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List Of Things Named After Niels Henrik Abel
This is the list of things named after Niels Henrik Abel (1802–1829), a Norwegian mathematician. Mathematics * Abel's binomial theorem * Abel elliptic functions * Abel equation * Abel equation of the first kind * Abel–Goncharov interpolation * Abel–Plana formula * Abel function * Abel's integral equation * Abel's identity * Abel's inequality * Abel's irreducibility theorem * Abel–Jacobi map ** Abel–Jacobi theorem * Abel polynomials * Abel's summation formula ** Abelian means * Abel's test * Abel's theorem **Abelian theorem * Abel–Ruffini theorem * Abel transform * Abel transformation * Abelian category ** Pre-abelian category ** Quasi-abelian category * Abelian group **Abelianization ** Metabelian group **Non-abelian group * Abelian extension * Abelian integral * Abelian surface * Abelian variety ** Abelian variety of CM-type ** Dual abelian variety * Abelian von Neumann algebra Geography * Abeltoppen, a mountain in Dickson Land at Spitsbergen, Svalbard. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" The Abel Prize in mathematics, originally proposed in 1899 to complement the Nobel Prizes (but f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Sum
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-abelian Group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metabelian Group
In mathematics, a metabelian group is a group (mathematics), group whose commutator subgroup is abelian group, abelian. Equivalently, a group ''G'' is metabelian if and only if there is an abelian normal subgroup ''A'' such that the quotient group ''G/A'' is abelian. Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms. Metabelian groups are solvable group, solvable. In fact, they are precisely the solvable groups of derived length at most 2. Examples * Any dihedral group is metabelian, as it has a cyclic normal subgroup of Index of a subgroup, index 2. More generally, any generalized dihedral group is metabelian, as it has an abelian normal subgroup of index 2. * If ''F'' is a field (mathematics), field, the group of affine maps x \mapsto ax+b (where ''a'' ≠ 0) acting on ''F'' is metabelian. Here the abelian normal subgroup is the group of pure translations x\mapsto x+b , and the abelian quotient group is isomorphic to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-abelian Category
In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. A quasi-abelian category is an exact category. Definition Let \mathcal A be a pre-abelian category. A morphism f is a kernel (a cokernel) if there exists a morphism g such that f is a kernel (cokernel) of g. The category \mathcal A is quasi-abelian if for every kernel f: X\rightarrow Y and every morphism h: X\rightarrow Z in the pushout diagram the morphism f' is again a kernel and, dually, for every cokernel g: X\rightarrow Y and every morphism h: Z\rightarrow Y in the pullback diagram the morphism g' is again a cokernel. Equivalently, a quasi-abelian category is a pre-abelian category in which the system of all kernel-cokernel pairs forms an exact structure. Given a pre-abelian category, those kernels, which a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pre-abelian Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism ''f'': ''A'' → ''B'' in C, the equaliser of ''f'' and the zero morphism from ''A'' to ''B'' exists (this is by definition the kernel of ''f''), as does the coequaliser (this is by definition the cokernel of ''f''). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(''A'',''B''), which is an abelian group by item 1; or as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Transformation
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Statement Suppose \ and \ are two sequences. Then, :\sum_^n f_k(g_-g_k) = \left(f_g_ - f_m g_m\right) - \sum_^n g_(f_- f_). Using the forward difference operator \Delta, it can be stated more succinctly as :\sum_^n f_k\Delta g_k = \left(f_ g_ - f_m g_m\right) - \sum_^ g_\Delta f_k, Summation by parts is an analogue to integration by parts: :\int f\,dg = f g - \int g\,df, or to Abel's summation formula: :\sum_^n f(k)(g_-g_)= \left(f(n)g_ - f(m) g_m\right) - \int_^n g_ f'(t) dt. An alternative statement is :f_n g_n - f_m g_m = \sum_^ f_k\Delta g_k + \sum_^ g_k\Delta f_k + \sum_^ \Delta f_k \Delta g_k which is analogous to the integration by parts formula for semimartingales. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Transform
In mathematics, the Abel transform,N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826). named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function ''f''(''r'') is given by : F(y) = 2 \int_y^\infty \frac \,dr. Assuming that ''f''(''r'') drops to zero more quickly than 1/''r'', the inverse Abel transform is given by : f(r) = -\frac \int_r^\infty \frac \,\frac. In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function. In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance ''y'' from the center of the flame, while the inve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel–Ruffini Theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means that the coefficients of the equation are viewed and manipulated as indeterminates. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824. ''Abel–Ruffini theorem'' refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from . Galois theory implies also that :x^5-x-1=0 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
