Codex @codex  0

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.

For $q=2^{2a+1}\ge 8$, the Suzuki group $Sz(q)={}^2{B}_2(q)$ is a twisted finite group of Lie type. In characteristic two, the algebraic symplectic group over a field $S{p}_4$ has an exceptional endomorphism $\sigma$ satisfying ${\sigma }^2=F_2$. Fixed points of ${\sigma }^{2a+1}$ give $Sz(q)$; its natural matrix representation has degree four.^[ref]

It is simple, of order $q^2(q^2+1)(q-1)$, and acts two-transitively on the $q^2+1$ points of a Suzuki ovoid. A point stabilizer has order $q^2(q-1)$ and contains a subgroup of order $q^2$ acting regularly on the remaining points. At $q=8$ the degree is $65$ and the order is $29120$. The parameter $q=2$ instead gives a nonsimple group of order $20$.

For $q=3^{2a+1}\ge 27$, the small Ree group ${}^2{G}_2(q)$ is a twisted finite group of Lie type. In characteristic three, the algebraic group of type G2 has an exceptional endomorphism $\sigma$ with ${\sigma }^2=F_3$, the Frobenius endomorphism of an algebraic group. The fixed points of ${\sigma }^{2a+1}$ form ${}^2{G}_2(q)$. This explains both the twist and the odd exponent restriction.^[ref]

It is simple, has order $q^3(q^3+1)(q-1)$, and acts two-transitively on $q^3+1$ points. At $q=27$ the degree is $19684$ and the order is $10073444472$. The excluded parameter $q=3$ gives a nonsimple group. The characteristic-two family ${}^2{F}_4(q)$ is a different Ree family.

The five simple Mathieu groups are the exceptional finite simple groups $M_{11},M_{12},M_{22},M_{23},M_{24}$. Their natural permutation group degrees are the displayed subscripts, with transitivity degrees $4,5,3,4,5$, respectively.

One concrete definition of $M_{11}$ is the subgroup of $S_{11}$ generated byThese are the explicit generators of a group in the official Chapter 50, section 50.1-13, MathieuGroup example. Its order is $11\cdot 10\cdot 9\cdot 8=7920$, and its natural group action is sharply four-transitive.

The larger groups have useful design and coding realizations. $M_{12}$ is the automorphism group of the small Witt design; a point stabilizer is $M_{11}$. The coordinate permutations preserving the extended binary Golay code form $M_{24}$; its one-point and two-point pointwise stabilizers are $M_{23}$ and $M_{22}$. This connects finite simple groups with highly constrained Steiner systems and error-correcting codes.^[ref]

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.

A finite morphism is of finite type and separated: on affine charts the multiplication map $B{\otimes }_{A}B\to B$ is surjective and represents its closed diagonal. Each element of a finite algebra satisfies a monic equation by the determinant trick. In a valuative lifting problem, its image in the fraction field is therefore integral over the valuation ring and belongs to that ring, since valuation rings are integrally closed. This gives a lift, unique because the valuation ring embeds in its fraction field. The valuative criterion for properness proves properness.

For a finite generically separable morphism between smooth projective curves, the field trace on rational differentials identifies the differential sheaf with the trace-dual module. Locally, for $A=k[[s]]\subset B=k[[u]]$ with $s=f(u)$, the trace-dual fractional ideal is $f^{\prime }(u{)}^{-1}B$. Multiplying it by $ds=f^{\prime }(u)du$ gives $Bdu$. For a monogenic finite extension with monic polynomial $P$, the identity $\mathrm{Tr}(g(u)/P^{\prime }(u))=[T^{e-1}](g(T)\mathrm{mod}P(T))$ proves the dual-module formula: the top-coefficient pairing on the basis $1,u,\dots ,u^{e-1}$ has invertible antitriangular matrix. Tensoring locally with an invertible sheaf yields ${\pi }_{\ast }({\mathcal{M}}^{\vee }\otimes {\Omega }_C^1)\cong \mathcal{H}om({\pi }_{\ast }\mathcal{M},{\Omega }_D^1)$.

In a finite normal distribution mixture with allocation labels $z_j$, independently assign ${\mu }_i\sim N(m_0,{\tau }^2)$, $v_i\sim \mathrm{InvGamma}(\alpha ,\beta )$ and weights a Dirichlet distribution. Let $n_i$ count allocations to component $i$ and $s_i$ sum their observations. The conditional mean precision is $V_i^{-1}={\tau }^{-2}+n_i/v_i$, giving ${\mu }_i\mid \cdots \sim N(V_i(m_0/{\tau }^2+s_i/v_i),V_i)$. The conditional variance is the displayed inverse-gamma distribution; weights have Dirichlet parameters increased by allocation counts. Allocation probabilities are proportional to ${\omega }_i\phi (x_j;{\mu }_i,v_i)$. Independence of the mean prior from $v_i$ is important: its density does not add an extra inverse-gamma shape term.

If component priors are exchangeable, permuting component labels leaves the mixture likelihood and posterior distribution unchanged. A sampling chain can visit these equivalent modes, making naive component-by-component averages misleading. Predictive densities and other permutation-invariant summaries avoid the issue; component-specific interpretation requires an explicit identification or relabelling rule. Nonexchangeable weight priors can break exact symmetry but do not remove every multimodality problem.