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A matched asymptotic expansion constructs approximations in regions with different distinguished limits and fixes their free constants by requiring agreement in an overlap region. The inner expansion resolves rapid variation; the outer expansion describes the surrounding region. Matching compares their common limiting behavior, rather than imposing distant boundary conditions directly on a local approximation.^[ref]

For the singular perturbation problem $ϵy^{\prime \prime }+y^{\prime }=0$, $y(0)=0$, $y(1)=1$, the leading outer solution is $y_0=1$. The inner variable $X=x/ϵ$ gives $Y_0^{\prime \prime }+Y_0^{\prime }=0$ and $Y_0(0)=0$, hence $Y_0=A(1-e^{-X})$. In the overlap region $ϵ\ll x\ll 1$, the inner variable tends to infinity while the outer coordinate tends to zero. The asymptotic matching condition isso $A=1$. One concrete path through the overlap region is $x=\sqrt{ϵ}$: then $x\to 0$ and $X={ϵ}^{-1/2}\to \infty$. Matching selects the free constant that the left boundary condition alone could not determine.

Why choose the right boundary condition for the outer solution? If instead one selects $y_0=0$ from the left, a proposed right boundary layer with $Z=(1-x)/ϵ$ satisfies $U_0^{\prime \prime }-U_0^{\prime }=0$. Its general solution is $A+B{e}^Z$. Matching to zero as $Z\to \infty$ forces $A=B=0$, contradicting $U_0(0)=1$. The direction in which the exponential function decays determines which endpoint can carry the boundary layer; endpoint selection is part of the calculation.

Let $K$ be a finite $n$-dimensional simplicial pseudomanifold, $n\ge 1$, without boundary. If its top-dimensional simplexes admit compatible orientations, their signed sum is a simplicial cycle $[K]$. Compatibility means that the two induced orientations on each shared codimension-one face of a simplex cancel. Since the top simplexes are connected through such faces, every integral top-dimensional cycle has one common coefficient, up to these orientation signs. Thereforegenerated by $[K]$. If compatible orientations do not exist, propagating a coefficient around an orientation-reversing chain of faces forces it to equal its negative; the integral top homology is then zero.

Finiteness matters for ordinary homology: a triangulated real line is orientable but has $H_1(\mathbb{R};\mathbb{Z})=0$. Summing all its edges is an infinite, locally finite chain, rather than an ordinary finite singular chain. Its orientation class belongs instead to Borel-Moore homology.

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

A simplicial set is a functor from the opposite of the simplex category to sets. Its face and degeneracy maps satisfy the simplicial identities.

The simplex category has objects the finite nonempty ordered sets $[n]=\{0<\cdots <n\}$ and morphisms the order-preserving maps.

The vertices of the barycentric subdivision of a simplicial complex are its nonempty faces, and its simplices are strict chains of faces.

The Euler characteristic compresses a finite CW complex's homology into one integer: count its topological cells with alternating signs, or count its Betti numbers with the same signs. A filled triangle has Euler characteristic one; its cycle graph boundary has Euler characteristic zero. The missing face changes the answer even though the vertices and edges are identical.

For a finite CW complex $X$, including the geometric realization of a finite simplicial complex,Here $b_i={\dim }_{\mathbb{Q}}{H}_i(X;\mathbb{Q})$ is the rational Betti number. More generally this ordinary homology definition makes sense whenever every rational homology group is finite-dimensional and only finitely many are nonzero. An arbitrary infinite CW complex need not satisfy these assumptions; its formal infinite alternating cell count is not a definition of this invariant.

The usefulness is that a finite description can reveal something about a space without computing every homology group. By the cell-count formula for Euler characteristic, count pieces rather than solve all the boundary operators. If two spaces give different answers, topological invariance of Euler characteristic rules out a homeomorphism, and even a homotopy equivalence. Equal answers are less decisive: the Euler characteristic of an annulus and the Euler characteristic of a torus are both zero, while their homology groups differ. This is a compact diagnostic, with a clearly defined limit on what it can tell us.

