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| − | Any equivalence relation $\sim$ on a set of objects $\mathscr M$ defines the [[quotient set]] $\mathscr M/\sim$ whose elements are equivalence classes: the equivalence class of an element $M\in\mathscr M$ is denoted $[M]=\{M'\in\mathscr M:~M'\sim M\}$. Description of the quotient set is referred to as the ''classification problem'' for $\mathscr M$ with respect to the equivalence relation. The ''normal form'' of an object $M$ is a "selected representative" from the class $[M]$, usually possessing some nice properties (simplicity, integrability etc). Often (although not always) one requires that two ''distinct'' representatives ("normal forms") are ''not equivalent'' to each other: $M_1\ne M_2\iff M_1\not\sim M_2$.
| + | #REDIRECT [[Normal form]] |
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| − | The equivalence $\sim$ can be an identical transformation in a certain formal system: the respective normal form in such case is a "canonical representative" among many possibilities, see, e.g., [[Disjunctive normal form|disjunctive normal form]] and [[Conjunctive normal form|conjunctive normal form]] for Boolean functions.
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| − | However, the most typical classification problems appear when there is a group $G$ acting on $\mathscr M$: then the natural equivalence relation arises, $M_1\sim M_2\iff \exists g\in G:~g\cdot M_1=M_2$. If both $\mathscr M$ and $G$ are finite-dimensional spaces, the classification problem is usually much easier than in the case of infinite-dimensional spaces.
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| − | Below follows a list (very partial) of the most important classification problems in which normal forms are known and very useful.
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| − | ==Finite-dimensional classification problems==
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| − | When the objects of classification form a finite-dimensional variety, in most cases it is a subvariety of matrices, with the equivalence relation induced by transformations reflecting the change of basis.
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| − | ===Linear maps between finite-dimensional linear spaces===
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| − | Let $\Bbbk$ be a field. A linear map from $\Bbbk^m$ to $\Bbbk^n$ is represented by an $n\times m$ matrix over $\Bbbk$ ($m$ rows and $n$ columns). A different choice of bases in the source and the target space results in a matrix $M$ being replaced by another matrix $M'=HML$, where $H$ (resp., $L$) is an ''invertible'' $m\times m$ (resp., $n\times n$) matrix of transition between the bases,
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| − | $$
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| − | M\sim M'\iff\exists H\in\operatorname{GL}(m,\Bbbk),\ L\in \operatorname{GL}(n,\Bbbk):\quad M'=HML.
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| − | \tag{LR}
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| − | $$
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| − | Obviously, this binary relation $\sim$ is an equivalence (symmetric, reflexive and transitive), called ''left-right linear equivalence''. Each matrix $M$ is left-right equivalent to a matrix (of the same size) with $k\leqslant\min(n,m)$ units on the diagonal and zeros everywhere else. The number $k$ is a complete invariant of equivalence (matrices of different ranks are not equivalent) and is called the [[rank]] of a matrix.
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| − | A similar question may be posed about homomorphisms of finitely generated modules over rings. For some rings the normal form is known as the [[Normal_form_(for_matrices)#The_Smith_normal_form|Smith normal form]].
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| − | ===Linear operators (self-maps)===
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| − | The matrix of a linear operator of an $n$-dimensional space over $\Bbbk$ ''into itself'' is transformed by a change of basis in a more restrictive way compared to (LR): if the source and the target spaces coincide, then necessarily $n=m$ and $L=H^{-1}$. The corresponding equivalence is called [[similarity]] (sometimes ''conjugacy'' or ''linear conjugacy'') of matrices, and the normal form is known as the [[Jordan normal form]], see also [[Normal_form_(for_matrices)#The_Jordan_normal_form|here]]. This normal form is characterized by a specific block diagonal structure and explicitly features the [[Eigen value|eigenvalues]] on the diagonal. Note that this form holds only over an algebraically closed field $\Bbbk$, e.g., $\Bbbk=\CC$.
