For a determinant-one lift of a hyperbolic holonomy representation, the squared matrix trace of each fixed group word is a polynomial in the finitely many generator-matrix entries. The displayed identity identifies equal positive lengths with equal squared traces. Lift signs have no effect. Finite choices of length matchings therefore give finite unions of affine algebraic sets.

An adjacency-preserving rooted-tree embedding is an injective vertex map preserving the root and every adjacency. Because each root-to-vertex path maps to a simple path of the same length, vertex depths are preserved. This is more restrictive than a homeomorphic embedding of a rooted tree, which may stretch edges into paths. The branching-depth antichain of rooted trees shows that the adjacency relation is not a well-quasi-ordering.

The cross-lattice sufficient condition for stochastic comparison of two measures:The join and meet are coordinatewise maximum and minimum on a Boolean configuration cube. Normalization factors cancel, so the condition can be checked using unnormalized positive weights.

Attach an orthogonal classical label register to an ensemble. Its quantum mutual information with the quantum system equals the ensemble Holevo quantity. A local quantum channel changes the ensemble states but preserves their labels and probabilities. Data processing for quantum mutual information then proves that the output Holevo quantity cannot exceed the input value.

If a finite extension $B/A$ of number fields has a totally ramified finite prime, the ordinary Hilbert class field $H_A$ is disjoint from $B$. Its base change $B{H}_A/B$ is abelian and unramified at finite primes. Its real places also split, since a split-real unramified extension retains this property under base change. It therefore lies in the ordinary Hilbert class field of $B$, proving $h_A\mid h_B$ even when $A$ has real places.

Selecting a baryonic mass $M_b$ from a normalized cumulative distribution $M_b(<j)\propto j$ fixes a cutoff specific angular momentum $j_d=(M_b/M_{b,\mathrm{tot}})j_{\mathrm{vir}}$. With circular support and dominant enclosed baryonic gravity, the outer radius is $r_d\simeq j_d^2/(G{M}_b)$ and speed $v_d\simeq G{M}_b/j_d$. The mean specific angular momentum is $j_d/2$, which defines a different one-zone radius. The estimate assumes energy dissipation but negligible redistribution of specific angular momentum.

The integrated halo star-formation efficiency is the fraction of a dark matter halo's reference baryonic mass converted into stars: ${ϵ}_{\ast }=M_{\ast }/(f_b{M}_h)$, with cosmic baryon fraction $f_b$. It is distinct from star-formation efficiency per free-fall time. Cooling, gas supply and retention, stellar feedback and active-galactic-nucleus feedback make it strongly dependent on dark matter halo mass. With stellar mass-to-light ratio ${{\rm Y}}_{\ast }$, a simple central-galaxy mapping is $L={ϵ}_{\ast }{f}_b{M}_h/{{\rm Y}}_{\ast }$. Changing this efficiency changes both the mapping and the logarithmic Jacobian between the dark matter halo mass function and the luminosity function.

Graphs in $\mathcal{C}(a,b)$ admit a partition into $a$ total classes, of which $b$ are designated cliques and $a-b$ are designated independent sets. Cross edges are arbitrary and empty classes are allowed. Balanced cross-edge choices give quadratic labelled speed coefficient $1-1/a$. Comparing this count with every $(a+1)$-class type shows that the hereditary colouring number of this class is exactly $a$.

This parameter measures the largest completely unrestricted clique-independent partition class contained in the property. It is zero for bounded-order hereditary classes and infinite for all graphs. For proper unbounded classes it is a positive integer; it differs from the degeneracy-based colouring number of an individual graph.

When a hemispherical wetted front loses liquid at fixed volumetric flux per unit area $F_e$, equilibrium balances inlet flow with $2\pi R^2{F}_e$. The displayed radius exceeds $R_s$ and is stable because inlet flow decreases while evaporating area increases with radius. Porosity affects the adjustment time but cancels from the equilibrium radius.

