germ

Dictionary: germ (jûrm) pronunciation

Biology. A small mass of protoplasm or cells from which a new organism or one of its parts may develop.
The earliest form of an organism; a seed, bud, or spore.
A microorganism, especially a pathogen.
Something that may serve as the basis of further growth or development: the germ of a project.

[Middle English, bud, from Old French germe, from Latin germen.]

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5min Related Video: germ

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Food Lover's Companion: germ

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In the food world, the word "germ" refers to a grain (like wheat) kernel's nucleus or embryo. Wheat germ is one of the more commercially popular types on the market. The nutritiously endowed germ furnishes thiamine, vitamin E, iron and riboflavin.

Thesaurus: germ

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noun

A minute organism usually producing disease: bug, microbe, microorganism. See beings.
A source of further growth and development: bud¹, embryo, kernel, nucleus, seed, spark¹. See start/end.

Health Dictionary: germs

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Microorganisms that can cause disease or infection.

Veterinary Dictionary: germ

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1. old-fashioned and lay term for a pathogenic microorganism.
2. living substance capable of developing into an organ, part or organism as a whole; a primordium. Commonly used to refer to the embryos of wheat grains which are removed during milling and sold separately as wheat germ.

g. cell — direct descendants of the primordial cells which originate from the yolk sac endoderm and migrate to the gonadal ridges of the embryo, where they give rise to either ova or spermatozoa. Called also gonocytes, sex cells.
g. cell tumor — a rare tumor in dogs, similar to more common lesions in humans. Similar to pituitary adenomas in distribution and cellular characteristics.
g. line — the genetic material as it is transferred via the gametes, before being modified by somatic recombination or mutation.
g. line cells — gametes.
g. line transmission — a mode of transmission, particularly of retroviruses, whereby the genome of the virus is integrated into the chromosomal DNA and transmitted via gametes to offspring.
g. plasma evaluation program — a planned investigative, large scale breeding program aimed at accumulating comparative information on the relative performance of various breeds and crossbreeds of agricultural animals.
g. theory — 1. all organisms are developed from a cell.
g. tube — a tube-like structure that develops during the growth of some fungi and becomes a hypha; a feature of the yeast, Candida albicans.
wheat g. — see wheat germ.

Essay: The germ theory of disease

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In the middle of the 19th century, some surgeons realized that doctors could spread disease from one person to another. Ignaz Semmelweiss in 1847 reduced the number of cases of childbed fever by having surgeons wash their hands between deliveries. Joseph Lister in 1865 went further and proposed using carbolic acid (also called phenol) on patients' wounds during surgery.

Doctors were not the only agents spreading disease, of course. John Snow, a doctor in London, noticed that cholera cases were clustered where people used water from particular sources. He persuaded the authorities to remove the pump handle from a well at the center of the largest outbreak area; the number of cases in that area dropped immediately. William Budd learned of Snow's success and repeated it, stopping a cholera epidemic in Bristol in 1866 by managing the town's water supply.

Louis Pasteur urged French army surgeons to sterilize their instruments and their patients during the Franco-Prussian War of 1870. After the war, Pasteur began an extensive study of the causes of disease. He started with anthrax--a disease that affects both farm animals and humans--because another proponent of the newly developing germ theory of disease, Robert Koch, had already found the bacterium that causes anthrax in 1876. Pasteur first helped stop the spread of anthrax by proposing sterilization and burial of the bodies of animals that had died from it. Later he was to find something even more effective.

In 1879 Pasteur discovered by accident that bacteria could be weakened. In their weakened state, they failed to cause disease but still provoked immunity. He soon learned that the anthrax bacterium could be weakened in this way and by 1881 was able to demonstrate that an effective immunization against anthrax resulted from injection with the weakened bacteria. Pasteur called the process of producing immunization in this way vaccination, in recognition of Edward Jenner's use of cowpox (vaccinia) to prevent smallpox. Pasteur, Koch, and others developed vaccines of varying degrees of effectiveness against many of the major communicable diseases of the time--cholera, tuberculosis, tetanus, diphtheria, rabies.