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| − | ===Quadratic forms on linear spaces===
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| − | A quadratic form $Q\colon\Bbbk^n\to\Bbbk$, $(x_1,\dots,x_n)\mapsto \sum a_{i,j}^n a_{ij}x_ix_j$ with a symmetric matrix $Q$ after a ''linear invertible'' change of coordinates will have a new matrix $Q'=HQH^*$ (the asterisk means the transpose):
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| − | $$
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| − | Q'\sim Q\iff \exists H\in\operatorname{GL}(n,\Bbbk):\ Q'=HQH^*.\tag{QL}
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| − | $$
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| − | The normal form for this equivalence is diagonal, but the diagonal entries depend on the field:
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| − | * Over $\RR$, the diagional entries can be all made $0$ or $\pm 1$. The number of entries of each type is an invariant of classification, called (or closely related) to the [[inertia index]].
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| − | * Over $\CC$, one can keep only zeros and units (not signed). The number of units is called the [[rank]] of a quadratic form; it is a complete invariant.
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| − | ===Quadratic forms on Euclidean spaces===
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| − | This classification deals with real symmetric matrices representing quadratic forms, yet the condition (QL) is represented by a more restrictive condition that the conjugacy matrix $H$ is orthogonal (preserves the Euclidean scalar product):
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| − | $$
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| − | Q'\sim Q\iff \exists H\in\operatorname{O}(n,\RR)=\{H\in\operatorname{GL}(n,\RR):\ HH^*=E\}:\ Q'=HQH^*.\tag{QE}
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| − | $$
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| − | The normal form is diagonal, with the diagonal entries forming a complete system of invariants.
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| − | A similar set of normal forms exists for self-adjoint matrices conjugated by Hermitian matrices.
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| − | ===Conic sections in the real affine and projective plane===
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| − | This problem reduces to classification of quadratic forms on $\RR^3$. An conic section is the intersection of the cone $\{Q(x,y,z)=0\}$ defined by a quadratic form on $\RR^3$, with the affine subspace $\{z=1\}$. Projective transformations are defined by linear invertible self-maps of $\RR^3$, respectively, the affine transformations consist of linear self-maps preserving the plane $\{z=0\}$ in the homogeneous coordinates (the "infinite line"). In addition, one can replace the form $Q$ by $\lambda Q$ with $\lambda\ne 0$. This defines two equivalence relations on the space of quadratic forms.
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| − | The list of normal forms for both classifications is follows from the normal form of quadratic forms:
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| − | {| class="wikitable"
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| − | ! Rank of $Q$ !! Projective curves !! Affine curves
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| − | | 3 || $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$ || $\varnothing_1=\{x^2+y^2=-1\}$, circle $\{x^2+y^2=1\}$, parabola $\{y=x^2\}$, hyperbola $\{x^2-y^2=1\}$
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| − | | 2 || point $\{x^2+y^2=0\}$, two lines $\{x^2-y^2=0\}$ || point $\{x^2+y^2=0\}$, two crossing lines $\{x^2-y^2=0\}$,
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| − | two parallel lines $\{x^2=1\}$, $\varnothing_2=\{x^2=-1\}$
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| − | | 1 || "double" line $\{x^2=0\}$ || $\varnothing_3=\{1=0\}$, "double" line $\{x^2=0\}$
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| − | |}
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| − | Note that the three empty sets $\varnothing_i$, are different from the algebraic standpoint: $\varnothing_1$ is an imaginary cicrle, $\varnothing_2$ is a pair of parallel imaginary lines which intersect "at infinity" (if these imaginary lines intersect at a finite point, this point is real), and $\varnothing_3$ is a double line "at infinity".
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| − | ===Families of finite-dimensional objects===
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| − | $\def\l{\lambda}$
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| − | In each of the above problems one can instead of an individual map $M$ (or a form $Q$) consider a ''local parametric family'' of objects $\{M_\lambda\}$, depending regularly (continuously, $C^k$- or $C^\infty$-differentiably, holomorphically) on finitely many real or complex parameters $\lambda$ varying near a certain point $a$ in the parameter space, $\l\in(\RR^p,a)$ or $\l\in(\CC^p,0)$ respectively. Two such local families $M_\lambda$ and $M'_\lambda$ are said to be equivalent by the action of a group $G$, if there exists a local parametric family of group elements, $\{g_\lambda\}$, also regular (although perhaps in a weaker or just different sense) that conjugates the two families: $g_\lambda\cdot M_\lambda=M_\lambda$ for all admissible values of $\lambda$.
