But in addition, statistical analysis indicates that lower-mass stars ( red dwarfs, of spectral category M ) are less likely to have planets massive enough to detect.

Stars of spectral category A typically rotate very quickly, which makes it very difficult to measure the small Doppler shifts induced by orbiting planets since the spectral lines are very broad.

Observations using the Spitzer Space Telescope indicate that extremely massive stars of spectral category O, which are much hotter than our Sun, produce a photo-evaporation effect that inhibits planetary formation.

* The category CohSp of coherent sober spaces ( and coherent maps ) is equivalent to the category CohLoc of coherent ( or spectral ) locales ( and coherent maps ), on the assumption of the Boolean prime ideal theorem ( in fact, this statement is equivalent to that assumption ).

Nevertheless, the DDO can still be used for spectral astronomy and is the site of a number of important studies, including pioneering measurements of the distance to globular clusters, providing the first direct evidence that Cygnus X-1 was a black hole, and the discovery that Polaris was stabilizing and appeared to be " falling out " of the Cepheid variable category.

In the ungraded situation described above, r < sub > 0 </ sub > is irrelevant, but in practice most spectral sequences occur in the category of doubly graded modules over a ring R ( or doubly graded sheaves of modules over a sheaf of rings ).

The category of spectral sequences is an abelian category.

BY Draconis variables are main sequence variable stars of late spectral types, usually K or M. The name comes from the archetype for this category of variable star system, BY Draconis.

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.

Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.

Fundamental group: Consider the category of pointed topological spaces, i. e. topological spaces with distinguished points.

We thus obtain a functor from the category of pointed topological spaces to the category of groups.

In the category of topological spaces ( without distinguished point ), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint.

Algebra of continuous functions: a contravariant functor from the category of topological spaces ( with continuous maps as morphisms ) to the category of real associative algebras is given by assigning to every topological space X the algebra C ( X ) of all real-valued continuous functions on that space.

Doing these constructions pointwise gives covariant and contravariant functors from the category of pointed differentiable manifolds to the category of real vector spaces.

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

In other words, we have a functor from the category of topological spaces with base point to the category of groups.

Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings that preserve all the topological properties of a given space.

In other words, in the category of normed vector spaces, the space K is an injective object.

* Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit.

The composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the set of measurable functions as the arrows.

Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.

It has been obvious to the assessors, particularly those in shore communities, that boats comprise the largest category of tangible personal property which they have been unable to reach.

Notable in this category are the Jupiter and Thor intermediate range ballistic missiles, which have been successfully developed, produced, and deployed, but the relative importance of which has diminished with the increasing availability of the Atlas intercontinental ballistic missile.

It is natural from the marksman's viewpoint to call a bull's-eye a success, but in the mice example it is arbitrary which category corresponds to straight hair in a mouse.

The first possibility results in a closed interval of tangent points in the f-plane, the end points of which fall into category ( B ) or ( C ).

It is the only county in the state so far this month reporting a possible shortage in GA category, for which emergency allotment can be given by the state if necessary.

The honored professionals were awarded for all the work done in a certain category for the qualifying period ; for example, Jannings received the award for two movies in which he starred during that period.

Verbs which fall into this category include sitzen ( to sit ), liegen ( to lie ) and, in parts of Carinthia, schlafen ( to sleep ).

There are several results in category theory which invoke the axiom of choice for their proof.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

Dialling these numbers will cause the local switch to announce which carrier your calls are being routed through for a specific category of calls.

They comprise one category of substance ( ousiae ) existing independently ( man, tree ) and nine categories of accidents, which can only exist in something else ( time, place ).

Jefferson is known to have done an unusual amount of traveling for the time in the American South, which is reflected in the difficulty of pigeonholing his music into one regional category.

For the second exam, called the Principles and Practices, Part 2, or the Professional Engineering exam, candidates may select a particular engineering discipline's content to be tested on ; there is currently not an option for BME with this, meaning that any biomedical engineers seeking a license must prepare to take this examination in another category ( which does not affect the actual license, since most jurisdictions do not recognize discipline specialties anyway ).

; Independent countries: This category has independent countries, which the CIA defines as people " politically organized into a sovereign state with a definite territory ".

Some strips which are still in affiliation with the original creator are produced by small teams or entire companies, such as Jim Davis ' Garfield, however there is some debate if these strips fall in this category.

Abstracting again, a category is itself a type of mathematical structure, so we can look for " processes " which preserve this structure in some sense ; such a process is called a functor.

It is a natural question to ask: under which conditions can two categories be considered to be " essentially the same ", in the sense that theorems about one category can readily be transformed into theorems about the other category?

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.

For example, a ( strict ) 2-category is a category together with " morphisms between morphisms ", i. e., processes which allow us to transform one morphism into another.

* Interactive Web page which generates examples of categorical constructions in the category of finite sets.