A module endomorphism is a module homomorphism from a module to itself. These maps form the endomorphism ring ${\mathrm{End}}_R(M)$ under pointwise addition and composition. An invertible module endomorphism is a module automorphism.

For a front written in $s=Z+c\theta$ connecting $0$ at $s=-\infty$ to nonzero $\beta$ at $s=+\infty$, one integration gives $ϵc^2{q}^{\prime }=q-c{q}^3/3$. The endpoint condition fixes $c=3/{\beta }^2$. Setting $w=q^2$ gives a logistic equation and the displayed profile, with the sign of $q$ selected by $\beta$. The propagation speed in the $\theta$ coordinate is $-{\beta }^2/3$, agreeing with the Rankine-Hugoniot condition for flux $-q^3/3$.

For a Poisson random variable $S_L$ of mean $L\lambda >0$ and $0<\beta <1$, the scaled cumulant-generating function is $\lambda L^{1-\beta }(e^{\theta L^{(\beta -1)/2}}-1-\theta L^{(\beta -1)/2})$. Its Taylor expansion tends to $\lambda {\theta }^2/2$ for every real $\theta$. The Gärtner–Ellis theorem applies because the limit is finite and differentiable everywhere. Its Legendre-Fenchel transform is the displayed quadratic good rate function.

If $A\equiv B$, the elementary diagram of $B$ is finitely satisfiable by interpreting its finitely many named elements in $A$. Choose those finite realizations as coordinates, extend the filter of diagram cones to an ultrafilter, and use the Łoś theorem to embed $B$ elementarily into an ultrapower of $A$. If $A\prec B$, work over the named elements of $A$ and keep their coordinates constant. The resulting embedding extends the diagonal map of $A$. This relative assertion allows an alternating chain of ultrapowers to keep every previously embedded substructure fixed.

The Frayne embedding lemma embeds $B$ into an ultrapower $A_1$ of $A$. Its relative version embeds $A_1$ into an ultrapower $B_1$ of $B$, fixing the diagonal copy of $B$. Continue alternately over the previously fixed elementary substructure. After coherent identifications one has $B⪯A_1⪯B_1⪯A_2⪯B_2⪯\cdots$, while each $A_{n+1}$ is an ultrapower of $A_n$ along its diagonal map, and similarly for $B$. The two cofinal subsequences have the same elementary union. This proves the ultralimit assertion without the stronger single-ultrapower isomorphism theorem.

For incidence from a solid with compressional/shear speeds $\alpha ,\beta$ into a fluid of sound speed ${\alpha }_f$, let $E={\cos }^{2}2{\theta }_S+({\beta }^2/{\alpha }^2)\sin 2{\theta }_P\sin 2{\theta }_S$. The displayed coefficient multiplies the transmitted displacement field potential relative to the incident P wave potential, using the same retarded-time waveform. Snell law for elastic and acoustic waves matches the tangential slowness. Continuity of normal displacement field, zero tangential traction, and continuity of normal traction give the coefficient. It is not a displacement-amplitude coefficient: longitudinal displacement field acquires an additional ratio $\alpha /{\alpha }_f$.

A nonidentity Möbius transformation of finite order $n$ is conjugate to multiplication by a primitive $n$th root of unity. Zero and infinity are fixed; for any other point, a return after $k$ iterates requires ${\mu }^k=1$, whose least positive solution is $n$. Thus its only orbit sizes are one and $n$. For the identity, every orbit has size one. This is a direct orbit computation, independent of an appeal to the orbit-stabilizer theorem.

A nonidentity Möbius transformation with only one fixed point is conjugate to a nonzero translation of the Riemann sphere. Its translation iterates are $z+nb$, tending to infinity in the topology of the Riemann sphere for finite $z$, with infinity itself fixed. Conjugating back proves that every point tends to the unique fixed point. This is convergence on the sphere; if the fixed point is finite, intermediate iterates can still pass through infinity.

The Tikhonov regularization of the Volterra integration operator replaces the unbounded differentiation gain $a_n$ by $a_n/(1+\alpha a_n^2)$. The series converges in $L^2$ for every datum and $\alpha >0$. On the range of the Volterra integration operator, it converges to the weak derivative of the datum as $\alpha \downarrow 0$. No pointwise differentiation of arbitrary noisy L2 space data is required.

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