A regular simplex has pairwise equidistant vertices. Conversely, every finite equidistant subset of Euclidean space is the vertex set of a regular simplex.

A face is the simplex spanned by a subset of the vertices.

Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

The quartic term is a specific anharmonicity of the oscillator potential.

Every finite nonabelian simple group has order at least 60; the alternating group A5 attains this bound.

The conjugacy classes of $A_5$ have sizesA normal subgroup is a union of these classes containing the identity. No proper nontrivial such union has size dividing $60$, so Lagrange's theorem proves that $A_5$ is simple.

The iteration $y_{n+1}=y_n-h\mathrm{sign}(y_n)$ agrees with the exact linear descent until the first step past zero. It then either remains at zero or alternates between two values of magnitude at most $h$. Consequently its uniform grid-point error is at most $h$ for arbitrarily long finite time intervals.

Lebesgue measure on the real numbers extends interval length: $\lambda ([a,b])=b-a$ for $a\le b$. Its restriction to the Borel sigma-algebra is the unique translation-invariant sigma-finite measure on that sigma-algebra normalized by $\lambda ((0,1])=1$. The full Lebesgue measure is the completion of a measure of this restriction; its domain is the Lebesgue measurable sets. Translation invariance follows by comparing a translated Borel measure on half-open intervals and applying the uniqueness theorem for measures. For the construction and interval properties, see MIT lecture 8.

Every singleton has Lebesgue measure zero: $\{x\}\subseteq [x-\epsilon ,x+\epsilon ]$ gives $0\le \lambda (\{x\})\le 2\epsilon$ for every $\epsilon >0$. Enumerating the countable set $\mathbb{Q}\cap [0,1]$ and applying countable additivity yieldsThus a dense set can be a null set, and a union of null sets indexed by an uncountable set can have positive Lebesgue measure. The distinction explains both the Dirichlet function example and why almost everywhere is weaker than everywhere.

A Dynkin system on $\Omega$ contains $\Omega$, is closed under complements relative to $\Omega$, and is closed under countable disjoint unions. Unlike a sigma-algebra, it need not be closed under intersections of arbitrary members.

An equivalent axiom list is: contain $\Omega$, contain $B\setminus A$ whenever $A\subseteq B$ are members, and be closed under increasing countable unions. For the forward direction, $B\setminus A=(B^c\cup A{)}^c$, using a disjoint union. For the reverse direction, differences from $\Omega$ give complements; finite disjoint unions follow by two differences, and increasing partial unions give the countable disjoint-union axiom.

For example, on $\Omega =\{1,2,3,4\}$, take the empty set, $\Omega$, and every two-element subset. This is a Dynkin system: a disjoint union of two nonempty proper members is $\Omega$. It is not a sigma-algebra, since two intersecting pairs can have a singleton intersection. The Dynkin lemma says that adjoining intersection closure through a generating pi-system forces the full generated sigma-algebra.

A pi-system is a nonempty family of subsets closed under finite intersections.

A Borel set in a topological space belongs to the smallest sigma-algebra containing every open set.

A sigma-algebra atom is set-theoretically minimal. An atom is instead indivisible only modulo sets of measure zero.

An atom of a sigma-algebra is a nonempty measurable set containing no proper nonempty measurable subset. A finite sigma-algebra partitions its underlying set into atoms, and each measurable set is a union of them.

A Pi-1 formula in the arithmetical hierarchy has the form $\forall nR(n,\mathbf{x})$, with $R$ a decidable predicate; several universal number quantifiers can be combined by coding tuples. In a formal arithmetic presentation, the matrix is taken to have bounded quantifiers, with equivalent computable-predicate normal forms obtained by coding computations.

For example, the assertion that a fixed Turing machine never halts is $\forall tR(t)$, where $R(t)$ checks that it has not halted in its first $t$ steps. A false such assertion has a finite counterexample, while a true one need not have a finite computational certificate. Negating a Pi-1 formula gives a Sigma-1 formula.