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An order in a number field $K$ is a subring $A\subseteq {\mathcal{O}}_K$ containing $1$ and having full rank $[K:\mathbb{Q}]$ as an additive abelian group. Its finite index in the ring of integers of a number field is related to its discriminant by the discriminant-index formula for an integral lattice.

A ring homomorphism from a ring to itself. Surjectivity need not imply injectivity in general, but a surjective ring endomorphism of a Noetherian ring is a ring automorphism.

The stabilizer is the automorphism group of a quiver representation, an open set in its endomorphism algebra. Thus $\dim {\mathcal{O}}_X={\sum }_i{n}_i^2-\dim {\mathrm{End}}_Q(X)$. The Ringel form gives orbit codimension $\dim {\mathrm{Ext}}_Q^1(X,X)$. Since orbits are locally closed and the representation space is irreducible, rigidity is equivalent to an open dense orbit.

If a line bundle on a smooth projective curve has both $h^0(\mathcal{L})>0$ and $h^0({\omega }_C\otimes {\mathcal{L}}^{-1})>0$, the displayed inequality holds. Write these dimensions as $r,s$. The product dimension bound for sections on a curve gives $r+s-1\le g$, and the Riemann-Roch theorem gives $r-s=\mathrm{deg}\mathcal{L}+1-g$. Adding gives $2r\le \mathrm{deg}\mathcal{L}+2$.

A function $g$ on $[0,\infty )$ is directly Riemann integrable when its upper and lower sums on an equal-width mesh are absolutely finite and converge to the same finite integral as the mesh width tends to zero. A locally absolutely continuous integrable function with integrable derivative has this property: the upper-minus-lower sum is bounded by mesh width times its total variation. The derivative bound therefore gives bounded variation as well as tail control. This stronger form of integrability is a hypothesis of the key renewal theorem.

A bounded function on $[0,1]$ is Riemann integrable if and only if its upper Darboux sum minus its lower Darboux sum on the equal-length partition $D_n$ tends to zero. Sufficiency follows directly from the Riemann integrability criterion. Necessity follows by comparing with a fixed partition whose gap is small and using the finite bad-cell estimate for Darboux sums. The equal-length partitions need not be nested.

If $f$ is Riemann integrable and $g$ is Lipschitz on an interval containing the range of $f$, then $g\circ f$ is Riemann integrable. With Lipschitz constant $K$, the oscillation on each partition cell satisfies $\mathrm{osc}(g\circ f)\le K\mathrm{osc}(f)$, and hence the gap between Darboux sums is multiplied by at most $K$. A continuously differentiable $g$ is Lipschitz on each closed bounded interval by the mean value theorem and the extreme value theorem applied to $g^{\prime }$.

For constant nonzero $c$, putting $y=x^{\prime }/(cx)$ in $y^{\prime }+c{y}^2+a(t)y+b(t)=0$ cancels the quadratic derivative terms and gives $x^{\prime \prime }+a(t)x^{\prime }+cb(t)x=0$. The logarithmic derivative is defined on intervals where $x\ne 0$, and multiplying $x$ by a nonzero scalar does not change $y$. For variable $c(t)\ne 0$, the derivative coefficient becomes $a-c^{\prime }/c$.

A return level answers the inverse question to a return period: which threshold has the specified rarity? Let $F(z)=F_{\mathrm{annual}}(z)=\mathbb{P}(M_j\le z)$ be the common cumulative distribution function of annual sample maxima in a model with stationarity. For a continuous, strictly increasing $F$ and $T>1$, the $T$-year return level isIt has the same units as $M_j$, whereas $T$ is a count of annual blocks. Independence between years is unnecessary for this one-year quantile definition; it is needed for the geometric waiting and multi-year formulas under return period. For a discrete probability distribution, use the generalized quantile function $F^{-1}(q)=\inf \{z:F(z)\ge q\}$. Then $\mathbb{P}(M_j>z_T)\le 1/T$, with equality not guaranteed at jumps; an exact requested exceedance probability may be unattainable.

For a concrete standard model, choose the maxima convention of the Gumbel distribution, with location $\mu$ and scale $b>0$:The second formula follows by taking logarithms twice and solving $F(z_T)=1-1/T$. The maxima cumulative distribution function and its inverse are given in NIST: extreme value type I distribution, maximum case. With the deliberately chosen toy values $\mu =10$ and $b=2$ in arbitrary units, $z_{10}=14.501$, $z_{100}=19.200$, and $z_{1000}=23.815$, rounded to three decimals. In particular, $19.200$ is approximately the annual quantile at cumulative probability $0.99$; it is not a prediction of the largest observation in the next 100 years. Under independence, the probability that all those annual sample maxima stay below the exact $z_{100}$ is $0.99^{100}\approx 0.366$.