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| − | The most instructive example is that of families of linear operators. A "generic" operator $M=M_0$ is diagonalizable with pairwise different eigenvalues $\mu_1(\lambda),\dots,\mu_n(\lambda)$ (depending, naturally, on $\lambda$). One can show that any finite-parametric family $\{M_\lambda|\lambda\in(\RR^p,0)\}$ can be diagonalized by a transformation $M_\lambda\mapsto H_\lambda M_\lambda H_\lambda^{-1}$ by the similarity transformation depending on $\l\in(\RR^p,0)$ with the same regularity. This follows from the [[Implicit function|Implicit function theorem]].
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| − | However, when some of the eigenvalues tend to a collision $\mu_i(0)=\mu_j(0)$, the diagonalizing transformation $H_\lambda$ may tend to a degenerate matrix so that $H_\lambda^{-1}$ diverges to infinity, while the transformation of a matrix to its Jordan normal form is far away from the family $\{H_\lambda\}$. However, a different choice of the normal form resolves these problems.
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| − | '''Example'''. Assume that the local family of matrices $\{M_\l|\l\in(\RR^p,0)\}$ is a deformation of the matrix $M_0$ whose normal form is a single Jordan block of size $n$. Then there exists a family of invertible matrices $\{H_\l|\l\in(\RR^p,0)\}$ such that
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| − | $$
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| − | H_\l M_\l H_\l^{-1}=
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| − | \begin{pmatrix}
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| − | \mu & 1&\\
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| − | &\mu& 1&\\
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| − | &&\mu&1&\\
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| − | &&&\ddots&\ddots\\
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| − | &&&&\mu&1\\
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| − | \alpha_1&\alpha_2&\alpha_3&\cdots&\alpha_{n-1}&\alpha_n
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| − | \end{pmatrix},\tag{SF}
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| − | $$
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| − | where $\mu=\mu(\l)$ and $\alpha_i=\alpha_i(\l)$, $i=1,\dots,n$ are regular (continuous, smooth, analytic,\dots) functions of the parameters $\l\in(\RR^p,0)$ of the same class as the initial family $\{M_\l\}$.
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| − | The normal form (SF) is called the Sylvester form, or sometimes the [[companion matrix]]. It is closely related to the transformation reducing a higher order linear ordinary differential equation to the system of first order equations, cf. [[Fuchsian_singular_point#Fuchsian_singularity_of_a_linear_.24n.24th_order_differential_equation|here]].
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| − | Deformation of a matrix which consists of several Jordan blocks with different eigenvalues can be reduced to a finite parameter normal form which involves $d$ constants which will depend regularly on $\l$, with
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| − | $$
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| − | d=\sum_\mu (\nu_1(\mu)+3\nu_2(\mu)+5\nu_3(\mu)+\cdots).
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| − | $$
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| − | Hhere $\nu_1(\mu)\geqslant n_2(\mu)\geqslant \nu_3(\mu)\geqslant\cdots~$ are the sizes of the Jordan blocks of $M_0$ with the same eigenvalue $\mu$ (arranged in the non-increasing order), and the summation is extended over all different eigenvalues of the matrix $M_0$ {{Cite|A71|Theorem 4.4.}}.
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| − | For a systematic exposition of this subject, see {{Cite|A83|Sect. 29, 30}}. Normal forms for parametric families of objects (mainly dynamical systems) belong to the area of responsibility of the [[Bifurcation|bifurcation theory]].
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| − | ==Singularities of differentiable mappings==
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| − | This area refers to classification of (germs of) maps $(\RR^m,0)\to(\RR^n,0)$, which constitute an infinite-dimensional space, with respect to the left-right equivalence: two germs $f,f':(\RR^m,0)\to(\RR^n,0)$ are equivalent, if there exist two germs of diffeomorphisms $h:(\RR^m,0)\to(\RR^m,0)$ and $g:(\RR^n,0)\to(\RR^n,0)$ such that $f=g^{-1}\circ f\circ h$. This left-right action corresponds to a change of local coordinates near the source and target points.
