For $V(r)=km{r}^2$, fixed nonzero angular momentum $h$ gives the effective potential $V_{\mathrm{eff}}=h^2/(2m{r}^2)+km{r}^2$. A positive-radius circular orbit exists when $k>0$, at the displayed radius, and $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8km>0$ proves radial stability by the effective potential stability criterion. Its angular speed has magnitude $\sqrt{2k}$ and its small radial oscillation frequency is $\sqrt{8k}$. If $k\le 0$, no such orbit with nonzero angular momentum exists; for $h=0,k>0$, the sole minimum is at the origin, not a positive-radius circular orbit.

A subset $U$ is chronologically saturated in both directions when $I^{+}(U)=U=I^{-}(U)$. If the ambient spacetime is connected and $U$ is nonempty, then $U$ is the whole spacetime.

For a composition free energy $F=\int [f(\varphi )+\kappa |\nabla \varphi {|}^2/2]$, the local chemical potential is the functional derivativeAt phase coexistence, exchanges between the bulk phases require equal chemical potentials. Mechanical equilibrium additionally requires equal pressure for a flat interface, or a Young–Laplace equation pressure jump for a curved interface.

For normal ordering of a fermion bilinear, charge conjugation sends $\bar{\psi }\Gamma \psi$ to $\bar{\psi }(C^{-1}\Gamma C{)}^T\psi$. In particular the Dirac current is odd and the axial current is even under charge conjugation.

The phase in a field transformation $\hat{C}{V}^{\mu }{\hat{C}}^{-1}={\eta }_C{V}^{\mu †}$. For a real vector relativistic quantum field, Hermitian conjugation restricts this phase to $\pm 1$.

For a Dirac field, the charge-conjugation matrix implements ${\psi }^c=C{\bar{\psi }}^T$. In the standard gamma matrix representation one can take $C=i{\gamma }^2{\gamma }^0$. An overall phase is conventional. The transpose intertwining relation is different from the entrywise complex conjugation involved in an antiunitary operator.

A theory violates charge conjugation when conjugating particles to antiparticles changes a physical interaction or its transition probabilities. This is different from CP violation: an interaction can violate charge conjugation and parity symmetry in quantum field theory separately while preserving their combination.

For $\mathcal{L}=D^3+a_2{D}^2+a_{1}D+a_0$ with smooth coefficients and leading coefficient one, a Green function solving $\mathcal{L}G={\delta }_{\xi }$ is continuous together with its first derivative at an interior source, and its second derivative jumps by one. Distributional differentiation shows why: a jump in $G$ would create a delta second derivative, a jump in $G^{\prime }$ a delta first derivative, and a jump in $G^{\prime \prime }$ the required delta. For a causal initial-value Green function, take the left branch zero and choose the right branch with initial values $0,0,1$ at the source.

The PC algorithm starts with a candidate undirected graph, removes edges using conditional independence tests over increasing conditioning-set sizes, records separating sets, then orients unshielded colliders and propagates orientations. Under oracle tests and faithfulness of a directed acyclic graph, it estimates the graph's Markov equivalence class.

A contour deformation changes the path of a contour integral through a region where its integrand is a holomorphic function. The Cauchy integral theorem preserves the integral when the endpoints are fixed and the connecting boundary terms vanish. Crossing a pole instead contributes the appropriately oriented residue; a branch cut limits which deformations are permitted on one analytic sheet.

For an entire function $f(z)={\sum }_j{c}_j{z}^j$, equality $|c_n|={\max }_{|z|=r}|f(z)|/r^n$ at a fixed $r>0$ holds exactly when $f(z)=c{z}^n$. The coefficient is the average of $r^{-n}f(r{e}^{i\theta })e^{-in\theta }$. Equality in its modulus bound forces the continuous integrand to have constant modulus and argument, hence to be constant. The identity theorem then extends the monomial identity from the circle to the whole plane. The zero function and the constant case $n=0$ are included.

A groupoid is a category in which every morphism is invertible. A group is a groupoid with one object, while an equivalence relation defines a groupoid with exactly one arrow between any related pair of objects. Every functor sends invertible arrows to invertible arrows.

A subcategory $\mathcal{S}$ of a category $\mathcal{C}$ specifies some objects and some morphisms between them, containing their identities and closed under composition. It is full if it contains every ambient morphism between its selected objects. A skeleton of a category is a full subcategory containing one object from each isomorphism class.

A skeletal category has no distinct isomorphic objects. A full and faithful functor with essential surjectivity between two skeletal categories is an isomorphism of categories: surjectivity on isomorphism classes becomes surjectivity on objects, and faithfulness and fullness reflect an equality of image objects to an isomorphism, hence an equality, of source objects.

The category of unital rings has identity-preserving ring homomorphisms as morphisms. The zero ring, allowing $0=1$, is terminal, and $\mathbb{Z}$ is initial. Nonzero rings admit no maps from the zero ring. Monomorphisms are injective: maps from $\mathbb{Z}[x]$ probe individual elements. It is a regular category, with surjective homomorphisms as regular epimorphisms; these are distinct from arbitrary categorical epimorphisms.

An effective Cartier divisor is cut out locally by one non-zero-divisor; it is a codimension-one closed subscheme. On an integral scheme a nonzero global section of a line bundle defines such a divisor, possibly empty. On a nonreduced scheme a nonzero section need not be regular, so its zero locus need not be a Cartier divisor.

Two Cartier divisors are linearly equivalent when their difference is a principal Cartier divisor. Their associated line bundles are then isomorphic, so their global section spaces have the same dimension. This definition works in every dimension.

The positivity of a Cartier divisor can be measured by sections, intersection numbers, or its numerical class. Ample divisors give projective embeddings after taking a multiple; nef divisors are their numerical limits; big divisors have the maximum possible order of section growth. Nefness and bigness are different conditions.

A real Cartier divisor is a finite real linear combination of Cartier divisors. A rational Cartier divisor is defined analogously with rational coefficients; after multiplication by a common denominator it is Cartier.