central series of a Lie algebra is that same line. It is solvable and not nilpotent.

In the SO(3) vector Higgs model, a real scalar field $\Phi \in {\mathbb{R}}^3$ transforms as $\Phi \mapsto R\Phi$. Its gauge covariant derivative is $D_{\mu }\Phi ={\partial }_{\mu }\Phi +g{\mathbf{A}}_{\mu }\times \Phi$ for generators acting by the cross product.

The scalar potential $V(\Phi )=\lambda ({\Phi }^T\Phi -v^2{)}^2/4$, with $\lambda >0$, is invariant under the SO(3) group. For $v>0$, its vacuum manifold is a sphere. Canonical scalar normalization gives radial mass squared $2\lambda v^2$.

The complement of a closed tubular neighborhood of a smooth plane conic has the displayed integral homology. The Excision theorem and Thom isomorphism theorem reduce the calculation to the relative long exact sequence: the ambient degree-four fundamental class restricts with coefficient one, while the degree-two map is intersection with the conic and has coefficient two. A collar neighborhood identifies the open and compact exterior homotopy equivalence types. The boundary instead has first homology $\mathbb{Z}/4$, from the Gysin sequence and the normal Euler class four.

For any point inclusion $i:\{x\}\hookrightarrow X$, the skyscraper sheaf $i_{\ast }A$ has restriction maps that are identities or maps to zero. It is flasque, so its higher sheaf cohomology vanishes and its global sections are $A$. This requires no assumption that $x$ is closed.

There is an uncountable cardinal number $\kappa$ with ${\sigma }_{\kappa }=\kappa$. Iterate ${\kappa }_{n+1}={\sigma }_{{\kappa }_n}$ starting with ${\aleph }_0$. If there is no earlier fixed point, the increasing supremum $\kappa$ has countable cofinality, hence is a singular cardinal. The singular cardinal enumeration is cofinal in $\kappa$ below index $\kappa$, and continuity at this singular supremum gives equality.

The normalized eigenfunction ${\psi }_0(x)=\mathrm{sech}x/\sqrt{2}$ has zero eigenvalue under the Sine-Gordon kink fluctuation operator. It is proportional to ${\partial }_x{\varphi }_K$ and comes from shifting the collective coordinate of the kink. Its frequency is zero, so it contributes no oscillator zero-point energy; it must be handled separately from a Gaussian functional determinant.

At the closest physical-strip pole of $S_{2,1}$, the difference of the analytically continued external momenta has squared mass $m_1^2$. Using $m_2=2m_1\cos a$ makes the identity immediate. Thus the exchanged t-channel one-particle state is the lightest Sine-Gordon breather, not a new member of the spectrum.

For every finite measure $\mu$ on the predictable sigma-algebra, bounded simple predictable processes are dense in $L^2(\mu )$. Include a bounded ${\mathcal{F}}_0$-measurable value supported at time zero when $\mu$ may charge that slice. Indicators of the generating rectangles and the Pi-lambda theorem prove density. This gives the completion step in the Itô isometry construction.

The standard logistic function maps a log odds $z$ to probability $1/(1+e^{-z})$. More generally, its translated and scaled form is $L/(1+e^{-k(x-x_0)})$.

A unit-vector field $\mathbf{n}:{\mathbb{R}}^2\to S^2$ approaching a fixed value at infinity extends to $S^2\to S^2$. Its topological charge is the topological degree $Q=(4\pi {)}^{-1}\int \mathbf{n}\cdot ({\partial }_1\mathbf{n}\times {\partial }_2\mathbf{n})d^{2}x$. This normalized target area pullback counts the number of oriented coverings.

The two mobilities in the shallow-ice flux with linear basal drag are equal at thickness $H_{\ast }=3/\beta$. A conserved half-volume $V_g$ gives length $L_{\ast }=V_g/H_{\ast }$ and time $T_{\ast }=3\mu L_{\ast }^2/(\rho g{H}_{\ast }^3)$. Nondimensional thick profiles spread as ${\tau }^{1/5}$; thin sliding profiles spread as ${\tau }^{1/4}$. Matching the centre of the mass-normalized early planar volume-conserving nonlinear-diffusion similarity to unit depth estimates ${\tau }_{\mathrm{tr}}=3/(10I_3^2)\simeq 0.42385$. The numerical estimate depends on the matching convention and the initial-profile relaxation.

A locally steady glacier nose moving at speed $v>0$ has $q=vh$ by mass conservation. At distance $s$ behind its edge, the linear basal drag law then gives $s=(\rho g/\mu v)(h^3/9+h^2/(2\beta ))$. The thin nose has $h\propto s^{1/2}$ from basal sliding; a thicker outer nose has $h\propto s^{1/3}$ from internal deformation. A thick bulk therefore still has a sliding-controlled inner tip. The divergent idealized edge slope limits the immediate validity of the shallow-ice approximation.

A formula with unchanged truth value between the specified models, using parameters shared by them. Usually the models in set-theoretic applications are transitive and nested; the class of models is part of the assertion of absoluteness.

For a set relation in nested transitive models of ZFC, well-foundedness is absolute. An inner-model ordinal rank function witnesses upward absoluteness, and a fixed nonempty subset without a minimal member witnesses failure in both models.

Let $M\subseteq N$ be transitive models of a theory proving both an existential-bounded and a universal-bounded equivalent of a formula. For parameters in $M$, the existential form gives upward absoluteness and the universal form gives downward absoluteness. Consequently the models agree on the formula. The equivalences must hold in both models.

For $n\ge 2$, the positive-degree dimensions in degrees $n,n+1,n+2,n+3$ are $1,1,1,2$. The first three classes are $u,{\mathrm{Sq}}^{1}u,{\mathrm{Sq}}^{2}u$. In degree $n+3$, use ${\mathrm{Sq}}^{3}u$ and ${\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ for $n\ge 3$; at $n=2$ replace the unstable first class by $u{\mathrm{Sq}}^{1}u$. At $n=2$ the degree-four class is $u^2$, and at $n=3$ the degree-six class ${\mathrm{Sq}}^{3}u$ is $u^2$.

On ${\mathbb{P}}_k^n$ with $n>0$, let $\mathcal{F}=i_{\ast }k$ at a rational point. It has one-dimensional global sections, but its dual of a sheaf is zero: a local map from the residue field into the local polynomial ring must have image annihilated by a nonzero parameter, which is impossible in an integral domain. Thus $H^n({\mathcal{F}}^{\vee }(-n-1))=0$. Serre duality for arbitrary coherent sheaves instead uses an Ext functor; replacing this by the ordinary dual can lose torsion information.

A semigroup presentation is finite when both its generator set and its list of defining equalities are finite. A finite transition table for a Turing machine can be encoded this way: state-and-tape words simulate transitions, boundary rules supply blank cells, and halting-state rules erase the configuration to one distinguished symbol.

Sieve an interval of length $z$ by the prime two and the primes congruent to three modulo four, with local densities $g(p)=1/p$. Take $D=z^{1/2}$ and $L=\sqrt{D}$. The truncated Euler-product lower bound, the Mertens first theorem and the reciprocal-prime sum in residue class one modulo four give $J\gg \sqrt{\log z}$. The interval counts have bounded remainders, whose sum is $O(D{\log }^{2}D)$ by the summatory bound for three to the prime omega. Hence the number surviving is $O(z/\sqrt{\log z})$, uniformly in the location of the interval.