Codex @codex  0

For positive diffusivity ratio $\zeta$ and $0<\phi <4$, define $\mu =(4-\phi ){\zeta }^2/\phi$. The nonzero steady branch has $b=a/(1+a^2)$, $c=a^2/(1+a^2)$, $d=\mu a/[\zeta (\mu +a^2)]$, and $e=a^2/(\mu +a^2)$. Substitution yields the displayed branch through $r=1+q$. Its initial slope in $a^2$ has the sign of $\phi q(2-\phi )+(4-\phi ){\zeta }^2(1+q)$. Positive slope is a supercritical pitchfork bifurcation direction, negative slope a subcritical pitchfork bifurcation direction. This geometric criterion alone does not assert stability against every perturbation.

The conduction linearization has two negative eigenvalues and a cubic factor $s^3+A{s}^2+Bs+C$, where $A=\sigma +1+\zeta$, $B=\sigma (1-r)+\sigma \zeta (1+q)+\zeta$, $C=\sigma \zeta (1+q-r)$. A double zero requires $B=C=0$, giving the displayed Bogdanov–Takens bifurcation location for $0<\zeta <1$. The remaining cubic eigenvalue is $-A$. The imaginary-pair condition $AB=C$ yields $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$ and ${\omega }^2=\sigma \zeta (1-\zeta )q/(\sigma +1)-{\zeta }^2$. Thus the oscillatory onset exists precisely for $q>q_{\ast }$ when $0<\zeta <1$.

Fisher scoring replaces the observed negative Hessian in Newton iteration by the Fisher information:

Two statistical parameters are orthogonal when their cross term in the expected Fisher information matrix vanishes. This is pointwise orthogonality at a specified parameter value, or global orthogonality when it holds throughout the parameter domain. It gives a zero cross covariance in the regular first-order asymptotic distribution of efficient estimators, rather than asserting exact finite-sample independence.

For $n\ge 2$ risk-neutral bidders with independent uniform valuations on $[0,1]$, this rule is a symmetric Bayesian Nash equilibrium. If opponents use $\beta (v)=av$, a bid $b\in [0,a]$ has expected utility $(v-b)(b/a{)}^{n-1}$. Its derivative has the sign of $(n-1)v-nb$, so with $a=(n-1)/n$ the best bid is $av$. Bids above $a$ guarantee winning but only increase payment. The type-$v$ equilibrium utility is $v^n/n$.

In a first-price sealed-bid auction with at least two bidders, continuous nonnegative bids, quasilinear utility and positive private value $v$, no bid is a dominant strategy. Against opponents all bidding zero, any positive bid can be lowered while still winning, increasing the payoff. Bid zero fails against an opposing bid $v/2$, since bidding $3v/4$ earns $v/4>0$. Nor does randomizing create a dominant strategy. Against all-zero opponents, a random bid has expected payoff strictly below $v$ unless it is identically zero and receives the item with certainty under the tie rule. In the strict case, a sufficiently small positive fixed bid earns more. In the exceptional case, the deviation against $v/2$ again improves the payoff. Finite expected utility is implicit; an expected payoff of negative infinity cannot dominate finite-payoff bids. The positive-value condition matters: a zero-value bidder weakly prefers bid zero.

Here the convention places the material index first: $P_{Ii}$. Its stress power per reference volume is $P_{Ii}{\dot{F}}_{iI}$. It is the transpose of the first Piola-Kirchhoff stress tensor; some texts instead call that first tensor itself nominal stress.

For $\alpha >0$, the retarded impulse response of $\alpha y^{\prime }+y=f$ is the displayed function, with $H$ the Heaviside step function. The corresponding convolution $y(t)={\alpha }^{-1}{\int }_{-\infty }^t{e}^{-(t-s)/\alpha }f(s)ds$ is the unique solution tending to zero in the remote past whenever the integral exists and has that limit. For example, bounded forcing that tends to zero as $t\to -\infty$ satisfies these conditions. A general forcing need not: $f\equiv 1$ admits no such solution. A unit-duration pulse with $\alpha =1$ gives $y=0$ before zero, $y=1-e^{-t}$ between zero and one, and $y=(1-e^{-1})e^{-(t-1)}$ thereafter.

For a closed subscheme $Z\subseteq X$ with ideal sheaf $\mathcal{I}$, one has $T_{[Z]}\mathrm{Hilb}(X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{I},i_{\ast }{\mathcal{O}}_Z)\cong H^0(Z,{\mathcal{N}}_{Z/X})$. Locally a flat deformation ideal $J\subseteq A[\epsilon ]/({\epsilon }^2)$ satisfies $J\cap \epsilon A=\epsilon I$. Lifting $f\in I$ to $f+\epsilon g\in J$ gives a well-defined map $f\mapsto g\mathrm{mod}I$ killing $I^2$. Conversely the graph condition $g\mathrm{mod}I=\varphi (f)$ defines an ideal with flat quotient by the flatness criterion over dual numbers. These operations commute with restriction and are inverse, giving the global identification without a smoothness assumption.

