Identify a regular orbit with the group using a chosen base point. A permutation commuting with all left translations satisfies $c(x)=xc(1)$. Thus the centralizer consists of right translations. Defining $R_g(x)=x{g}^{-1}$ makes $R_g{R}_h=R_{gh}$ and gives an isomorphism to $G$, rather than an unmentioned reversal of multiplication.

For disjoint nonempty vertex sets $U,V$ of a graph, an $\epsilon$-irregular pair is one that is not an $\epsilon$-regular pair of vertex sets. Equivalently there are witnesses $U^{\prime }\subseteq U$, $V^{\prime }\subseteq V$ with $|U^{\prime }|\ge \epsilon |U|$, $|V^{\prime }|\ge \epsilon |V|$ and $|d(U^{\prime },V^{\prime })-d(U,V)|>\epsilon$. Splitting both cells by these witnesses increases their contribution to the regularity energy by at least ${\epsilon }^4|U||V|/n^2$: the witness rectangle has relative weight at least ${\epsilon }^2$, its mean discrepancy has magnitude greater than $\epsilon$, and the Cauchy-Schwarz inequality bounds the weighted sum of squared discrepancies below by the rectangle weight times its squared mean. This is the local increment in the energy-increment proof of Szemerédi regularity.

The Szemerédi regularity lemma gives equal cells. If every regular cross-cell pair had density at most $1/4$, their total edge contribution would be at most $n^2/8$. Exceptional vertices, within-cell edges and irregular pairs contribute at most $(2\eta +1/(2m_0))n^2$. Choosing $\eta =1/100$ and $m_0=100$ contradicts $e(G)=n^2/6$. The dense pair is $1/8$-regular since it is already $\eta$-regular.

Let $P\omega =\mathcal{P}(\mathbb{N})$ ordered by subset. Encode a finite subset $u$ and output $n$ by a natural number $⟨u,n⟩$. Define $a\cdot b=\{n:\exists u\subseteq b\text{ finite},⟨u,n⟩\in a\}$ and $e(f)=\{⟨u,n⟩:n\in f(u)\}$. Every Scott continuous map satisfies $f(b)={\bigcup }_{u\subseteq b\text{ finite}}f(u)$, proving that application is the projection of a retraction. Lambda abstraction has graph $\{⟨u,n⟩:n\in [[M{]]}_{\rho [x:=u]}\}$.

Near a nilpotent origin with $\sigma =1$, $\mu =0$, the weighted scaling $u_{\mathrm{old}}={\epsilon }^{2}u$, $v_{\mathrm{old}}=\epsilon v$, $\tau =\epsilon t$, ${\mu }_{\mathrm{old}}={\epsilon }^2\mu$, $\sigma =1+{\epsilon }^{2}s$ makes the leading cubic confinement flow $u^{\prime }=-sv+v^3/2$, $v^{\prime }=2u$. Direct differentiation proves conservation of $H=u^2+s{v}^2/2-v^4/8$. For $s>0$ the origin is a centre and $(0,\pm \sqrt{2s})$ are saddles. Their heteroclinic connections have $H=s^2/2$ and $u=\pm (2s-v^2)/(2\sqrt{2})$ for $|v|\le \sqrt{2s}$. Positive lower energy levels surround the centre; higher levels escape the potential well.

For a reflection-symmetric vector field $\dot{u}=v+f_{30}{u}^3+f_{21}{u}^{2}v+\cdots$, $\dot{v}=g_{30}{u}^3+g_{21}{u}^{2}v+\cdots$, introduce $x=u$, $y=\dot{u}$. Differentiating gives $\dot{x}=y$ and a cubic acceleration with $x^{2}y$ coefficient $b=g_{21}+3f_{30}$. The contribution $3f_{30}$ comes from differentiating the first equation's cubic term. Further cubic near-identity transformations can remove nonresonant terms but cannot turn $b=0$ into a nonzero coefficient by a nonsingular scaling. This is a useful check before assigning a generic reflection-symmetric Bogdanov–Takens bifurcation to a calculated Taylor expansion.