Originally the germ theory applied only to bacteria, but about the same time as Pasteur was writing his major exposition of the theory, Charles Laveran discovered that the organism causing malaria is a protist (sometimes called a protozoan), not a bacterium. The agents of some diseases, especially skin diseases, may be fungi. In other communicable diseases, such as measles, neither a bacterium, protist, nor fungus could be found in the late 19th or early 20th centuries. Because of the success of the germ theory, organisms too small to be seen or trapped in filters--filterable viruses--were postulated to explain these diseases. With the invention of the electron microscope in the 20th century, such viruses were finally observed directly.

Today the germ theory of disease is so entrenched that even diseases that are not communicable--such as cancer, diabetes, and multiple sclerosis--are suspected of being caused by some "germ."

Wikipedia: Germ (mathematics)

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In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the sets or maps in question will have specific properties, such as being analytic or smooth, but in general this is not needed (the maps or functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local have some sense.

The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.

1 Formal definition
2 Relation with sheaves
3 Examples
- 3.1 Notation
4 Applications
5 See also
6 References
7 External links

Formal definition

Basic definition

Given a point x of a topological space X, and two maps f, g : X → Y (where Y is any set), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal. That is,

$f|_U \,= \, g|_U$

(meaning that $\scriptstyle\forall x'\in U,\; f(x')=g(x')).$ Similarly, if S,T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that

$S\,\cap \,U = T \,\cap\, U.$

It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written $\scriptstyle f \sim_x g$ or $\scriptstyle S \sim_x T$ .

Given a map f on X, then its germ at x is usually denoted [f]_x. Similarly, the germ at x of a set S is written [S]_x. Thus,

$[f]_x = \{g:X\to Y \mid g \sim_x f\}.$

To denote a map germ at x in X which maps the point $\scriptstyle x\in X$ to the point $\scriptstyle y\in Y$ , one writes

$f:(X,x) \to (Y,y),$

such an f is then an entire equivalence class of maps, and it is usual to use the same letter f for any representative map.

Notice that two sets are germ equivalent at x if their characteristic functions are germ-equivalent at x:

$S\sim_x T \; \Longleftrightarrow \; \mathbf{1}_S \sim_x \mathbf{1}_T.$

More generally

Maps need not be defined on all of X, and in particular they don't need to have the same domain. However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say $\scriptstyle S\,\cap\,U\;=\;T\,\cap\,U$ , and then moreover $\scriptstyle f|_{S\cap V} = g|_{T\cap V}$ , for some smaller neighbourhood V with $\scriptstyle x\in V \subset U$ . This is particularly relevant in two settings:

f is defined on a subvariety V of X, and
f has a pole of some sort at x, so is not even defined at x, as for example a rational function, which would be defined off a subvariety.

Basic properties

If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbouhood of x.

Moreover, if the target Y is a vector space, then it makes sense to add germs: to define [f]_x + [g]_x, first take representatives f and g, defined on neighbourhoods U and V respectively, then [f]_x + [g]_x is the germ at x of the map f + g (where f + g is defined on $\scriptstyle U\cap V$ ). (In the same way one can define more general linear combinations.)

The set of germs at x of maps from X to Y does not have a useful topology, except for the discrete one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if X and Y are manifolds, then the spaces of jets $\scriptstyle J_x^k(X,Y)$ (finite order Taylor series at x of map(-germs)) do have topologies as they can be identified with finite-dimensional vector spaces.

Relation with sheaves

The idea of germ is behind the definition of sheaves and presheaves. A presheaf $\scriptstyle\mathcal{F}$ on a topological space X is an assignment of an Abelian group $\scriptstyle\mathcal{F}(U)$ to each open set U in X. Typical examples of Abelian groups here are: real valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex space), constant functions on U and differential operators on U.