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| − | One can consider several parallel flavors of the classification theory:
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| − | *''holomorphic'' (or ''real analytic''), when both the germ $f$ and the conjugacies $g,h$ are assumed/required to be sums of the convergent Taylor series;
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| − | * ''smooth'', more precisely, $C^\infty$-smooth;
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| − | * ''formal'' theory, where all objects are represented by formal Taylor series without any assumptions on their convergence.
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| − | However, for the left-right classification, the three classifications usually coincide. In particular, if two holomorphic germs are conjugated by a pair of formal self-maps, then they also can be conjugated by a pair of holomorphic self-maps. If two $C^\infty$ germs are formally conjugated, then they are also $C^\infty$ conjugated, etc. The finite smoothness category is not as developed as the three flavors above: one could expect that the differentiability class of the conjugacies will in general be lower than that of the maps, but the sharp estimates are mostly unknown.
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| − | For more detailed exposition see [[Singularities of differentiable mappings]]. Here we give only a brief summary of available results.
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| − | ===Maps of full rank===
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| − | With each smooth germ $f:(\RR^m,0)\to(\RR^n,0)$ one can associate a linear map $M:\RR^m\to\RR^n$ which is the linearization of $f$ ($M$ is also called the tangent map to $f$, the Jacobian matrix or the differential of $f$ at the origin). In coordinates one can write this as follows,
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| − | $$
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| − | \forall x\in (\RR^m,0)\quad f(x)=Mx+\phi(x)\in (\RR^n,0),\qquad M=\biggl(\frac{\partial f_i}{\partial x_j}(0)\biggr)_{\begin{aligned} i&=1,\dots,n, \\ j&=1,\dots, m,\end{aligned}}\quad \|\phi(x)\|=O(\|x\|^2).
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| − | $$
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| − | If the operator $M$ has the full rank, then $f$ is right-left equivalent to the linear germ $g'(x)=Mx$ {{Cite|GG|Corollaries 2.5, 2.6}}.
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| − | These assumptions hold in two cases: where $m\le n$ and $M$ is injective, and where $m\ge n$ and $M$ is surjective. The conclusion reduces the classification of nonlinear germs to that of linear maps, which was already discussed [[#Linear_maps_between_finite-dimensional_linear_spaces|earlier]].
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| − | This result is equivalent to the [[Implicit function]] theorem. In particular, it shows that the image of an [[immersion]] locally looks like a coordinate subspace, and the preimages of points by a [[submersion]] locally look like a family of parallel affine subspaces of the appropriate dimension.
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| − | The obvious reformulation of this theorem is valid also for real-analytic and complex holomoprhic germs.
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| − | ===Germs of maps in small dimension===
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| − | When the rank condition fails, the normal form is nonlinear and is known in small dimensions. The corresponding theory is known by the name [[Singularity theory]] of differential maps, or the [[Catastrophe theory]].
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| − | The classification is organized along a tree: the normal forms depend on the rank of the Jacobian matrix, but also on some relationships between higher order Taylor coefficients of $f$ at the origin, introducing deeper and deeper degeneracy. Each such set of conditions is characterized by its [[codimension]], the number of algebraically independent conditions imposed on the initial segment of the Taylor series of $f$ (in the invariant terms, on the [[jet]] of $f$). By the Thom's [[Transversality]] theorem, singularities of codimension $k$ and higher generically do not occur in generic families of maps involving less than $k$ parameters.
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| − | ====Holomorphic curves====
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| − | A nonconstant holomorphic (or real analytic) germ $f:(\C^1,0)\to(\C^1,0)$ is biholomorphically left-right equivalent to the monomial map $g:z\mapsto z^\mu$, $\mu\in\NN$; the number $\mu=1$ corresponds to a full rank map and the normal form is linear, for $\mu>1$ nonlinear. In a similar way a nonconstant (germ of a) [[holomorphic curve]] $f:(\CC^1,0)\to(\CC^n,0)$ is monomial of the form
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| − | $$
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| − | z\mapsto (z^{\mu_1},z^{\mu_2},\dots,z^{\mu_k},0,\dots,0),\qquad 1\le \mu_1<\mu_2<\cdots<\mu_k,\ k\le n.
