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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 values of a sequence at different indices, or its finite differences. A first-order autonomous system has the displayed form, with a scalar or vector state. Writing $z_{n+1}-z_n=F(z_n)-z_n$ gives its equivalent finite-difference form. Its fixed point stability is determined by the Jacobian matrix of the update and the unit circle, whereas a continuous-time differential equation uses the real parts of its linearized 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.

A nontrivial stationary front of the real-coefficient cubic-quintic amplitude equation joining a nonzero pattern to zero must have constant phase. Writing $A=R{e}^{i\varphi }$ gives $R^2{\varphi }^{\prime }=J$; the finite first integral contains $J^2/(2R^2)$, so approaching zero forces $J=0$. Equal endpoint potentials then give $R_{-}^2=3\alpha /4$ and the unique parameter ${\mu }_M=-3{\alpha }^2/16$. The explicit profile is $A=e^{i{\varphi }_0}\sqrt{(3\alpha /4)/(1+e^{\sqrt{3}\alpha (x-x_0)/2})}$. This proves existence as well as the necessary equal-potential condition. The zero solution is excluded from the term front.

With initially independent actions, same-round independence is preserved by reactive updates: each next action uses the other independent previous action and fresh independent randomization. Marginal cooperation follows $x^{\prime }=q_1+(p_1-q_1)y$, $y^{\prime }=q_2+(p_2-q_2)x$. If the product of response slopes has modulus below one, these recurrences converge and the joint distribution is their product. Averaging $R,S,T,P$ over that distribution gives the long-time payoffs. At deterministic response slopes of modulus one, initial-memory-specific cycles must instead be averaged directly.

Tit for tat cooperates initially and then repeats the opponent's previous action. Two such players cooperate forever without mistakes under this initial convention. It retaliates against defection and returns to cooperation after the opponent cooperates.

Always cooperate chooses cooperation independently of the opponent's actions.

Always defect chooses defection independently of the opponent's actions.