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### DefinitionsUpDw('Definitions','-Abz-');

1. The quotient is a complex manifold whose first Betti number is one, so by the Hodge theorem, it cannot be Kähler.
2. The quotient is isomorphic (via) to a submodule of the Noetherian module, so is generated by finitely many elements.
3. The quotient is (antilinearly) isomorphic to the space of holomorphic modular forms of weight 2 − k.
4. This quotient is realized explicitly by the famous Hopf fibration S 1 → S 2 n +1 → CP n, the fibers of which are among the great circles of. (Web site)
5. The quotient is a hyperbolic triangle orbifold, with a cusp.

### First Isomorphism TheoremUpDw('FIRST_ISOMORPHISM_THEOREM','-Abz-');

1. The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S). (Web site)
2. The first isomorphism theorem for groups states that this quotient group is naturally isomorphic to the image of f (which is a subgroup of H). (Web site)

### Quotient AlgebraUpDw('QUOTIENT_ALGEBRA','-Abz-');

1. The first isomorphism theorem for Mal'cev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B).
2. Returns a quotient algebra defined via the action of a free algebra A on a (finitely generated) free module.

### Quotient ObjectsUpDw('QUOTIENT_OBJECTS','-Abz-');

1. This is an equivalence relation, and the equivalence classes are defined to be the quotient objects of X. (Web site)
2. Furthermore, you don't get to choose your sub- and quotient objects; they are imposed by the category.
3. Subobject) and quotient objects (cf.

### Rayleigh Quotient IterationUpDw('RAYLEIGH_QUOTIENT_ITERATION','-Abz-');

1. Specifically, this is the basis for Rayleigh quotient iteration. (Web site)

### Quotient RingsUpDw('QUOTIENT_RINGS','-Abz-');

1. It is the same construction used for quotient groups and quotient rings.
2. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. (Web site)

### Quotient ModuleUpDw('QUOTIENT_MODULE','-Abz-');

1. A quotient representation is a quotient module of the group ring.

### Quotient FieldUpDw('QUOTIENT_FIELD','-Abz-');

1. The set, together with the addition and multiplication operations shown above, forms a field, the quotient field of the integers.
2. Note that in this quotient field, every number can be uniquely written as p −n u with a natural number n and a p -adic integer u. (Web site)

### Rayleigh QuotientUpDw('RAYLEIGH_QUOTIENT','-Abz-');

1. It is shown that a complete (non-degenerate) solution of the Rayleigh quotient flow visits each of the eigenvectors of A in ascending order. (Web site)
2. The scaling factor ensures that the rate of variation of the Rayleigh quotient is constant and positive along solutions. (Web site)
3. In other words, it is the range of the Rayleigh quotient. (Web site)

### CoequalizerUpDw('COEQUALIZER','-Abz-');

1. Similarly, If is a coequalizer of, then is the "largest" quotient object of.

### Quotient TopologyUpDw('QUOTIENT_TOPOLOGY','-Abz-');

1. The topology on the quotient space is the quotient topology.
2. Dually, the coequalizer is given by placing the quotient topology on the set-theoretic coequalizer. (Web site)
3. An alternate version of the same construction is to define the quotient topology on using a surjection.

### CompactlyUpDw('COMPACTLY','-Abz-');

1. Quotient space s of locally compact Hausdorff spaces are compactly generated. (Web site)

### Quotient SpacesUpDw('QUOTIENT_SPACES','-Abz-');

1. Quotient spaces of locally compact Hausdorff spaces are compactly generated. (Web site)
2. There is, however, no reason to expect such quotient spaces to be manifolds.
3. Subspace s and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff.

### ParticularUpDw('PARTICULAR','-Abz-');

1. In particular, it is shown that complete solutions of the Rayleigh quotient flow visits the eigenvectors of \$A\$ in ascending order. (Web site)

### RespectUpDw('RESPECT','-Abz-');

1. The space is called the quotient space of the space with respect to.
2. On the other hand, the matrix of the quotient with respect to a basis for integral homology has determinant 10.
3. This can be made into a normed vector space in a standard way; one simply takes the quotient space with respect to the kernel of · p.

### Equivalence ClassUpDw('EQUIVALENCE_CLASS','-Abz-');

1. Taking the quotient with respect to the degenerate subspace gives a Hilbert space H A, a typical element of which is an equivalence class we denote by [ x].

### ElementsUpDw('ELEMENTS','-Abz-');

1. Together, these equivalence classes are the elements of a quotient group.

### DomainUpDw('DOMAIN','-Abz-');

1. By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. (Web site)
2. This is the dual notion to the kernel: just as the kernel is a sub space of the domain, the co-kernel is a quotient space of the target.

### StructureUpDw('STRUCTURE','-Abz-');

1. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups. (Web site)

### AlgebrasUpDw('ALGEBRAS','-Abz-');

1. As a quotient of two C* algebras, the Calkin algebra is a C* algebra itself.

### PointUpDw('POINT','-Abz-');

1. Let be the composite of the Hopf bundle map and the quotient map, which collapses the 2-skeleton of the 3-torus to a point.