If $\scriptstyle V \subset U$ then there is a restriction map $\scriptstyle\mathrm{res}_{VU}:\mathcal{F}(U)\to \mathcal{F}(V)$ , satisfying certain compatibility conditions. For a fixed x, one says that elements $\scriptstyle f\in\mathcal{F}(U)$ and $\scriptstyle g\in \mathcal{F}(V)$ are equivalent at x if there is a neighbourhood $\scriptstyle W\subset U\cap V$ of x with res_WU(f) = res_WV(g) (both elements of $\scriptstyle\mathcal{F}(W)$ ). The equivalence classes form the stalk $\scriptstyle\mathcal{F}_x$ at x of the presheaf $\scriptstyle\mathcal{F}$ . This equivalence relation is an abstraction of the germ equivalence described above.

Examples

If $X$ and $Y$ have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf $\scriptstyle\mathcal{F}$ and corresponding germs: some notable examples follow.

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.

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.

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.

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.

Notation

The stalk of a sheaf $\scriptstyle\mathcal{F}$ on a topological space $X$ at a point $x$ of $X$ is commonly denoted by $\scriptstyle\mathcal{F}_x$ . 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 $x$ .
$\mathcal{C}_x^\infty\,$ 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. 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 $x = 0$ .

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.

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.

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.

References

Nicolas Bourbaki (1989). 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 Narasimhan (1973). Analysis on Real and Complex Manifolds (2^nd ed. 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

Chirka, Evgeniǐ Mikhaǐlovich (2001), Hazewinkel, Michiel, ed., Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Germ of smooth functions at PlanetMath.
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.

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Translations: Germ

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Dansk (Danish)
n. - bakterie, smittekim, mikrobe, spire

idioms:

germ warfare biologisk krigsførelse
the germ of kimen til

Nederlands (Dutch)
bacil, (ziekte)kiem, grondgedachte

Français (French)
n. - microbe, (lit, fig) germe

idioms:

germ warfare guerre bactériologique
the germ of le germe de

Deutsch (German)
n. - Keim, Krankheitserreger

idioms:

germ warfare biologische Kriegsführung
the germ of der Ansatz für

Ελληνική (Greek)
n. - (βιολ.) σπέρμα, μικρόβιο, (μτφ.) αρχική αιτία, σπέρμα

idioms:

germ warfare (στρατ.) μικροβιολογικός πόλεμος
the germ of το σπέρμα (π.χ. μιας ιδέας)

Italiano (Italian)
germe, bacillo

idioms:

germ warfare guerra batteriologica
the germ of l'inizio di

Português (Portuguese)
n. - micróbio (m), embrião (m), semente (f), origem (f)

idioms:

germ warfare guerra (f) bacteriológica (Mil.)
the germ of a origem (f) de

Русский (Russian)
микроб, эмбрион, завязь, давать ростки

idioms:

germ warfare бактериологическая война
the germ of зародыш, зарождение чего-л.

Español (Spanish)
n. - germen, bacilo, microbio, bacteria

idioms:

germ warfare guerra bacteriológica
the germ of algo que está por desarrollarse

Svenska (Swedish)
n. - embryo, bakterie, frö

中文（简体）(Chinese (Simplified))
细菌, 根源, 种子

idioms:

germ warfare 细菌战争
the germ of 一种...的起源

中文（繁體）(Chinese (Traditional))
n. - 細菌, 根源, 種子

idioms:

germ warfare 細菌戰爭
the germ of 一種...的起源

한국어 (Korean)
n. - 세균, 배[종], 기원

idioms:

the germ of ~의 기원으로

日本語 (Japanese)
n. - 胚, 微生物, 細菌, 病原菌, 芽生え, 根源, 芽
v. - 芽を出す, 開く

idioms:

germ warfare 細菌戦
the germ of 初期に

العربيه (Arabic)
‏(الاسم) جرثومه‏

עברית (Hebrew)
n. - ‮חיידק, נבט, התהוות, ראשית‬

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Copyrights:

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more
		Food Lover's Companion. Food Lover's Companion. Copyright © 2001 by Barron's Educational Series, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Health Dictionary. The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Essay. History of Science and Technology, edited by Bryan Bunch and Alexander Hellemans. Copyright © 2004 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Germ (mathematics)". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

germ

Contents

Formal definition

Basic definition

More generally

Basic properties

Relation with sheaves

Examples

Notation

Applications

See also

References

External links