$#C+1 = 31 : ~/encyclopedia/old_files/data/A013/A.0103750 Asymptotic power series
...ries may be added, multiplied, divided and integrated just like convergent power series.
4 KB (542 words) - 06:35, 14 April 2024
==Hypotheses on the distribution of power residues and non-residues.==
(Cf. [[Power residue|Power residue]]; [[Quadratic residue|Quadratic residue]].)
3 KB (433 words) - 09:08, 2 January 2021
The derivative of a polynomial, rational function or formal power series, which can be defined purely algebraically (without using the concep
\sum _ {i = 0 } ^ { n }
2 KB (246 words) - 19:39, 5 June 2020
Power series in one complex variable $ z $.
s(z) \ = \ \sum _ { k=0 } ^ \infty b _ {k} (z-a) ^ {k} ,
16 KB (2,404 words) - 13:34, 4 November 2023
...e of series of complex numbers, used often to determine the convergence of power series at the radius of convergence
If $\sum a_n$ is a convergent series of real numbers and $\{b_n\}$ is a bounded mono
2 KB (382 words) - 12:44, 10 December 2013
$#C+1 = 97 : ~/encyclopedia/old_files/data/P074/P.0704200 Power function
is an integer, the power function is a particular case of a [[Rational function|rational function]].
6 KB (860 words) - 17:32, 6 January 2024
''on power series''
If a [[Power series|power series]]
7 KB (1,065 words) - 09:52, 11 November 2023
...n element of an analytic function is the circular element in the form of a power series
f (z) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
4 KB (679 words) - 18:34, 5 April 2020
1 + \sum _ {n = 1 } ^ \infty
\sum _ {n = 0 } ^ \infty {
3 KB (504 words) - 14:33, 10 March 2024
...(see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
...| < 1$ the sum $\phi(x)$ of the series \eqref{eq1} can be represented as a power series:
2 KB (368 words) - 07:06, 29 March 2024
...t $e \neq 0$, then $A$ can be decomposed according to Peirce into a direct sum of vector subspaces:
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Albert, "Power-associative rings" ''Trans. Amer. Math. Soc.'' , '''64''' (1948) pp. 552
2 KB (369 words) - 10:19, 16 March 2023
The sum of the power series
F ( x , w ) = \sum _ { n= 0} ^ { \infty} a _ {n} ( x) w^ {n}
3 KB (451 words) - 08:25, 16 March 2023
\right ) + \sum
\right ) = \sum _
4 KB (524 words) - 04:11, 6 June 2020
1) If for a power series
f ( z) = \sum _ {n=1} ^ \infty a _ {n} z ^ {\lambda _ {n} }
3 KB (438 words) - 12:39, 6 January 2024
and if the power series
\sum _ {k = 0 } ^ \infty
4 KB (577 words) - 19:43, 5 June 2020
...\alpha = P / Q$, that is, $\alpha$ is equal to the formal expansion of $P \sum _ { n = 0 } ^ { \infty } ( Q - 1 ) ^ { n }$. For instance, if $K \subseteq
...with coefficients in $A$, and let $A ( ( X ) )$ denote the set of Laurent power series, that is,
5 KB (828 words) - 11:51, 24 December 2020
u \cdot v = \sum _ {i = 1 } ^ { n } u _ {i} v _ {i} .
\sum _ {j = 0 } ^ { {n } - \nu } \left ( \begin{array}{c}
6 KB (890 words) - 04:11, 6 June 2020
A generalization of a [[power series]] in non-negative integral powers of the difference $ z - a $
\sum _ {k = - \infty } ^ {+\infty }
8 KB (1,203 words) - 10:36, 20 January 2024
...ial estimates of trigonometric sums (cf. [[Trigonometric sum|Trigonometric sum]]) of the form
S( f ) = \sum _ {1 \leq x \leq P } e ^ {2 \pi i f( x) } ,
5 KB (759 words) - 08:29, 6 June 2020
are two formal power series, then, by definition,
The set $A[[T_1,\ldots,T_N]]$ of all formal power series forms a ring under these operations.
6 KB (1,093 words) - 08:26, 16 March 2023
Thus, the generating function $ \sum _ {h \geq 0 } a _ {h} X ^ {h} $
are given by a generalized power sum $ a _ {h} = a ( h ) = \sum _ {i = 1 } ^ {m} A _ {i} ( h ) \alpha _ {i} ^ {h} $ ($ h = 0,1, \dots $
6 KB (908 words) - 06:04, 12 July 2022
For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute valu
2 KB (302 words) - 10:06, 15 April 2014
is summable by the Hölder method $(H,k)$ to sum $s$ if
...mable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is eq
2 KB (280 words) - 13:46, 14 February 2020
Consider a complex power series
\sum _ {k = 0 } ^ \infty
4 KB (625 words) - 15:35, 4 June 2020
The formula for the expansion of an arbitrary positive integral power of a [[Binomial|binomial]] in a polynomial arranged in powers of one of the
\sum _ {k = 0 } ^ { m } \left ( \begin{array}{c}
2 KB (262 words) - 08:02, 6 June 2020
\sum _ {m , n = 1 } ^ \infty u _ {mn} ,
S _ {mn} = \sum _ {i = 1 } ^ { m }
7 KB (1,082 words) - 19:36, 5 June 2020
and if the sequence of partial sums of a series $ \sum _ {n = 1 } ^ \infty b _ {n} ( x) $
may take complex values), then the series $ \sum _ {n = 1 } ^ \infty a _ {n} ( x) b _ {n} ( x) $
1 KB (212 words) - 07:38, 1 November 2023
$#C+1 = 41 : ~/encyclopedia/old_files/data/D032/D.0302740 Direct sum
...[[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $ \mathfrak A $
4 KB (680 words) - 19:35, 5 June 2020
is summable by means of the Euler summation method ($(E,q)$-summable) to the sum $S$ if
...eries. Thus, the series $\sum_{n=0}^\infty z^n$ is $(E,q)$-summable to the sum $1/(1-z)$ in the disc with centre at $-q$ and of radius $q+1$.
2 KB (358 words) - 17:36, 14 February 2020
A power series of the form
\sum _ { n=0 } ^ \infty
3 KB (470 words) - 08:17, 26 March 2023
\sum _ { s=0 } ^ { n }
\sum _ { s=0 } ^ { n-1 }
6 KB (828 words) - 10:58, 29 May 2020
th exterior power $ \wedge ^ {r} V $.
The direct sum of the spaces of skew-symmetric $ r $-
1 KB (155 words) - 19:38, 5 June 2020
\sum _ {k=0} ^ \infty u _ {k} $$
is summable by the Lindelöf summation method to the sum $ s $
3 KB (417 words) - 08:17, 6 January 2024
f ( x + h ) = \sum _ {n = 0 } ^ \infty
P ( x + \xi h ) = \sum _ {\nu = 0 } ^ { m } P _ \nu ( x , h ) \xi ^ \nu ,
6 KB (901 words) - 16:08, 1 April 2020
Abel's theorem on power series: If the power series
S ( z ) \ = \ \sum _ {k = 0} ^ \infty a _ {k} ( z - b ) ^ {k} ,
6 KB (894 words) - 06:14, 26 March 2023
be its power function (cf. [[Power function of a test|Power function of a test]]), which gives for every $ \theta $
the corresponding sequence of power functions $ \{ \beta _ {n} ( \theta ) \} $
7 KB (902 words) - 17:46, 4 June 2020
...r any $x_0\in I$ there is a neighborhood $J$ of $x_0$ and a power series $\sum a_n (x-x_0)^n$ such that
An analytic function is infinitely differentiable and its power expansion coincides with the [[Taylor series]]. Namely, the coefficients $a
6 KB (1,048 words) - 21:19, 14 January 2021
..., where $ \lambda $ is a limit ordinal number and $ n $ is an integer, the sum being understood in the sense of addition of [[Order type|order types]].
