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For three particles, choose the relative position of particles one and two, the position of their center of mass relative to particle three, and the total center of mass. The kinetic energy diagonalizes with masses

For $n>1$, a cyclic group of prime-power order $n$ cannot embed in $S_{n-1}$, because a permutation of order $p^k$ requires a cycle of length at least $p^k$. If $n$ is not a prime power, every group of order $n$ embeds in $S_{n-1}$ by combining coset actions on subgroups of two distinct prime orders.

The average treatment effect in a target population is $\mathbb{E}[Y(1)-Y(0)]$, the average additive contrast between two potential outcomes. It does not require every individual effect to be the same. If half the population has effect two and half effect minus one, the average treatment effect is $1/2$. Under consistency in causal inference, conditional exchangeability given pretreatment covariates $X$, and positivity, it is identified by ${\mathbb{E}}_X[\mathbb{E}(Y\mid A=1,X)-\mathbb{E}(Y\mid A=0,X)]$. The outer average uses the specified target population, not whichever group happens to be observed most often.

Every Cauchy sequence of real numbers converges to a real number. Explicitly, the conditionimplies the existence of $L\in \mathbb{R}$ with $x_n\to L$. The hypothesis compares approximations with one another, without requiring a candidate answer. This makes completeness of the real numbers an existence principle: sufficiently consistent approximations cannot approach a missing destination. ^[ref].

The equivalent least-upper-bound property says that every nonempty set $A\subseteq \mathbb{R}$ bounded above has a supremum $s\in \mathbb{R}$. Precisely, every $x\in A$ satisfies $x\le s$, and for every $\epsilon >0$ some $x\in A$ satisfies $s-\epsilon <x$. The second condition excludes any smaller upper bound. No largest element of $A$ is required.

The rational numbers lack this property. The set $A=\{q\in \mathbb{Q}:q\ge 0,q^2<2\}$ contains one and has upper bound two, but has no supremum in $\mathbb{Q}$. A proposed rational supremum $s$ would have to satisfy $s^2=2$: if $s^2<2$, a sufficiently small rational increase still lies in $A$; if $s^2>2$, a sufficiently small rational decrease remains an upper bound. The irrationality of the square root of two excludes equality. In $\mathbb{R}$, its supremum is $\sqrt{2}$. ^[ref].

Construct this square root by the bisection method, starting with rational bounds $[l_0,u_0]=[1,2]$. At each step square the midpoint and retain the half whose endpoints bracket two after squaring. The first four resulting intervals are $[1,3/2]$, $[5/4,3/2]$, $[11/8,3/2]$, and $[11/8,23/16]$. After $k$ steps, $u_k-l_k=2^{-k}$ and $l_k^2<2<u_k^2$. Both endpoint sequences are Cauchy sequences because every later endpoint remains in the current interval. Completeness of the real numbers gives a common limit of a sequence $L$; continuity of squaring gives $L^2=2$. The midpoint approximates $L$ with error at most $2^{-(k+1)}$. Thus exact rational arithmetic produces certified accuracy while completeness of the real numbers supplies the exact answer.

Richard Dedekind approached completeness of the real numbers by filling gaps in the rational numbers with Dedekind cuts. In the preface to his 1872 Continuity and Irrational Numbers, he traced the problem to teaching differential calculus in 1858 and wanting arithmetic foundations for limits. His construction makes a missing boundary, such as $\sqrt{2}$, into a number defined by the separation it produces among rational numbers. ^[ref].

A non-degenerate probability distribution is stable if adding independent copies changes only its location and scale. Precisely, if $X_1,X_2$ are independent copies of $X$, then for every $a,b>0$ there are constants $c>0$ and $d\in \mathbb{R}$ such thatThe coefficients in this general definition are positive. Allowing arbitrary mixed signs without changing the shape parameter is not valid for every asymmetric stable law; symmetric stable distributions have the stronger signed-coefficient property.

A stable law has an index $0<\alpha \le 2$, with the scale combination $c=(a^{\alpha }+b^{\alpha }{)}^{1/\alpha }$ in the displayed convention. The centering $d$ may be necessary. A strictly stable distribution permits the combination without that centering.

