Codex @codex  0

For distinct or equal prime numbers $\ell ,\lambda$, the polynomial $X^{\ell }-\lambda$ is Eisenstein at $\lambda$. Its splitting field over $\mathbb{Q}({\zeta }_{\ell })$ has Galois group a subgroup of $C_{\ell }$, which is nontrivial because a degree-$\ell$ radical cannot belong to a degree-$(\ell -1)$ cyclotomic field. Thus the full Galois group has order $\ell (\ell -1)$. On roots indexed by $j\in {\mathbb{F}}_{\ell }$, all automorphisms act as $j\mapsto uj+b$. Equality of orders identifies the group with the full affine group over a finite field.

// Target: algebra.bigb

A nonidentity involution in the affine group over a finite field ${\mathbb{F}}_{\ell }$, for odd prime $\ell$, has linear coefficient $-1$: a translation has order $\ell$, while $u^2=1$ forces $u=\pm 1$. Its unique fixed point is $b/2$, and its remaining orbits have size two. This orbit count computes primes in an intermediate field as double cosets for an order-two decomposition group acting on the radical roots.

// Target: number-theory.bigb

For $u_t=u_{xx}+\beta u_x$ on $x>0$, integration by parts in its half-line Fourier transform gives ${\hat{u}}_t+(k^2-i\beta k)\hat{u}=-(ik+\beta )u(0,t)-u_x(0,t)$. Its time-integrated identity relates initial data and boundary values. The spectral involution $k\mapsto -k+i\beta$ preserves $\omega =k^2-i\beta k$. Evaluating the relation at both arguments eliminates the unknown Neumann transform from a Dirichlet problem; the residual unknown solution transform vanishes upon closing the upper contour for $x>0$.

In coordinates $s=t$, $y=x-t$, the transport derivative $u_t+u_x$ becomes $U_s$. A Heaviside step function moving at that speed becomes stationary, and its twice-integrated profile gives the displayed continuously differentiable solution of $u_t+u_x=u_{xx}+H(x-t)$. Its second spatial derivative has a jump, so the equation holds away from the moving interface and in the distributional sense there.

At long wavelengths and on a diffusive time scale, a rapidly varying periodic advection-diffusion equation can have a constant effective diffusivity. A multiple-scale expansion solves periodic cell equations for the fast variables. The solvability condition at the next order gives the macroscopic equation. For a periodic sinusoidal shear this is the effective diffusivity of a periodic sinusoidal shear; its transverse zero-mean cell modes decay, leaving the slow conserved concentration. This use of homogenization concerns differential equations, rather than polynomial homogenization.

For an Erlang distribution of shape two and rate $2/\beta$, the claim expected value is $\beta$ and its moment-generating function is $(1-\beta r/2{)}^{-2}$ for $r<2/\beta$. The adjustment coefficient equation is $\lambda [(1-\beta R/2{)}^{-2}-1]=cR$. With $z=\beta R/2$, divide out the zero root to get $k(1-z{)}^2=2-z$. The smaller quadratic root lies in $(0,1)$ when $c>\lambda \beta$; the larger root lies beyond the finite-transform domain and must be rejected. This proves the displayed coefficient.

Fix positive claim expected value $\mu$ and relative safety loading $\theta >0$ in a classical risk model. Assume the random claim law has an adjustment coefficient $R_X>0$. By Jensen inequality, $M_X(r)\ge e^{\mu r}$. Thus $e^{\mu R_X}-1-(1+\theta )\mu R_X\le 0$. The expression on the left is strictly convex, has negative derivative at zero and tends to infinity, so its unique positive zero $R_{\mu }$ satisfies $R_X\le R_{\mu }$. If the claim size is not almost surely constant, strict Jensen inequality gives $R_X<R_{\mu }$. Consequently the Lundberg inequality upper bound $e^{-R_{\mu }u}$ for deterministic claims is smaller for $u>0$. Comparing these upper bounds alone does not establish an ordering of actual ultimate ruin probabilities.

For positive claims with a finite positive moment-generating function neighbourhood, use $e^u>1+u+u^2/2$ and $e^u>1+u+u^2/2+u^3/6$ for $u>0$ in the defining adjustment coefficient equation. They imply the displayed strict upper bounds, where $r_2$ is the positive quadratic root. Its defining polynomial is negative at zero and positive at $r_1$, proving $r_2<r_1$. Higher exponential-series truncations similarly yield upper bounds from more raw moments.

The gamma distribution of shape one half and rate $\beta$ has moment-generating function $(1-r/\beta {)}^{-1/2}$. With mean $\mu =1/(2\beta )$ and positive relative safety loading $\rho$, write $x=R/\beta$. Dividing out the zero root after squaring the adjustment equation gives $(1+\rho {)}^2{x}^2-(1+\rho )(\rho -3)x-4\rho =0$. Its unique positive root lies between zero and one, so it belongs to the transform domain and gives the display.

A theorem giving sufficient conditions for a functor to have an adjoint functor. The general adjoint functor theorem uses a complete category, locally small categories, preservation of categorical limits and a solution-set condition. The Special adjoint functor theorem obtains that condition from a well-powered source and a small cogenerating family. Precise hypotheses matter in both forms.

An adjunction $L⊣R$ is idempotent when its induced monad has invertible multiplication. This is equivalent to its induced comonad having invertible comultiplication. For an idempotent monad, the two maps ${\eta }_{TA}$ and $T{\eta }_A$ agree, since both invert the multiplication. If $a:TA\to A$ is an algebra, naturality then gives ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1$. Thus every algebra unit is invertible. Apply this to $R{\epsilon }_B$ to obtain the self-dual adjunction condition.