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| − | $$
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| − | For a "generic" germ of a holomorphic curve $\mu_i=i$, for all $i=1,\dots,n$. The codimension of the degeneracy corresponding to the collection $\mu=(\mu_1,\dots,\mu_n)$ is equal to $\mu_n-n$; if $k<n$, the corresponding codimension is infinite (the curve is hyperplanar).
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| − | ====Nondegenerate critical points of functions and the Morse lemma====
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| − | A smooth map $f:(\RR^n,0)\to(\RR,0)$ which is not of the full rank, has a [[critical point]] at the origin: $\rd f(0)=0$. In this case the quadratic approximation $Q:\RR^n\to\RR$, $(x_1,\dots,x_n)\mapsto\sum_{i,j=1}^n q_{ij}x_ix_j$ provided by the [[Hessian matrix]] $\rd ^2f(0)=\|q_{ij}\|$, $q_{ij}=\frac{\partial^2 f}{\partial x_i\partial x_j}(0)$, is the normal form for the left-right equivalence, assuming that the rank of this form is full. This assertion is famous under the name of the [[Morse lemma]] {{Cite|M}}, {{Cite|AVG}}:
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| − | $$
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| − | \rd f(0)=0,\ \operatorname{rank}\rd^2 f(0)=n\implies f(x)\sim Q(x).
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| − | $$
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| − | The known [[#Quadratic_forms_on_linear_spaces|classification of quadratic forms]] allows to bring $f(x)$ to the normal form $f(x)=x_1^2+\cdots+x_k^2-x_{k+1}^2-\cdots-x_n^2$. It is worth mentioning that one can transform a germ to its normal form by applying the change of variables in the source only: change of the variable in the target space is unnecessary for critical points.
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| − | ====Degenerate critical points of smooth functions====
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| − | If the critical point of a function is degenerate and its corank $\delta=\operatorname{corank}Q=n-\operatorname{rank}Q>0$, the normal forms become more complicated, although the initial steps are still simple.
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| − | If $\delta=1$, then the classification reduces to that of (smooth or analytic) functions of one variable. Except for an "infinitely degenerate" subcase, a function with Hessian of corank 1 can be brought to the normal form denoted by "class $A_\mu$":
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| − | $$
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| − | \rd f(0)=0,\ \operatorname{corank} \rd^2f(0)=1\implies f\sim x_1^{\mu+1}+\sum_{k=2}^n \pm x_k^2.
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| − | $$
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| − | ===="Elementary catastrophes"====
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| − | Smooth germs between two ''different'' spaces $f:(\RR^2,0)\to(\RR^2,0)$ have polynomial normal forms for the case $\rd f(0)=0$, if the higher order terms are not too degenerate. The rank condition means that the determinant (Jacobian) $\det \rd f(x)$ vanishes on a curve $\varSigma\subseteq(\RR^2,0)$ passing through the origin. The curve $\varSigma$, called the ''discriminant'' (the critical locus of $f$) is generically smooth at the origin and has a tangent line $\ell=T_0\varSigma\subseteq T_0\RR^2$. Position of this line can be compared with another line $\ell'=\operatorname{Ker} \rd f(0)\subseteq T_0\RR^2$.
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| − | If the two lines are transversal (cross each other by a nonzero angle), $T_0\varSigma\pitchfork \operatorname{Ker}\rd f(0)$, then the corresponding singular point is called ''fold'' and is right-left equivalent to the quadratic map
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| − | $$
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| − | f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}x^2\\y\end{pmatrix}.
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| − | $$
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| − | This map is a two-fold cover of the right half-plane $\{u\geqslant0\}$ in the targer plane. The line $\{u=0\}$ is the visible contour of the map.
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| − | If the two lines coincide, one needs an additional nondegeneracy assumption<ref>The angle between the directions $\ell$ and $\ell'$, measured along the curve $\varSigma$, should have a simple root at the origin.</ref>, yet under this condition the singular point is called [[cuspidal singularity]] and is right-left equivalent to the cubic map
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| − | $$
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| − | f:\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}u\\v\end{pmatrix}=\begin{pmatrix}xy+x^3\\y\end{pmatrix}.