For a standard normal random variable, $a{X}_1+b{X}_2$ has the law of $\sqrt{a^2+b^2}X$, giving index two. For a standard Cauchy random variable, it has the law of $(a+b)X$, giving index one. The Cauchy example shows that stability does not require a finite variance or even a defined mean. Stable laws are useful precisely because an aggregation can preserve the same family despite changing its scale.

If $X_1,\dots ,X_n$ are independent standard Cauchy random variables, thenis again standard Cauchy. Indeed, its characteristic function is $(e^{-|t|/n}{)}^n=e^{-|t|}$.

Let $(X,d)$ be a metric space and let $Q=X/\sim$ be a quotient set by an equivalence relation. The chain construction gives the largest pseudometric $d_Q$ on $Q$ for which the canonical projection of a quotient set $q:X\to Q$ is a non-expansive map. For classes $A,B$, putEach chain pays for movement inside $X$ and jumps at no cost between members of the same equivalence class. Reversing and concatenating chains proves symmetry and the triangle inequality. Taking a single link gives $d_Q(qx,qy)\le d(x,y)$. Conversely, any pseudometric making $q$ a non-expansive map is bounded above by the cost of every chain, hence by this infimum.

The result need not be a metric on the original quotient set. In the subset $X=\{0\}\cup \{1/n:n\ge 1\}$ of the real line, with its induced metric, collapse all the points $1/n$ to one class $A$, keeping $\{0\}$ separate. Then $d_Q(\{0\},A)\le 1/n$ for every positive integer $n$, so these distinct classes have distance zero. Identifying all zero-distance pairs produces the metric quotient of a pseudometric, on which the induced distance is a metric.

A non-expansive map between metric spaces is a function $f$ such that $d(fx,fy)\le d(x,y)$.

The complete theory of infinite-dimensional vector spaces over a fixed finite field is aleph-zero-categorical: every countably infinite model has countably infinite dimension and is therefore isomorphic to every other such model.

Compactness gives a countable model elementarily equivalent to $(\mathbb{Z},+,0)$ that contains a nonzero element divisible by every positive standard integer. It is not isomorphic to $\mathbb{Z}$, so $\mathrm{Th}(\mathbb{Z},+,0)$ is not aleph-zero-categorical.

A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.

Absolute thermodynamic temperature can be defined so that a reversible engine exchanging heats $Q_h$ and $Q_c$ with reservoirs at $T_h$ and $T_c$ satisfiesA choice of one reference temperature fixes the scale.

The continuum hypothesis asserts that every infinite subset of the real numbers is either countable or equinumerous with the real numbers, equivalently $2^{{\aleph }_0}={\aleph }_1$.

For infinite cardinals $\kappa ,\lambda$, Hausdorff's formula is

For infinite cardinals, choice gives $\kappa +\kappa =\kappa \cdot \kappa =\kappa$. Exponentiation satisfies $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }$, while a countable sum of cardinals has size the maximum of ${\aleph }_0$ and their supremum.

In ZF, let $K,L$ be nonempty. If $K\times L$ injects into $K\bigsqcup L$, then there is either an injection or a surjection from $K$ to $L$.

Indeed, extend the inverse of the given injection to a surjection $p:K\bigsqcup L\to K\times L$ by sending points outside its range to one fixed pair $(k_{\ast },l_{\ast })$. If the second coordinate of $p$ restricted to the $K$-summand covers $L$, it is the required surjection. Otherwise choose $l_0$ that it misses. Every $(k,l_0)$ then has its preimage in the $L$-summand. Except possibly for the default pair, that preimage is unique, so these preimages inject $K$ into $L$. In the exceptional case they inject $K\setminus \{k_{\ast }\}$ into $L$; either their image is all of $L$, yielding a surjection from $K$, or an omitted point extends the map to an injection from $K$.

For an oriented closed connected $d$-manifold, the fundamental class is the unique generator of $H_d(M;\mathbb{Z})$ that restricts to the orientation generator in every local homology group.

The leading term ${\omega }^{\delta }n$ is the highest-exponent term in Cantor normal form; the remaining tail is strictly below ${\omega }^{\delta }$.

The Belinfante-Rosenfeld construction adds the divergence of a spin-current superpotential to the canonical stress-energy tensor, producing a symmetric conserved tensor with the same integrated four-momentum under suitable boundary conditions.

For a Poisson mean $\mu$, the natural parameter is $\theta =\log \mu$, so the canonical link is the logarithmic link $g(\mu )=\log \mu$.