### ProjectionUpDw('PROJECTION','-Abz-');

1. Then the kernel of this projection is another group, so it's the quotient of another free group.

### DimensionUpDw('DIMENSION','-Abz-');

1. The matrix induces thus an Euclidean structure on the quotient space which is isomorphic to since the kernel of has dimension 3. (Web site)

### SpheresUpDw('SPHERES','-Abz-');

1. The quotient of π n S by the image of the J-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres (Adams 1966).

### WayUpDw('WAY','-Abz-');

1. To do this, we should first find a better way to define the quotient of S by G when the action fails to be free.
2. This way, the quotient space "forgets" information that is contained in the subspace W.
3. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient. (Web site)

### PolycyclicUpDw('POLYCYCLIC','-Abz-');

1. In this case the field DerivedLength will denote the biggest integer k such that the quotient of G by the k+1 -st term in the derived series is polycyclic.
2. It also returns the polycyclic presentation and the appropriate homomorphism and map if the quotient is polycyclic. (Web site)

### CyclicUpDw('CYCLIC','-Abz-');

1. All subgroups and quotient groups of cyclic groups are cyclic. (Web site)
2. AbelianQuotientInvariants(F7); [ 29 ] The maximal abelian quotient of F(7) is cyclic of order 29. (Web site)

### ImpliedUpDw('IMPLIED','-Abz-');

1. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easily derived.

### RelationUpDw('RELATION','-Abz-');

1. The quotient group of under this relation is often denoted (said, " mod "). (Web site)

### NotionUpDw('NOTION','-Abz-');

1. Building bigger groups by smaller ones, such as D 4 from its subgroup R and the quotient is abstracted by a notion called semidirect product. (Web site)

### DualUpDw('DUAL','-Abz-');

1. Alternatives include the split-complex plane and the dual numbers, as introduced by quotient rings. (Web site)

### GeneralUpDw('GENERAL','-Abz-');

1. Compatibility with other topological notions Separation In general, quotient spaces are ill-behaved with respect to separation axioms. (Web site)

### FunctionsUpDw('FUNCTIONS','-Abz-');

1. The functions in this section enable the user to construct the radical, its quotient and an elementary abelian series.
2. In the quotient space, two functions f and g are identified if f = g almost everywhere.

### Kolmogorov QuotientUpDw('KOLMOGOROV_QUOTIENT','-Abz-');

1. The Kolmogorov quotient of X (which identifies topologically indistinguishable points) is T 1. Every open set is the union of closed sets.
2. When we form the Kolmogorov quotient, the actual L 2(R), these structures and properties are preserved.
3. Taking Kolmogorov quotient s, we see that all normal T 1 space s are Tychonoff. (Web site)

### ConverselyUpDw('CONVERSELY','-Abz-');

1. Conversely, a space is R 0 if and only if its Kolmogorov quotient (which identifies topologially indistinguishable points) is T 1.

### NotionsUpDw('NOTIONS','-Abz-');

1. I think we have to abandon the notions of sub and quotient in the homotopical world and stick to notions like (homotopy) limit and colimit.

### ContinuityUpDw('CONTINUITY','-Abz-');

1. The adic topology of the integers is deﬁned by the requirement of continuity of the projection into the quotient space modulo every nontrivial ideal.

### Discrete GroupUpDw('DISCRETE_GROUP','-Abz-');

1. Equivalently, think of as a quotient of hyperbolic space by a discrete group of isometries.

### DeterminantsUpDw('DETERMINANTS','-Abz-');

1. Cramer's rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two determinants.

### End ResultUpDw('END_RESULT','-Abz-');

1. The end result is the lightest Disney film in many a moon, a joyous romp akin to Aladdin in its quotient of laughs for kids and adults.

### PlaneUpDw('PLANE','-Abz-');

1. All parabolic surfaces can be obtained as a quotient of the plane.

### Hausdorff SpaceUpDw('HAUSDORFF_SPACE','-Abz-');

1. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.
2. Conversely, if X is a Hausdorff space and ker f is a closed set, then the coimage of f, if given the quotient space topology, must also be a Hausdorff space. (Web site)

### Difference QuotientUpDw('DIFFERENCE_QUOTIENT','-Abz-');

1. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient. (Web site)
2. In practice, the continuity of the difference quotient Q(h) at h = 0 is shown by modifying the numerator to cancel h in the denominator. (Web site)
3. If a finite difference is divided by b − a, one gets a difference quotient. (Web site)

### DerivativeUpDw('DERIVATIVE','-Abz-');

1. The derivative is the value of the difference quotient as the secant lines get closer and closer to the tangent line.

### Symmetry GroupUpDw('SYMMETRY_GROUP','-Abz-');

1. GENUS2 symmetry group This is a symmetry group on the Klein model of hyperbolic space whose quotient group is a genus 2 hyperbolic manifold. (Web site)

### ResultUpDw('RESULT','-Abz-');

1. A quotient can also mean just the integer part of the result of dividing two integers. (Web site)
2. The result will be a quotient module of the target of f. (Web site)

### LatticeUpDw('LATTICE','-Abz-');

1. The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. (Web site)

### CategoriesUpDw('Categories','-Abz-');

1. Information > Science > Mathematics > Subgroup
2. Science > Astronomy > Universe > Space
3. Group
4. Subspace
5. Science > Mathematics > Numbers > Integers

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