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Let $X$ be a real Hausdorff topological vector space, and let $U$ be a bounded, convex, open neighbourhood of zero with $U=-U$. Its Minkowski functionalis a norm and $U=\{x:p_U(x)<1\}$. Convexity and symmetry give a real balanced set, so the gauge is a seminorm. Openness gives the displayed unit-ball identity.

To prove definiteness, suppose $p_U(x)=0$. For any zero neighbourhood $V$, boundedness of $U$ gives $\epsilon U\subseteq V$ for a sufficiently small $\epsilon >0$. Since $x\in \epsilon U$, it belongs to every zero neighbourhood. The Hausdorff property forces $x=0$. The same boundedness comparison shows that the norm topology agrees with the original topology.

For a complex vector space, require $U$ to be balanced under complex scalars; symmetry under minus one alone only supplies the real homogeneity. The argument is independent of finite dimension and requires no Euclidean inner or outer balls.^[ref]

Split independent observational blocks into two folds. Use one fold to estimate the combination weight for the two sample means in the other fold, and conversely. Each held-out combination is an unbiased estimator conditionally on its training fold because its two held-out sample means have the same expectation. A deterministic average of the two combinations remains unbiased. Under consistency of the moment estimates and balanced fold sizes, the weights approach those of the minimum-variance combination of unbiased sample means. Paired observations must stay together when splitting or bootstrapping so their covariance is preserved.

On $\{0,\dots ,m{\}}^2$, give an adjacent edge capacity equal to the larger endpoint distance $|i_1-i_2|$ from the diagonal. The set $S=\{i_1<i_2\}$ has $2m$ outgoing edges: two at each interior diagonal vertex and one at each endpoint. Every such edge has capacity one. Thus any flow from above to below the diagonal is at most $2m$; a feasible path decomposition attaining this value certifies optimality by weak duality.

Let the first $k$ columns of a matrix be the first dependent prefix, with no zero column. Their relation space has dimension one, and every nonzero relation $b$ has at least two nonzero coordinates. If both $Ab=0$ and $A\mathrm{diag}({\lambda }_j)b=0$, one-dimensionality gives ${\lambda }_j{b}_j=c{b}_j$ for all $j$. Two nonzero coordinates would force two of the distinct weights ${\lambda }_j$ to agree, a contradiction.

Every entire classical minimal graph over ${\mathbb{R}}^n$ is affine for $1\le n\le 7$. The dimension restriction is essential. This theorem applies to the minimal surface equation for a graph, without assuming a global bound on its gradient.

// Target: analysis.bigb

A stable minimal surface has nonnegative second variation of surface area for every admissible normal variation $fN$ fixing the boundary. For a two-sided minimal surface in three-dimensional Euclidean space, the quadratic form isHere ${\nabla }_{M}f$ is the gradient for the induced Riemannian metric. The gradient term penalizes rapid changes in the displacement; the principal curvature term can make the second variation negative. A plane has only the nonnegative gradient term. Nonnegativity alone, especially with zero modes, does not prove a strict local minimum or a global minimum.

For a finite catenoid between fixed rings, the answer depends on the ring separation, waist radius, and allowed variations. The axisymmetric second variation of a catenoid tests a specified restricted class. Comparing with two disconnected disks tests a different class of competitors, including a change in topology. Consequently, a stationary catenoid can have greater surface area than two disks without immediately collapsing under every small perturbation.

With $A(X,Y)=({\overline{\nabla }}_{X}Y{)}^{\perp }$ and $\mathbf{H}=\mathrm{tr}A$, a variation with velocity $V$ satisfiesThus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.

Let a boundaryless minimal surface be closed in ${\mathbb{R}}^3$, and let its distance to a plane tend to infinity along every sequence escaping bounded sets. If it has no planar point, it must meet the plane. Otherwise absolute signed height has a positive attained minimum by properness. Its Hessian there is semidefinite and equals the second fundamental form up to sign because the plane is tangent. Minimality makes its trace zero, so the entire form vanishes, contradicting the absence of planar points. This argument uses curvature and attained extrema, rather than an unproved contact comparison for Gaussian curvature.

