$#C+1 = 12 : ~/encyclopedia/old_files/data/D032/D.0302900 Dirichlet principle
...nals in certain classes of functions. In the narrow sense of the term, the Dirichlet principle reduces the first boundary value problem
6 KB (847 words) - 19:35, 5 June 2020
''Dirichlet–Laplace operator''
...$C _ { 0 } ^ { \infty } ( \Omega )$) is given by the [[Dirichlet integral|Dirichlet integral]]
4 KB (579 words) - 19:34, 7 February 2024
A name referring to several theorems associated with Peter Gustav Lejeune Dirichlet (1805-1859).
===Dirichlet's theorem in the theory of Diophantine approximations===
7 KB (1,065 words) - 14:05, 17 March 2020
...s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Dirichlet density. Here, $N(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the nu
i) The set of all prime ideals of $K$ is regular with Dirichlet density $1$.
2 KB (310 words) - 09:04, 26 November 2023
$#C+1 = 151 : ~/encyclopedia/old_files/data/D032/D.0302910 Dirichlet problem
...this task were studied as early as 1840 by C.F. Gauss, and then by P.G.L. Dirichlet [[#References|[1]]].
17 KB (2,472 words) - 08:26, 21 March 2022
The Dirichlet convolution of two [[arithmetic function]]s $f(n)$ and $g(n)$ is defined as
...over the positive divisors $d$ of $n$. General background material on the Dirichlet convolution can be found in, e.g., [[#References|[a1]]], [[#References|[a6]
4 KB (627 words) - 09:28, 19 March 2023
...\gamma$ is the [[Euler constant]], $\gamma \approx 0.577$. Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points $(x,y)$ wi
951 bytes (154 words) - 08:27, 30 December 2015
$#C+1 = 120 : ~/encyclopedia/old_files/data/D032/D.0302810 Dirichlet character
In other words, a Dirichlet character $ \mathop{\rm mod} k $
10 KB (1,462 words) - 11:49, 26 March 2023
''Voronoi tesselation, Dirichlet–Voronoi tesselation, Dirichlet–Voronoi decomposition, Thiessen tesselation''
...is also called the Dirichlet–Voronoi tiling. The straight-line dual of the Dirichlet tesselation is the [[Delaunay triangulation|Delaunay triangulation]].
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...a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras were introduced by A.M. Gleason [[#References|[a4]]].
...ine{\mathbf D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, t
7 KB (1,114 words) - 19:36, 23 December 2023
$#C+1 = 13 : ~/encyclopedia/old_files/data/D032/D.0302880 Dirichlet kernel
It was shown by P.G.L. Dirichlet [[#References|[1]]] that the partial sum $ S _ {n} $
3 KB (508 words) - 23:39, 29 December 2021
-
783 bytes (125 words) - 21:31, 18 November 2017
$#C+1 = 131 : ~/encyclopedia/old_files/data/D032/D.0302920 Dirichlet series
one obtains the so-called ordinary Dirichlet series
11 KB (1,638 words) - 11:32, 16 April 2023
...Dirichlet polynomials are partial sums of corresponding [[Dirichlet series|Dirichlet series]].
...Dirichlet $L$-function]]), as well as their powers, can be approximated by Dirichlet polynomials, mostly with $\lambda _ { m } = \operatorname { log } m$. For e
5 KB (706 words) - 17:03, 1 July 2020
$#C+1 = 104 : ~/encyclopedia/old_files/data/D110/D.1100210 Dirichlet process
...tained in the collection of normal distributions. The large support of the Dirichlet process accounts for its use in non-parametric Bayesian analysis. General r
11 KB (1,528 words) - 19:35, 5 June 2020
$#C+1 = 38 : ~/encyclopedia/old_files/data/D032/D.0302870 Dirichlet integral
A functional connected with the solution of the [[Dirichlet problem]] for the Laplace equation by the variational method. Let $ \Omeg
4 KB (557 words) - 10:41, 20 January 2024
...Dirichlet boundary value problem (cf. also [[Dirichlet boundary conditions|Dirichlet boundary conditions]]):
...is bounded and the boundary $\partial \Omega$ is sufficiently regular, the Dirichlet Laplacian has a discrete spectrum of infinitely many positive eigenvalues w
12 KB (1,753 words) - 10:12, 16 March 2023
$#C+1 = 16 : ~/encyclopedia/old_files/data/D032/D.0302840 Dirichlet distribution
...e Dirichlet distribution: the [[Beta-distribution|beta-distribution]]. The Dirichlet distribution plays an important role in the theory of order statistics. For
2 KB (306 words) - 08:57, 8 April 2023
...inuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L.