...omega $ is the least initial ordinal number. The initial ordinal number of power $ \tau $ is denoted by $ \omega(\tau) $. The set $ \{ \omega(\delta) \mid \
9 KB (1,404 words) - 18:33, 4 December 2017
...uestion must be posed, not what the sum is equal to, but how to define the sum of a divergent series, and he found an approach to the solution of this pro
\sum _ {n = 0 } ^ \infty
4 KB (679 words) - 19:36, 5 June 2020
th symmetric power of $ E $(
th exterior power of the module $ E $(
4 KB (590 words) - 07:45, 7 January 2024
...\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined by
...dratic reciprocity law]]). This relation resembles the reciprocity law for power-residue symbols. Several elementary proofs of this relation can be found in
3 KB (398 words) - 21:39, 23 December 2015
''integro-power series''
The finite sum of Volterra terms (of all types) of degree $ m $
5 KB (717 words) - 08:28, 6 June 2020
A divided power structure on $ R $(
A divided power sequence in a co-algebra $ ( C, \mu ) $
4 KB (663 words) - 08:28, 20 January 2024
...eries completions of $ A $ and $ L $, i.e., $ \widehat{A} $ is the ring of power series in the associative but non-commutative variables $ u $ and $ v $, an
...e Campbell-Hausdorff formula provides an expression for $ u \circ v $ as a power series in $ u $ and $ v $:
6 KB (1,020 words) - 17:41, 4 May 2017
...ivisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.
The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$.
2 KB (317 words) - 19:43, 17 November 2023
1 + \sum _ {1 \leq i \leq \infty }
} \sum
3 KB (465 words) - 08:28, 6 June 2020
...differentiable at $x_0$, its Taylor series at $x_0$ is the [[Power series|power series]] given by
...function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coinc
4 KB (710 words) - 06:13, 13 June 2022
\sum _ {n=0 } ^ \infty
f (x) = \sum _ {n=0 } ^ { N }
4 KB (660 words) - 19:29, 13 April 2024
\sum _ { i } {\mathcal H} ^ { i } ( X ; h ^ {n-i} ( \mathop{\rm pt} ) \otim
is isomorphic to the ring of formal power series $ \Omega _ {u} ^ {*} [ [ u ] ] $,
5 KB (632 words) - 11:51, 21 March 2022
h _ {n} = \sum _ {k = 1 } ^ { n } f _ {k} B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } )
Y _ {n} ( g _ {1} \dots g _ {n} ) = \sum _ {k = 1 } ^ { n } B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } ) .
12 KB (1,714 words) - 10:58, 29 May 2020
\sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k}
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function
3 KB (417 words) - 07:38, 26 February 2022
such that any non-zero ideal is generated by some power of the element $ \pi $;
of formal power series in one variable $ T $
5 KB (800 words) - 19:36, 5 June 2020
\sum _ {n = 0 } ^ \infty
where $ A(t) = \sum _ {k=0 } ^ \infty a _ {k} t ^ {k} $
11 KB (1,595 words) - 17:13, 2 January 2021
and radius $ | z ^ {0} - a | = ( \sum _ {\nu = 1 } ^ {n} | z _ \nu ^ {0} - a _ \nu | ^ {2} ) ^ {1/2} $
...of points of absolute convergence (i.e. the domain of convergence) of some power series in $ z _ {1} - a _ {1} \dots z _ {n} - a _ {n} $,
4 KB (586 words) - 08:10, 6 June 2020
An arithmetical [[Fraction|fraction]] with an integral power of 10 as its denominator. The following notation has been accepted for a de
is the sum of such a series, i.e.
2 KB (331 words) - 17:32, 5 June 2020
...tic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from $ k [[ X _ {1} \dots X _ {n} ]] $
in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in $ X _ {1} \dots X _ {n} $
5 KB (787 words) - 22:10, 5 June 2020
A Cartesian power $ \mathbf R ^ {n} $
\langle x, y \rangle = \sum _ {i=1 } ^ { n }
2 KB (254 words) - 08:25, 4 March 2022
''transfinite number, power in the sense of Cantor, cardinality of a set $ A $
...et of real numbers, then $ \mathsf{card}(\mathbf{R}) = \mathfrak{c} $, the power of the continuum. The set $ 2^{A} $ of all subsets of $ A $ is not equivale
9 KB (1,402 words) - 11:57, 10 April 2018
N ^ {-1} \sum _ { j=1 } ^ { N-k } x _ {j} x _ {j+k} ,\ \
r _ {k} ^ {*} + \sum _ { j=1 } ^ { q } \beta _ {j} r _ {| k - j | } ^ {*} = 0 ,\ k = 1 \do
6 KB (747 words) - 18:22, 14 January 2021
\begin{equation}a_k=\sum^n_{i=1}c_ia_{k-i}\end{equation}
...sequence $\mathbb{a}=(a_k)$ over $F$ with the [[Formal power series|formal power series]]
9 KB (1,477 words) - 09:28, 17 May 2021
...By the complexity of a diagram of functional elements one understands the sum of the weights of all functional elements that are present in this diagram.
...ts are equal to 1; or 2) the weight of a threshold element is equal to the sum of the absolute values of all coefficients $ w _ {i} $(
12 KB (1,820 words) - 11:43, 26 March 2023
} \cdot \sum _ {n \leq x } \left | {f ( n ) } \right | ^ {q} \right ) ^ { {1 / q } } .
} \cdot \sum _ {n \leq x } f ( n )
5 KB (715 words) - 10:00, 9 January 2021
The permutation is chosen so that the power of Wilcoxon's test for the given alternative is highest. The statistical di
cf. [[Rank sum test|Rank sum test]]; [[Mann–Whitney test|Mann–Whitney test]]). See also [[Van der Wa
3 KB (399 words) - 10:34, 7 February 2021
\sum _ {k = 0 } ^ { {n } - 1 } e _ {k} u _ {k} ,
\sum _ {k = 0 } ^ { {n } - 1 } w ^ {(} k) e _ {k} = 0,
5 KB (724 words) - 22:11, 5 June 2020
with all coordinates equal, but not a power of an element — a concept that is not defined in an arbitrary vector spac
\sum _ {\begin{array}{c}
2 KB (316 words) - 22:10, 5 June 2020
...problem|Goldbach problem]]); and the representation of a given number as a sum of one prime number and two squares (cf. [[Hardy–Littlewood problem|Hardy
...atic results in this field were obtained in 1748 by L. Euler, who employed power series to study the partition of integers into positive summands; in partic
10 KB (1,609 words) - 13:03, 8 February 2020
\begin{equation*} S = \sum _ { n \in A } e ^ { 2 \pi i f ( n ) }, \end{equation*}
...$f$ is a real-valued function (cf. also [[Trigonometric sum|Trigonometric sum]]). The basic problem is to show, under suitable circumstances, that $S = o
10 KB (1,528 words) - 07:11, 24 January 2024
\sum _ { i = 1 } ^ { k } \
\sum _ {\nu = 1 } ^ { {\alpha _ i } }
9 KB (1,382 words) - 08:27, 6 June 2020
...$ is summable by the Cesàro method of order $k$, or $(C,k)$-summable, with sum $S$ if
...eries is summable by the method $(C,k)$, then it is summable with the same sum by the method $(C,k')$ for $k' > k > -1$. This property does not hold for $
3 KB (519 words) - 15:44, 7 May 2012
$ \sum m _ {i} = | m | \geq 2 $
...case one says that one has [[Small denominators|small denominators]]), the power series $ x = y + \dots $
7 KB (1,085 words) - 08:13, 6 June 2020
\sum _ {r = 0 } ^ \infty
...on-central "chi-squared" distribution appears as the distribution of the sum of squares of independent random variables $ X _ {1} \dots X _ {n} $
4 KB (556 words) - 18:53, 24 January 2024
...bo looked for an algebra that is $8$-dimensional over the complex numbers, power-associative and, unlike the [[octonion]] algebra, has the [[Lie algebra|Lie
\begin{equation*} e _ { j } * e _ { k } = \sum _ { l = 1 } ^ { 8 } ( \sqrt { 3 } d _ { j k l } - f _ { j k l } ) e _ { l }
7 KB (944 words) - 10:21, 8 March 2021
...pediaofmath.org/legacyimages/b/b015/b015210/b01521051.png" />; the ordinal sum is defined uniquely up to an isomorphism, by specifying the components and
10 KB (1,447 words) - 17:26, 7 February 2011
and the Chern polynomial as an element of the formal power series ring $ H ^ {**} ( \mathop{\rm BU} _ {n} ) [ [ t ] ] $.
in other words $ c _ {k} ( \xi \oplus \eta ) = \sum _ {i} c _ {i} ( \xi ) c _ {k-i} ( \eta ) $
13 KB (1,910 words) - 19:40, 7 January 2024
''of a power series
\sum _ {k = 0 } ^ \infty
4 KB (602 words) - 19:35, 5 June 2020
\sum _ { n=0 } ^ \infty a _ {n} $$
is summable by Abel's method to a sum $ S $
11 KB (1,603 words) - 10:19, 7 May 2021
associated with a graded ring $ R = \sum _ {n=0} ^ \infty R _ {n} $(
does not contain $ \sum _ {n=1} ^ \infty R _ {n} $.
