Suppose $X_n\ge 0$ is an integrable adapted process, $Y_n\ge 0$ is finite and adapted, and $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]\le (1+Y_n)X_n$. The displayed correction is a nonnegative supermartingale: its next denominator is measurable at time $n$, and its reciprocal is at most one. If ${\sum }_n{Y}_n<\infty$ almost surely, the denominators converge to a finite positive limit since $\log (1+y)\le y$. The almost sure supermartingale convergence theorem then gives finite almost sure convergence of $X_n$. No integrability of the infinite product or of ${\sum }_n{Y}_n$ is needed.

A probability measure-preserving system is compact if every $L^2$ observable has a relatively compact Koopman operator orbit in the global $L^2$ norm. For an invertible system one may use all integer times; for a general transformation use nonnegative times. This is a measure-theoretic definition, stronger than compactness of the underlying topological space.

If the forward orbit of $f$ under an isometry $U$ is totally bounded, choose a finite net whose centers are $U^{n_j}f$ and let $M={\max }_j{n}_j$. For every sufficiently large $n$, some center is within $\epsilon$ of $U^{n}f$, so cancellation gives $n-n_j\in R_{\epsilon }$. Thus $R_{\epsilon }$ is a syndetic set. Telescoping also gives $\Vert U^{kn}f-f{\Vert }_2\le k\Vert U^{n}f-f{\Vert }_2$.

For a matrix in the Gamma 0 congruence subgroup, argument dilation conjugates its action to an integral determinant-one matrix for each divisor $M$ of $N$. The anomaly of $E_2(M\tau )$ is $6c(c\tau +d)/(M\pi i)$, so the displayed linear condition cancels it. Necessity follows by testing the matrix with rows $(1,0)$ and $(N,1)$. The combinations $E_2(\tau )-M{E}_2(M\tau )$, with $M>1$ dividing $N$, span this space.

The finite Laurent coefficient $C(z)$ in the Kronecker limit formula is invariant under the modular group. Its Wirtinger derivative satisfies ${\partial }_{z}C=-\pi i(E_2(z)-3/(\pi y))/12$, by differentiating the convergent product in that formula. Differentiating $C(\gamma z)=C(z)$ gives weight-two covariance for $E_2^{\ast }$. Since $\Im (\gamma z)=y/|cz+d{|}^2$, substitution gives the displayed transformation law. This proof needs no prior modularity assertion for the eta product.

This is the degree-one convention for an almost commutative algebra: the exhaustive filtration is generated by a finite-dimensional first step containing $1$, the degree-zero part is $k$, and its associated graded algebra is commutative. The first step is a finite-dimensional Lie algebra under commutators. Its inclusion into $A$ extends by the universal property to a surjection from its universal enveloping algebra. The element $1$ of the first step is retained as a central Lie generator and its image is the associative identity; quotienting the first step by $k1$ before taking the bracket can lose scalar commutators. This convention is stronger than merely requiring a finitely generated commutative associated graded algebra for an arbitrary weighted filtration.

// Destination: noncommutative-algebra.bigb

For a unital irreducible action and $\xi \ne 0$, the annihilator $L_{\xi }=\{a:\pi (a)\xi =0\}$ is a maximal left ideal and is closed. Thus $A/L_{\xi }$ supplies the displayed complete norm on the module, with contractive action. Every module endomorphism is bounded for this quotient norm: its value at the cyclic vector is $a\xi$, so it is induced by bounded right multiplication by $a$ on the quotient. The Gelfand-Mazur theorem then makes the module's commutant consist of scalar operators. This constructs one compatible norm; it does not by itself establish continuity in a different preassigned norm.

For countably many linearly independent vectors in an infinite-dimensional irreducible complex Banach module, there are elements $a_n$ satisfying the displayed annihilation conditions while the tail images $\{\pi (a_n\cdots a_1){\xi }_j:j\ge n\}$ remain linearly independent. Use the quotient norm on an irreducible Banach module to make each finite annihilator a closed Banach subspace. The Jacobson density theorem makes each condition of independence of finitely many tail images open and dense there; the Baire category theorem intersects these countably many conditions. This is stronger than prescribing only finitely many images once.

An algebraically stable Runge-Kutta method is B-stable: for a dissipative vector field, a well-defined step cannot increase the distance between two starting values. This assumes that the compared stage equations have solutions, as required by stage solvability of an implicit Runge-Kutta method. The Runge-Kutta contractivity identity expresses the distance change as a nonpositive weighted term coming from the dissipative vector field minus a nonnegative quadratic form involving the matrix defining algebraic stability. Positive weights also rule out stage poles in the open left half-plane on the scalar test equation, giving A-stability there; zero-weight redundant stages need separate solvability attention.

