A finite-dimensional module $X$ degenerates to $Y$ when $Y$ has a representative in the closure of $X$'s change-of-basis orbit in the representation variety of an associative algebra. The full orbit of $Y$ then lies in that closure. The relation is transitive because orbit closures are closed and invariant under change of basis. A split extension as a degeneration shows that a module degenerates to the direct sum of its composition factors.

For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

The compact quotient for the Gamma 0 congruence subgroup at level two has projective index three, two modular cusps of cusp widths one and two, one order-two elliptic orbit and no order-three elliptic orbit. It has genus zero. It differs from the level-two principal-congruence curve for $\Gamma (2)$, which has three modular cusps.

For $k>0$, a nonzero spectral parameter for a linear boundary value problem givesThe identity $A^2+B^2=k^2$ proves that this solves the modified Helmholtz equation. The reciprocal companion is $v_{1/\lambda }=\overline{v_{\overline{\lambda }}}$.

For $m>0$, the decaying fundamental solution of $-\Delta +m^2$ in two dimensions is $K_0(mr)/(2\pi )$, with $K_0$ the Modified Bessel function of the second kind. Green second identity reconstructs an interior solution from its Dirichlet boundary condition $f$ and outward normal derivative $h$ as $q(r)={\int }_{\partial \Omega }({\Gamma }_{m}h-f{\partial }_n{\Gamma }_m)ds$.

A model companion of $T$ is a model-complete theory $T^{\ast }$ in the same language with $T_{\forall }^{\ast }=T_{\forall }$. Equivalently each model of either theory embeds into a model of the other. Alternating these embeddings and using the elementary chain theorem proves uniqueness up to logical equivalence. A model companion need not be a syntactic extension of its original theory.

A morphism in the chosen weak-equivalence class of a model category. In the projective model structure on nonnegative chain complexes these are quasi-isomorphisms.

A morphism in the chosen fibration class of a model category. In the projective model structure on nonnegative chain complexes, it is surjective in strictly positive degrees; degree zero is not required.

A morphism in the chosen cofibration class of a model category. In the projective model structure on nonnegative chain complexes it is a degreewise split injection with degreewise projective cokernel. This usage is distinguished from the topological cofibration.

The model category of nonnegative complexes of left modules has quasi-isomorphisms as model weak equivalences, positive-degree surjections as model fibrations, and degreewise injections with projective cokernel as model cofibrations. All objects are fibrant; cofibrant objects are degreewise projective.

Attach cells for all lifting squares at each successive stage, then take a suitable colimit. Smallness of the generating domains makes every eventual lifting square occur at an earlier stage. This factors a map into a relative cell complex followed by a map with the prescribed right lifting property.

For a free field in the annihilation-operator vacuum, the coincident two-point function is its mode-power integral. Unsmeared vacuum products generally have an ultraviolet divergence. A finite band or spatial smearing defines a regulated variance without altering the meaning of a mode-by-mode dimensionless power spectrum.

Put $A_j(x)={\sum }_{d\le x}\mu (d){\log }^j(x/d)/d$. The elementary estimates areThe identities ${\sum }_d\mu (d)\lfloor x/d\rfloor =1$ and ${\sum }_d\mu (d)H_{\lfloor x/d\rfloor }/d=1$ give the first two bounds. Convolve the harmonic divisor sum with $\mu (d)/d$: because $\mu \ast \tau =1$, its left side is $H_{\lfloor x\rfloor }$. Thus $\frac{1}{2}{A}_2+2\gamma A_1+c{A}_0=\log x+O(1)$. The summed error is $O(x^{-1/2}{\sum }_{d\le x}{d}^{-1/2}\log (2x/d))=O(1)$.

This elementary smoothing argument is used in Selberg's original proof and Ramaré's sieve lectures, Lemma 4.2.

For every positive integer $n$, the divisor sum of the Möbius function isThe sum includes all positive integer divisors, but only squarefree divisors contribute. Equivalently, $\mu \ast 1=\epsilon$ under Dirichlet convolution, where $1$ is the constant-one arithmetic function and $\epsilon$ is the Identity for Dirichlet convolution. ^[ref]

Each squarefree divisor chooses a subset of the distinct prime factors of $n$, contributing $(-1{)}^r$ when that subset has $r$ elements. Expanding a product therefore givesFor $n>1$ at least one prime factor occurs, so the product is zero. At $n=1$ it is an empty product, equal to one. This is the same alternating cancellation that drives inclusion-exclusion.

For $30=2\cdot 3\cdot 5$, all eight positive integer divisors are squarefree integers, givingPair each subset not containing $2$ with the subset obtained by inserting $2$: $(1,2),(3,6),(5,10),(15,30)$. Each pair cancels. The same nonzero terms occur for $60$, because its squarefree divisors are still the integer divisors of $30$; its other integer divisors have Möbius function value zero.

The Möbius divisor-sum identity is the cancellation behind Möbius inversion and reciprocal Dirichlet series: aggregation over integer divisors has an exact inverse.

With a fully observed covariate $X$ taking finitely many values of positive probability, response $Y$ of finite variance, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.

When a response $Y$ is missing at random conditional on a fully observed category $X$, with positive response probability in every occupied category, estimate its conditional probabilities using respondents within each category and average them with the category frequencies from all patients. For a binary response, $\hat{P}(Y=1)={\sum }_x\hat{P}(X=x)\hat{P}(Y=1\mid X=x,R=1)$. The weighting uses the full cohort rather than only complete cases.

For mostly-plus signature, $x^{+}=(x^{D-1}+x^0)/\sqrt{2}$ and $x^{-}=(x^{D-1}-x^0)/\sqrt{2}$ give $d{s}^2=2d{x}^{+}d{x}^{-}+d{x}^{I}d{x}^I$. In this convention raising a plus or minus derivative exchanges the two labels.

Let $j(\mathfrak{b})$ be the Minkowski embedding of a number field applied to a nonzero fractional ideal. Under the metric $|x{|}^2={\sum }_{v\text{ real}}{x}_v^2+2{\sum }_{v\text{ complex}}|x_v{|}^2$, its covolume is $\sqrt{|D_K|}N(\mathfrak{b})$. For ordinary coordinate Lebesgue measure $\prod d{x}_v\prod d\mathrm{Re}{z}_{v}d\mathrm{Im}{z}_v$, its covolume is instead $2^{-r_2}\sqrt{|D_K|}N(\mathfrak{b})$. Here $D_K$ is the field discriminant, and the absolute norm of a fractional ideal is positive and multiplicative. The formula follows by taking the embedding determinant of an integral basis, then using the index of an integral ideal and scaling to handle a fractional ideal.

In Euclidean space, a Minkowski sum of a compact set $X$ and a closed set $Y$ is closed. From any convergent sequence $x_j+y_j$, extract a convergent subsequence of $x_j$ using compactness. The corresponding $y_j$ then converges to the difference of the two limits, which lies in $Y$. If $Y$ is also bounded, the sum is bounded and therefore compact.

In $\mathbb{R}$, the sets $X=\{n:n\ge 1\}$ and $Y=\{-n+1/n:n\ge 1\}$ are closed because their points have no finite accumulation point. Their Minkowski sum contains $1/n\to 0$ but does not contain zero, since $m-n+1/n=0$ has no positive-integer solution. Compactness of one summand is a useful sufficient condition missing from this counterexample.