3 KB (468 words) - 09:09, 6 January 2024
denotes the algebra of all power series $ \sum a _ \alpha z _ {1} ^ {\alpha _ {1} } \dots z _ {n} ^ {\alpha _ {n} } $
$ | \alpha | = \sum \alpha _ {i} $).
8 KB (1,161 words) - 08:25, 6 June 2020
A power series which offers a complete solution to the problem of local inversion o
\sum _ {n = 1 } ^ \infty {
5 KB (731 words) - 06:29, 30 May 2020
represented by a power series
\sum _ {v = 0 } ^ \infty
5 KB (790 words) - 08:29, 20 January 2024
is represented as the sum of a convergent power series (also denoted by $ f _ {i} $
...expressed by the following properties of these series, regarded as formal power series in $ 2n $
10 KB (1,553 words) - 22:16, 5 June 2020
is the [[pointwise operation|pointwise]] sum and
...ormal Dirichlet series over $\mathbb{C}$ is isomorphic to a ring of formal power series in countably many variables.
2 KB (358 words) - 17:25, 11 November 2023
...residue formula one usually understands an integral representation for the sum of the values of a holomorphic function at all the zeros of a holomorphic m
...n - 1 ) ! } { ( 2 \pi i ) ^ {n } } \int _ { \partial D } \varphi \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } w _ { k } d w [ k ] \wedge d f } {
7 KB (1,090 words) - 10:25, 16 March 2024
...[[Fabry theorem|Fabry theorem]] on gaps; [[Lacunary power series|Lacunary power series]].
\sum _ { j= 1} ^ { k } L _ {ij} u _ {j} ,\ 1 \leq i \leq k ,
6 KB (898 words) - 14:30, 8 January 2022
...number and a left-part number of the other, whereas the right part of the sum consists of the sums of one number and a right part of the other.
...presentations of the surreal numbers which can be given in terms of formal power series over an index "set" which is isomorphic to the surreal numbers the
8 KB (1,226 words) - 16:11, 3 July 2016
in the form of a power series $ \Pi ( z _ {1} ; r ) = \sum _ {\nu = 0 } ^ \infty c _ \nu ( z - z _ {1} ) ^ \nu $
3 KB (399 words) - 22:17, 5 June 2020
...lane (except, possibly, at the point at infinity). It can be expanded in a power series
\sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} ,\ \
8 KB (1,251 words) - 20:13, 10 January 2021
X = \sum _ {i = 1 } ^ { m }
...B.L. van der Waerden, "Order tests for the two-sample problem and their power" ''Proc. Kon. Nederl. Akad. Wetensch. A'' , '''55''' (1952) pp. 453–45
3 KB (401 words) - 08:27, 6 June 2020
\sum _ { n=1 } ^ \infty a _ {n} e ^ {- \lambda _ {n} s } ,
\sum _ { n=1 } ^ \infty
11 KB (1,638 words) - 11:32, 16 April 2023
...are [[elementary symmetric function]]s and $p_k(x_1,x_2,\ldots)$ are power sum symmetric functions. The algebraic structure underlying both identities is
...ence $(x_1,x_2,\ldots)$, then the $(k+1)$-st term in $P(x^n)$ is the power sum symmetric function $x_1^n+\cdots+x_k^n$ and the $k$-th term in $P(xP(\ldots
6 KB (960 words) - 07:40, 18 November 2023
or is a power of the characteristic $ p $
[ L:K ] \geq \sum _ {i = 1 } ^ { m } e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) .
5 KB (827 words) - 17:32, 5 June 2020
the coefficients of these series are convergent power series in $ q $,
\sum _{n=1} ^ \infty \left [ 2 ^ {n+1}
3 KB (379 words) - 16:31, 6 January 2024
and is called the $ p $-th Pontryagin power $ {\mathcal P} _ {p} $.
\left ( \sum _ { i= 1} ^ { p- 1 }
6 KB (870 words) - 10:04, 11 July 2022
...e function $\phi(m)/m$ is a strongly multiplicative arithmetic function, a power function $m^k$ is a totally multiplicative arithmetic function.
3 KB (419 words) - 20:15, 19 November 2017
f ( z ) = \sum _ { k=0 } ^ \infty c _ {k} ( z - a ) ^ {k}
as a power series along all possible rays from the centre $ a $
5 KB (749 words) - 19:11, 19 June 2020
th exterior power of the space $ V $.
th exterior power $ \wedge ^ {r} M $,
7 KB (1,013 words) - 19:38, 5 June 2020
$ \sum _ {i} \pi _ {i} = 1 $,
where $ a _ {i} = \sum _ {j} n _ {i.ij } $
5 KB (733 words) - 06:29, 30 May 2020
...'(0,0)$ is a natural number) has a unique solution in the form of a formal power series:
...solution $y=\xi(x)$ of equation \eqref{1} cannot be proved by any partial sum of the Taylor series of $f$ (cf. [[#References|[2]]], [[#References|[3]]]).
2 KB (376 words) - 17:27, 14 February 2020
...s an injective object, and each injective object is isomorphic to a direct sum of indecomposable injective objects; this representation is moreover unique
...pos]] an object is injective if and only if it occurs as a retract of some power-object, and injective objects are used in the study of the associated sheaf
4 KB (643 words) - 22:12, 5 June 2020
Taking the direct sum of the internal space $ S _ {k} ( U) $
and the Hilbert sum of the central spaces there results a triplet
8 KB (1,190 words) - 08:16, 20 January 2024
...nce sets, the families using higher-order residues and also the twin prime power series, see [[#References|[a1]]], Sect. VI.8. The families of difference se
...exist whenever $q$ is a power of $2$ or $3$ or the square of an odd prime power; see [[#References|[a1]]], Sect. VI.9–12, for the construction of all the
9 KB (1,331 words) - 19:36, 13 February 2024
...functions in combinatorics considerable use is made of algebras of formal power series and of various symbolic methods (cf. [[#References|[1]]], [[#Referen
...that take real values and are such that $f(x,y) = 0$ unless $x \le y$. The sum of two such functions and multiplication by a number are defined in the usu
8 KB (1,343 words) - 08:52, 12 November 2023
of formal power series in $ n $
formal power series in $ 2n $
17 KB (2,537 words) - 22:38, 15 December 2019
In mathematical analysis, the need arises to generalize the concept of the sum of a series (limit of a sequence, value of an integral) to include the case
If the sum of the series is defined as
10 KB (1,530 words) - 08:24, 6 June 2020
The field concerned with best rational approximation to power series. Let
f( z) = \sum _ {k=0}^ \infty f _ {k} z ^ {k}
15 KB (2,227 words) - 13:32, 11 November 2023
...[[#References|[a46]]], and all papers by Turán mentioned below) the power sum method, by which one can investigate certain minimax problems described bel
Bohr norm: $M _ { 0 } ( k ) = \sum _ { j = 1 } ^ { n } | b _ { j } \| z _ { j } | ^ { k }$;
40 KB (5,729 words) - 17:51, 5 May 2024
w _ {k} ( \xi \oplus \eta ) = \sum _ { i } w _ {i} ( \xi ) w _ {k- i} ( \eta ),\ \
are rings of formal power series in the Stiefel–Whitney classes:
5 KB (692 words) - 17:18, 6 January 2024
...}$ in place of ${\cal R }_ { q ^ { 2 } }$ above (and now take an infinite sum in the exponential). It does not, however, live in the algebraic tensor pro
\begin{equation*} \phi h = \sum h _{( 2 )} \phi_{ ( 2 )} \langle S h _ { ( 1 ) } , \phi _ { ( 1 ) } \rangle
14 KB (2,048 words) - 18:24, 3 August 2020
\sum _ { k= 0} ^ \infty u _ {k} ( z),
then the sum $ s ( z) $
22 KB (3,120 words) - 01:51, 24 February 2022
then the series \eqref{2} is absolutely summable by the method $A$ to the sum $s$. Condition \eqref{1} is an additional requirement which makes absolute
...ries. If each such series is summable by this method to a sum equal to the sum to which it converges, then the method is called absolutely regular. For in
5 KB (850 words) - 15:41, 14 February 2020
of a power series
f _ \infty = f _ \infty ( z) = \sum _{k=0} ^ \infty c _ {k} z ^ {-k}
13 KB (1,972 words) - 20:02, 15 January 2024
...butive lattice]] (see also [[FKG inequality|FKG inequality]]), such as the power set of a finite set ordered by proper inclusion. For subsets $ A $,
let $ f ( A ) = \sum _ {a \in A } f ( a ) $.