The ratio of the terminal value of a standard Brownian motion to its running maximum at time one has density $(2-r{)}^{-2}$ for $r<1$ and zero otherwise. The Brownian reflection principle gives joint density $2(2m-w)\phi (2m-w)$ for $m>0,w<m$. Substituting $w=rm$ and integrating the Jacobian factor $m$ proves the density; its integral on $(-\infty ,1)$ is one.

Apply the Skorokhod reflection map on the half-line to Brownian motion with negative constant drift and positive variance parameter ${\sigma }^2$. The result is kept nonnegative by a regulator acting only at zero. It has a stationary distribution, unlike zero-drift reflection on the whole half-line.

For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

A finite group of Lie type is obtained from rational points of a connected reductive algebraic group with a suitable Frobenius endomorphism of an algebraic group, often followed by central quotient or passage to a derived subgroup. Its complex representations combine finite-group character theory with the geometry of Borel subgroups, maximal algebraic tori, and the Weyl group.

If a nonzero nilpotent element $a$ existed in $A$, its proper annihilator would lie in a maximal ideal $\mathfrak{m}$. No element outside $\mathfrak{m}$ annihilates $a$, so $a/1$ is nonzero in the localization at a prime ideal $A_{\mathfrak{m}}$. Conversely, if $(a/s{)}^n=0$ in a localization, some $u$ in the multiplicative subset satisfies $u{a}^n=0$. Then $(ua{)}^n=0$, so reducedness of $A$ gives $ua=0$ and hence $a/s=0$.

For odd $n$ prime to 35, proper representation by a positive definite binary quadratic form of discriminant $-35$ is equivalent to $-35$ being a square modulo every prime divisor of $n$. A primitive representing vector extends to a determinant-one integer matrix, producing a form with leading coefficient $n$ and middle coefficient $b$ satisfying $b^2\equiv -35(\mathrm{mod}4n)$. Conversely this congruence constructs such a form. Its local roots lift to prime powers and combine by the Chinese remainder theorem. Every class reduces to one of $x^2+xy+9y^2$ and $3x^2+xy+3y^2$. The condition is local at each prime divisor; a product Jacobi symbol equal to one alone is insufficient.

In ideal reduced magnetohydrodynamics with constant background sound and Alfvén speeds, perpendicular total-pressure balance gives $\delta p+{\rho }_0{v}_A^2\delta B_{\parallel }/B_0=0$. Put $s=\delta p/p_0-\gamma \delta \rho /{\rho }_0$, $r=\delta \rho /{\rho }_0+s/\gamma$, $c_T=v_A{c}_s/(v_A^2+c_s^2{)}^{1/2}$ and $h=c_s^{2}r/c_T$. Then the parallel momentum, continuity and induction equations yieldThus $f^{\pm }=u_{\parallel }\pm h$ satisfy $D{f}^{\pm }=\mp c_T{\nabla }_{\parallel }{f}^{\pm }$. The two slow-wave populations propagate along bent magnetic field lines and are mixed by the perpendicular Alfvénic turbulence, while the entropy fluctuation is simply advected. They do not force the perpendicular Alfvénic system at this leading order.

For ideal reduced magnetohydrodynamics with periodic or vanishing-flux boundaries, the five quadratic cascade channels are $I_A^{\pm }=\int |{\mathbf{z}}_{\perp }^{\pm }{|}^{2}dV$, $I_S^{\pm }=\int (f^{\pm }{)}^{2}dV$ and $I_e=\int s^{2}dV$. The perpendicular Elsässer variables, slow-wave amplitudes $f^{\pm }$, and entropy fluctuation $s$ each obey an equation with divergence-free advection. Multiplying each equation by its own field and integrating makes both advection and propagation terms boundary fluxes, while the perpendicular pressure term vanishes by solenoidality. Each integral is therefore separately conserved. The two Alfvén cascades distort one another without exchanging their integrated invariants; the other three are passively cascaded by the Alfvénic motion without feedback at this order. This count concerns quadratic fluctuation-energy channels: an advected scalar also conserves arbitrary integrals $\int F(s)dV$, which are not extra independent quadratic modal channels.

