Germ (mathematics) - Citizendia

In mathematics, a germ of (continuous, differentiable or analytic) functions is an equivalence class of (continuous, differentiable or analytic) functions from a topological space to another (often from the real line to itself), grouped together on the basis of their equality on the neighborhood of a fixed reference point in their domain of definition. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change This article is about both real and complex analytic functions The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In the same way, a germ of sets is an equivalence class of subsets of a given topological space, grouped together on the basis of their equality on the neighborhood of a fixed reference point belonging to all of them. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is the "heart" of a function, as it is for a grain. The germ is the "heart" of the Cereal kernel the Embryo of the Seed. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

1 Formal definition
2 Applications
3 See also
4 References
5 External links

Formal definition

Basic definition

Two functions $f$ and $g$ between the same topological space $X$ and a set $Y$ are said to be equivalent near a point $x$ in their domain, if there is some open neighborhood $U$ of $x$ in $X$ on which they agree, i. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. e.

$f(x') = g(x') \quad \forall x'\in U\iff f|_U=g|_U \iff f \sim_x g$

This is an equivalence relation on the space $Y X = Hom(X, Y)$ of maps between $X$ and $Y$ . In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. For the proof, it is sufficient to note that equality is used in its definition: then reflexivity and symmetry are immediate consequences. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that For transitivity, given functions $f, g, h$ such that $f = g$ on $U$ and $g = h$ on $V$ , then $f = g = h$ on $U$ ∩ $V$ . In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b

The equivalence classes have the following form

$[f]_x=\left\{g \in Y^X\mid f \sim_x g\right\}\,$

Then the space of germs of functions at $x$ ∈ $X$ , is the quotient set

$\Gamma_x=Y^X/\!\!\sim_x\;=\left\{[f]_x \mid f\in Y^X\right\}\,$

As it can be easily seen, the germ at $x$ is the stalk of the sheaf of functions at $x$ . In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

The basic definition of germ does not require a topology on the codomain: a topology is only necessary to define neighborhoods of points in the domain. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, the codomain, or target, of a function f: X → Y is the set In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Similarly, it does not require any continuity or smoothness condition on the functions. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability With even more generality, functions can only be defined on a neighborhood $U$ of the given point $\scriptstyle x \in X$ and need not be restrictions of globally defined functions, so only a presheaf is needed to define germs. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. Since the stalks of a presheaf agree with the stalks of its sheafification (indeed, a construction of sheafification uses the sheaf of stalks of the given presheaf), the wording stalks of a sheaf is commonly used. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, the gluing axiom is introduced to define what a sheaf F on a Topological space X must satisfy given that it is a In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Germs of various classes of functions

If $X$ and $Y$ have additional structure, it is possible to define subsets of $Y X$ , or more generally sub-presheaves of a given presheaf $\scriptstyle\mathcal{F}$ and corresponding germs: some notable examples follow. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

If $X, Y$ are both topological spaces, the subset

$C^0(X,Y) \subset \mbox{Hom}(X,Y)\,$

of continuous functions defines germs of continuous functions. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

If both $X$ and $Y$ admit a differentiable structure, the subset

$C^k(X,Y) \subset \mbox{Hom}(X,Y)\,$

k

-times continuously differentiable functions, the subset

$C^\infty(X,Y)=\bigcap_k C^k(X,Y)\subset \mbox{Hom}(X,Y)\,$

of smooth functions and the subset

$C^\omega(X,Y)\subset \mbox{Hom}(X,Y)$

of analytic functions can be defined ( $ω$ here is the ordinal for infinity; this is an abuse of notation, by analogy with $C k$ and $C$ ^∞), and then spaces of germs of (finitely) differentiable, smooth, analytic functions can be constructed. In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability This article is about both real and complex analytic functions In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. In Mathematics, abuse of notation occurs when an author uses a Mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition

If $X, Y$ have a complex structure (for instance, are subsets of complex vector spaces), holomorphic (complex analytic) functions between them can be defined, and therefore spaces of germs of holomorphic functions can be constructed. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

If $X, Y$ have an algebraic structure, then regular (and rational) functions between them can be defined, and germs of regular functions (and likewise rational) can be defined. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Complex analysis, see Holomorphic function. In Mathematics, a regular function in the sense of Algebraic geometry In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In

Germs of various classes of sets

Two subsets $V, W$ of a topological space $X$ are said to be equivalent near a point $x$ belonging to them if there is some open neighborhood $U$ of $x$ in $X$ such that

$U \cap V = U \cap W \iff V \cong_x\! W$

This is equivalent to say that the germs of the characteristic functions of the two subsets are equal, i. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of e.

$V \cong_x\! W \iff \chi_V \sim_x \chi_W$

In more abstract terms, the contravariant functor which maps a set to its power set is representable by the set $Y = {0,1}$ , i. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, especially in Category theory, a representable functor is a Functor of a special form from an arbitrary category into the e. is naturally isomorphic to the hom functor $Hom( - , Y)$ . In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, specifically in Category theory, Hom-sets ie sets of Morphisms between objects give rise to important Functors to the Category Consequently, for a given set $X$ , it is possible to alternatively analyze or its subsets or functions belonging to $Hom(X, Y)$ where $Y = {0,1}$ , since those objects are isomorphic. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Notably, if $x$ is in the interior of both $V$ and $W$ , then they are equivalent near $x$ .