The identity $(BA{)}^{j+1}=B(AB{)}^{j}A$ gives $m_{BA}(t)\mid t{m}_{AB}(t)$. If $AB$ is diagonalizable, its minimal polynomial has simple roots. Hence all nonzero Jordan blocks of $BA$ have size one and its zero blocks have size at most two. Squaring kills these nilpotent blocks, making both squared products diagonalizable. Their eigenvalue multisets agree by Characteristic polynomials of AB and BA, so the squared products are similar matrices.

The descendant of a face with cyclic vertices $p_0,\dots ,p_{n-1}$ has vertices $p_i^{\prime }=(p_i+p_{i+1})/2$. Their arithmetic mean is unchanged. On the cyclic Fourier basis, a mode of frequency $j$ is multiplied by $(1+e^{2\pi ij/n})/2$. Every nonconstant mode has modulus less than one, so the descendant face shrinks to the original vertex centroid. This puts that centroid on the subdivision limit, without requiring the original face to be planar.

For rigid walls $a<r<b$, a nonzero vertical wavenumber $k$ and radial Lagrangian fluid displacement $\xi$, set $y=r\xi$ and $\mathcal{D}={\kappa }^2-r(d/dr)(v_A^2/r^2)$. Axisymmetric modes satisfyThe operator on the left is a positive self-adjoint operator. Its Rayleigh quotient is ${\omega }^2=k^2\int \mathcal{D}|y{|}^2/rdr/\int (|y^{\prime }{|}^2+k^2|y{|}^2)/rdr$. All mode squares are real; a growing mode exists exactly when the continuous coefficient $\mathcal{D}$ is negative somewhere. Sufficiency follows by a test function supported in that negative interval and the compact operator and self-adjoint operator eigenproblem obtained by conjugating the multiplication operator with the inverse square root of the positive differential operator.

// Target: astrophysical-fluid-dynamics.bigb

For $B_{\varphi }\propto r^p$, constant density and Keplerian ${\kappa }^2={\Omega }^2$, the Michael criterion for axisymmetric toroidal-field interchange becomes ${\Omega }^2-2(p-1)v_A^2/r^2$. Instability therefore requires $p>1$ and $v_A^2>r^2{\Omega }^2/[2(p-1)]$. For a field varying on the radial scale $r$, this is an orbital-speed field, much stronger than a gas-pressure-scale field with $v_A\lesssim c_s\sim H\Omega$ in a thin disk. Steep radial field gradients require the exact derivative criterion rather than this scale estimate. The instability is powered by the toroidal field distribution and is distinct from weak vertical-field magnetorotational instability.

// Target: fluid-mechanics.bigb

A Metropolis-within-Gibbs sweep updates one coordinate or block at a time by a Metropolis–Hastings algorithm transition whose invariant law is that block's full conditional distribution. For a proposal changing only block $j$, the acceptance ratio uses the full joint target with the other coordinates fixed, together with the reverse/forward proposal ratio. Each block kernel preserves the joint posterior, so their composition does as well. Conjugate blocks can instead use exact Gibbs sampling draws. Systematic composition need not be reversible, even when every block kernel is reversible.

For the unit ball of $C^p({\mathbb{T}}^n)$ measured in the uniform metric, $c{\epsilon }^{-n/p}\le H(\epsilon )\le C{\epsilon }^{-n/p}\log (1/\epsilon )$. The lower bound uses disjoint bump functions of spatial scale $h$, height proportional to $h^p$ and independently chosen zero/one coefficients: there are $\exp (c{h}^{-n})$ functions separated at scale $h^p$. For the upper bound, Multivariable Jackson approximation gives a trigonometric polynomial of degree $O({\epsilon }^{-1/p})$. Its $O({\epsilon }^{-n/p})$ coefficients are bounded and can be rounded with coefficient accuracy proportional to $\epsilon$ divided by the coefficient count. Counting those rounded coefficient arrays supplies the logarithmic factor. The continuous-function unit ball without derivative bounds has infinite entropy for $\epsilon <1/2$, by an infinite sequence of disjoint height-one bumps.

