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A finite mixture distribution of exponential distributions with rates $r_i>0$ and nonnegative weights summing to one. Its expected value is ${\sum }_i{w}_i/r_i$ and its moment-generating function is ${\sum }_i{w}_i{r}_i/(r_i-t)$ on the common finite domain. The mixture chooses one component for each observation; it is not a sum of independent exponential variables.

Suppose $\Psi (x)=x+O(xE(x))$ and ${\sum }_{j\ge 0}{\sup }_{2^j\le u<2^{j+1}}|E(u)|<\infty$. The Möbius divisor-sum identity gives ${\sum }_{d\le x}\mu (d)/d=O(1)$. The exact Dirichlet convolution identity $\mu \log =-\mu \ast \Lambda$ then gives ${\sum }_{n\le x}\mu (n)\log n=O(x)$, because its error is a convergent dyadic harmonic sum. Since ${\sum }_{n\le x}\log (x/n)=O(x)$, the displayed Mertens function bound follows. The classical Prime number theorem error $E(x)=e^{-c\sqrt{\log x}}$ meets the summability condition. This is a deduction from the Chebyshev error itself, without a separate estimate for the reciprocal zeta function.

For a finite first absolute moment, the minimizers of expected absolute loss are the medians. With a density, differentiating the loss gives $2F(a)-1$, so every point where the cumulative distribution function equals one half minimizes it. A density gap may give a whole minimizing interval. In contrast, finite squared loss is uniquely minimized by the expectation.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

A measure on a topological space is outer regular on its measurable sets when the displayed open-cover identity holds. Lebesgue measure has this property. It allows a null set to be placed in arbitrarily small open neighborhoods, and a compact small set to receive a smooth cutoff of similarly small integral.

For a finite positive measure on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x{)}^{-1}$ instead, changing the sign. The transform of an empirical spectral measure equals the normalized trace of the Stieltjes matrix resolvent.

For real functions continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $\xi \in (a,b)$ withApply Rolle's theorem to a linear combination of the functions with equal endpoint values. If $g^{\prime }$ never vanishes, $g(b)\ne g(a)$ and the equality can be divided into a ratio. This form directly proves the zero-over-zero endpoint case of L'Hôpital's rule, without continuity of the derivatives.

A shifted exponential distribution has location estimate equal to the sample minimum and rate estimate given above, for a nondegenerate sample of at least two observations. The minimum exceeds the true location on average by $1/(n\lambda )$. The residual sum above the minimum is gamma with shape $n-1$, so the rate estimate has expectation $n\lambda /(n-2)$ for $n>2$ and infinite expectation at $n=2$. The estimated location plus reciprocal rate is the unbiased sample mean.

The determinant is a smooth Lie group homomorphism from the real general linear group to ${\mathbb{R}}^{\times }$. Its derivative at the identity is the trace, and the flow-defined exponential on ${\mathbb{R}}^{\times }$ is the ordinary scalar exponential. Naturality of the Lie group exponential therefore proves the identity without diagonalizability assumptions. The series ${\sum }_{k\ge 0}{t}^k{A}^k/k!$ is the flow-defined matrix exponential because it solves $E^{\prime }=EA$, $E(0)=I$, and has inverse $E(-t)$.

At rest the two complex supercharges have anticommutators $2m{\delta }_{ab}$. A normalized scalar Clifford vacuum generates four states at occupation levels $0,1,2$: two spin-zero bosonic states and one spin-half pair. Products of three creation charges vanish. The one-creation normalization is $(2m{)}^{-1/2}$ and the two-creation normalization is $(2m{)}^{-1}$. The resulting on-shell content is a complex scalar and a massive fermion with equal masses.

A mass eigenstate diagonalizes the Hermitian mass operator or mass matrix and has a definite mass. In a flavor basis with off-diagonal mass terms, it is a linear combination of flavor states. For unstable particles, decay propagation instead involves an effective complex matrix; its eigenvectors need not inherit the orthogonality of a Hermitian mass problem.

On a graph, a binary configuration has probability proportional to $\exp (\alpha {\sum }_v{x}_v+\beta {\sum }_{\{v,w\}}{x}_v{x}_w)$, with each edge counted once. The full conditional probability of a one is logistic with linear predictor $\alpha +\beta {\sum }_{w\sim v}{x}_w$. Local neighbor sums eliminate the need to evaluate the normalizing constant.

Reversible-jump Markov chain Monte Carlo samples a posterior on a disjoint union of model-specific parameter spaces. A dimension-matching bijection between old parameters plus auxiliary variables and new parameters plus reverse auxiliary variables supplies a reversible proposal. Its acceptance ratio includes posterior-density, model-selection, proposal-density, and Jacobian determinant factors. The equal-dimension reversible-jump acceptance probability is the ordinary density-based special case.

A nonnegative row vector $x$, not necessarily of total mass one, is an invariant measure for a Markov chain if $x_j={\sum }_i{x}_i{p}_{ij}$. On an irreducible Markov chain, every nonzero invariant measure has positive coordinates: iterate stationarity and use a positive transition probability along a path.

An ice-edge band is an elongated concentration of ice floes separated from neighbouring ice by open water. In one mechanism, an off-ice wind opens polynyas; short wind waves then exert wave radiation stress on their downwind floes. Wave forcing compacts floes into bands, opposed by incoming longer-period swell. Collisions and mergers can produce fewer, larger bands.

Compute observed-minus-expected event contributions within each stratum's risk set, then add them across independent strata. Under the null, the corresponding conditional variances add. Standardizing gives an approximate normal distribution test, or its square gives a one-degree-of-freedom chi-squared distribution test. The comparison requires the usual exchangeability and independent censoring assumptions within strata.

Two local frames of a vector bundle differ on their overlap by an invertible matrix-valued function. These functions satisfy the Čech cocycle condition. For a holomorphic vector bundle they are holomorphic, and for a holomorphic line bundle they are scalar holomorphic units.

A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

Let a nontrivially valued local field have compact valuation ring $\mathcal{O}$, and choose any $t$ with $0<|t|<1$. The quotient $\mathcal{O}/t\mathcal{O}$ is finite. A fixed set of representatives containing zero gives every field element a unique convergent digit expansion, with finitely many nonzero negative-index terms. Recursive reduction modulo $t\mathcal{O}$ gives existence; comparing the first differing digit modulo this ideal gives uniqueness. The element $t$ need not be a uniformizer.

Scalar advection preserves every finite Lp norm if its complete differentiable characteristic flow preserves volume and the initial datum belongs to the corresponding Lp space. In general the $p$th power of the norm is the initial $|f_0{|}^p$ weighted by the phase-space flow Jacobian. Completeness alone preserves the essential supremum, but does not preserve finite-$p$ norms for compressible transport.