Dirichlet in his studies on the attraction of ellipsoids [[#References|[1]]]. The int
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.G.L.
Dirichlet, "Werke" , '''1''' , Chelsea, reprint (1969)</TD></TR><TR><TD valign="to
799 bytes (121 words) - 14:14, 14 February 2020
''Dirichlet conditions, Dirichlet data, boundary conditions of the first kind''
are called Dirichlet boundary conditions.
810 bytes (116 words) - 14:55, 16 October 2014
$#C+1 = 21 : ~/encyclopedia/old_files/data/D032/D.0302950 Dirichlet variational problem
The problem of finding the minimum of the [[Dirichlet integral|Dirichlet integral]]
3 KB (503 words) - 17:53, 6 January 2024
#REDIRECT [[Dirichlet L-function]]
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$#C+1 = 4 : ~/encyclopedia/old_files/data/D032/D.0302800 Dirichlet box principle
elements comprises at least one set with at least two elements. Dirichlet's box principle can be formulated in a most popular manner as follows: If
928 bytes (141 words) - 19:35, 5 June 2020
A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive in
is the [[Dirichlet convolution]] of $a$ and $b$.
2 KB (358 words) - 17:25, 11 November 2023
$#C+1 = 182 : ~/encyclopedia/old_files/data/D032/D.0302890 Dirichlet \BMI L\EMI\AAhfunction,
''Dirichlet $ L $-
15 KB (2,181 words) - 11:50, 26 March 2023
...)$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]].
...roximation of functions of a complex variable]]). Then $R ( X )$ is a hypo-Dirichlet algebra [[#References|[a3]]].
3 KB (474 words) - 19:56, 27 February 2021
...a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in {{Cite|Di}}. If a sequence of real numbers $a
|valign="top"|{{Ref|Di}}|| P.G.L. Dirichlet, "Démonstration d’un théorème d’Abel", ''J. de Math. (2)'' , '''7'''
730 bytes (127 words) - 19:46, 16 January 2016
$#C+1 = 27 : ~/encyclopedia/old_files/data/D032/D.0302930 Dirichlet series for an analytic almost\AAhperiodic function
...erent almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a $ 2 \pi $-
3 KB (422 words) - 19:35, 5 June 2020
$#C+1 = 4 : ~/encyclopedia/old_files/data/D032/D.0302800 Dirichlet box principle
elements comprises at least one set with at least two elements. Dirichlet's box principle can be formulated in a most popular manner as follows: If
928 bytes (141 words) - 19:35, 5 June 2020
''Dirichlet conditions, Dirichlet data, boundary conditions of the first kind''
are called Dirichlet boundary conditions.
810 bytes (116 words) - 14:55, 16 October 2014
''Dirichlet principal character''
...erve to define the concepts of primitive and imprimitive characters (cf. [[Dirichlet character]]).
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#REDIRECT [[Dirichlet series]]
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#REDIRECT [[Dirichlet L-function]]
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#REDIRECT [[Dirichlet L-function]]
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...ependent functions with zero mathematical expectations; [[Dirichlet series|Dirichlet series]] with exponents that are independent over the field of rational num
452 bytes (57 words) - 17:11, 7 February 2011
''Voronoi tesselation, Dirichlet–Voronoi tesselation, Dirichlet–Voronoi decomposition, Thiessen tesselation''
...is also called the Dirichlet–Voronoi tiling. The straight-line dual of the Dirichlet tesselation is the [[Delaunay triangulation|Delaunay triangulation]].