4 KB (523 words) - 19:25, 4 April 2020
\begin{equation*} D_ { n } ( x , a ) = \sum _ { i = 0 } ^ { \lfloor n / 2 \rfloor } \frac { n } { n - i } \left( \begin
\begin{equation*} \sum _ { n = 0 } ^ { \infty } D _ { n } ( x , a ) z ^ { n } = \frac { 2 - x z }
15 KB (2,207 words) - 16:45, 1 July 2020
...functions, up to an arbitrary order, are summable to a given degree with a power weight. For such cases the imbeddings of weighted spaces have been studied
\sum _ {| k | = l } | ( 1 + | x |) ^ {- \alpha } D ^ {k} f ;
9 KB (1,435 words) - 08:13, 13 January 2024
\begin{equation*} s _ { \lambda } = \sum _ { T } \mathbf{x} ^ { T }, \end{equation*}
where the sum is over all semi-standard Young tableaux of shape $\lambda$ with entries be
14 KB (2,001 words) - 10:09, 11 November 2023
The sum of the powers of any term of a polynomial is called the ''degree'' of that
...ower of $x_1$ is higher, and if these powers are equal, that for which the power of $x_2$ is higher, etc. If all terms of a polynomial are ordered so that e
9 KB (1,497 words) - 10:44, 27 June 2015
a Riemann sum, $ \sum _ \varpsi f = \sum _ {i = 1 } ^ {n} f ( x _ {i} ) ( a _ {i} - a _ {i - 1 } ) $.
\left | {\sum _ \varpsi f - I } \right | < \epsilon.
11 KB (1,718 words) - 22:15, 5 June 2020
where the sum is over all strings $i_1,\ldots,i_k$, $i_j \ge 1$, such that $i_1+\cdots+i_
where the sum is over all $r \in \mathbf{N}$ and pairs of order-preserving injective mapp
4 KB (709 words) - 13:46, 20 March 2023
...negative. A line bundle is said to be big if the sections of some positive power of $L$ give a [[Birational mapping|birational mapping]] of $X$ into project
...line bundle $E$ plus an effective divisor (cf. [[Divisor|Divisor]]) $D = \sum _ { k = 1 } ^ { r } a _ { k } D _ { k }$, where $a_k$ are positive integers
8 KB (1,253 words) - 15:30, 1 July 2020
is endowed with the natural structure of a Hopf algebra. Let $ A = \sum _ {n \in \mathbf Z} A _{n} $
modules. Then $ A ^{*} = \sum _ {n \in \mathbf Z} A _{n} ^{*} $ ,
11 KB (1,576 words) - 08:51, 16 December 2019
...representation of a finite group in a topological vector space is a direct sum of irreducible representations. On the other hand, there are some fundament
the sum of the squares of the dimensions of representations representing the differ
10 KB (1,488 words) - 19:39, 5 June 2020
where addition of sets is taken as the vector sum.
is the element $ \sum _ {s \in G } sv $(
8 KB (1,112 words) - 08:24, 6 June 2020
...nctioning of the adder leads, as a rule, to a change in the result by some power of 2. In this connection, a (single) arithmetic error in the ring of intege
N = \sum _ {i = 0 } ^ { k }
8 KB (1,189 words) - 09:46, 26 March 2023
be a [[Formal power series|formal power series]] over the field $ \mathbf Q $.
...since under such a definition the usual formula for the total class of the sum of two vector bundles, even after including $ 1/2 $
14 KB (1,987 words) - 19:15, 16 January 2024
or [[power associativity]] (i.e. every element generates an associative subalgebra), o
Beginning with Albert's problem of classifying all power-associative FLA algebras $ \mathfrak A $
22 KB (3,154 words) - 19:10, 26 March 2023
...s of local rings. Any field or valuation ring is local. The ring of formal power series $ k [ [ X _ {1} \dots X _ {n} ] ] $
of formal power series, where $ S $
9 KB (1,456 words) - 22:17, 5 June 2020
...s are Abelian, in particular, the additive group of integers. All [[direct sum]]s of cyclic groups are Abelian. Also the additive group of rational number
...n general group theory). Every torsion group splits uniquely into a direct sum of primary groups that correspond to distinct prime numbers.
11 KB (1,810 words) - 22:12, 29 August 2015
- n - n ^ {- 1 } \sum _ { i } \{ ( 2i - 1 ) { \mathop{\rm ln} } z _ {i} + ( 2n + 1 - 2i ) { \math
...TD valign="top"> F.C. Drost, W.C.M. Kallenberg, J. Oosterhoff, "The power of EDF tests to fit under non-robust estimation of nuisance parameters" ''
5 KB (709 words) - 06:49, 26 March 2023
...direct sum of finitely many quasi-cyclic groups and cyclic groups of prime-power order (cf. [[Quasi-cyclic group]]; [[Group-of-type-p^infinity|Group of type
4 KB (589 words) - 18:19, 15 January 2017
\cong \sum _ { j= 1} ^ { N }
...(1) exists. The sum on the right-hand side of (1) is called the quadrature sum, the numbers $ x _ {j} $
11 KB (1,719 words) - 01:56, 4 January 2022
\sum _ { i=1 } ^ { n } ( f , f _ {i} ) ( f _ {1} \otimes
\sum _ { i=1 } ^ { n } ( - 1 ) ^ {i-1} ( f , f _ {i} ) ( f _ {i} \otimes \dots
8 KB (1,124 words) - 18:47, 5 April 2020
...ntial polynomial which vanishes in all solutions of this system. A certain power of $ G $
\omega _{V} = \sum _ {0 \leq i \leq m} a _{i} \binom{x+i}{i} ,
30 KB (4,468 words) - 18:44, 17 December 2019
...[order type]] is $ \alpha $. The definitions of the sum, the product and a power of alephs are obvious. One has
4 KB (603 words) - 09:05, 2 January 2021
\sum _ {n = 1 } ^ \infty
\sum _ {n = 0 } ^ \infty
11 KB (1,537 words) - 19:39, 5 June 2020
...[Analytic number theory|analytic number theory]] (cf. also [[Weyl sum|Weyl sum]]). General application to the first such class, exponential sums within th
The original branch of the Bombieri–Iwaniec method deals with a sum
14 KB (2,083 words) - 17:01, 1 July 2020
\sum _ { k= 1} ^ \infty
\sum _ { k= 1} ^ { m } a _ {k} T _ {k} ( X)
16 KB (2,261 words) - 06:40, 10 May 2022
\sum (x^i - x_0^i)^2 = R^2
...o which the points of a straight line (the radical axis) have an identical power (different for different points), forms a [[Net|net]] of spheres. The total
9 KB (1,500 words) - 21:06, 8 November 2014
$ \sum _ {i=1} ^ {n} m _ {i} \geq 2 $.
is a power series of which all monomials are resonant. Here a monomial $ y ^ {m} e
11 KB (1,659 words) - 05:14, 24 February 2022
y _ {j+ 1} \ = y _ {j} + \theta \sum _ { i= 1} ^ { q } A _ {i} k _ {i} ,
...n (2) on the exact solution of (1) should have the highest order (''i.e.'' power of $ \theta $,
9 KB (1,315 words) - 20:04, 15 January 2024
...quivalently, one may consider the algebras of [[Formal power series|formal power series]] or the algebras of germs of smooth functions at the origin $0 \in
\begin{equation*} f _ { A } ( x + h ) = f ( x ) + \sum _ { | \alpha | \geq 1 } \frac { 1 } { \alpha ! } \frac { \partial ^ { | \al
7 KB (1,051 words) - 06:27, 15 February 2024
...beta$ of the circle $T$ consisting only of regular boundary points for the sum of the series.
...\beta$, it does not follow that the points of this arc are regular for the sum of the series.