For stage states $Y_i=y_n+h{\sum }_j{a}_{ij}{k}_j$ and a vector field with $Y_i^T{k}_i=0$, expansion of the new squared norm gives $\Vert y_{n+1}{\Vert }^2-\Vert y_n{\Vert }^2=-h^2{\sum }_{i,j}{M}_{ij}{k}_i^T{k}_j$, where $M_{ij}=b_i{a}_{ij}+b_j{a}_{ji}-b_i{b}_j$. Hence $M=0$ preserves the quadratic form exactly for every well-defined step. The same argument with a fixed symmetric matrix in the inner product preserves any quadratic invariant of the vector field. In particular it preserves the Euclidean norm for a skew-symmetric matrix differential equation.

Neighboring magnetic surfaces with different Alfvén speeds carry the same axial waveform at different rates. For a traveling profile $f(R,z+v(R)t)$, differentiation produces the displayed term, so transverse gradients grow even without axial wave steepening. Small magnetic diffusivity or viscosity can eventually act strongly on these gradients. The ideal process redistributes phase rather than directly dissipating wave energy.

A kinetic Alfvén wave is a dispersive extension of the Alfvén wave in a plasma when perpendicular kinetic or two-fluid effects, including a pressure response, matter. Its particular dispersion relation depends on the ordering and plasma parameters. A similar product of parallel and perpendicular wavenumbers can occur in a whistler wave, but agreement of a scaling power does not identify the underlying physical equations.

For an incompressible shear flow $U(z){\mathbf{e}}_x$ with uniform magnetic field $B{\mathbf{e}}_x$, define $S=\sigma +k_{x}U$ for time dependence $e^{i\sigma t}$ and $\zeta =i{k}_{x}v-i{k}_{y}u$. Eliminating the magnetic perturbations from the horizontal momentum equations gives the displayed product for $S\ne 0$. Away from the Alfvén wave resonance, it yields $\zeta =k_y{U}^{\prime }w/S$. At resonance one must retain the original system: for constant $U$, $k_y=0$, $S^2=k_x^2{B}^2/\rho$, any smooth transverse amplitude $v(z)$ with $b_y=k_{x}Bv/S$ and all other amplitudes zero is an Alfvén wave. Its nonzero vorticity disproves the unrestricted zero-vorticity reduction. Exponentially growing modes with real background coefficients cannot encounter these real-frequency resonances.

On the unit interval, the weak form is $a(u,v)={\int }_0^{1}fv$, with $a(u,v)={\int }_0^1(u^{\prime }{v}^{\prime }+xuv)$ on the zero-boundary Sobolev space. The nonnegative potential makes this form coercive in the derivative norm, and the Poincaré inequality makes that norm equivalent to the Sobolev norm. The Lax-Milgram theorem gives a unique solution. A conforming Ritz method minimizes the corresponding strictly convex quadratic energy and produces a symmetric positive-definite tridiagonal matrix for piecewise-linear hats.

The nearest cross-pair distance determines the next merge. At a threshold, clusters are connected components of the graph joining pairs below that threshold. This handles nonconvex shapes but can join otherwise separate groups through a chain of intermediate observations.

The farthest cross-pair distance determines the next merge. It favors groups with small diameter and reduces the chaining effect of single-linkage clustering, but can be sensitive to isolated extreme observations. Ties can affect intermediate partitions.

A dendrogram displays successive cluster merges, with the height of each merge equal to its intercluster dissimilarity. Cutting at a height gives a partition. Single-linkage clustering measures the closest cross-pair and can form chains, while complete-linkage clustering measures the farthest cross-pair and favors smaller within-cluster diameters.

The min and max cross-pair definitions of single-linkage clustering and complete-linkage clustering remain meaningful for a symmetric nonnegative dissimilarity matrix that violates the triangle inequality. Such a calculation gives a valid algorithmic dendrogram, but does not turn the original matrix into Euclidean distances. A triangle-inequality counterexample identifies that source defect without requiring invented replacement data.

Average linkage merges the pair with the smallest mean cross-pair dissimilarity. This differs from the minimum used by single-linkage clustering and the maximum used by complete-linkage clustering. The convention here weights individual cross-pairs equally; weighting already formed clusters equally is a different algorithm.

Ward clustering merges the pair causing the smallest increase in within-cluster sum of squares. Expanding each cluster's squared deviations about the merged mean gives the displayed increase. Its Euclidean sum-of-squares interpretation must not be asserted for arbitrary non-Euclidean dissimilarities.