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Algebraic irreducibility excludes every proper nonzero invariant vector subspace, including those that are not closed. It is stronger than topological irreducibility, which excludes only closed invariant subspaces. For a nonzero irreducible action, every nonzero vector is cyclic in the algebraic sense: $A\xi =X$.

A convex terminal payoff has a convex pricing extension in the binomial market, because risk-neutral pricing is a positive weighted sum of rescaled copies of the payoff. Let the stock factors be $u,d$, the riskless gross factor be $R$, and $q=(R-d)/(u-d)$. The replicating portfolio in a binomial market satisfiesThe weights sum to one. Convexity orders the secant slopes on the adjacent intervals $[d^{2}s,uds]$ and $[uds,u^{2}s]$, proving the displayed inequalities. The inequalities need not be strict: an affine payoff has constant stock holding.

For a finite commuting family in a unital Banach algebra, divide each $x_j$ by $r(x_j)+\epsilon /2$. The spectral radius formula makes each normalized element's powers bounded. Commutativity lets every product be reordered into one power of each generator, so the entire generated semigroup is bounded. Renorming a bounded multiplicative semigroup gives the displayed bounds simultaneously. Applying this to two commuting elements proves submultiplicativity and subadditivity of their spectral radius.

For $1<s\le \sqrt{2}$, set $h_s(x)=-(s-1)x$ and $J_s=[-(s-1),s-1]$. The tent map sends $J_s$ to $[1+s-s^2,1]$, with disjoint interiors, and $T_s^2(J_s)\subseteq J_s$. Directly,Successive squaring eventually brings every $s>1$ into $(\sqrt{2},2]$. A horseshoe for an interval map of an iterate in the renormalized coordinate lifts to a horseshoe of an iterate of the original map.

// Target: dynamical-systems.bigb

A standard thermal Cosmic neutrino background with one populated neutrino and one antineutrino helicity per species has $n_{\nu }+n_{\bar{\nu }}=3n_{\gamma }/11$ after electron-positron annihilation in cosmology. If this species is nonrelativistic today and dominates a spatially flat matter universe, equate $m_{\nu }(n_{\nu }+n_{\bar{\nu }})$ to the critical density $3H_0^2{M}_P^2/(8\pi )$, using the unreduced Planck mass. The photon number density gives the displayed conditional mass. Extra populated states or several massive species change the abundance count; vacuum-dominated cosmology changes the density available for matter.

Suppose $0<p_L\le p\le p_U<1$ and $0<a_L\le a\le a_U<1$. The formula for relative risk from group compositions increases with $a$ and decreases with $p$, since $a/(1-a)$ increases and $(1-p)/p$ decreases. This monotonicity proves the displayed bounds. If percentages are rounded to the nearest whole percentage point, each corresponding proportion lies within $0.005$ of its displayed value. These bounds describe rounding uncertainty; they are not a confidence interval for sampling uncertainty.

Compare expected utility of $W(1+\varphi )$, where $\mathbb{E}\varphi =0$, with the certainty equivalent $W(1-\psi )$. A second-order Taylor expansion gives $U(W)+W^2{U}^{\prime \prime }(W)\mathbb{E}{\varphi }^2/2$ for the risky side and $U(W)-W{U}^{\prime }(W)\psi$ for the sure side. Equality gives the displayed proportional risk premium. This requires a genuinely small-shock family with controlled remainders; small variance alone does not control rare large wealth shocks.

Modification of the interaction between electric charges by a medium's electrical response. In the simplest Wannier-Mott exciton model, the Coulomb potential is reduced by a constant relative permittivity; actual screening can depend on spatial and temporal scales.

For a based CW triad with connected intersection $A$, if $(B,A)$ is $p$-connected and $(C,A)$ is $q$-connected, $p,q\ge 1$, the comparison map is an isomorphism for $i<p+q$ and a surjection for $i=p+q$. This connectivity form of the Blakers-Massey theorem controls the failure of ordinary relative homotopy to satisfy unrestricted excision.

If a pointed CW complex is covered by $l$ contractible subcomplexes containing the basepoint, every reduced class lifts to a relative class for each appropriate subcomplex. Their relative cup product lies in $h^{\ast }(X,{\bigcup }_j{A}_j)=h^{\ast }(X,X)=0$, so every product of $l$ reduced classes vanishes. The conclusion also holds if $X$ is path connected. Without either condition a two-point space covered by its two points is a counterexample in ordinary degree-zero cohomology.

A relative cycle for $A\subset B$ is a chain $b\in C_n(B)$ whose boundary lies in $A$. Equivalently its coset is a homological cycle in $C_{\ast }(B,A)$. Unlike an absolute cycle, its boundary need not be zero. It represents a relative homology class.

The coordinate hyperplane is an embedded submanifold, as standard projective charts flatten its defining homogeneous coordinate. Its complement is the affine chart ${\mathbb{R}}^n$ and is connected. The lemma regular zero hypersurfaces have disconnected complements therefore excludes its being the zero set of a smooth real function with zero a regular value.

Classically, a regular formula sequent that is not provable has a set-valued countermodel, with empty sorts allowed. Start with the finite diagram of its antecedent, successively adjoin witnesses to the existential consequences of each axiom, and identify elements when equality axioms require it. Every new fact about the original parameters has a finite derivation, so the nonderivable consequent remains false. The filtered union of this witness construction is a model. The conservativity property of this chase construction is studied in Forssell and Lumsdaine's paper.

A chosen regular system of parameters gives the displayed completion isomorphism by sending $T_i$ to $t_i$. The associated graded ring of a regular local ring is the polynomial ring on their initial forms. Successively removing the homogeneous initial form of each remainder gives a unique formal series for every local function. The Krull intersection theorem embeds the local ring in its completion. The power-series characterization is also given in Stacks Project Lemma 10.160.10.

The depth of a Noetherian local ring is the maximal length of an $R$-regular sequence in its maximal ideal. It is at most its Krull dimension. Quotienting by a nonzero divisor lowers depth by one. This invariant measures how many independent successive nonzero-divisor equations the local algebra supports.

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.