4 KB (706 words) - 19:19, 27 July 2012
which corresponds to the formal pair of power series,
...m _ { k = 1 } ^ { \infty } c _ { - k } ( - z ) ^ { k } , L _ { \infty } = \sum _ { k = 0 } ^ { \infty } c _ { k } ( - z ) ^ { - k }. \end{equation}
9 KB (1,334 words) - 19:52, 6 February 2024
...mial $p ( t )$ is normalized or monic, i.e. the coefficient at the highest power $t ^ { N }$ is equal to $1$. Algebraic operators with characteristic polyno
\begin{equation*} \{ f ( t ) \} _ { ( k ; t _ { i } ) } = \sum _ { m = 0 } ^ { k } \frac { ( t - t _ { i } ) ^ { m } } { m ! } \frac { d ^
10 KB (1,502 words) - 22:40, 12 December 2020
\sum _ {m = - \infty } ^ \infty
&=& 1 + 2 \sum _ {m = 1 } ^ \infty (- 1) ^ {m} e ^ {i \pi m ^ {2} \tau } \cos ( 2 \p
18 KB (2,349 words) - 02:00, 20 June 2022
of the Nile," is principally known for a remarkable power-law
sum of river discharges. The
9 KB (1,398 words) - 20:37, 22 September 2016
The sum and the difference of two fractions $a/b$ and $c/b$ having a common denomin
...ction. A fraction is said to be a decimal fraction if its denominator is a power of the number 10 (cf.
4 KB (649 words) - 17:15, 9 December 2013
\sum _ {f ( x) = x } i ( x) = \Lambda ( f , X ) .
is equal to the sum of the indices of the zeros of a global $ C ^ \infty $-
10 KB (1,491 words) - 19:04, 26 March 2023
...e = \frac { | U | } { | G | } \left( \sum _ { b \in B } b \right) \left( \sum _ { w \in W } \operatorname { sign } ( w ) w \right) \end{equation*}
...ends to the $\overline { G }$-module with highest weight $q - 1$ times the sum of the fundamental weights, which is accordingly also denoted $\operatornam
9 KB (1,369 words) - 13:31, 25 November 2023
A power-associative (in particular, an associative) algebra (cf. [[Algebra with ass
...its quotient algebra has no non-zero nil ideals. The question whether the sum of one-sided nil ideals is a one-sided nil ideal is Köthe's problem. It is
6 KB (1,020 words) - 16:48, 25 March 2023
always belongs to one and the same residue class, called the sum $ A+B $
\sum _ { j=1 } ^ s a _{ij} x _{j} \ \equiv \ b _{i} \ ( \mathop{\rm mod}\nol
20 KB (3,011 words) - 09:59, 26 March 2023
...ob Bernoulli [[#References|[1]]] in connection with the calculation of the sum of equal powers of natural numbers:
...as the coefficients of the expansions of certain elementary functions into power series. Thus, for example,
4 KB (684 words) - 18:44, 5 October 2023
S _ {n} ( x _ {1} \dots x _ {n} ) = \sum _ {\sigma \in S _ {n} }
\sum _ {\sigma \in S _ {n} } ( - 1 ) ^ \sigma y _ {1} x _ {\sigma ( 1) } \do
15 KB (2,252 words) - 08:04, 6 June 2020
have power singularities has also been studied: for example, $ f = x ^ \alpha f _
...op{\rm exp} [ i \lambda S( x ^ {0} )] \sum _ { k= 0} ^ \infty \left ( \sum _ { l= 0} ^ { N } a _ {kl} \lambda ^ {- r _ {k} } ( \mathop{\rm ln} \lam
8 KB (1,104 words) - 08:08, 4 March 2022
...ers can be obtained if one adjoins to the language, for every $n\geq 2$, a power predicate $P_n$ interpreted by $P_n(x)\Leftrightarrow\exists y(y^n=x)$.
...quare operator in the real case, and the above form is the analogue of the sum of squares.
8 KB (1,400 words) - 06:24, 21 December 2020
as the sum of a semi-simple and a nilpotent endomorphism that commute with each other:
...r some extension of the ground field, a nilpotent endomorphism is one some power of which is the zero endomorphism.) If in some basis of the space the matri
10 KB (1,474 words) - 02:51, 25 February 2022
...estion of how to judge the quality of the other members and their relative power positions before an acceptable compromise solution emerges.
\begin{equation*} f _ { j } = \sum _ { i } c _ { i } g _ {i j }. \end{equation*}
18 KB (2,526 words) - 15:30, 1 July 2020
...l{H} = - \sum _ { i < j = 1 } ^ { N } J _ { i j } S _ { i } S _ { j } - H \sum _ { i = 1 } ^ { N } S _ { i }. \end{equation}
...interpretation of the model to describe magnetic ordering in solids ($M = \sum _ { i = 1 } ^ { N } S _ { i }$ is "magnetization" , the Zeeman energy $- H
16 KB (2,407 words) - 19:47, 17 February 2024
...that, together with their derivatives, vanish at infinity faster than any power of $\frac{1}{|x|}$. Formula (1) still holds for $\phi\in S$, and $(F \phi)(
denotes the scalar product $ \sum _{ {i = 1}^{n}} x _{i} \xi _{i} $.
6 KB (971 words) - 12:28, 1 February 2020
\begin{equation*} p _ { m } ( z ) = m ! \sum _ { 0 \leq n \leq m - 1 } b _ { m } ( n + 1 ) z ^ { n } , \quad z \in \math
...that the Euler–Frobenius polynomials provide the coefficients of the local power series expansion of the function
5 KB (742 words) - 07:27, 25 January 2024
\alpha = \sum _ {i _ {1}, \dots, i _ {p} } a _ {i _ {1} \dots i _ {p} } dx ^ {i _ {1} }
converts $ \Omega ^ {*} ( M) = \sum _ {p = 0 } ^ {n} \Omega ^ {p} ( M) $
27 KB (4,062 words) - 01:31, 7 May 2022
...n the parameter for sufficiently small values of it, either in the form of power series in $ \mu $(
see [[#References|[1]]], Chapt. IX) or in the form of power series in $ \mu $
14 KB (1,929 words) - 14:54, 7 June 2020
...per vector space, is a [[Vector space|vector space]] $V$ which is a direct sum of two components, $V = V _ { \bar{0}} \oplus V _ { \bar{1} }$. Vectors bel
...inants below.) The symmetric groups $\sum _ { n }$ act on the $n$th tensor power of a super vector space and there exist super analogues of the usual repres
18 KB (2,722 words) - 17:47, 1 July 2020
..., $\beta$ and $\gamma$ will be used for multi-indices, with $| \alpha | = \sum _ { j = 1 } ^ { N } \alpha _ { j }$ and $D ^ { \alpha } = D _ { 1 } ^ { \al
\begin{equation*} \| u \| _ { p , m , T } = \sum _ { | \alpha | \leq m } \| D ^ { \alpha } u \| _ { p , T }, \end{equation*}
11 KB (1,716 words) - 17:52, 3 January 2021
| style="text-align:left" | $\sum(y-\mu)^2$
| style="text-align:left" | $\sum\{(y\mu-\mu^2/2)/\sigma^2-y^2/2\sigma^2-5\ln(2\pi\sigma^2)\}$
30 KB (4,314 words) - 13:25, 30 May 2016
and $ {\mathcal P} ^ {i} $ (cf. [[Steenrod reduced power|Steenrod reduced power]]); the [[Pontryagin square|Pontryagin square]] $ {\mathcal P} _ {1} $;
the [[Postnikov square|Postnikov square]]; raising to the $ k $-th power, $ \mu _ {k} $:
29 KB (4,197 words) - 09:49, 26 March 2023
\sum _ {i = 1} ^ {p-1} {
w _{n} (X) = \sum _{d\mid n} dX _{d} ^{n/d} .
