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For $F,G:{\mathcal{C}}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{D}$, a dinatural transformation has components ${\alpha }_C:F(C,C)\to G(C,C)$ satisfying, for $f:C\to D$,Both composites have domain $F(D,C)$ and codomain $G(C,D)$. It is the compatibility used for categorical ends and categorical coends, rather than ordinary naturality on just one variance.

The interval dependency problem is the loss of relationships between repeated occurrences of the same input. For $X=[0,1]$, interval subtraction gives $X-X=[-1,1]$: it permits $u-v$ with independent choices $u,v\in X$. But the function $x\mapsto x-x$ has actual image of a function $\{0\}=[0,0]$. Giving both operands the name $X$ does not tell an interval arithmetic calculation to enforce $u=v$.

For the polynomial $f(x)=x(1-x)$ on $X=[0,1]$, the natural interval extension isThis allows the impossible combination $x=1$ in the first factor and $x=0$ in the second, producing the spurious upper bound $1$. In the actual function, both occurrences use the same $x$. Completing the square gives $x(1-x)=1/4-(x-1/2{)}^2$, which lies between $0$ and $1/4$ on $[0,1]$; every value between these bounds occurs, by continuity and the intermediate value theorem. The maximum $1/4$ occurs at $x=1/2$, while either endpoint gives the minimum $0$.

A natural exponential family is full when its parameter set is the entire natural parameter space, with no restrictions on the allowable canonical parameters beyond finiteness of its normalizing integral. Fullness must not be confused with minimality or regularity.

A natural exponential family is regular when its parameter space is open. In a full regular family, differentiation under the normalizing integral is justified locally by exponential moments in a neighbourhood. Regularity does not guarantee that every possible data set has a finite maximum-likelihood estimator.

The convex support is the closed convex hull of the canonical sufficient-statistic support. For a minimal exponential family it has nonempty interior. Means at finite interior natural parameters lie in that interior, because a supporting hyperplane at a boundary mean would force the sufficient statistic itself onto that hyperplane.

For each player in a finite bimatrix game, add the positive pure-strategy gains to its current mixed strategy and normalize. This gives a continuous self-map of the product of probability simplices. At a Brouwer fixed-point theorem fixed point, the gain coordinates equal the current strategy times their total gain. If that total were positive, every supported action would beat the current average payoff, contradicting that average. Thus every pure deviation has nonpositive gain for both players, proving a Nash equilibrium.

Define $M_2=K_2$ and $M_{k+1}=\mu (M_k)$ for $k\ge 2$. By the Mycielskian chromatic increase and Mycielskian triangle preservation, each $M_k$ is a finite triangle-free graph with chromatic number $k$. The first steps are $K_2,C_5$, the Grötzsch graph, and then a 23-vertex graph needing five colours. The vertex recurrence $n_{k+1}=2n_k+1$ gives $n_k=3\cdot 2^{k-2}-1$. These explicit examples show how arbitrarily strong graph coloring constraints can accumulate without any three-vertex clique.

A finite simple graph $G$ is a triangle-free graph if and only if its Mycielskian is a triangle-free graph. No triangle in a graph can contain the Mycielski apex, whose neighbours are an independent set, or two Mycielski shadow vertices. A triangle in a graph containing one shadow $u_i$ and two originals $v_j,v_{\ell }$ requires all three original edges $v_i{v}_j,v_i{v}_{\ell },v_j{v}_{\ell }$, hence an original triangle in a graph. Conversely every original triangle in a graph remains. In fact each original triangle in a graph gives four in $\mu (G)$: itself and the three choices of replacing one of its vertices by its shadow. Thus triangle-freeness is preserved, but a positive triangle count is multiplied by four.

For a nonempty finite simple graph, the upper bound copies an optimal proper vertex colouring to its Mycielski shadow vertices and gives the Mycielski apex a fresh colour. For the lower bound, suppose $\mu (G)$ uses $k=\chi (G)$ colours, with $w$ coloured $k$. Shadows avoid $k$. Replace colour $k$ at each original $v_i$ by $u_i$'s colour. Changed originals form an independent set; unchanged neighbours differ because $u_i{v}_j$ is an edge. This proper vertex colouring of $G$ uses $k-1$ colours, contradicting its chromatic number.^[ref]

The nonempty condition means $V(G)\ne \varnothing$, not $E(G)\ne \varnothing$. An edgeless $n$-vertex graph with $n\ge 1$ has chromatic number one; its Mycielskian is a star graph on $w,u_1,\dots ,u_n$ together with $n$ isolated vertices, and needs two colours. If $G$ has no vertices, $\mu (G)$ is a single vertex, so the same formula also holds with the convention $\chi (G)=0$.

The Mycielski apex is the added vertex adjacent to every Mycielski shadow vertex and to no original vertex. Its graph neighbourhood is precisely $\{u_1,\dots ,u_n\}$.

A Mycielski shadow vertex $u_i$ has graph neighbourhood $N_G(v_i)\cup \{w\}$ in $\mu (G)$. Its neighbours other than the Mycielski apex are original vertices. All Mycielski shadow vertices form an independent set.

The Grötzsch graph is the Mycielskian of the five-vertex cycle graph. Take original edges $v_i{v}_{i+1}$, with indices interpreted modulo five; add $u_i{v}_{i-1},u_i{v}_{i+1}$ and $w{u}_i$ for every $i$. There are eleven vertices and twenty edges: five original, ten between originals and Mycielski shadow vertices, and five from the Mycielski apex.

