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An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.

A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.

For a connected regular covering corresponding to $H\le {\pi }_1(X,x_0)$, the subgroup $H$ is normal andThe quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.

For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.

For a connected covering corresponding to $H\le {\pi }_1(X)$, every fibre has cardinality $[{\pi }_1(X):H]$.

A support vector machine chooses a separating decision function by trading a wide geometric margin against hinge-loss violations.

For binary classification with posterior estimate ${\hat{p}}_1$, the plug-in classifier satisfies

The L-nearest-neighbour classifier estimates each class probability by its empirical frequency among the $L$ nearest training covariates and predicts a class of greatest estimated probability.

In the classical dilute regime, a monatomic ideal gas has entropy

For energy $pc+U(x)$,If $U(r)=r^{2n}/V^{2n/3}$, thenConsequently $pV=N{k}_{B}T$, while the mean energy and variance are $NA{k}_{B}T$ and $NA(k_{B}T{)}^2$ with $A=3+3/(2n)$.

Let $d$ be a positive odd integer with $k\ge 1$ distinct prime factors. Thenwhere $h$ is the class number of a negative discriminant.

Assign each complete prime power in the prime factorization of $d$ to one of two factors $a,c$. There are $2^k$ ordered integer factorizations $d=ac$ with coprime factors. Interchanging $a$ and $c$ pairs them without a fixed pair, because $d>1$ and coprime equal factors would both be one. Thus there are $2^{k-1}$ unordered pairs.

Each pair gives the primitive binary quadratic form $[\min (a,c),0,\max (a,c)]$ of discriminant of a binary quadratic form $-4d$. It is a reduced positive definite binary quadratic form, so distinct pairs give distinct equivalence classes under proper equivalence by uniqueness of the reduced positive definite binary quadratic form. This proves the lower bound. For the excluded case $d=1$, the separate value $h(-4)=1$ also satisfies the weaker numerical inequality $1\ge 2^{-1}$.

For an odd prime $q$, the formsare primitive binary quadratic forms of discriminant of a binary quadratic form $-8q$. Their diagonal polynomial coefficients are positive, so they are positive definite, and $q\ge 3$ makes both satisfy the reduced positive definite binary quadratic form inequalities.

Their proper equivalence classes are distinct. The first represents one at $(1,0)$, but the second has minimum positive value two: if $y\ne 0$, $2x^2+q{y}^2\ge q\ge 3$, while $y=0$ and $x\ne 0$ give at least two. Since proper equivalence preserves represented integers, the two forms cannot be equivalent.

Thus $h(-8q)\ge 2$. If the class number of a negative discriminant is exactly two, these are all its primitive proper classes. That completeness hypothesis is the reason a local square test can select one specific form in prime representation when h of minus eight q equals two.

The Jacobi constant is a conserved quantity for the circular restricted three-body problem. In barycentric coordinates rotating at constant angular frequency $n$, defineThe equation of motion in a rotating frame is ${\dot{\mathbf{v}}}_{\mathrm{rot}}+2\mathbf{\Omega }\times {\mathbf{v}}_{\mathrm{rot}}=\nabla U$. Taking its dot product with ${\mathbf{v}}_{\mathrm{rot}}$ proves $d(|{\mathbf{v}}_{\mathrm{rot}}{|}^2/2-U)/dt=0$: the Coriolis force does no work.

Only positions satisfying $2U-C\ge 0$ are accessible. The boundary gives a zero-velocity surface; openings near Lagrange points allow transit between regions but do not force it. Ordinary inertial-frame energy can change even while $C$ remains constant.^[ref]

A no-signalling box is a conditional probability distribution $P(x,y|a,b)$ whose marginal distribution for either party is independent of the other party's input. It can therefore exhibit correlations stronger than quantum correlations without transmitting information superluminally.

Attaching a new vertex that is a leaf of a graph multiplies the chromatic polynomial by $t-1$: after colouring the old graph, the new vertex may receive any colour except its neighbour's. Starting with one vertex and repeatedly adding a leaf of a graph constructs any finite tree. Thus a tree with $n\ge 1$ vertices hasIn contrast, every pair of vertices in a complete graph must receive different colours, so the multiplication principle gives the falling factorialFor positive integers $t<n$ this product vanishes, expressing the impossibility of assigning distinct colours to all $n$ vertices.

Deleting an edge removes one colour constraint. Edge contraction describes exactly the newly admitted colourings that violate that constraint. If $e$ joins distinct vertices $u,v$ of a simple graph $G$, write $G-e$ for the graph with that edge deleted and $G/e$ for the graph with $u,v$ identified and $e$ removed. Parallel edges produced by the contraction may be reduced to one edge, since they impose the same condition on a proper vertex coloring. ThenTo see why, divide the proper vertex colorings of $G-e$ into two classes. Those with different colours at $u,v$ are exactly the colourings of $G$. Those with equal colours at $u,v$ correspond one-to-one to colourings of $G/e$: the merged vertex receives their common colour. Subtracting the second class leaves the first.

For the triangle, delete $AC$ to get a three-vertex path graph, or contract $AC$ to get two vertices joined by an edge after removing the redundant parallel edge. Consequently,At $t=3$, this says $6=12-6$. Of the twelve path colourings pictured above, six give $A,C$ the same colour; these are precisely the ones ruled out when we restore $AC$.

This recurrence is useful because it reduces a constrained counting problem to smaller graphs. A graph on $n$ vertices with no edges has $P_G(t)=t^n$, since every vertex chooses its colour independently. Repeated edge deletion and edge contraction reduce to this case and show that the counting function is a polynomial. The same recurrence gives its leading coefficients. The method is conceptually simple, although the number of recursive cases can still grow rapidly.

A connected graph of maximum degree $\Delta$ has chromatic number at most $\Delta$ unless it is a complete graph or an odd cycle.

Chi-squared distributions are stochastically increasing in their statistical degrees of freedom. For any real $0<r<s$, take independent $Q_r\sim {\chi }_r^2$ and $W\sim {\chi }_{s-r}^2$. Gamma additivity givesFor integer $r,s$, this is the familiar construction by adding extra independent squared standard normal variables. For noninteger degrees of freedom, multiplying the gamma moment-generating functions proves the same statement.

Under this coupling the larger-degree variable is larger on every sample path. Hence $F_s(x)\le F_r(x)$ for every $x$, and in fact the inequality is strict for $x>0$: average $F_r(x-W)$ and use $W>0$ and the strictly positive chi-squared density. Therefore every quantile at a fixed probability in $(0,1)$ strictly increases with the degrees of freedom. Comparing chi-squared critical values must account for this change; the distribution is not fixed while only its label changes.

On projective space, the Euler sequence is

The total Chern class is $c(E)=1+c_1(E)+c_2(E)+\cdots$.