17 KB (2,502 words) - 17:25, 22 December 2019
...ons, $\phi _ { n } ( x )$, $n = 0 , \ldots , N$, in the form $u _ { N } = \sum _ { n = 0 } ^ { N } a _ { n } \phi _ { n } ( x )$, where the coefficients $
...l equation into spectral space $R ( x ; a _ { 0 } , \dots , a _ { N } ) = \sum _ { n } r _ { n } ( a _ { 0 } , \dots , a _ { N } ) \phi _ { n } ( x )$ and
9 KB (1,381 words) - 16:55, 1 July 2020
\begin{equation} \tag{a1} B _ { 2 n } = A _ { 2 n } - \sum _ { p - 1 | 2 n } \frac { 1 } { p }, \end{equation}
...tion*} k ^ { n } B _ { n } \left( \frac { h } { k } \right) = G _ { n } - \sum \frac { 1 } { p }, \end{equation*}
8 KB (1,206 words) - 16:55, 1 July 2020
...esolution procedure that not only is capable of refuting but has computing power at the same time.
which enable a unique evaluation of every sum of two numbers expressed in terms of $ \mathbf o $
15 KB (2,311 words) - 04:11, 6 June 2020
''Wick monomial, Wick power''
\sum _{G} \prod _ {e \in G} < f _{ {e _ 1}} f _{ {e _ 2}} > : \prod _ {i \notin
13 KB (1,983 words) - 18:50, 11 December 2020
...= ( s _ { i + j - 1} )$ one naturally associates the function $r ( z ) = \sum _ { k = 1 } ^ { \infty } s _ { k } z ^ { - k }$, which is called its symbol
...{ - i }$ can uniquely be extended to a power series $g ( z ) = r ( z ) + \sum _ { i = 1 } ^ { \infty } s _ { 2 m + i } z ^ { - ( 2 m + i ) }$ in such a w
14 KB (2,126 words) - 16:45, 1 July 2020
divisor $ D = \sum ( t ) - \sum ( r ) $.
Since $ \sum t - \sum r = 0 $
16 KB (2,293 words) - 19:37, 5 June 2020
is a prime power. A perfect $ e $-
in such a way that the sum of the $ n + 1 $
5 KB (772 words) - 11:10, 23 March 2023
...natural exponential family. A more elementary example is given by $\mu = \sum _ { x = 1 } ^ { \infty } n ^ { - 3 } \delta _ { n }$.
...name { cos } t ) ^ { - 1 }$), at least up to an affinity and a convolution power. Similarly, in [[#References|[a8]]], the classification in $6$ types of the
10 KB (1,596 words) - 15:30, 1 July 2020
...intended to serve several users; for example, a central source of electric power. One is seeking the minimum of a function
...cility location problem, for which the function $f$ to be minimized is the sum of the weighted distances from $k$ new facilities to $n$ given facilities a
5 KB (888 words) - 14:13, 21 August 2014
\begin{equation} \tag{a1} \sum _ { j \geq 1 } | e _ { j } | ^ { \gamma } \leq L _ { \gamma , n } \int _ {
...ociated with the inequality (a1) is the semi-classical approximation for $\sum | e | ^ { \gamma }$, which serves as a heuristic motivation for (a1). It is
15 KB (2,262 words) - 19:26, 26 March 2023
I + \int\limits _ { s } ^ { t } A ( t _ {1} ) dt _ {1} + \sum
t \dot{u} = \left ( \sum _ { k=0} ^ \infty A ^ {(} k) t ^ {k} \right ) u ,
32 KB (4,803 words) - 17:49, 6 January 2024
...[[Padé approximation]]). Finite matrices whose entries depend only on the sum of the coordinates were studied first by H. Hankel [[#References|[a8]]]. On
...+ k} ) _ { j , k \geq 0}$ determines a compact operator if and only if $\sum _ { j \geq 0 } \alpha _ { j } z ^ { j } \in \operatorname{VMO}$.
18 KB (2,757 words) - 00:48, 15 February 2024
\sum _ {\overline \lambda }
where the sum is over all partitions $ \overline \lambda $
18 KB (2,500 words) - 13:04, 18 February 2022
...thfrak{a} )$, where $\wedge ^ { k } (\mathfrak{a} )$ is the $k$th exterior power of $\frak a$, $k = 0 , \ldots , n = \operatorname { dim } \mathfrak{a}$. Le
\begin{equation*} = \sum _ { i = 1 } ^ { k } ( - 1 ) ^ { i + 1 } X X _ { i } \bigotimes X _ { 1 } \b
12 KB (1,752 words) - 07:39, 10 February 2024
\sum _ {k = 1 } ^ { N }
...he point at infinity. The residue theorem implies the theorem on the total sum of residues: If $ f( z) $
16 KB (2,407 words) - 07:56, 11 January 2022
...$ of $\Gamma$. Let $\Lambda = \mathcal{O} [ [ T ] ]$ be the ring of formal power series in an indeterminate $T$ with coefficients in $\mathcal{O}$. $P ( T )
...\operatorname { deg } ( f _ { i } ( T ) ^ { l _ { i } } ) , \ \mu ( X ) = \sum _ { j = 1 } ^ { t } m _ { j }. \end{equation*}
19 KB (2,876 words) - 05:38, 15 February 2024
\sum _{j=1} ^ n (-1) ^{j}
algebra splits into the direct sum $ L = L _{0} \oplus L _{1} $
10 KB (1,443 words) - 10:14, 17 December 2019
if it can be expanded in a power series in a neighbourhood of each point $ z _ {0} \in D $:
\sum _ {k = 0 } ^ \infty
13 KB (1,922 words) - 19:55, 1 February 2022
...1]]]. Note that an arbitrary lift will lead to multiplying the period by a power of $ p $,
...eld admits the following expansion: $ \mathbf Z _ {p} [ \zeta _ {n} ] = \sum _ {i = 0 } ^ \infty p ^ {i} T ^ {i} $,
7 KB (1,070 words) - 19:40, 5 June 2020
...spaces of $A$, together with the operations of join given by closed linear sum, product given by closed linear product of subspaces, and involution by inv
...ann quantale exactly if it is a complete atomic Boolean algebra, hence the power set of its set of points. A homomorphism $X \rightarrow Q$ of Gel'fand quan
13 KB (2,065 words) - 18:59, 12 December 2023
\sum _ {i _ {1} + \dots + i _ {n} \leq m }
\sum (- 1) ^ {i _ {1} + \dots + i _ {n} }
25 KB (3,768 words) - 09:07, 14 June 2022
The logarithm of the likelihood ratio as given in (a1) is a sum of independent, identically distributed random variables $Z_i = \log f_1(X_
...analysis]] (see [[#References|[a2]]] for details) to find the approximate power functions and average sample number functions for such procedures. However,
8 KB (1,217 words) - 12:30, 13 February 2024
...er way to formulate the nondegeneracy is to require that the highest wedge power $\omega\land\cdots\land\omega$ ($n$ times) is a nonvanishing volume form.</
...\to (\R^{2n},0)$, such that in these coordinates $\omega$ takes the form $\sum \rd x_i\land\rd p_i$.
5 KB (856 words) - 05:41, 24 February 2022
\sum a _ {k _ {1} \dots k _ {m} }
\sum a _ {k _ {1} \dots k _ \nu }
21 KB (3,060 words) - 20:29, 10 January 2021
...e "infinitesimal" (cf. [[Analytic function|Analytic function]]), namely: power series expansions, the [[Cauchy-Riemann equations]], and even the [[Morera
...\{ z \in \mathbf{C} ^ { n + 1 } : \operatorname { Im } z _ { n + 1 } > \sum ^ { n _ { j = 1 } } | z _ { j } | ^ { 2 } \right\}, \end{equation*}
6 KB (961 words) - 16:45, 1 July 2020
\begin{equation*} q_Q ( x ) = \sum _ { j \in Q _ { 0 } } x _ { j } ^ { 2 } - \sum _ { i , j \in Q _ { 0 } } d _ { i j } x _ { i } x _ { j }, \end{equation*}
...radical|Jacobson radical]] $\operatorname{rad} R$ of $R$ and containing a power of $\operatorname{rad} R$. Assume that $Q$ has no oriented cycles (and henc
18 KB (2,636 words) - 06:50, 15 February 2024
together with all their derivatives faster than any power of $ | t | ^ {- 1} $.
\phi ( x) d x \approx \sum _ { k= 1} ^ { n }
19 KB (2,911 words) - 18:54, 26 March 2023
\sum _{j} \oplus e _ {j _{1} \dots j} D ^{j} ( U ) ,
denotes the matrix direct sum, and the direct sum on the right-hand side contains each irreducible representation $ D ^{j}
46 KB (6,595 words) - 13:20, 27 January 2020
...standard local properties of holomorphic functions, for example, the local power series representation. Under suitable hypothesis, an analogous formula (the
\begin{equation*} = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } ( \overline { \zeta _ { j } } - \ov
15 KB (2,167 words) - 16:10, 11 February 2024
\begin{equation*} Q _ { \lambda } = \frac { 1 } { n ! } \sum _ { \pi \in O ( n ) } 2 ^ { ( r ( \lambda ) + r ( \pi ) + \epsilon ( \lambd
...}$ is the order of that class and $p _ { \pi }$ is the corresponding power-sum symmetric function and $\epsilon ( \lambda ) = 0$ or $1$ according as $n -
11 KB (1,601 words) - 09:48, 10 November 2023
...is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.
