A left monoid action assigns each $m\in M$ a map $a\mapsto m\cdot a$ with $1\cdot a=a$ and $(mn)\cdot a=m\cdot (n\cdot a)$. It need not be invertible or injective. A left M-set is equivalently a covariant functor from the one-object category determined by $M$.

For a commutative Noetherian ring, the distinct radicals in a minimal primary decomposition are exactly the associated primes of a module $R/I$, and hence are invariant. The injection into the sum of primary quotients proves one inclusion; an element supported on one component, supplied by irredundancy, proves the other. This does not imply uniqueness of embedded primary components.

For a positive-definite Hermitian operator $A={\sum }_j{t}_j|u_j⟩⟨u_j|$ and a unit vector $v$, the weights $|⟨v|u_j⟩{|}^2$ form a probability distribution. Scalar logarithmic concavity bounds their average of $\log t_j$ by the logarithm of their average of $t_j$. Positive semidefinite operators follow by a limiting convention. Using the eigenbasis of a density operator $\rho$ gives $S(\rho \Vert \sigma )\ge D(r\Vert q)$, where $r$ is the spectrum of $\rho$ and $q$ the diagonal of $\sigma$ in that basis. This yields nonnegativity of quantum relative entropy without commutativity.

On complex symmetric matrices with positive-definite real part, the determinant has a nonzero analytic square root normalized positively on real positive matrices. It is continued within this accretive domain. For $A=G+iB$, put $C=G^{-1/2}B{G}^{-1/2}$ with real eigenvalues $b_j$; then $d(A)=\sqrt{\det G}{\prod }_j\sqrt{1+i{b}_j}$, using positive-real-part roots for each factor. The principal scalar square root of the product determinant need not equal this continued root: accumulated determinant arguments can cross the scalar branch cut. The principal matrix logarithm instead tracks all factors consistently.

A martingale problem is well-posed for a specified test domain and class of paths if it admits a solution for every specified starting point and that solution is unique in law. Existence on one fixed probability space is not required. Test-domain and true-versus-local conventions must be stated. For bounded diffusion coefficients and $C_b^2$ test functions, boundedness on every finite horizon upgrades the local identities to true martingale identities.

When the knot polynomial is normalized as ${\psi }_i(x)={\prod }_{\ell =1}^{k-1}(x-t_{i+\ell })/(k-1)!$, the displayed formula has no additional factorial prefactor. Applied to Marsden's identity, it extracts the coefficient of the corresponding B-spline in any polynomial of degree at most $k-1$. Differentiating the finite sum makes adjacent terms cancel, proving independence of $x$. For linear polynomials the coefficient is their value at the Greville abscissa.

For a diffusion with generator $({\sigma }^2/2){\partial }_{xx}+b{\partial }_x$ and increasing scale function of a one-dimensional diffusion $s$, its speed density is $m=2/({\sigma }^2{s}^{\prime })$. Scale records hitting probabilities; speed records expected occupation and exit times. Multiplying scale by a positive constant divides speed by the same constant, leaving their Green-kernel product unchanged.

A positive normalized continuous local martingale deflator requires a locally square-integrable Brownian diffusion coefficient. If the forced market price of risk is $-t^{-1/2}$, the coefficient has magnitude $Z_t/\sqrt{t}$. Since $Z_0>0$, continuity makes its squared integral diverge near zero. The bank account can nevertheless exist because $t^{-1/2}$ itself is integrable; finite integrated interest is weaker than square-integrability of the required risk compensation.

If a compact H-hull lies in a disc of radius $r$ centred at $\xi \in \mathbb{R}$, the displayed estimate holds for $|z-\xi |>2r$, with an absolute constant $C$. It refines the Laurent expansion uniformly in the hull and supplies the first-order increment needed to derive the Chordal Loewner equation from the Loewner local growth property.

For steady axisymmetric ideal flow with only poloidal components, induction implies $\mathbf{u}\times \mathbf{B}=C{\mathbf{e}}_{\varphi }/R$. Axis regularity, or zero toroidal electromotive force, sets $C=0$. Continuity then gives $\rho \mathbf{u}=k(\psi )\mathbf{B}$, locally on connected magnetic surfaces of the poloidal magnetic flux function. Without the zero-circulation condition, annular domains can support steady cross-field flow.

