In a normal linear model with an intercept and factor effects, set the effect of one factor level to zero. The intercept then denotes that reference level's mean, and the remaining effects are contrasts to it. The constraint removes redundant statistical parameters without altering the fitted mean space.

In a shared zero-inflated Gamma-Poisson count model, write $m_j=(1-\pi ){\mu }_j$ and $\kappa =(\tau +\pi )/(1-\pi )$. Its first two moments are $\mathrm{Var}(Y_j)=m_j+\kappa m_j^2$ and $\mathrm{Cov}(Y_j,Y_k)=\kappa m_j{m}_k$. These moments do not separately identify $\pi$, $\tau$, and the component-mean intercept: taking ${\pi }^{\prime }=0$, ${\tau }^{\prime }=\kappa$ and ${\mu }_j^{\prime }=m_j$ gives the same first two moments. The full count distribution can carry information absent from the moments. A consistent estimator of a marginal mean ratio therefore need not consistently estimate a structural-zero fraction from a misspecified moment parameterization.

A snowflake is an ice crystal or an aggregate of ice crystals grown in atmospheric conditions. Its morphology depends on vapour supersaturation, transport, temperature and crystal anisotropy. Not every snowflake is dendritic.

Suppose a degree-$N$ polynomial obeys $|{\partial }^{\alpha }Q(\xi )|\le C_{\alpha }|\xi {|}^{-\delta |\alpha |}|Q(\xi )|$ at large real frequency, with $\delta >0$. A nonzero constant derivative of total order $N$ gives $|Q|\gtrsim ⟨\xi {⟩}^{\delta N}$. Reciprocal differentiation then bounds ${\partial }^{\alpha }(1/Q)$ by $⟨\xi {⟩}^{-\delta N-\delta |\alpha |}$. The high-frequency reciprocal parametrix kernel therefore gains $\delta N$ Sobolev derivatives. Localize the distribution by a cutoff function equal to one near the target; the commutator lies outside that target and the reciprocal kernel makes its contribution smooth. This proves the displayed gain without demanding global regularity of the original distribution.

For a marked closed hyperbolic surface, the representation of its fundamental group whose image acts as the deck group on the hyperbolic plane. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives $X={\mathbb{H}}^2/\rho ({\pi }_1{\Sigma }_g)$. For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the Riemannian holonomy group.

A geodesic hexagon in the hyperbolic plane with all six interior angles $\pi /2$. Any three positive alternating side lengths determine it up to isometry. If those sides are ${\ell }_i/2$, the opposite seam satisfies $\cosh s_i=(\cosh ({\ell }_i/2)+\cosh ({\ell }_j/2)\cosh ({\ell }_k/2))/(\sinh ({\ell }_j/2)\sinh ({\ell }_k/2))$. Two congruent such hexagons glued along the other three sides form a pair of pants with boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3$.

For two Grassmann spin components, $\mathbf{j}=\bar{\psi }\mathbf{\sigma }\psi$ satisfies ${\mathbf{j}}^2=-6{\bar{\psi }}_{\uparrow }{\bar{\psi }}_{\downarrow }{\psi }_{\downarrow }{\psi }_{\uparrow }$. A real vector auxiliary field with cost ${\mathbf{M}}^2/(2U)$ and coupling $-\mathbf{M}\cdot \mathbf{j}$ therefore reproduces the repulsive $3U$ contact action. All three spin components are essential to that normalization.

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

For a single spinful fermion orbital, ${\mathbf{S}}^2=3(n-2n_{\uparrow }{n}_{\downarrow })/4$. The contraction term is $3n/4$, so $:{\mathbf{S}}^2:={\mathbf{S}}^2-3n/4=-3n_{\uparrow }{n}_{\downarrow }/2$. Empty and doubly occupied states are spinless while singly occupied states have spin one half, which also proves the identity directly.

A colouring is computable when an algorithm sorts the finite input subset, computes its colour and terminates. Its colour classes are uniformly computable sets. The distinction between existence of an infinite homogeneous set for a colouring and effective construction of one is a basic phenomenon in computability theory.

An even-degree polynomial with all roots in $|z|<R$ admits a holomorphic square root throughout $|z|>R$. Factor out its even power of $z$ and use the convergent logarithm of each $1-a_j/z$ to construct it. Even degree removes the annular winding obstruction. The two choices differ by a global sign.