...nentials, that is, of the elements of the form $ \mathop{\rm exp} a = \sum _ {0} ^ \infty a ^ {n} / n ! $.
18 KB (2,806 words) - 03:47, 25 February 2022
\begin{equation*} \mu ( d x ) = \sum _ { k = 0 } ^ { n } \left( \begin{array} { l } { n } \\ { k } \end{array} \
\begin{equation*} \mathsf{P} ( \theta , \mu ) ( d x ) = \sum _ { k = 0 } ^ { n } \left( \begin{array} { l } { n } \\ { k } \end{array} \
12 KB (1,855 words) - 17:45, 1 July 2020
...nsider some category $\mathcal{C}$ which admits a direct "product" (or "sum" ) operation $\times$ on its objects. Suppose that this operation $\times$
...of this kind being simply the various [[cyclic group]]s $C_{p^n}$ of prime-power order $p^n$.
24 KB (3,738 words) - 07:41, 7 February 2024
...ordan, etc., algebras), nil algebras are defined as algebras in which some power of each element equals zero; in the case of anti-commutative algebras (i.e.
...top">[10]</TD> <TD valign="top"> G.P. Kukin, "Subalgebras of a free Lie sum of Lie algebras with an amalgamated subalgebra" ''Algebra and Logic'' , ''
16 KB (2,433 words) - 21:48, 5 January 2016
...a number $k$ such that any natural number $n\geq4$ can be represented as a sum of at most $k$ prime numbers.
...eorem]], which he declared to have solved, has turned out to be beyond the power of elementary methods.
13 KB (2,043 words) - 20:28, 13 October 2014
universities. For 25 years, he was to reign without sharing of power over
the minutest detail. One can well imagine that this concentration of power
11 KB (1,710 words) - 19:08, 6 March 2024
...] that is additive, meaning that in each of its arguments it preserves the sum/join operation of $ \mathbf B $.
on the power set $ 2 ^ {X} $,
8 KB (1,220 words) - 19:12, 11 December 2020
...modules that cannot be shifted into an injective or projective vertex by a power $\tau _ { A } ^ { j }$ for some integer $j$. In [[#References|[a3]]], Riedt
...tegory of $\Gamma _ { A }$. Forming the endomorphism algebra of the direct sum of all projective objects in this mesh category yields $A$ (up to [[Morita
9 KB (1,240 words) - 08:04, 25 November 2023
\sum _ {j, k = 1 } ^ { n }
...ator-theoretic aspects of the Nevanlinna–Pick interpolation problem" S.C. Power (ed.) , ''Operators and function theory'' , Reidel (1984) pp. 279–314</
8 KB (1,230 words) - 08:02, 6 June 2020
is the derivative of the $ m $-th power of $ f $
\sum _ { i= 1} ^ { k }
15 KB (2,177 words) - 16:07, 5 February 2022
\| A \| = \max _ { i } \sum _ { k= 1} ^ { m } | a _ {ik} | ,
z ( t _ {n+1} ) = z ( t _ {n} ) + \sum _ {\nu = 1 } ^ { p } \left . (- 1) ^ {\nu + 1 }
30 KB (4,292 words) - 05:40, 24 February 2022
...rse sequence may be defined as the sequence $u = (u_n)_n$ which counts the sum modulo $2$ of the digits of $n$ in base $2$ ($u_n$ gives the parity of the
is finite or, in the case $k$ is a prime power, to the fact that the series $\sum u_n Z^n$ is algebraic over $\mathbf{F}_k(Z)$. Note that, on the other hand,
10 KB (1,633 words) - 08:05, 26 November 2023
...ne the [[Thompson–McKay series|Thompson–McKay series]] $T _ { g } ( z ) = \sum _ { k = - 1 } ^ { \infty } \chi _ { k } ( g ) q ^ { k }$, where $q = \opera
...ue polynomial for which $Q _ { n } ( T _ { g } ( z ) ) - q ^ { - n }$ is a power series with only strictly positive powers of $q$. The resemblance of (a1) w
12 KB (1,765 words) - 09:46, 10 November 2023
T ^ {-1} \sum _ {\tau = 0 } ^ { T-1 } b( \tau ) = 0,
define the distribution of the average energy (or, more accurately, of the power) of the harmonic oscillations with frequency spectrum $ \lambda $
24 KB (3,614 words) - 08:52, 21 January 2024
...$D$ is a non-empty set of voter response profiles, and $2 ^ { X }$ is the power set of $X$. One requires $F ( A , d ) \subseteq A$ when $A \in \mathcal{X}$
...ght $w _ { i } \geq 0$ for voter $i$, $i = 1 , \dots , n$, and $f ( d ) = \sum w _ { i } d _ { i }$ for all $d \in D$. The $2 / 3$-majority rule for $a$'s
13 KB (2,002 words) - 09:34, 10 November 2023
...ak solutions to the [[Heat equation|heat equation]] $( \partial _ { t } - \sum _ { j = 1 } ^ { n } \partial _ { x _ { j } } ^ { 2 } ) u = 0$ are $C ^ { \i
\begin{equation*} P ( x , D ) = \sum _ { | \alpha | \leq m } p _ { \alpha } ( x ) D _ { x } ^ { \alpha } \end{eq
21 KB (3,112 words) - 08:01, 6 February 2024
...y approximation yields expressions of the type $ \mathop{\rm exp} \{ i \sum _ {j} \omega _ {j} n _ {j} t + \phi \} $,
...various types of motion. If the frequencies are almost-commensurable, the sum appearing in the exponent may be small, as a result of which the appropriat
35 KB (5,181 words) - 07:24, 8 June 2020
...ology, the direct sum, tensor product, dual bundle, symmetric and exterior power, induced algebraic vector bundle, etc., are defined for algebraic vector bu
...kind ring|Dedekind ring]]), then any algebraic vector bundle is the direct sum of a trivial and a line bundle. This also applies to algebraic vector bundl
14 KB (2,169 words) - 08:12, 14 December 2016
...two-player conflicts (cf. also [[Two-person zero-
sum game|Two-person zero-
sum game]]). Suppose there exists a set of possible strategies <img align="absm
...pediaofmath.org/legacyimages/e/e110/e110130/e11013088.png" /> denoting the
power set. Then <img align="absmiddle" border="0" src="https://www.encyclopediaof
28 KB (3,866 words) - 06:43, 27 September 2016
...lgebras of square matrices, algebras of polynomials and algebras of formal power series over fields.
...ebra without nilpotent elements over the field of real numbers is a direct sum of fields that are isomorphic either to $ \mathbf R $
21 KB (3,225 words) - 09:25, 13 July 2022
...{ n } )$. One can then assign a coordinate $t \in ( 0 , \infty )$ and use power series about $r = 0 \in ( - 1 , + 1 )$ to describe the Riemannian metric $\
...$( q _ { 1 } , \dots , q _ { m } )$ of non-negative integers with an even sum one can use a smooth partition of unity subordinate to the covering of $M$
40 KB (5,895 words) - 17:45, 1 July 2020
\sum _ { d }
\zeta ( s) = \sum _ {n = 1 } ^ \infty n ^ {-s}
36 KB (5,530 words) - 08:29, 14 January 2024
$ \sum i _ {s} = n $,
is a power series in two variables, here denoted by $ F _ {h} ( X , Y ) = \sum _ {i,j} a _ {ij} X ^ {i} Y ^ {j} $,
42 KB (6,290 words) - 19:33, 17 January 2024
\begin{equation*} f ( z ) = \sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n } \end{equation*}
with square-summable coefficients (i.e., $\sum _ { n = 0 } ^ { \infty } | a _ { n } | ^ { 2 } < \infty$). The shift ope
12 KB (1,802 words) - 17:01, 1 July 2020
...ectrum of the sum of two commuting operators is contained in the algebraic sum of their spectra) or very subtle (describing the spectra of functions of no
...m under small perturbations, and to express analytically (in the form of a power series in the parameter $ \mu $)
16 KB (2,424 words) - 08:22, 6 June 2020
\begin{equation*} R = \sum _ { i = 0 } ^ { n - 1 } Z ^ { i } G J G ^ { * } Z ^ { * i } = \end{equation
...= \sum _ { i = 0 } ^ { p - 1 } L ( x _ { i } ) L ^ { * } ( x _ { i } ) - \sum _ { i = 0 } ^ { q - 1 } L ( y _ { i } ) L ^ { * } ( y _ { i } ). \end{equat
21 KB (3,255 words) - 20:20, 13 January 2021
$ \sum _ {i=1} ^ {m} d _ {i} = n $,
as an orthogonal sum of multiplication operators by the independent variable.