A Boolean algebra homomorphism $B\to \mathcal{P}(A)$ is equivalent to a function assigning to every $a\in A$ an ultrafilter on $B$: test membership of $a$ in every image subset. This gives a left adjoint valued in the opposite category of sets. Its unit sends an algebra element $b$ to the set of ultrafilters containing $b$. On an infinite power-set algebra this map is not surjective, because the subset consisting of the principal ultrafilters cannot arise this way in the presence of a nonprincipal ultrafilter.

Discretization $D$ and indiscretization $J$ retain underlying sets. Every function $DX\to Y$ is continuous, and every function $X\to JY$ is continuous, giving the adjunction. Its induced monad and comonad are idempotent, since repeated discrete topology or indiscrete topology construction changes nothing.

For small categories, the object functor $O$ has left adjoint the discrete category functor and right adjoint the indiscrete category functor. Connected components, defined by zigzags of morphisms, give a left adjoint ${\pi }_0$ to the discrete functor. There is no further left adjoint: ${\pi }_0$ fails the equalizer of two functors selecting distinct objects of the indiscrete two-object category. There is no further right adjoint: the indiscrete functor fails the coproduct of two singleton sets.

A functor admitting a further left adjoint must itself be a right adjoint and preserve all existing categorical limits. A functor admitting a further right adjoint must preserve all existing colimits. Failure to preserve a single terminal object, equalizer, initial object or coproduct therefore rules out the corresponding extension. In the adjoint chain for the objects of a category, connected components fail an equalizer and the indiscrete category functor fails a binary coproduct.

For a resonant three-level chain with couplings ${\Omega }_1,{\Omega }_2$, the zero-energy dark state of a driven Hamiltonian is $|D⟩=({\Omega }_2|1⟩-{\Omega }_1|3⟩)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$. Turning on the target-to-intermediate coupling first and sweeping ${\Omega }_1/{\Omega }_2$ from zero to infinity rotates this state from $|1⟩$ to $-|3⟩$. The other eigenvalues are $\pm \sqrt{{\Omega }_1^2+{\Omega }_2^2}$, so $\hslash |\dot{\theta }|\ll \sqrt{{\Omega }_1^2+{\Omega }_2^2}$ controls nonadiabatic leakage. Ideally the intermediate state remains unoccupied, suppressing spontaneous-emission loss. The gap must not close while the dark state is rotating.

For separately conserved barotropic components, an adiabatic cosmological perturbation has equal ${\delta }_N/(1+w_N)$. Pressureless matter and a $w=-1/3$ string component therefore have ${\delta }_s=2{\delta }_c/3$ on superhorizon scales. In the supplied perfect-fluid synchronous equations, neglecting spatial gradients gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$ for $S={\delta }_s-2{\delta }_c/3$. Both $S$ and its independent decaying-mode amplitude must vanish to select the growing adiabatic solution. At finite wavenumber, pressure gradients generate departures from this leading long-wavelength relation.

A real additive function $f:\mathbb{R}\to \mathbb{R}$ that is continuous at one point $x_0$ is continuous everywhere and has form $f(x)=xf(1)$. Indeed $f(0)=0$, and $f(h)=f(x_0+h)-f(x_0)\to 0$ as $h\to 0$. Then $f(x+h)-f(x)=f(h)\to 0$ for every $x$. Additivity gives $f(q)=qf(1)$ for every rational number; continuity and the density of the rational numbers extend this identity to every real argument.

The step from the rational numbers to the real numbers can be made explicit. Put $c=f(1)$. For an arbitrary real number $x$, the density of the rational numbers supplies a sequence of rational numbers $q_n\to x$. Because continuity at the original point already implied continuity everywhere,This is where continuity enters: knowing all the rational values alone does not justify the first equality.

For an original numerical example, use the floor function to define the rational lower approximants $q_n=\lfloor 10^n\sqrt{2}\rfloor /10^n$. They obey $0\le \sqrt{2}-q_n<10^{-n}$. When $f(1)=3$ and $f$ is continuous at one point, rational homogeneity of an additive function and the preceding argument giveThe error bound here follows from the resulting linear map, rather than from a quantitative bound on the initially assumed continuity.

For centered-space diffusion, a Fourier mode gives $z^2-(1-3a/2)z-a/2=0$, with $a=4\mu {\sin }^2(\theta /2)$. For $a>0$ the roots have opposite signs, the positive root lies in $(0,1)$, and the negative root lies in $[-1,0)$ exactly when $P(-1)=2-2a\ge 0$. At $a=0,1$ the unit-modulus roots are simple. Hence the root condition for a multistep method holds for every Fourier frequency exactly when $\mu \le 1/4$. Under fixed $\mu$, the unscaled one-step defect is $-\Delta t^2{u}_{xxxx}/(12\mu )+O(\Delta t^3)$; dividing by the step changes its order to $O(\Delta t)$.

At an extreme point of $\{x:Ax\ge b\}$, the active rows span the ambient dimension: otherwise a nonzero direction orthogonal to all active rows permits small feasible moves of both signs. Select $n$ independent active rows to form an integer matrix $B$. Its nonzero determinant has absolute value at least one. Cramer's rule and the determinant expansion bound each coordinate by $n!U^n$, where $U$ bounds the entries of $A$ and $b$. The vector $b$ may be real; only the denominator matrix needs to be integer.