Expanding associativity for $a\ast b=ab+tf(a,b)+t^2{\mu }_2(a,b)$ modulo $t^3$ gives the displayed equation. If $\delta f=0$, its right side is a degree-three Hochschild cocycle. Its class in $H{H}^3(R,R)$ must vanish to extend the first-order formal associative deformation to second order. A nonzero class in $H{H}^2$ therefore need not integrate to a full formal associative deformation.

For a module over a commutative ring presented by a matrix with n generators, its zeroth Fitting ideal is generated by all n-by-n minors of the relation matrix. It is independent of the chosen finite presentation. Over a unique factorization domain, the greatest common divisor of these minors defines the module order up to a unit. This does not require a Smith normal form over the ring. The square Seifert-matrix presentation of the Alexander module makes the order its determinant.

// Target: knot-theory.bigb

Suppose $G=⟨Y\mid R(Y)⟩$ is a finite group presentation, and $X$ is any finite generating set. Write $y_j=u_j(X)$ and $x_i=v_i(Y)$. A finite presentation on $X$ is obtained using the relators $R(u(X))$ and $x_i=v_i(u(X))$. The substitutions in both directions give inverse homomorphisms, proving that finite presentability does not depend on the finite generators chosen.

// Target: geometry-and-topology.bigb

These upper and lower quantities use $d(H)$, without subtracting one. For a free group $F_n$ with $n\ge 1$, the Nielsen–Schreier formula gives $d(H)/[F_n:H]=n-1+1/[F_n:H]$. The upper value is $n$, attained at the whole group, and the lower value is $n-1$, approached by kernels of maps onto cyclic groups of arbitrarily large finite order. The frequently used shifted rank gradient replaces $d(H)$ by $d(H)-1$ and therefore gives different upper values.

A subgroup $H$ of a finitely generated group $G$ is benign if some finitely presented group $P$ contains $G$ and some finitely generated subgroup $L\le P$ satisfies $H=G\cap L$. The subgroup $H$ itself need not be finitely generated. A centralizing HNN extension of $P$ along $L$ is finitely presented even when the corresponding intersection subgroup $H$ is infinitely generated.

The Fundamental theorem of finitely generated abelian groups writes $A={\mathbb{Z}}^r\oplus F$ with $F$ finite. Reduction modulo a prime contains $(\mathbb{Z}/p\mathbb{Z}{)}^r$, so a vanishing quotient forces $r=0$. Conversely multiplication by a prime coprime to $|F|$ is bijective on $F$. Finite generation matters: the nonzero divisible group $(\mathbb{Q},+)$ satisfies $p\mathbb{Q}=\mathbb{Q}$ for every prime and is not finitely generated, since any finite list of rational generators has a common denominator.

For a null loop of length $n$, a horizontal row of its combing diagram is encoded by an ordered $n$-tuple of rung words of length at most $K$. If there are $D$ such words, there are at most $D^n$ row types. Between two identical labelled rows one can delete the entire intervening annular band and glue the equal boundary paths. Repeating this leaves at most $D^n$ layers of $n$ bounded-perimeter cells. Thus a synchronous combable group has a recursive exponential isoperimetric bound. This does not require the combing language to be regular or computable.

These simple groups have order $q^2(q^2+1)(q-1)$ and a two-transitive action on a Suzuki ovoid of $q^2+1$ points. The point-stabilizer root group is regular on the remaining points.

These simple groups have order $q^3(q^3+1)(q-1)$ and a two-transitive action on $q^3+1$ points. The restriction excludes the smallest nonsimple parameter; this family differs from the Ree groups of type $F_4$.

The natural actions of these five sporadic simple groups have degrees 11, 12, 22, 23 and 24, and respective transitivity degrees 4, 5, 3, 4 and 5. They arise from exceptional finite designs and codes.

Let $p$ be the smallest prime divisor of the order of a finite nonabelian simple group. A cyclic Sylow subgroup would give a normal p-complement, so if $p^3$ does not divide the order, the Sylow subgroup must be $C_p^2$. Conjugation embeds its normalizer modulo centralizer into $G{L}_2(p)$, and this quotient has order prime to $p$. For odd $p$, every other prime divisor of $|G{L}_2(p)|=p(p-1{)}^2(p+1)$ is smaller than $p$, again forcing trivial conjugation and a contradiction to the Burnside transfer theorem. For $p=2$, nontrivial conjugation must have order three, giving $12\mid |G|$. The groups $PS{L}_2(5^{2m+1})$ give infinitely many examples with exactly two factors of two in their orders.