In a regular category, a relation $R\hookrightarrow A\times B$ is total if its projection to $A$ is a regular epimorphism, and single-valued if two pairs with the same first component have the same second component. The latter makes that projection a monomorphism. A regular epimorphism that is monic is invertible, so $f={\pi }_B{\pi }_A^{-1}$ has graph exactly $R$. No axiom of choice or set-based element selection is used.

The category of relations on sets has sets as objects and binary relations as morphisms. Relational composition declares $aRc$ when some intermediate $b$ satisfies both component relations; the identity is the diagonal relation.

For a set-valued covariant functor on a small category, the Yoneda lemma sends the element $x\in F(c)$ to the transformation $\mathcal{C}(c,-)\to F$. These transformations form a cocone indexed by the opposite category of elements. At $a$, send the representative $(c,x,u:c\to a)$ to $F(u)x$. It is surjective using $(a,z,1_a)$; two representatives of $z$ are identified through that same object of the category of elements. This proves the displayed presentation pointwise.

The ordinary generating function $C(x)={\sum }_{n\ge 0}{C}_n{x}^n$ packages the Catalan recurrence. Its Cauchy product gives the formal power series identityThe constant term is $C_0=1$. Solving the quadratic equation chooseswith the formal power series square root having constant term one. The apparent quotient at $x=0$ has constant term one after cancellation; the other quadratic solution has a pole and cannot be this ordinary generating function. The equation records the empty object or an outer unit with two ordered smaller objects.

A Cartesian second-rank tensor invariant under all rotations about an oriented unit axis $n$ has the displayed form. A half-turn about the axis eliminates the mixed axial-transverse entries, and a quarter-turn makes the transverse block a scalar matrix plus an antisymmetric planar rotation generator. Conversely both transverse terms commute with every rotation about the axis. The antisymmetric term is allowed unless symmetry or an additional axis-reversing rotation is imposed. This differs from an axially invariant second-rank tensor for an unoriented axis.

For a constant orthogonal Cartesian coordinate change $x^{\prime }=Rx$, a Cartesian second-rank tensor obeys $T_{ij}^{\prime }=R_{ik}{R}_{jl}{T}_{kl}$. The chain rule gives ${\partial }_j^{\prime }=R_{jm}{\partial }_m$. Consequently ${\partial }_j^{\prime }{T}_{ij}^{\prime }=R_{ik}{R}_{jl}{R}_{jm}{\partial }_m{T}_{kl}=R_{ik}{\partial }_l{T}_{kl}$, the transformation rule for a vector. Under non-Cartesian position-dependent coordinate changes, partial derivatives must be replaced by the appropriate covariant derivative.

Every three-dimensional Cartesian second-rank tensor splits into scalar, symmetric-traceless, and antisymmetric parts. Their coefficients are $P=P_{kk}/3$, $S_{ij}=(P_{ij}+P_{ji})/2-P{\delta }_{ij}$, and $A_k={ϵ}_{kij}{P}_{ij}/2$. The epsilon contraction identity reconstructs the antisymmetric part and proves uniqueness. Under proper rotations $A$ behaves as a vector, but under reflections it is an axial vector.

A Cartesian second-rank tensor in three dimensions invariant under quarter-turn rotations about all three coordinate axes is a scalar multiple of the identity matrix. Squaring the quarter-turns gives coordinate half-turns; invariance under these forces every off-diagonal component to vanish. The quarter-turns then equate the three diagonal components. This conclusion does not require the original tensor to be symmetric.

An object isomorphic to its own exponential supplies mutually inverse folding and unfolding maps. Evaluation defines application, and folding after currying defines abstraction. The exponential universal property gives the beta rule, while the full isomorphism gives eta conversion. A mere retraction of the function object generally suffices for beta but not automatically eta.

A monoid encoding a one-object product and exponential structure: pairing is inverse to the map $z\mapsto (\pi z,{\pi }^{\prime }z)$, and currying is inverse to $y\mapsto \epsilon ⟨y\pi ,{\pi }^{\prime }⟩$. These bijections imply pairing naturality $⟨x,y⟩z=⟨xz,yz⟩$ and currying naturality $(x⟨z\pi ,{\pi }^{\prime }⟩{)}^{\ast }=x^{\ast }z$. The element $e=({\pi }^{\prime }{)}^{\ast }$ then satisfies $ez=e$ for every $z$, hence is idempotent. Splitting $e$ supplies a terminal object and a two-object Cartesian closed category whose endomorphisms at the original object are $M$. For the original object to be nonterminal, $M$ must be nontrivial.

A category with finite limits is locally Cartesian closed when every slice category is a Cartesian closed category. Covariant set-valued functor categories on small categories have this property, using the equivalence between their slices and functor categories on categories of elements.

Write $D{e}_i={\sum }_j{\theta }_{ij}\otimes e_j$, with the differentiated input index first. The graded Leibniz rule gives $D^2{e}_i={\sum }_j(d{\theta }_{ij}-{\sum }_k{\theta }_{ik}\wedge {\theta }_{kj})\otimes e_j$. For frames $e^{\prime }=ge$, the connection matrix transforms as ${\theta }^{\prime }=dg{g}^{-1}+g\theta g^{-1}$ and curvature by ${\Theta }^{\prime }=g\Theta g^{-1}$. Transposing to the usual coefficient-column convention gives $A={\theta }^t$ and $F=dA+A\wedge A={\Theta }^t$; the opposite sign is an index convention, not a different curvature.