Let $R>0$ and let $f$ be a holomorphic function on an open neighbourhood of the closed disk $\{z:|z|\le R\}$. Assume $f(0)\ne 0$ and that $f$ has no zeros of a function on the boundary circle $|z|=R$. List its interior zeros of a function as $a_1,\dots ,a_N$, repeating each according to its multiplicity of a zero. With the natural logarithm, Jensen's formula statesThe empty sum is zero. This boundary-zero-free version is the holomorphic special case of the general identity in the source.^[ref]

Every interior zero of a function makes a positive contribution; a zero nearer the centre contributes more. Consequently boundary growth relative to $|f(0)|$ limits how many zeros of a function can lie well inside the disk. The Jensen zero-count bound makes this quantitative, while the integrated zero count retains the individual radial weights. The boundary statistic here is the circle log-mean of a holomorphic function; exponentiating gives a circle geometric mean of a holomorphic function.

For a Hermitian manifold with the pointwise Hermitian inner product linear in the first variable, define the conjugate-linear star by $\alpha \wedge \ast \beta =⟨\alpha ,\beta ⟩{\omega }^n/n!$. It is the usual real Hodge star operator extended complex-linearly and then composed with complex conjugation. It has the displayed bidegree and satisfies ${\ast }^2=(-1{)}^{p+q}$ on $(p,q)$-forms.

A vector subspace closed in the norm topology of a Hilbert space is complete in the inherited norm and inner product. Its orthogonal complement gives a decomposition of the ambient Hilbert space into an orthogonal direct sum. This closedness permits orthogonal projection and ensures limits used in range arguments remain in the subspace. Every finite-dimensional vector subspace is closed.

The closed unit ball of a Hilbert space is compact in norm exactly when the space is finite-dimensional. Finite-dimensional compactness follows from the Heine-Borel theorem. In infinite dimension, an orthonormal sequence has pairwise distance $\sqrt{2}$ and no convergent subsequence. Restricting a compact operator that equals a nonzero multiple of the identity on a subspace therefore forces that subspace to be finite-dimensional.

The technical hierarchy problem asks why a small mass scale, once chosen, remains stable under quantum corrections from much larger scales. A symmetry can protect this radiative stability without explaining the numerical origin of the scale separation. For a fundamental scalar mass, heavy thresholds often give additive corrections proportional to the heavy mass squared; supersymmetric cancellation of quadratic divergences supplies one protection mechanism when supersymmetry breaking is sufficiently soft.

Fix a number field $K$, a positive integer $d$, and a finite set $S$ of its finite places. There are finitely many field extensions $L/K$ of degree at most $d$ unramified outside $S$, inside a fixed algebraic closure. To see this, view them as extensions of the rational numbers of bounded degree. Their ramified rational primes belong to a fixed finite set. At each of these primes, the different exponent is bounded in terms of the local degree and ramification index, giving a bound for the absolute field discriminant. The Hermite–Minkowski theorem gives finitely many fields up to isomorphism, and each has only finitely many embeddings into the fixed closure. This useful form of arithmetic compactness is stated and proved for $K=\mathbb{Q}$ in Andrew Sutherland's number-theory notes, Theorem 14.25; the preceding reduction gives the general case.

Let $M$ be a connected noncompact Riemannian manifold without boundary, and let $u$ be a smooth real harmonic function. Assume every set $\{|u|\ge \epsilon \}$ is compact for $\epsilon >0$, expressing decay to zero at infinity. Then $u=0$.

For a positive regular value $\epsilon$, the superlevel set $D=\{u>\epsilon \}$ has compact closure and smooth boundary. Integration by parts givesHence the gradient vanishes on $D$. A nonempty component would then be constant with value greater than $\epsilon$, incompatible with boundary value $\epsilon$. Such a component cannot be all of the connected noncompact $M$, since its closure is compact. Thus no positive value occurs; applying the argument to $-u$ excludes negative values.

The noncompactness and decay hypotheses matter. On a compact connected manifold without boundary, every constant has compact superlevel sets, and a nonzero constant disproves the vanishing conclusion. Alternatively, under the stated noncompact decay hypotheses, the strong maximum principle for harmonic functions proves vanishing by an attained positive maximum or negative minimum.

For $r<1$, average the real part of the convergent power series for $\log (1+r{e}^{i\theta })$: every nonconstant Fourier mode vanishes. For $r>1$, factor out $r$ and apply the same argument to $r^{-1}$. The boundary singularity at $r=1$ is logarithmically integrable. This identity computes the expected logarithmic distance after an isotropic Gaussian displacement.

Invariant theory studies functions and tensors unchanged by a group action. For a linear representation, the polynomial invariant ring records invariant polynomial functions on the representation space. Tensor invariants and commuting algebra actions connect it with Schur–Weyl duality.