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A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N ( x )< sup > β / 2 </ sup >, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.
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constructible and sheaf
Every torsion sheaf is a filtered inductive limit of constructible sheaves.
) More generally the cohomology with coefficients in any constructible sheaf is the same.
A perverse sheaf is an object C of the bounded derived category of sheaves with constructible cohomology on a space X such that the set of points x with
constructible and on
It is based on the Koch curve, which appeared in a 1904 paper titled " On a continuous curve without tangents, constructible from elementary geometry " ( original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire ) by the Swedish mathematician Helge von Koch.
There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle.
The angles that are constructible form an abelian group under addition modulo 2π ( which corresponds to multiplication of the points on the unit circle viewed as complex numbers ).
If x is any set, then x < sup >#</ sup > is defined analogously to 0 < sup >#</ sup > except that one uses L instead of L. See the section on relative constructibility in constructible universe.
It is called constructible if X can be covered by a finite family of subschemes on each of which the restriction of F is finite locally constant.
If f is a separated morphism of finite type then R < sup > q </ sup > f < sub >!</ sub > takes constructible sheaves on X to constructible sheaves on S. If in addition the fibers of f have dimension at most n then R < sup > q </ sup > f < sub >!</ sub > vanishes on torsion sheaves for q > 2n.
Defeasibility as an anytime algorithm: Here, it is assumed that calculating arguments takes time, and at any given time, based on a subset of the potentially constructible arguments, a conclusion is defeasibly justified.
Recently announced by Capcom, Rocketmen: Axis Of Evil, a downloadable arcade style video game based on the constructible strategy game, was slated to be released Fall 2007 for the PlayStation Network and Xbox Live Arcade.
constructible and variety
The image of a variety under the Veronese map is again a variety, rather than simply a constructible set ; furthermore, these are isomorphic in the sense that the inverse map exists and is regular – the Veronese map is biregular.
constructible and over
In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC ( and even in much weaker theories ).
Because constructible models can be easily damaged, augmented, or destroyed with the addition or removal of construction elements, a unit's abilities can vary greatly over the course of a battle.
constructible and finite
Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.
constructible and field
The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension of which is closed under square root and complex conjugation.
Given any such interpretation of a set of points as complex numbers, the points constructible using valid compass and straightedge constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations ( to avoid ambiguity, we can specify the square root with complex argument less than π ).
The set of ratios constructible using compass and straightedge from such a set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots.
In abstract terms, using these more powerful tools of either neusis using a markable ruler or the constructions of origami extends the field of constructible numbers to a larger subfield of the complex numbers, which contains not only the square root, but also the cube roots, of every element.
The arithmetic formulae for constructible points described above have analogies in this larger field, allowing formulae that include cube roots as well.
The field extension generated by any additional point constructible in this larger field has degree a multiple of a power of two and a power of three, and may be broken into a tower of extensions of degree 2 and 3.
One can show that any number constructible in one step from a field is a solution of a polynomial of degree 2, and therefore any number which is constructible by a series of steps is the solution of a minimal polynomial whose degree is a power of 2.
constructible and is
Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system ( or a fixed line segment of unit length ), the point can be constructed with unruled straightedge and compass.
A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x-and y-coordinate axes.
It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length | r | can be constructed with compass and straightedge.
It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible.
The geometric definition of a constructible point is as follows.
* March 30 – Carl Gauss obtains conditions for the constructibility by ruler and compass of regular polygons, and is able to announce that the regular 17-gon is constructible by ruler and compasses.
As a corollary of this, one finds that the degree of the minimal polynomial for a constructible point ( and therefore of any constructible length ) is a power of 2.
In particular, any constructible point ( or length ) is an algebraic number, though not every algebraic number is constructible ( i. e. the relationship between constructible lengths and algebraic numbers is not bijective ); for example, is algebraic but not constructible.
constructible and called
The numbers that can be constructed are called the origami or pythagorean numbers, if the distance between the two given points is 1 then the constructible points are all of the form where and are Pythagorean numbers.
It is called torsion if F ( U ) is a torsion group for all étale covers U of X. Finite locally constant sheaves are constructible, and constructible sheaves are torsion.
It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF ( which is also an inner model of ZFC + GCH ), called the constructible universe, or L.
constructible and if
The Whitehead conjecture is true if all sets are constructible.
* Statements true if all sets are constructible
Roughly speaking, if 0 < sup >#</ sup > exists then the universe V of sets is much larger than the universe L of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets.
Kunen showed that 0 < sup >#</ sup > exists if and only if there exists a non-trivial elementary embedding for the Gödel constructible universe L into itself.
On the other hand, if 0 < sup >#</ sup > does not exist, then the constructible universe L is the core model — that is, the canonical inner model that approximates the large cardinal structure of the universe considered.
Or if there is a measurable cardinal, then iterating the definable powerset operation rather than the full one yields Gödel's constructible universe, L, which does not satisfy the statement " there is a measurable cardinal " ( even though it contains the measurable cardinal as an ordinal ).
constructible and for
( Note that by designating cardinal directions for 1 ,-1,, and, complex numbers such as are considered constructible.
Some regular polygons, like the heptagon, become constructible ; and John H. Conway gives constructions for several of them ; but the 11-sided polygon, the hendecagon, is still impossible, and infinitely many others.
It is open whether there are infinitely many primes p for which a regular p-gon is constructible with ruler, compass and an angle trisector.
For example, for any constructible angle, the angle can be trivially trisected by ignoring the given angle and directly constructing an angle of measure.
This is close to being best possible, because the existence of 0 < sup >#</ sup > implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0 < sup >#</ sup >.
This is in some sense the simplest possibility for a non-constructible set, since all Σ and Π sets of integers are constructible.
Totally indescribable cardinals remain totally indescribable in the constructible universe and in other canonical inner models, and similarly for Π and Σ indescribability.
Compass and straightedge constructions are known for all constructible polygons.
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.
* The constructible topology and Zariski topology for coincide.
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