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When two indicial roots differ by a positive integer, the recurrence for the smaller-root Frobenius solution can have a zero coefficient at the index of the larger root. If the corresponding right-hand side also vanishes, that coefficient is free and adds the larger-root solution; if it does not vanish, a pure Frobenius series is obstructed and a logarithmic term may be required. For $x^2{y}^{\prime \prime }-2x{y}^{\prime }-x^{2}y=0$, the recurrence $n(n-3)a_n=a_{n-2}$ has $a_1=0$ and leaves $a_3$ free. Choosing $a_3=0$ selects the even solution, not the only solution with exponent zero.

A Frobenius complement is the nontrivial proper subgroup $H$ in a Frobenius group with the displayed trivial-intersection property. In the action on cosets of $H$, it is a point stabilizer; no nonidentity element fixes two distinct points.

The Frobenius kernel consists of the identity and the elements lying in no conjugate of a Frobenius complement. The Frobenius kernel theorem proves that this conjugacy-invariant set is a normal subgroup, rather than assuming closure as part of its definition.

The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.

Write the suspension of an $(n-2)$-connected based CW complex as two cones meeting in the original space. Each cone pair is $(n-1)$-connected because its relative groups shift the original homotopy groups by one. Homotopy excision theorem compares the cone pair with the suspension relative to the other cone, giving the suspension homomorphism as an isomorphism for $i<2n-2$ and a surjection at $i=2n-2$ in the indexing ${\pi }_{i-1}X\to {\pi }_i\Sigma X$.

A Freiman 2-isomorphism is a bijection between subsets of abelian groups preserving additive quadruple relations in both directions. It need not extend to an ambient group homomorphism. Multiplication by a nonzero scalar in a finite field and translation are examples; reduction of an integer set modulo a prime is an example only when it creates no new additive relations.

An eight-term Freiman isomorphism extends injectively to the two-sum/two-difference set by sending its representations to the corresponding integer differences. Independence of representation uses four-term relations, and additivity whenever $y,z,y+z$ lie in that difference set uses six-term relations. Thus a proper symmetric progression in the modeled difference set lifts, one coordinate step at a time, to a proper integer progression of the same rank and cardinality.

For a finite subset $B\subseteq {\mathbb{F}}_2^N$ containing zero, choose $m=\lceil {\log }_2(2|12B|)\rceil$ and a uniformly random linear map $L:{\mathbb{F}}_2^N\to {\mathbb{F}}_2^m$. Each fixed nonzero vector has zero image with probability $2^{-m}$. The union bound gives a map whose kernel meets $12B$ only at zero. It is injective on $6B$, and $2^m<4|12B|$. The Plünnecke-Ruzsa inequality makes the image of $B$ have density at least $1/(4C^{12})$ if $|B+B|\le C|B|$.

Let $A$ be a nonempty subset of ${\mathbb{F}}_2^N$ with doubling constant at most $C\ge 1$. Translate to $B=A+a$ containing zero. The Petridis minimal-growth lemma and Ruzsa triangle inequality bound $|kB|\le C^k|B|$. In injective linear modelling of a small sumset, the image has density at least $1/(4C^{12})$. The Finite-field Bogolyubov lemma finds a subspace of codimension at most $\lceil 32C^{24}\rceil$ in its fourfold sumset.

After lifting a subspace through a Freiman model, obtain $\tilde{W}\subseteq 4B$ with $|\tilde{W}|\ge 2^{-\lceil 32C^{24}\rceil }|B|$. The set $B+\tilde{W}\subseteq 5B$ uses at most $C^5{2}^{\lceil 32C^{24}\rceil }$ cosets. Their representatives generate a quotient group of size at most two to this number. Since $|\tilde{W}|\le C^4|B|$, adjoining those representatives and $a$ proves the displayed bound. Its constants are deliberately coarse, but depend only on $C$. The nonempty assumption is necessary: no vector subspace has cardinality at most a constant times the size of the empty set.