Give $v_1,\dots ,v_5$ colours $(1,2,1,2,3)$, give each $u_i$ the colour of $v_i$, and give $w$ colour 4. This is a proper vertex colouring. The Mycielskian chromatic increase proves that three colours are impossible, so the chromatic number is exactly four. The Mycielskian triangle preservation proves that this is a triangle-free graph. In particular its clique number is only two, even though its chromatic number is four.

The Mycielski apex has an independent set as its graph neighbourhood, yet all three old colours are needed there in any three-colour proper vertex colouring of the subgraph obtained by deleting it. Thus a locally easy group of mutually nonadjacent neighbours can carry globally incompatible colour requirements. Absence of triangles in a graph alone cannot bound the chromatic number.

A mechanism that waits for ownership by repeatedly trying while consuming CPU execution time. A spinlock built from test-and-set can enforce mutual exclusion without guaranteeing starvation freedom. Progress also requires the owner to run and release: if a waiting software thread monopolizes the only CPU while the owner is suspended, exclusive entry alone cannot help.^[ref]

Ordinary negative-muon decay is ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$; the positive-muon channel has the conjugate particles. The process proceeds through a weak charged current. At energies much smaller than the W mass, a four-fermion interaction with the Fermi constant describes it.

At leading order, neglecting the electron mass and radiative corrections, the width is $\Gamma =G_F^2{m}_{\mu }^5/(192{\pi }^3)$ in natural units. The fifth power reflects the interaction dimension together with three-body phase space. The electron has a continuous energy distribution because two neutrinos share the missing energy and momentum. Precision extraction of the Fermi constant uses the measured lifetime and includes electron-mass and radiative corrections. A stopping experiment measures disappearance or decay timing, so nuclear capture must be distinguished from free decay when relevant.

For $Z_t=W_t+\Theta W_{t-1}$ and vector white noise of covariance matrix $D$, the only nonzero covariance lags are $\Gamma (0)=D+\Theta D{\Theta }^T$, $\Gamma (1)=D{\Theta }^T$ and $\Gamma (-1)=\Theta D$. This follows by retaining only matching noise times in $\mathbb{E}[Z_t{Z}_{t+h}^T]$. Its spectral matrix, with this lag convention, is $(I+\Theta e^{i\lambda })D(I+{\Theta }^T{e}^{-i\lambda })/(2\pi )$. In two dimensions, with $D=\mathrm{diag}(a,b)$, its cross-spectrum is $[a{\theta }_{11}{\theta }_{21}+b{\theta }_{12}{\theta }_{22}+a{\theta }_{21}{e}^{-i\lambda }+b{\theta }_{12}{e}^{i\lambda }]/(2\pi )$.

For centered jointly Gaussian random variables, an odd product has zero expectation, and an even product's expectation is the sum over all pairings of the products of the pair covariances. In particular,For a proof, the moment-generating function is $\exp (t^T\Sigma t/2)$. Differentiating once in each variable at zero leaves precisely products of covariance entries indexed by complete pairings. No independence assumption between the coordinates is needed.

This criterion holds even for reducible nonnegative matrices. If a fixed point $p\ne 0$ exists, $1-e^{-z}<z$ for $z>0$ gives $(\Lambda p{)}_i>p_i$ on its support. Some $\alpha >1$ therefore satisfies $\Lambda p\ge \alpha p$, forcing the spectral radius above one by the Perron–Frobenius theorem. Conversely take $v\ge 0$, $v\ne 0$, with $\Lambda v=\rho v$ and $\rho >1$. For sufficiently small $\eta >0$, $1-e^{-\eta \rho v_i}\ge \eta v_i$. Thus $\eta v$ is a subsolution; monotone iteration under $T$ gives a nonzero nonnegative fixed point below $1$. Strictness of $1-e^{-z}<z$ excludes survival at the critical value for this Poisson offspring law.

If a region is invariant under one coordinate reflection and an integrable function changes sign under that reflection, its multiple integral is zero. Apply the change of variables formula to the reflection, whose absolute Jacobian determinant is one: the integral equals its negative. This applies to odd coordinate powers in symmetric volumes and to the off-diagonal entries of an inertia tensor when the mass density shares the coordinate-reflection symmetries.

Counts of several categories before a fixed number of stopping-category outcomes have a negative multinomial distribution. For one stopping outcome of probability $r=1-\alpha -\beta$, the two-count mass function is $r\left(\genfrac{}{}{0ex}{}{x+y}{x}\right){\alpha }^x{\beta }^y$. Its probability generating function is $r/(1-\alpha s-\beta t)$ and its means are $\alpha /r$ and $\beta /r$.

For cell means ${\mu }_j$ with ${\sum }_j{\mu }_j=n$, the independent Poisson distribution likelihood and the multinomial distribution likelihood with probabilities ${\mu }_j/n$ differ by a parameter-independent factor. Thus a maximum-likelihood estimator for the unrestricted Poisson model whose fitted total is $n$ is also an estimator for the constrained multinomial model. Equality of named parameter estimates presumes identifiability and uniqueness; without these, the conclusion concerns the sets of maximizers. A free intercept in a log-linear Poisson model enforces the fitted-total equation.