The time block matters. If a toy year contains $m$ independent daily blocks, each with exceedance probability $q$ at one fixed threshold, its annual exceedance probability is $p=1-(1-q{)}^m$, giving annual return period $1/[1-(1-q{)}^m]$. The reciprocal $1/q$ counts daily blocks. Dependence among daily exceedances requires their joint law instead; a threshold crossed on several successive days still produces only one exceedance indicator for that year's sample maximum. The peaks-over-threshold method and declustering of extremes address a different choice of extreme-event sampling.

A once-punctured genus-$h$ closed orientable surface retracts to a wedge of $2h$ circles. A retraction from ${\Sigma }_g$ would inject its degree-one cohomology as an isotropic subspace of a symplectic vector space for the Poincare duality pairing, since the punctured surface has zero degree-two cohomology. The maximal isotropic dimension is $g$, giving $2h\le g$. Equality is realized by folding the double of the punctured surface onto one copy. Additional handles in the other copy can be pinched away while fixing the joining boundary, proving sufficiency for every embedding of such a one-boundary subsurface when $g\ge 2h$.

For a solenoidal vector field $\mathbf{B}$ and constant magnetic diffusivity $\eta$, the magnetic induction equation isIt combines Faraday's law with the moving-conductor relation for electrical conductivity, neglecting displacement current. With incompressible flow, its advective form is ${\partial }_t\mathbf{B}+(\mathbf{u}\cdot \nabla )\mathbf{B}=(\mathbf{B}\cdot \nabla )\mathbf{u}+\eta {\nabla }^2\mathbf{B}$.

A normal-field wall imposes zero tangential magnetic field: $\mathbf{B}\times \mathbf{n}=0$ for the wall unit normal $\mathbf{n}$. In a channel with imposed normal field $B_0\hat{\mathbf{z}}$, an induced tangential component $b(z)\hat{\mathbf{x}}$ therefore satisfies $b=0$ on each wall. This condition is distinct from the no-slip boundary condition on fluid velocity.

Hartmann flow is a fully developed viscous conducting-fluid flow driven along a planar channel while a uniform magnetic field crosses the walls. Induction and the Lorentz force density couple its velocity and tangential induced field. For no-slip, normal-field walls separated by $2L$, the velocity is a hyperbolic cosine profile that approaches plane Poiseuille flow at zero imposed field and develops a uniform core with Hartmann layers at large Hartmann number.

Dynamo action is growth or sustained maintenance of a magnetic field by conducting-fluid motion despite magnetic diffusion. In the kinematic problem the velocity is prescribed and the resistive induction equation is linear in the magnetic field. Magnetic feedback through the Lorentz force density matters once the field becomes dynamically important.

A triangular localized potential energy has forces $+F$ behind and $-F$ ahead of the trap centre. With $v_c=F/\zeta =L{f}_c$, $\alpha =2X_0/L$ and $\beta =f_T/f_c$, its circular response is $f_p/f_c=\alpha \beta /({\beta }^2-1+\alpha )$ for $\beta >1$. A trapped particle has $f_p/f_c=\beta$ for $0\le \beta \le 1$, interpreting the discontinuous force at the cusp by a sticking or rounded-well limit.

A repulsive renormalization-group trajectory leaves a renormalization-group fixed point along its unstable manifold as the coarse-graining length increases. Its tangent directions are relevant operators. The critical surface is instead the stable manifold, whose irrelevant coordinates approach the critical fixed point after relevant fields are tuned. For a discrete step, relevance is determined by multipliers larger than one in magnitude, not merely by their being positive.

A defective renewal equation is $z(u)=g(u)+{\int }_{[0,u]}z(u-x)Q(dx)$ where the nonnegative kernel measure has total mass less than one. Its renewal representation is a convergent sum of convolutions. Exponential tilting can turn the kernel into a probability measure, enabling the key renewal theorem.

For positive interarrivals with law $K$, the unique locally bounded solution of $z=g+z\ast K$ is $z=g\ast U={\sum }_{j\ge 0}g\ast K^{\ast j}$, where $U$ is the renewal measure. Iteration proves the identity: the residual is bounded on each compact interval by a constant times the probability that the sum of many positive interarrivals stays in that interval, which tends to zero.

If encoding and decoding quantum channels factor through a $k$-dimensional space, their composite Kraus operators all have matrix rank at most $k$. The rank bound for a weighted operator trace and the Kraus formula for entanglement fidelity implyThus high operation fidelity requires the leading $k$ eigenvalues of the source density operator to carry almost all its probability. The bound applies to arbitrary completely positive trace-preserving encoders and decoders.

The factorial-normalized quartic vertex gives an angle-independent tree amplitude of magnitude $|\lambda |$. For equal incoming and outgoing masses, the momentum ratio in the two-body cross-section cancels. The event density for identical outgoing particles over a full sphere is ${\lambda }^2/(128{\pi }^{2}s)$. On a hemisphere representing each event once, the density is ${\lambda }^2/(64{\pi }^{2}s)$. Both conventions give total event cross-section ${\lambda }^2/(32\pi s)$.