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A monoidal functor whose tensor and unit comparison maps are isomorphisms.

A lax monoidal functor $\phi$ and opmonoidal functor $\psi$ on the same underlying functor, satisfying ${\psi }_{A\otimes B,C}{\phi }_{A,B\otimes C}=({\phi }_{A,B}\otimes 1)(1\otimes {\psi }_{B,C})$ and ${\psi }_{A,B\otimes C}{\phi }_{A\otimes B,C}=(1\otimes {\phi }_{B,C})({\psi }_{A,B}\otimes 1)$ in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

A morphism $f:A\to B$ satisfying $f{m}_A=m_B(f\otimes f)$ and $f{j}_A=j_B$.

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

A probability distribution is moment-determinate if its moments uniquely specify it among probability distributions. A moment-generating function finite in a neighbourhood of zero is a sufficient condition. The standard normal distribution is moment-determinate.

The fourth moment is $\mathbb{E}[X^4]$. Its finiteness is equivalent to square integrability of $X^2$. For centered variables, dividing it by the squared variance gives kurtosis.

For a modulus of a number field $\mathfrak{c}={\mathfrak{c}}_0{\mathfrak{c}}_{\infty }$, the multiplicity at a finite place is the exponent of its prime ideal in ${\mathfrak{c}}_0$. At a real Archimedean place it is one if that place occurs in ${\mathfrak{c}}_{\infty }$ and zero otherwise. Complex places have multiplicity zero. These effective modulus multiplicities encode congruences and positivity requirements in the ray class group.

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

A module is finitely presented when it admits the displayed exact sequence for finite integers $r,s$. Thus it has finitely many generators and finitely many relations. On a Noetherian ring, every finitely generated module has a finite presentation.

For a finite-index inclusion of modular groups and a form with trivial multiplier, multiply its slash operator for modular forms translates over left coset representatives. Right multiplication permutes the factors, producing a form of weight $k[{\Gamma }^{\prime }:\Gamma ]$ for the larger group. modular cusp cusp widths determine the orders contributed by the factors; the product is nonzero if the original form is nonzero.

On a square, a normalized modified Helmholtz adjoint plane wave with positive normal growth toward one side has magnitude one there, magnitude $e^{-2\eta }$ on the opposite side, and an adjacent-side magnitude integral bounded by $1/\eta$. Thus large tangential spectral frequencies suppress remote-side normal-trace coefficients. Paired sine collocation of square modified Helmholtz global relations further cancels adjacent-side unknown traces exactly.

A càdlàg modification of a stochastic process is a modification of a stochastic process whose paths are càdlàg on one event of probability one. Equality almost surely at every separately fixed time does not imply indistinguishability of stochastic processes when the original process lacks path regularity. A stochastic process starting at zero, with independent increments, stationary increments, and stochastic continuity, has a càdlàg modification and is a Lévy process under the intrinsic definition.

Testing the number of components in a finite mixture model is nonregular: absent components have zero weights on the parameter boundary and their means or other parameters are unidentified. The usual chi-squared limit for a likelihood-ratio test therefore need not apply. Information criteria are useful working comparisons, while parametric bootstrap and predictive cross-validation can provide more appropriate evidence. Neither a mixture fit nor a selected component count proves genuine latent populations.

Variable selection chooses which predictor variables enter a statistical model. The Lasso produces exact zero slopes along its Lasso regularization path; ridge regression generally shrinks without excluding variables. Selection for prediction should use predictive cross-validation or another stated criterion, rather than equating a coefficient p-value with practical usefulness.

For Bayesian deviance $D(\theta )=-2\log L(\theta )$ under a common likelihood convention, define $\overline{D}=\mathbb{E}[D(\theta )\mid y]$ and $p_D=\overline{D}-D(\mathbb{E}[\theta \mid y])$. Then $\mathrm{DIC}=\overline{D}+p_D$ trades average fit against effective complexity. Lower values suggest better penalized predictive fit in comparable models; differences are not Bayes factors or model posterior probabilities. The choice of stochastic parameters and latent-variable representation matters, especially in a hierarchical Bayesian model.

A finite abelian subgroup of the Möbius group is cyclic or isomorphic to $C_2\times C_2$. If it contains an element of order greater than two, conjugate that element's two fixed points to $0,\infty$. Every commuting map must preserve these points individually: an interchange would conjugate the element to its inverse. All maps are then scalings, forming a cyclic finite subgroup of ${\mathbb{C}}^{\times }$. If every nonidentity element has order two, choose one with fixed points $0,\infty$; the subgroup acts on this pair with image of size at most two and kernel contained in $\{z,-z\}$. Its size is therefore at most four, giving the asserted possibilities.

A Gaussian scale mixture is a random variable $X=RZ$, where $Z$ has a standard normal distribution and is independent of a positive random scale $R$. Its probability density function is the average of the conditional normal densities. Random scales can produce larger kurtosis than a single normal law.

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.

In $k[u,v]$, the ideal $I=(u^2,uv)$ has radical of an ideal $(u)$, a prime ideal, and hence a unique minimal prime ideal. Nevertheless $uv\in I$, $u\notin I$, and $v^j\notin I$ for all $j\ge 1$, so $I$ is not a primary ideal. The same example works after adjoining inverses of $1+u$ and $1+v$; thus it also gives an example in a two-variable Laurent polynomial ring.