This is an equivalence relation on the power set of the topological space $X$ , to which $V$ and $W$ both belong: the equivalence classes have the following form

$V_x=\left\{W \in \mathcal{P}(X)|\, V \cong_x\! W\right\}\,$

Then the space of germs of sets at $x$ in $X$ is the quotient set

$X_x=X/\!\!\cong_x\;=\left\{V_x|V\in \mathcal{P}(X)\right\}\,\,$

Various spaces of germs of sets can be defined in the same way as it can be done for germs of functions. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X However, the space of germs of a variety is the most frequently encountered in standard mathematical research: the subset of the power set of the topological space $X$ used in its construction is the class of analytic varieties. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, specifically Geometry, an analytic variety is defined locally as the set of common solutions of several equations involving Analytic functions

Notation

The stalk of a sheaf $\scriptstyle\mathcal{F}$ on a topological space $X$ at a point $x$ of $X$ is commonly denoted by $\scriptstyle\mathcal{F}_x$ . In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. As a consequence germs, being stalks of sheaves of various kind of functions, borrow this scheme of notation:

$\mathcal{C}_x^0\,$ is the space of germs of continuous functions at $x$ .
$\mathcal{C}_x^k\,$ for each natural number $k$ is the space of germs of $k$ -times-differentiable functions at $x$ . In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an
$\mathcal{C}_x^\infty\,$ is the space of germs of infinitely differentiable ("smooth") functions at $x$ .
$\mathcal{C}_x^\omega\,$ is the space of germs of analytic functions at $x$ .
$\mathcal{O}_x\,$ is the space of germs of holomorphic functions (in complex geometry), or space of germs of regular functions (in algebraic geometry) at $x$ .

For germs of sets and varieties, the notation is not so well established: some notations found in literature include:

$\mathfrak{V}_x\,$ is the space of germs of analytic varieties at $x$ .

When the point $x$ is fixed and known (e. g. when $X$ is a topological vector space and $x = 0$ ), it can be dropped in each of the above symbols: also, when dim $X = n$ , a subscript before the symbol can be added. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. As example

${_n\mathcal{C}^0},{_n\mathcal{C}^k},{_n\mathcal{C}^\infty},{_n\mathcal{C}^\omega},{_n\mathcal{O}},{_n\mathfrak{V}}\,$ are the spaces of germs shown above when $X$ is a $n$ -dimensional vector space and $x = 0$ . In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

Applications

The key word in the applications of germs is locality: all local properties of a function at a point can be studied analyzing its germ. They are a generalization of Taylor series, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

Germs are useful in determining the properties of dynamical systems near chosen points of their phase space: they are one of the main tools in singularity theory and catastrophe theory. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System For other mathematical uses see Mathematical singularity. For non-mathematical uses see Gravitational singularity. This article refers to the study of dynamical systems For other meanings see Catastrophe.

When the topological spaces considered are Riemann surfaces or more generally analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, specifically Geometry, an analytic variety is defined locally as the set of common solutions of several equations involving Analytic functions Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. This article is about both real and complex analytic functions

References

Nicolas Bourbaki (1989). In Mathematics, specifically Geometry, an analytic variety is defined locally as the set of common solutions of several equations involving Analytic functions This article refers to the study of dynamical systems For other meanings see Catastrophe. In Mathematics, the gluing axiom is introduced to define what a sheaf F on a Topological space X must satisfy given that it is a Here we define the mathematical concept of a map germ. Given the set of all maps from one Manifold to another we can collect maps together under an Equivalence In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition General Topology. Chapters 1-4, paperback ed. , Springer-Verlag. ISBN 3-540-64241-2. , chapter I, paragraph 6, subparagraph 10 "Germs at a point".
Raghavan Narsimhan (1973). Analysis on Real and Complex Manifolds, 2^nd ed. , North-Holland Elsevier. ISBN 0-7204-2501-8. , chapter 2, paragraph 2. 1, "Basic Definitions".
Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall. , chapter 2 "Local Rings of Holomorphic Functions", especially paragraph A "The Elementary Properties of the Local Rings" and paragraph E "Germs of Varieties".
Giuseppe Tallini (1973). Varietà differenziabili e coomologia di De Rham (Differentiable manifolds and De Rham cohomology). Edizioni Cremonese. ISBN 88-7083413-1. , paragraph 31, "Germi di funzioni differenziabili in un punto $P$ di $V n$ (Germs of differentiable functions at a point $P$ of $V n$ )" (in Italian).

External links

Evgeniǐ Mikhaǐlovich Chirka "Germ", Springer-Verlag Online Encyclopaedia of Mathematics.
"Germ of smooth functions", Planethmath. org Encyclopedia.
Dorota Mozyrska, Zbigniew Bartosiewicz"Systems of germs and theorems of zeros in infinite-dimensional spaces", arxiv.org e-Prints server (Primary site at Cornell University). A research preprint dealing with germs of analytic varieties in an infinite dimensional setting. In Mathematics, specifically Geometry, an analytic variety is defined locally as the set of common solutions of several equations involving Analytic functions

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