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| − | $$
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| − | The image of the curve $\varSigma$, the visible contour of the map, is a semicubic parabola $4u^2-9v^3=0$, also referred to as the [[cusp]]. For the detailed exposition see {{Cite|GG|Ch. VI, Sect. 2}}.
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| − | <small>
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| − | ----
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| − | <references/></small>
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| − | ==Classification of dynamical systems==
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| − | This is the classification of (usually invertible) self-maps $f\:(\RR^n,0)\to(\RR^m,0)$; two self-maps are considered as equivalent if there exists a germ of diffeomorphism $h:(\RR^n,0)\to(\RR^n,0)$ such that $f'=h^{-1}\circ f\circ h$. This equivalence respects iteration, i.e., extends as the equivalence of cyclic subgroups of $\operatorname{Diff}(\RR^n,0)$:
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| − | $$
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| − | f\sim f'\iff \underbrace{f\circ \cdots\circ f}_{n\text{ times}} \sim \underbrace{f'\circ \cdots\circ f'}_{n\text{ times}}\qquad\forall n=1,2,\dots
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| − | $$
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| − | Such subgroups are naturally identified with discrete-time dynamical systems. A closely related classification of one-parametric subgroups $\{f^t|t\in\RR\}\subseteq\operatorname{Diff}(\RR^n,0)$ reduces to classification of germs of [[vector field]]s with a singular point at the origin. Two vector fields $v,v'$ are called equivalent, if there exists a diffeomorphism $h$ as above, such that
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| − | $$
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| − | h_*v=v'\circ h,\qquad h_*=\biggl(\frac{\partial h}{\partial x}\biggr)(0)=\rd f(0).
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| − | $$
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| − | As with the left-right equivalence of maps, one can first attempt to conjugate a vector field $v$ (or a self-map $f$) to its linear part $A=\rd v(0)$ (resp., $M=\rd f(0)$) and reduce the classification to that of linear operators. However, unlike the previous theory, the possibility of such linearization depends on the arithmetic nature of eigenvalues of $A$ (resp., $M$), in particular, on the presence of [[resonance]]s between them.
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| − | Besides, again in contrast with the left-right classification, the three parallel theories (analytic, $C^\infty$-smooth and formal) in the case of dynamic equivalence differ very much: the unavoidable divergence of the formal series conjugating an object with its normal form is a typical phenomenon.
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| − | For more information see [[Local normal forms for dynamical systems]].
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| − | ===Notes===
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| − | == References and basic literature ==
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| − | {| class="sortable"
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| − | |-
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| − | |valign="top"|{{Ref|M}} || valign="top"|J. W. Milnor, ''Morse theory''. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. '''51''' Princeton University Press, Princeton, N.J. 1963, {{MR|0163331}}.
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| − | |-
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| − | |valign="top"|{{Ref|A71}} || valign="top"|Arnold V. I., Matrices depending on parameters. ''Russian Math. Surveys'' '''26''' (1971), no. 2, 29--43, {{MR|0301242}}
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| − | |valign="top"|{{Ref|GG}}||valign="top"| M. Golubitsky, V. Guillemin, ''Stable mappings and their singularities'', Graduate Texts in Mathematics, Vol. '''14'''. Springer-Verlag, New York-Heidelberg, 1973, {{MR|0341518}}.
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| − | |-
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| − | |valign="top"|{{Ref|A83}}||valign="top"| Arnold V. I., ''Geometrical methods in the theory of ordinary differential equations''. Grundlehren der Mathematischen Wissenschaften, '''250'''. Springer-Verlag, New York-Berlin, 1983, {{MR|0695786}}
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| − | |valign="top"|{{Ref|AVG}}||valign="top"| V. I. Arnold, S. M. Guseĭn-Zade, A. N. Varchenko, ''Singularities of differentiable maps'', Vol. I, The classification of critical points, caustics and wave fronts. Monographs in Mathematics, '''82'''. Birkhäuser Boston, Inc., Boston, MA, 1985, ISBN: 0-8176-3187-9, {{MR|0777682}}.
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| − | |}
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