In aligned steady ideal flow with $p=c_s^2\rho$ and constant isothermal sound speed, the Lorentz force density has no field-parallel component. Projecting momentum along the magnetic field makes $u^2/2+\Phi +c_s^2\ln (\rho /{\rho }_0)$ constant on each connected magnetic surface. The arbitrary reference density changes only the additive constant.

For a cold corotating field line anchored in a circular point-mass orbit, the effective potential has meridional Hessian ${\omega }^2\mathrm{diag}(-3,1)$. Along a straight line inclined by $\alpha$ to the vertical its quadratic curvature is ${\omega }^2(1-4{\sin }^2\alpha )$. Negative curvature gives the strict local launching criterion. Equality is marginal and global escape requires additional information.

For a logarithmically homogeneous barrier on a proper cone, the convex conjugate with reversed argument defines a barrier on the interior dual cone. The inverse gradient relation makes $y=-\mu \nabla F(s)$ equivalent to $s=-\mu \nabla F_{†}(y)$. This gives a joint primal-dual central path without assuming the barrier is identical on a self-dual cone.

The probability chain rule makes a sum of sequential predictive log scores equal the log Bayesian model evidence when the forecasts are coherent Bayesian one-step predictions under proper priors. Differences of total scores are therefore log Bayes factors.

Evaluate the real logarithmic derivative of Riemann xi at real part two, where every summand is positive and comparable to this kernel. The gamma logarithmic derivative and the bounded zeta logarithmic derivative give the logarithmic bound. It implies the local unit-interval count, and conversely such local counts imply this smoothed bound by summing the decaying tails.

Put $F(s)=(s-1)\zeta (s)$. The fractional-part continuation formula for the Riemann zeta function gives $|F(s)|\ll (|s|+1{)}^2$ for $\mathrm{Re}s\ge 1/4$. At $2+ij$, the reciprocal Euler product bounds $|\zeta (2+ij)|$ below by $1/\zeta (2)$. Applying the Jensen zero-count bound with outer radius $7/4$ and inner radius $8/5$ gives $O(\log (|j|+2))$ zeros in the latter disk. Those disks cover the half-strip $1/2\le \mathrm{Re}s\le 1$. Reflecting zeros using the functional equation of the Riemann zeta function completes the $O(T\log T)$ bound without requiring left-half-plane growth estimates.

A local principal connection form $A$ and a fiber coordinate $\gamma$ reconstruct the connection on a principal bundle in a local section. Changing that section by $s^{\prime }=su$ changes $\gamma$ to $u^{-1}\gamma$ and $A$ to $u^{-1}Au+u^{-1}du$. Substitution cancels the two terms involving $du$, so the reconstructed forms agree on overlaps. Here $A$ is Lie-algebra-valued and matrix conjugation denotes the adjoint action. Agreement across trivializations differs from the right-action equivariance $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$.

Under local asymptotic normality, a fixed $h/\sqrt{n}$ shift has a likelihood ratio limit $\exp (W-v/2)$ with $W\sim N(0,v)$. This is positive with mean one. Le Cam first lemma gives mutual contiguity of the product laws, so consistency at the central parameter transfers to these local alternatives.

For a local field $F$ containing all $m$ distinct $m$th roots of unity, with $m$ prime to the field characteristic, the symbol is ${\mathrm{rec}}_F(b)(a^{1/m})/a^{1/m}\in {\mu }_m$, after fixing a convention for Local Artin reciprocity. It is independent of the chosen root and bilinear. Its kernel in the second variable is the norm subgroup of $F(a^{1/m})/F$. The quadratic Hilbert symbol is the sign-valued case relevant to quadratic forms.

The vertical Liouville vector field on a cotangent bundle has complete flow $D_s(x,p)=(x,e^{s}p)$. Its pullbacks scale both the Liouville one-form and the canonical symplectic form by $e^s$. As $s\to -\infty$, each orbit tends to its base point on the zero section. This contraction proves rigidity of cotangent Liouville-form preservation.