51 KB (7,267 words) - 07:39, 14 January 2024
is conjugate to a power of some unique generator $ \gamma _ {i} $.
the sum of all remaining angles is $ 2 \pi $.
14 KB (2,109 words) - 20:03, 15 March 2023
...y ideal|primary ideal]], which replaces, for the multi-dimensional case, a power of a prime ideal. He also defined the prime ideal associated to a primary i
Finally, Lasker extended a number of his results to the ring of convergent power series by considering these from an algebraic point of view.
16 KB (2,400 words) - 17:45, 4 June 2020
...t problem [[#References|[a8]]]. All of these problems generalize classical power moment problems on the real line, whose study was initiated by Th.J. Stielt
...olynomial, non-negative on the real plane, that cannot be represented as a sum of squares of polynomials (cf. [[#References|[a3]]], [[#References|[a14]]],
23 KB (3,450 words) - 17:45, 1 July 2020
..., Y _ { d } ]]$, with $d \geq 1$, denotes the [[Formal power series|formal power series]] ring in $2 d$ variables over a [[Field|field]] $k$. Then $A$ is a
\begin{equation*} \operatorname {I} ( M ) = \sum _ { i = 0 } ^ { s - 1 } \left( \begin{array} { c } { s - 1 } \\ { i } \end{
27 KB (4,003 words) - 17:43, 1 July 2020
\sum L ( E),
by the power method, which is based on the following consideration. The number of all po
27 KB (4,142 words) - 14:55, 7 June 2020
the sum of two complex numbers $z=(x,y)$ and $z'=(x',y')$ is the complex number $(x
respect to the real axis. The sum and the product of two conjugate
14 KB (2,281 words) - 09:55, 26 March 2023
where $u ( x , t ) = i \
sum _ { k } \hat { u } _ { k } ( t ) \operatorname { exp } ( i k x )$, $k = n q
...in the light of the characteristic shape of the normalized (time-averaged)
power spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofm
21 KB (3,050 words) - 17:43, 1 July 2020
...wing Volterra, Fredholm replaced the integral in (3) by a Riemann integral sum and considered the integral equation (3) as a limiting case of a finite sys
...]]) independently of the Fredholm theory by representing the kernel as the sum of a degenerate and a "small" kernel. T. Carleman achieved a substantial we
14 KB (2,157 words) - 17:38, 3 September 2013
th power of some other characteristic function:
as a sum of $ n $
11 KB (1,489 words) - 09:00, 13 January 2024
sequence $\,\{T_n\}$ in order to attain the power $\,\beta$
Wilcoxon rank sum test with respect to the corresponding Student
18 KB (2,622 words) - 20:30, 6 March 2024
...ctions that, together with all their derivatives, decrease faster than any power of $ | x | ^ {- 1} $
for which $ | \xi | _ {p} = \sum _ {n = 0 } ^ \infty | \xi _ {n} | a _ {np} < \infty $
26 KB (3,852 words) - 07:00, 6 May 2022
...diaofmath.org/legacyimages/i/i051/i051410/i051410106.png" /> is the direct
sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.or
...ek, "Minimal factorization, linear systems and integral operators" S.S.
Power (ed.) , ''Operators and function theory'' , Reidel (1985) pp. 41–86</TD
32 KB (4,357 words) - 17:14, 7 February 2011
A function that can be locally represented by power series. Such functions are usually divided into two important classes: the
...d by Weierstrass, is based on the possibility of representing functions by power series; it is thus connected with the analytic apparatus by means of which
61 KB (9,850 words) - 19:04, 20 January 2022
can be written as a finite sum $ P = \sum a _ {i _ {1} \dots i _ {n} } \partial _ {1} ^ {i _ {1} } \dots \partial
...} ( M , {\mathcal O} _ {X} ) = \{ {f \in {\mathcal O} _ {X} ^ {q} } : {\sum _ {j=1} ^ {q} P _ {ij} f _ {j} = 0 } \} $.
24 KB (3,511 words) - 07:03, 10 May 2022
is attained for the partial sum of the Fourier series \ref{eq2} of $f$:
...the Fourier series of a function $f$ at a point $x$, and the value of the sum when it converges, depends only on the behaviour of $f$ in an arbitrarily
21 KB (3,473 words) - 19:50, 26 April 2012
then one has mean-power approximation, or $ L _ {p} $-
\sum _ { k=1 } ^ { N }
29 KB (4,328 words) - 18:47, 5 April 2020
...\lambda _ { n } )$ is a decreasing sequence of non-negative integers whose sum is $r$. If the requirement "decreasing" in the definition above is dropped,
...a $ of the group algebra $K \mathfrak { S } _ { r }$ to be the alternating sum
33 KB (5,081 words) - 10:26, 11 November 2023
consisting of a power series $$ \tag{1}
\sum _{k=0} ^ \infty c _{k} (z - \zeta ) ^{k} $$
66 KB (9,825 words) - 01:45, 23 June 2022
\sum _ { j,k } a _ {jk} z ^ {j} w ^ {k} = 0,
is a convergent power series in two variables, considered with all possible analytic continuation
14 KB (2,068 words) - 02:08, 18 July 2022
...the description of systems with a variable number of particles, the direct sum of the spaces $ \mathcal{F}_{n} $, namely the Fock space $ \displaystyle \m
The Fourier transform of the sum $ \phi(f) = a(\widetilde{f}) + {a^{*}}(\widetilde{f}) $ of the creation and
35 KB (5,270 words) - 23:26, 6 December 2016
\begin{equation*} d _ { \chi } ^ { G } ( A ) : = \sum _ { \sigma \in G } \chi ( \sigma ) \prod _ { i = 1 } ^ { n } a _ {i \sigma
\begin{equation*} d _{( n )} ( A ) = \operatorname { per } ( A ) = \sum _ { \sigma \in S _ { n } } \prod _ { i = 1 } ^ { n } a _ { i \sigma ( i ) }
19 KB (2,837 words) - 05:34, 15 February 2024
...tive algebra $\def\cO{\mathcal{O}}\cO_n(k) = k[[X_1,\dots,X_n]]$ of formal power series over a field $k$, then instead of $\Vect(M)$ one obtains the Lie alg
be the lower central series of $G$. Then $L$ is the direct sum of the additively written quotient groups $G_i/G_{i+1}$ and, by definition,
25 KB (4,037 words) - 07:06, 23 April 2016
===Whitney sum of bundles===
...the vector bundles one usually says about the ''direct sum'', or ''Whitney sum'' and denoted by $\pi_1\oplus \pi_2$.
32 KB (5,299 words) - 12:27, 12 December 2020
purpose to demonstrate the power of the new algebra created by
A presupposition for justness is that the sum of the
16 KB (2,633 words) - 12:03, 1 November 2023
...z \overline { w } )$ is non-negative on the unit disc in the sense that $\sum _ { i , j = 1 } ^ { n } \overline { c } _ { i } K _ { S } ( w _ { j } , w _
...6]</td> <td valign="top"> L. de Branges, J. Rovnyak, "Square summable power series" , Holt, Rinehart&Winston (1966)</td></tr><tr><td valign="top">
14 KB (2,163 words) - 09:53, 11 November 2023
...tion. Let $X\subset {\mathbb P}^2$ be a smooth plane cubic curve. Then the sum of
and $Q$. In other words, the sum of three points on $X$ vanishes if
19 KB (3,251 words) - 20:37, 19 September 2017
...r{\alpha}$ means that the unit $\bar{\alpha}$ is not to be multiplied by a power of the unknown).
...t y\, dx$, while hinting at the actual process of constructing an integral sum, also includes explicit indication of the integrand and the variable of int
18 KB (2,697 words) - 13:11, 13 December 2013