A modular Hecke algebra is the algebra generated by Hecke operators acting on a specified space of modular forms, with level, weight and coefficient ring fixed. Its Eisenstein ideal encodes congruences with Eisenstein eigenvalues. This arithmetic algebra should not be confused with the generic Hecke algebra of a Coxeter group.

On a closed manifold with a Riemannian metric, a smooth solution of ${\partial }_{t}u+\Delta u=0$ for the positive Laplace-Beltrami operator satisfies the displayed identity by integration by parts. Zero initial data force the solution to vanish, proving uniqueness and the semigroup property for heat evolution.

On a closed manifold with a Riemannian metric, the minimum of a smooth heat solution cannot fall below its initial minimum. At a spatial minimum, $\Delta u\le 0$ for the positive Laplace-Beltrami operator. Adding a small increasing function of time and considering the first crossing gives the assertion. Applying the result to nonnegative initial functions shows that the Riemannian heat kernel is nonnegative.

The backward heat equation $u_t=-D{u}_{xx}$, $D>0$, amplifies a spatial Fourier mode by $e^{D{k}^{2}t}$. On a periodic domain, fix $T>0$ and take initial data $f_n=e^{-D{n}^{2}T/2}\sin (nx)$. Their L2 norms tend to zero, while the corresponding solutions at $T$ have L2 norms growing without bound. Thus the solution map is not continuous in the ordinary L2 norm; this is an ill-posed problem.^[ref]

Real analyticity alone does not resolve the issue. The periodic analytic function $f(x)={\sum }_{n\ge 1}{e}^{-n}\cos (nx)$ has exponentially decaying coefficients, but after any positive time the proposed coefficients $e^{D{n}^{2}t-n}$ are not square summable. The required data restriction must control growth on the Gaussian frequency scale. For example, a fixed-horizon normgives $\Vert u(T){\Vert }_2=\Vert f{\Vert }_{X_T}$ when the Fourier basis is normalized. This very strong data norm changes the stability question. In inverse heat reconstruction, an explicit regularization or frequency cutoff is needed when measured data carry ordinary small-scale noise.

Classical heat equation solutions with identical continuous initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\epsilon {\Phi }_b$, where ${\Phi }_b=(1-4bt{)}^{-n/2}\exp (b|x{|}^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The heat equation maximum principle bounds both signs of the difference by $\epsilon {\Phi }_b$ inside. Increasing the cylinder radius and then sending $\epsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.

At ordered possible failure times $t_j$, the discrete hazard is $h_j=\mathbb{P}(T=t_j\mid T>t_{j-1})$. Its survival function satisfies $S(t_j)=S(t_{j-1})(1-h_j)$, and hence $S(t)={\prod }_{t_j\le t}(1-h_j)$. Estimating $h_j$ by the observed number of failures divided by the risk set size under independent censoring yields the Kaplan–Meier estimator. Unlike a continuous-time hazard rate, a discrete hazard is a probability between zero and one.

The core disk of a $k$-handle attachment $D^k\times D^{n-k}$ is $D^k\times \{0\}$. Together with its cap in the preceding stage, it often represents a homology class; for a surgery trace, capped two-handle cores give the basis of its surgery linking matrix.

On binary words of length $n$, normalized Hamming distance is the fraction of differing coordinates. Unlike unscaled Hamming distance, it makes the Hamming cubes a Lévy family under uniform probability measures.

A machine's halting times admit a total computable function as a bound on all inputs in its domain if and only if that domain is a computable set. A bound decides the domain by finite simulation; conversely, a domain decider allows returning zero outside the domain and the exactly simulated halting time inside it. In particular every machine computing a total computable function has an exact computable running-time function, while nontotal machines may or may not have total bounds.

The usual half-level NS claim that all physical vector polarizations have nonnegative norm for $a\le 1/2$ assumes nonzero particle momentum. At zero momentum and limiting intercept, all positive-mode constraints and transversality allow a timelike polarization with negative norm, and $A\sim A+\kappa p$ removes nothing. Excluding this exceptional momentum restores the ordinary null-vector argument; retaining it makes the unrestricted elementary criterion strict $a<1/2$.