A difference equation relates sequence values at different indices. A first-order autonomous system iswith a scalar or vector state; equivalently $z_{n+1}-z_n=F(z_n)-z_n$. A fixed point $z_{\ast }$ satisfies $F(z_{\ast })=z_{\ast }$, and a differentiable update has linearized errors $e_{n+1}=DF(z_{\ast })e_n+o(\Vert e_n\Vert )$.

In finite dimensions, if every eigenvalue of the Jacobian matrix has modulus below one, the fixed point is locally asymptotically stable. An eigenvalue outside the unit circle implies instability. An eigenvalue on the unit circle leaves nonlinear terms relevant. For example, $F_{-}(x)=x-x^3$ and $F_{+}(x)=x+x^3$ both have derivative one at zero. For $0<|x|<1$, iteration by $F_{-}$ decreases $|x|$ to zero; iteration by $F_{+}$ increases a nonzero positive input until it leaves any sufficiently small neighborhood. Their identical linearization does not determine their different stability.

For a continuous-time ordinary differential equation, the corresponding linear test uses real parts of eigenvalues, with a similar need for higher-order analysis at marginal eigenvalues.

For fixed $0<E<V_0$ and $\kappa a\gg 1$, the exact rectangular barrier transmission probability givesThis is the leading term of an asymptotic expansion: the wavefunction amplitude scale is $e^{-\kappa a}$, and the transmission probability scale is $e^{-2\kappa a}$, with an interface prefactor. Increasing width by $\Delta a$ multiplies this approximation by $e^{-2\kappa \Delta a}$. At fixed incident energy, increasing barrier height increases the tunnelling decay constant as $\sqrt{V_0-E}$; heavier nonrelativistic particles also have stronger suppression. For the numerical example in rectangular barrier transmission probability, the approximation gives $0.01397$, compared with the exact $0.01388$.

The regime condition matters. Near $E=V_0$ at fixed width, $\kappa a$ tends to zero and the opaque barrier approximation fails; use the exact threshold value. For $\kappa a\ll 1$, expansion of the hyperbolic sine givesThis tends to one as $a\to 0$ at fixed $E,V_0$, but the condition $\kappa a\ll 1$ alone does not imply nearly perfect transmission if the remaining factor is large. At $E>V_0$, use the oscillatory formula and its scattering resonances, rather than an exponential decay law.

Take the increasing event that a finite square torus graph contains some long horizontal or vertical rectangle crossing. Translations and a quarter turn make this a transitive increasing event on its bonds. A positive critical crossing bound and the Friedgut-Kalai sharp threshold theorem make its supercritical probability tend to one. A bounded family of shorter, thicker rectangles intercepts every possible long crossing. The square-root trick for positively associated events transfers the high probability to each fixed intercepting rectangle. Using every possible translate would lose the useful bounded exponent.

A rectangle with all four side lengths equal. It is a regular four-sided polygon. Rotations and translations preserve this definition; its sides need not be parallel to the coordinate axes.

Each real projective plane has cells $e_0,e_1,e_2$ with $d{e}_2=2e_1$ and $d{e}_1=0$. Their product cellular chain complex has ranks $1,2,3,2,1$. Its degree-two cycle $e_1\times e_1$ has order two in homology. The degree-three cycle $e_2\times e_1+e_1\times e_2$ also has order two, because its double is $d(e_2\times e_2)$. Consequently $H_0=\mathbb{Z}$, $H_1=(\mathbb{Z}/2{)}^2$, $H_2=H_3=\mathbb{Z}/2$ and all other groups vanish. The degree-three torsion is also the Tor functor term in the Künneth theorem.

A degree-$a$ homogeneous real polynomial defines a section of the $a$th tensor power of the dual real tautological line bundle over ${\mathbb{RP}}^2$. Its first Stiefel–Whitney class is $au$, where $u$ generates $H^1({\mathbb{RP}}^2;{\mathbb{F}}_2)$. By parity of zeros of a real line-bundle section, a generic zero curve has mod-two dual class $au$. Two transverse zero curves of degrees $a,b$ therefore have intersection parity $⟨ab{u}^2,[{\mathbb{RP}}^2{]}_2⟩=ab$, since two distinct projective lines intersect once. For example, the line $z=0$ and the conic $x^2+y^2-z^2=0$ are disjoint in the real projective plane, although their degree product is two.