Suppose outer functions in a family $B$ obey $|f(x)-f(y)|\le L{\max }_j|x_j-y_j|$, and compose them with $m$ inner functions drawn from $E$. Approximate the outer function within $\epsilon /2$ and each inner function within $\epsilon /(2L)$. The triangle inequality bounds the composed error by $\epsilon$. Multiplying the numbers of possible net choices and taking logarithms gives the stated inequality. A uniform bound on first partial derivatives supplies $L=m$ for the $C^p$ unit ball, so the exponent of a common entropy bound for the outer and inner classes is retained under function composition.

For a constant-coefficient linear differential equation $P(D)y=e^{rx}$, where $D$ denotes differentiation, a simple root $P(r)=0$ forces a resonant trial. Factor $P(z)=(z-r)Q(z)$; since $(D-r)(x{e}^{rx})=e^{rx}$, commuting the differential operators gives $P(D)(x{e}^{rx})=Q(r)e^{rx}=P^{\prime }(r)e^{rx}$. Thus the displayed expression is a particular solution. More generally, if the root has multiplicity $m$, the same factorization gives $P(D)(x^m{e}^{rx})=P^{(m)}(r)e^{rx}$, so $y_p=x^m{e}^{rx}/P^{(m)}(r)$. This is the extra polynomial factor required by resonance in a differential equation.

For $y^{\prime \prime }+ϵ(y^{\prime }{)}^n+y=0$, the method of multiple scales gives no first-order phase drift. If $n$ is odd, the amplitude obeys the displayed damping law with $C_n=2^{-(n+1)}\left(\genfrac{}{}{0ex}{}{n+1}{(n+1)/2}\right)$. If $n$ is even, the average is zero and the amplitude is unchanged on the first slow time $T=ϵt$. The odd case dissipates mechanical energy, whereas the even case is reversible and its force is not damping of both velocity signs.

For $x^{\prime \prime }+3ϵy{x}^{\prime }+x=2(y^{\prime }{)}^2$ and $y^{\prime }=ϵ(1+x-y)$ with leading oscillation $x=A(\tau )\sin t$, $\tau =ϵt$, averaging the auxiliary equation gives $Y^{\prime }=1-Y$. Cancellation of resonant forcing in the oscillator gives $2A^{\prime }+3YA=0$. With $Y(0)=3$ and $A(0)=1$, $Y=1+2e^{-\tau }$ and the displayed amplitude follows. Bounded oscillatory corrections repair the instantaneous auxiliary equation; secular terms in a fixed-time expansion show why the slow scale is needed.

For homogeneous Dirichlet endpoints, the centered first-difference matrix is skew-Hermitian and the centered second-difference matrix is negative definite. Thus $\dot{U}=(D_{xx}-\alpha D_x)U$ is contractive in the discrete L2 norm for every real $\alpha$ and every mesh width. The displayed identity follows by summation by parts. Nonnegative off-diagonal entries additionally require $|\alpha |h\le 2$; that maximum-principle condition is not necessary for the L2 energy estimate. numerical consistency and Duhamel's formula then give second-order numerical convergence for smooth solutions and compatible initial approximations.

For periodic indexing let $S$ be the unitary forward shift. The operator is $A=-3I/2+2S-S^2/2$. A discrete Fourier transform diagonalizes it with eigenvalue $\lambda (\theta )=-3/2+2e^{i\theta }-e^{2i\theta }/2$ and real part $-(1-\cos \theta {)}^2$. Consequently $\Vert e^{tA/h}{\Vert }_2\le 1$ for every $t\ge 0$, uniformly in the grid. Equivalently $A+A^{\ast }=-\frac{1}{2}(2I-S-S^{\ast }{)}^2$, proving decay of the discrete energy directly. The result is semidiscrete stability; a further time discretization must also have adequate absolute stability.