is the set of all characters $ \alpha $
...l group]] of the complex algebra $ \mathfrak g $[[#References|[7]]]. For characters in "general position" the representation $ e ( \alpha ) $
32 KB (4,602 words) - 04:46, 7 January 2022
Let $\def\t{\tau}\t_b := \s_{n-m-b,n-m+1,\dots,n}$ be a special Schubert cycle (cf.
...e Giambelli formula for expressing an arbitrary Schubert cycle in terms of special Schubert cycles. Define $\t_b = 0$ if $B<0$ or $B>m$, and $\t_0 = 1$. Then
8 KB (1,263 words) - 08:49, 30 March 2012
of rational characters of the torus $ T $ ,
in the special case of finite-dimensional semi-simple Lie algebras over the field of compl
9 KB (1,394 words) - 16:55, 17 December 2019
in the sense of the Pontryagin theory of characters, when the compact group $ G ^ {*} $
...ontryagin duality theorem (see [[Alexander duality|Alexander duality]]). A special case of these dualities is the Steenrod duality theorem (see [[Duality|Dual
10 KB (1,511 words) - 16:36, 13 January 2024
...process|stochastic process]]) by a series or integral with respect to some special system of functions, such that the coefficients in this expansion are pairw
to be some special space of entire analytic functions, then one arrives at a generalized stati
17 KB (2,406 words) - 20:00, 12 January 2024
...tation|Unitary representation]]), which has many applications. A number of special properties simplify their study. In particular, the orthogonal complement t
...dying dual objects, and the problems of the theory of spherical functions, characters and harmonic analysis related to them, including the study of various group
20 KB (2,775 words) - 16:40, 31 March 2020
...roups. A classification of reduced Abelian groups is known only in certain special cases. Thus, Ulm's theorem ([[#References|[1]]]) gives the classification o
...applied in many modern mathematical theories. Thus, the duality theory of characters for finite Abelian groups has been considerably extended to the duality the
11 KB (1,810 words) - 22:12, 29 August 2015
....png" />. The arithmetic-mean geometric-mean inequality corresponds to the
special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.
...<TD valign="top"> R.A. Liebler, M.R. Vitale, "Ordering the partition
characters of the symmetric group" ''J. of Algebra'' , '''25''' (1973) pp. 487–48
34 KB (4,631 words) - 18:28, 30 November 2016
...hic form|Automorphic form]]) involve their Fourier coefficients. Here, the special case of holomorphic modular forms $f$ of weight $k$ for the full [[Modular
...tau ( p ) | \leq 2 p ^ { 11 / 2 }$. P. Deligne proved this conjecture as a special case of a much more general result holding for all cusp forms. This result
13 KB (1,907 words) - 07:36, 22 March 2023
...ongruence on arbitrary semi-groups; 2) to describe all congruences on some special semi-groups, belonging to some class of importance. The first category incl
...mi-group of complex numbers, forms the subject of the theory of semi-group characters (cf. [[Character of a semi-group]]).
17 KB (2,435 words) - 09:18, 2 April 2023
The special position of cyclotomic fields among all algebraic number fields is illustra
runs through all non-trivial primitive multiplicative characters modulo $ n $.
12 KB (1,769 words) - 11:19, 26 March 2023
...the functional equations of $L$-series associated to automorphic forms. In special settings, global root numbers are known to have deep connections to the van
...h, J. Queyrut, "On the functional equation of the Artin L-function for characters of real representations" ''Invent. Math.'' , '''20''' (1973) pp. 125–1
12 KB (1,811 words) - 17:46, 1 July 2020
...f the letters of the Roman alphabet (which may be replaced or augmented by characters from a national set), digits, pairs of delimiters (parentheses), separators
12 KB (1,804 words) - 16:56, 7 February 2011
...resentations of finite groups (for example, the orthogonality relation for characters, or for matrix entries), and also the [[Peter–Weyl theorem|Peter–Weyl t
...roperty have been studied; for example, the condition of finiteness of the special rank, different variants of the maximum and minimum conditions for subgroup
14 KB (2,197 words) - 16:40, 31 March 2020
this has only been accomplished for special fields. The case when
characters of a torus. The duality theorem states that the cup-product
17 KB (2,785 words) - 22:37, 23 December 2014
To state Beilinson's conjectures on special values of $ L ( h ^ {i} ( X ) ,s ) $
Some special cases are as follows.
25 KB (3,559 words) - 19:33, 7 February 2024
[[#References|[9]]] are applicable to special cases of such cohomology theories.
...opology duality manifest itself: in duality (in the sense of the theory of characters) between the homology and cohomology groups of the same dimension with dual
64 KB (9,418 words) - 12:44, 8 February 2020
More special cases of bases of a set $ X $
...nd, in particular, to employ the well-developed apparatus of the theory of characters.
28 KB (4,564 words) - 07:37, 26 March 2023
...e of an $A$-module. The direct product and direct sum may be considered as special cases of the notions of a projective and an inductive limit.
...ntation]] of $G$ and are in one-to-one correspondence with the irreducible characters of the group. Modules over principal ideal rings and over Dedekind rings al
23 KB (3,918 words) - 04:31, 23 July 2018
...ns $nk_l$, $(k,l)=1$, $n=0,1,\ldots,$ leads to the problem on the zeros of special zeta-functions — the so-called Dirichlet $L$-series, of the form
...re $a_n$ depends on $n$ and on the difference $k$ of the series (Dirichlet characters modulo $k$).
27 KB (4,516 words) - 18:38, 18 October 2014