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For a uniform random sign $X\in \{-1,1\}$, $\psi (\lambda )=\log \cosh \lambda$ andfor $|x|\le 1$, with value $+\infty$ outside that interval.

For twice differentiable scalar fields,

For a twice differentiable scalar field on the plane,

The l2 sequence space is a uniformly convex Banach space. If $\Vert x{\Vert }_2=\Vert y{\Vert }_2=1$, the parallelogram law givesThus $\Vert x-y{\Vert }_2\ge \epsilon$ permits the modulus $\delta (\epsilon )=1-\sqrt{1-{\epsilon }^2/4}$.

Among $k$-Sperner subfamilies of $\mathcal{P}([n])$, the maximum cardinality is the sum of the $k$ largest binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. The union of the corresponding levels attains the bound. For the upper bound, the Lubell mass is at most $k$, and assigning its available mass to the levels with largest binomial coefficients maximizes cardinality.

An ordinary differential equation $\Delta (x,u,u^{\prime },\dots ,u^{(n)})=0$ defines a solution manifold in jet space. The prolonged graph of every classical solution lies in this manifold.

Inside the completed rational Iwasawa algebra, the convergent seriessends a finite-rank p-saturated group to a ${\mathbb{Z}}_p$-Lie algebra. The Baker--Campbell--Hausdorff formula and its commutator expansion prove closure under addition and the commutator bracket.

If $G$ is a complete finite-rank p-valued group with center $Z(G)$, thenThe key facts are that the finite-conjugacy center of a p-valued group equals its center and that compatibility through the finite group-algebra quotients eliminates coefficients on infinite conjugacy classes.

For a p-valued group $(G,\omega )$, give $p^r(g_1-1)\cdots (g_s-1)$ weight $r+{\sum }_i\omega (g_i)$. The spans of elements of weight at least $\lambda$ form a multiplicative filtration on ${\mathbb{Z}}_p[G]$ and its completions.

A filtration on a group $G$ is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is separated when $\omega (x)=\infty$ only for $x=1$.

For the double covering $\pi :S^{2n}\to {\mathbb{RP}}^{2n}$ and the antipodal map $a$, every pulled-back top form satisfies $a^{\ast }{\pi }^{\ast }\omega ={\pi }^{\ast }\omega$. Since $\mathrm{deg}a=-1$, its integral over the sphere is its own negative and hence vanishes. It is therefore exact on the sphere. Averaging a primitive under $a$ and descending it proves that $\omega$ is exact on real projective space.

Every closed communicating class of a finite Markov chain has its own invariant probability distribution. All invariant distributions are convex combinations of those class distributions. In particular, a finite irreducible chain has a unique invariant distribution, while a reducible chain may have many.

The integral curves of $V(x)=a\times x+b$ are lines when $a=0$, and helices around an axis when $a\ne 0$. The helix is closed exactly when its axial translation component vanishes, in which case every nonfixed orbit is a circle.

Let a commutative $k$-algebra $R$ be generated by $x_1,\dots ,x_n$, and let $R_j$ be the span of words of total degree at most $j$ in those generators. The associated graded ring of this filtration is a standard graded algebra, andThe Hilbert-Serre theorem therefore makes ${\dim }_k{R}_j$ eventually a polynomial in $j$. Its degree is $\dim R$; this follows by applying the Rees-ring deformation, whose special fibers are $R$ and the associated graded ring, so they have the same Krull dimension.

For planar Brownian motion started at $(x,y)$ with $y>0$, the probability that its first hit of the real axis lies in $(0,\infty )$ isThis is the bounded harmonic function in the upper half-plane with boundary values zero on the negative half-axis and one on the positive half-axis.

A Riemannian manifold is geodesically incomplete when some maximal geodesic has a finite endpoint in its affine-parameter interval.

If a smooth degree-$d$ projective plane curve had genus two, the genus-degree formula would give $(d-1)(d-2)=4$, equivalently $d^2-3d-2=0$. Its discriminant is $17$, so it has no integral solution for the degree $d$.

Because Gaussian white noise is not an ${\ell }^2$ vector, least squares over $\Theta \subset {\ell }^2$ is defined by maximizingover $\theta \in \Theta$, with the noise pairing interpreted through an isonormal Gaussian process. Differences of this contrast equal the formal differences of squared residual norms.

A Gaussian net test covers a composite alternative by finitely many metric balls and takes the maximum of the likelihood-ratio tests against their centers. Gaussian tail bounds control each test, while a union bound costs the logarithm of the covering number.

A Bernstein-von Mises theorem says that a suitably centered and scaled posterior distribution converges to the normal distribution prescribed by the local likelihood. For a fixed linear functional in a Gaussian sequence model and a prior with locally flat positive density, the limiting posterior variance is the squared ${\ell }^2$ norm of the functional's coefficient vector.