Write a group element as $\exp (v+a[e_1,e_2])$. The step-two Baker-Campbell-Hausdorff formula gives the displayed multiplication and inverse $(-v,-a)$. The scalar $a$ is the antisymmetric second signature coordinate, which equals the signed area for a closed planar path. The Carnot-Carathéodory distance is comparable to $|v|+|a{|}^{1/2}$ near the identity, with the same homogeneous scaling globally.

A finite linear combination of iterated Lie brackets in formal variables. It evaluates naturally in every Lie algebra, and every Lie algebra homomorphism preserves its evaluation. The homogeneous terms of the Baker--Campbell--Hausdorff formula are rational Lie polynomials; this permits the group law to be reconstructed from the bracket wherever the series converges.

In the nested partial unary operation category, suppose the lower-operation diagram $A\rightrightarrows B\stackrel{q}{\to }Q$ has a split coequalizer with $qs=1$, $ft=1$ and $gt=sq$. The section $s$ takes eligible fixed points to eligible fixed points, so the displayed next operation is defined on exactly the required domain. For eligible $x\in B$, the equations giveThus $q$ preserves the next operation. Every factored morphism preserves it by evaluating at $sz$, giving creation of the specified coequalizer. Together with reflection of isomorphisms, the Beck monadicity theorem proves one-step monadicity.

Nonzero bending modes have the sine, cosine, hyperbolic sine and hyperbolic cosine form. Free conditions at the first end identify the cosine and hyperbolic cosine coefficients, and likewise the sine coefficients. The remaining boundary determinant is $2(1-\cos q\cosh q)$. Its first positive roots are about $4.73004,7.85320,10.99561$. The complete basis additionally includes two zero-energy filament modes, constant translation and affine tilt; these are absent from the finite-coefficient formula at $k=0$.

A chiral Majorana fermion with modes $\{b_r,b_s\}={\delta }_{r+s,0}$ and $b_r^{†}=b_{-r}$ has $T=-:\psi \partial \psi :/2$ and central charge $1/2$. The even and odd fermion-number parts of its Neveu–Schwarz sector give unitary modules with lowest weights $0$ and $1/2$. In its Ramond sector, a choice of ground-state zero-mode eigenvalue gives the module of weight $1/16$; the other choice gives an isomorphic Virasoro algebra module. Their cyclic highest-weight submodules are irreducible: a positive-energy invariant proper submodule would have a lowest vector orthogonal to every descendant of the original highest vector, which is impossible in a positive inner-product cyclic module.

Suppose $u_t=D{u}_{xx}+f_0(u)+\epsilon h(u)$ has a balanced stationary front $U_0(z)$, where $z=(x-q)/\sqrt{D}$ and $U_0^{\prime \prime }+f_0(U_0)=0$. Take $\dot{q}/\sqrt{D}=\epsilon c_1+\cdots$ and $u=U_0+\epsilon U_1+\cdots$. At first order, $L{U}_1=-c_1{U}_0^{\prime }-h(U_0)$ with $L=d^2/d{z}^2+f_0^{\prime }(U_0)$. Differentiating the front equation proves $L{U}_0^{\prime }=0$. Integration by parts against the decaying translation mode givesThis is the Fredholm solvability condition determining the velocity, derived from the self-adjoint differential operator rather than assumed. An orthogonality condition or a fixed midpoint removes the freedom to add a multiple of $U_0^{\prime }$ to $U_1$. For the bistable cubic reaction-diffusion equation, $h=-a{U}_0(1-U_0)$ gives $c_1=-\sqrt{2}a$, and its first-order forcing vanishes pointwise, so the phase-fixed shape correction is zero.

