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...ayson, Daniel |chapter=Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry}}</ref> ==Finite fields== ...

{{DISPLAYTITLE:''K''-groups of a field}} ...[algebraic K-theory|algebraic ''K''-theory]], the '''algebraic ''K''-group of a field''' is important to compute. For a finite field, the complete calcul ...

...og of [[projective geometry]] in the setting of [[noncommutative algebraic geometry]]. *The quantum plane, the most basic example, is the [[quotient ring]] of the free ring: ...

...ntinuous multilinear alternating forms on the Lie algebra of smooth vector fields where the latter is given the <math>C^{\infty}</math> topology. ...irst2=D. B. |title=Cohomologies of Lie algebra of tangential vector fields of a smooth manifold |journal=Funct Anal Its Appl |volume=3 |pages=194–210 |ye ...

...nifold|smooth manifold]] <math>M</math>, is a generalization of the notion of a [[vector field]] on a manifold. ...>X</math> of the ''k''th [[exterior power]] <math>\wedge^k TM \to M</math> of the [[tangent bundle]], i.e. <math>X</math> assigns to each point <math>p \ ...

{{Short description|Derivative in differential geometry and vector calculus}} ...erivative of its derivative with respect to another two [[tangent vector]] fields. ...

...(LRT) is a generalization of the [[X-ray transform]] to symmetric tensor fields <ref name="sharaf"> V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,{{isbn|90-6764-165-0}}. Chapter 2.[http://www.math.nsc.ru/~sharaf ...

...n [[Morse theory]], a '''gradient-like vector field''' is a generalization of [[gradient vector field]]. ...els. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function. ...

...there is a fundamental system of neighborhoods (in the classical topology) of ''x'' whose intersection with ''Y'' is connected. ...and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result: ...

{{short description|Algebraic number fields are determined by their absolute Galois groups}} ...orlink=Florian Pop|year1=1990|year2=1994}} extended the result to infinite fields that are [[Finitely generated algebra|finitely generated]] over [[prime fie ...

...book.....C |author=Subrahmanyan Chandrasekhar |series=International Series of Monographs on Physics |publisher=Oxford: Clarendon |year=1961 |at=See discu this <math>\mathbf{F}</math> can be expressed as the sum of a toroidal field <math>\mathbf{T}</math> and poloidal vector field <math>\m ...

...variantly spread across space.|thumb|All vectors radiating from the center of this isotropic vector field are uniformly, symmetrically, and invariantly s In [[differential geometry]], an '''isotropic vector field''' is a concept that refers to a [[vector f ...

In [[differential geometry]], the '''last geometric statement of Jacobi''' is a [[conjecture]] named after [[Carl Gustav Jacob Jacobi]], whi ...I | mr = 1733567 | pages = 39–54 | publisher = Amer. Math. Soc. | series = Fields Inst. Commun. | title = The Arnoldfest (Toronto, ON, 1997) | volume = 24 | ...

{{Short description|Type of algebraic surface}} ...for such surfaces; in other words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these c ...

...ooth vector fields <math>v</math> and all smooth sections <math>s,t</math> of <math>E</math>. This is called the '''Chern connection''' on <math>E</math>. The curvature of the Chern connection is a (1, 1)-form. For details, see [[Holomorphic vect ...

...[ordered field]] that does not satisfy the [[Archimedean property]]. Such fields will contain infinitesimal and infinitely large elements, suitably defined. ...nt {{math|1}}, so that {{math|''nx''}} is the sum of {{math|''n''}} copies of {{math|''x''}}. ...

{{distinguish|Noncommutative geometry}} ...ura, Akio Tamagawa, "The Grothendieck conjecture of the fundamental groups of algebraic curves", ''Sugaku Exposition (AMS English Translation)'' (14) 1, ...

{{Short description|Concept in algebraic geometry}} ...d to prove the existence of [[semistable reduction|semistable reductions]] of algebraic varieties over one-dimensional bases. ...

{{For|more|Riemannian geometry}} Parallel vector fields (<math>\nabla X = 0</math>) are examples of recurrent tensors that find importance in mathematical research. For exampl ...

...-48116-6_4 | chapter = Excursion Probabilities | title = Random Fields and Geometry | url=https://archive.org/details/randomfieldsgeom00adle_474 | url-access=l ...andom vibration | doi = 10.1016/0022-460X(73)90059-X | journal = [[Journal of Sound and Vibration]]| volume = 27 | issue = 2 | pages = 165–182 | year = 1 ...