Let $N_i=R_i-{\sum }_{s=i+1}^J{m}_{is}$ count individuals never recaptured after release $i$. Partition them into death-interval counts $n_{ij}$ and terminal survivors $n_{iJ}$. Their unnormalized weights are $w_{ij}=(1-{\varphi }_j){\prod }_{\ell =i}^{j-1}{\varphi }_{\ell }(1-P_{\ell +1})$ for $j<J$, and $w_{iJ}={\prod }_{\ell =i}^{J-1}{\varphi }_{\ell }(1-P_{\ell +1})$. Conditional counts are multinomial with normalizer $C_i={\sum }_{j=i}^J{w}_{ij}$. The EM algorithm replaces all missing counts by the displayed conditional expectations, then maximizes the resulting complete-data log likelihood. Survival and detection updates are expected successes divided by expected opportunities. A flat observed-likelihood ridge remains nonidentifiable even if each conditional complete-data maximization is explicit.

A closed-population capture-recapture model assumes the same individuals remain throughout the observations: no births, deaths, immigration, or emigration. Its elementary two-sample version additionally requires retained, correctly recognized marks and a second sample chosen by simple random sampling independently of first-sample membership. Mixing and unchanged capture behavior help make this plausible. Equal individual capture probabilities alone do not guarantee that every size-$C$ subset is equally likely; that stronger sampling assumption gives the hypergeometric distribution. See Barker, §§1.1.1 and 1.1.3: closure, marks, and catchability assumptions.

Unequal catchability, mark loss, and population change can invalidate this capture-recapture model. For example, if especially catchable individuals appear repeatedly, overlap can be inflated and the Lincoln-Petersen estimator can underestimate population cardinality. Lost marks can decrease recorded overlap and increase the same ratio estimator, all else equal. Two samples alone cannot establish that the sampling assumptions hold.

In a locally small capital regular category, $\Gamma$ is a regular functor and reflects terminal objects. A well-supported object $A$ has a point: otherwise $A\times A$ is well-supported with no points, so its diagonal is invertible by well-pointedness; then $A$ is subterminal and its regular epimorphism to $1$ is invertible, a contradiction. Pulling a regular epimorphism back along a point gives a well-supported object, so the point lifts. This proves preservation of regular epimorphisms, while representability gives limit preservation. If $\Gamma X$ is a singleton, its unique point $1\to X$ is monic and contains all points; $X$ is well-supported, so that point is invertible. Global sections need not reflect arbitrary isomorphisms between objects of proper support.

Choose one regular functor witnessing the regular-logic separation of a proper subobject for each proper subobject represented by a monomorphism of the small regular category. Their product is regular and reflects invertibility of monomorphisms. If an arbitrary morphism becomes invertible, its image inclusion becomes invertible and so does the diagonal of its kernel pair. The original morphism is both a regular epimorphism and a monomorphism, hence invertible. The target is a power of sets as a capital regular category. This proves an isomorphism-reflecting regular representation without asserting full faithfulness.

Here $I$ is a set regarded as a discrete category. Finite limits and regular epimorphisms are coordinatewise. A well-supported object is a family of nonempty sets. With the axiom of choice, every coordinate element extends to a global point of its product. Any subobject containing all global points must therefore contain every coordinate element, and be the whole family. Thus the category is capital. Families with some empty coordinates explain why global points need not reflect arbitrary isomorphisms.

Adding positive quadratic confinement $t\int h^2/2$ to the membrane energy changes its height spectrum to $k_{B}T/(\sigma q^2+t)$. The continuous height-shift symmetry is explicitly broken. Connected correlation function decay on the finite length $\xi$ and the height-difference variance saturates for every internal dimension, although its saturation value can depend on a microscopic cutoff. A negative confinement coefficient gives an unstable quadratic model rather than pinning.

For $a>0$ and ${\sigma }^2>0$, place probabilitiesTheir weighted deviations cancel, giving expected value $\mu$; their weighted squared deviations give variance ${\sigma }^2$. Only the upper point belongs to $X-\mu \ge a$, so its tail probability attains Cantelli's inequality. With $a=2$, ${\sigma }^2=1$, $\mu =0$, the locations are two and $-1/2$, with masses $1/5$ and $4/5$. The optimized shift makes the lower point zero before squaring, explaining equality in the Markov's inequality step.

The threshold convention matters: this example has $\mathbb{P}(X-\mu >a)=0$. For a strict threshold, placing the upper point at $a+\epsilon$ instead gives probabilities approaching the same bound as $\epsilon \downarrow 0$, with the prescribed mean and variance retained. Extra restrictions on the support of a probability distribution can change which extremal models are allowed.

The difference between the two normalized splitting degrees of the canonical scroll of a trigonal curve measures its imbalance. Since a smooth trigonal curve cannot contain the negative section, its intersection with that section is nonnegative. This gives $3a\le g+2$, and parity gives $a\equiv g(\mathrm{mod}2)$.