A nonzero integer polynomial of bidegree at most $N$ can cancel the first $(N+1{)}^2-1$ Taylor coefficients of the Fredholm series. Functional transcendence makes the resulting analytic function nonzero. If $\alpha$ and $f_l(\alpha )$ were both algebraic, the relation $f_l({\alpha }^{l^r})=f_l(\alpha )-{\sum }_{j<r}{\alpha }^{l^j}$ would keep the auxiliary values in one fixed number field. Their Absolute logarithmic Weil height is at most $N{l}^{r}h(\alpha )(1+1/(l-1))+O_N(r+1)$, whereas their chosen absolute value is $O_N(|\alpha {|}^{T_N{l}^r})$. The Liouville height inequality forces $T_N=O(N)$, contradicting its quadratic lower bound for sufficiently large $N$.

For $0<p<q$, truncating the Fredholm series at $n$ gives a rational with denominator at most $Q_n=q^{l^n}$ and positive error at most $2Q_n^{-l(1-\log p/\log q)}$ for large $n$. The displayed strict inequality first rules out a rational value by the rational separation bound, and then rules out an algebraic irrational value by Roth theorem. Reduced denominators are at most $Q_n$ and tend to infinity because the truncations are distinct. In particular $l\ge 3$ and $p<q^{\delta }$, $\delta <1/3$, imply transcendence.

Near a Fredholm operator, split the domain into its kernel and orthogonal complement and the target into its range and orthogonal complement. The large invertible block stays invertible under small norm perturbations. Invertible block elimination reduces the perturbed operator to that block and a map between the finite-dimensional kernel and cokernel spaces. Its index is their dimension difference regardless of its rank. Thus the Fredholm set is open and its index is locally constant.

An equivariant Fredholm operator between unitary representations of a compact group has finite-dimensional invariant kernel and cokernel. Their formal difference is its index in the representation ring of a compact group. If domain and target decompose into irreducibles with finite multiplicities $m_{\pi },m_{\pi }^{\prime }$, Schur's lemma makes each intertwining block a matrix on the multiplicity spaces. Its rank cancels between kernel and cokernel, giving ${\mathrm{ind}}_{G}T={\sum }_{\pi }(m_{\pi }-m_{\pi }^{\prime })[\pi ]$, with finitely many nonzero terms.

Here $C^{\infty }([0,1])$ means the real- or complex-valued functions continuous on $[0,1]$, smooth on $(0,1)$, with every interior derivative having a continuous extension to the endpoints; endpoint values of derivatives are these extensions. On this space of smooth functions on a compact interval, defineThese finite-valued seminorms increase with $k$ and separate points. In fact every $p_k$ is a norm, since it includes $p_0(f)=\Vert f{\Vert }_{\infty }$. The terminology seminorm includes norms; it does not assert that every measurement has a nontrivial kernel of a seminorm. Smooth convergence means $p_k(f_n-f)\to 0$ for every fixed $k$, equivalently uniform convergence of every derivative separately. It does not demand one tolerance or one convergence index that works for all orders simultaneously. The maximum convention is equivalent to the derivative supremum norm convention using a finite sum, since $p_k(f)\le {\sum }_{j=0}^k\Vert f^{(j)}{\Vert }_{\infty }\le (k+1)p_k(f)$.

For a worked example, $f(t)=t^3$ gives $p_0(f)=1$, $p_1(f)=3$, $p_2(f)=6$, and $p_k(f)=6$ for $k\ge 3$. The linear functional $\ell (f)=f^{\prime \prime }(1/2)$ obeys $|\ell (f)|\le p_2(f)$ and is continuous. Similarly, differentiation is a continuous linear operator $D:f\mapsto f^{\prime }$ for smooth convergence, because $p_k(Df)\le p_{k+1}(f)$.

The space of smooth functions on a compact interval is complete for this seminorm topology. A Cauchy sequence has uniform limits $g_j$ for its derivatives of each order. Passing to the limit in the fundamental theorem of calculus identitygives $g_j(t)-g_j(0)={\int }_0^t{g}_{j+1}(s)ds$, so $g_j^{\prime }=g_{j+1}$ and $g_0$ belongs to the same space. This is the completeness of the smooth-function metric argument; the metric there and the increasing-$p_k$ metric above induce the same smooth topology and the same Cauchy sequences.