509 bytes (70 words) - 19:53, 29 April 2014
$#C+1 = 21 : ~/encyclopedia/old_files/data/D032/D.0302950 Dirichlet variational problem
The problem of finding the minimum of the [[Dirichlet integral|Dirichlet integral]]
3 KB (503 words) - 17:53, 6 January 2024
...s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Dirichlet density. Here, $N(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the nu
i) The set of all prime ideals of $K$ is regular with Dirichlet density $1$.
2 KB (310 words) - 09:04, 26 November 2023
$#C+1 = 13 : ~/encyclopedia/old_files/data/D032/D.0302880 Dirichlet kernel
It was shown by P.G.L. Dirichlet [[#References|[1]]] that the partial sum $ S _ {n} $
3 KB (508 words) - 23:39, 29 December 2021
...inuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L.
Dirichlet in his studies on the attraction of ellipsoids [[#References|[1]]]. The int
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.G.L.
Dirichlet, "Werke" , '''1''' , Chelsea, reprint (1969)</TD></TR><TR><TD valign="to
799 bytes (121 words) - 14:14, 14 February 2020
.... It follows from Dirichlet's "box" principle (cf. [[Dirichlet principle|Dirichlet principle]]) that the following always holds:
2 KB (342 words) - 12:47, 20 December 2014
$#C+1 = 27 : ~/encyclopedia/old_files/data/D032/D.0302930 Dirichlet series for an analytic almost\AAhperiodic function
...erent almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a $ 2 \pi $-
3 KB (422 words) - 19:35, 5 June 2020
...emann zeta function]] which may be used to exhibit properties of various [[Dirichlet L-function]]s.
722 bytes (101 words) - 09:28, 19 March 2023
$#C+1 = 12 : ~/encyclopedia/old_files/data/D032/D.0302900 Dirichlet principle
...nals in certain classes of functions. In the narrow sense of the term, the Dirichlet principle reduces the first boundary value problem
6 KB (847 words) - 19:35, 5 June 2020
...a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in {{Cite|Di}}. If a sequence of real numbers $a
|valign="top"|{{Ref|Di}}|| P.G.L. Dirichlet, "Démonstration d’un théorème d’Abel", ''J. de Math. (2)'' , '''7'''
730 bytes (127 words) - 19:46, 16 January 2016
...ue problem|First boundary value problem]]; [[Dirichlet boundary conditions|Dirichlet boundary conditions]]; [[Third boundary value problem|Third boundary value
885 bytes (129 words) - 07:13, 23 August 2014
A name referring to several theorems associated with Peter Gustav Lejeune Dirichlet (1805-1859).
===Dirichlet's theorem in the theory of Diophantine approximations===
7 KB (1,065 words) - 14:05, 17 March 2020
$#C+1 = 38 : ~/encyclopedia/old_files/data/D032/D.0302870 Dirichlet integral
A functional connected with the solution of the [[Dirichlet problem]] for the Laplace equation by the variational method. Let $ \Omeg
4 KB (557 words) - 10:41, 20 January 2024
...Dirichlet polynomials are partial sums of corresponding [[Dirichlet series|Dirichlet series]].
...Dirichlet $L$-function]]), as well as their powers, can be approximated by Dirichlet polynomials, mostly with $\lambda _ { m } = \operatorname { log } m$. For e
5 KB (706 words) - 17:03, 1 July 2020
A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive in
is the [[Dirichlet convolution]] of $a$ and $b$.
2 KB (358 words) - 17:25, 11 November 2023
...convergence criterion is the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]].
860 bytes (157 words) - 20:29, 9 December 2013
...ates on the hyperbola $xy = n$. The average value of $\tau(n)$ is given by Dirichlet's asymptotic formula (cf. [[Divisor problems]]).
...etic function|average value]] of the number of divisors was obtained by P. Dirichlet in 1849, in the form
2 KB (276 words) - 08:15, 4 November 2023
[[Dirichlet series|Dirichlet series]] or an
...finite fields, etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf.
2 KB (347 words) - 21:23, 9 January 2015
...ed by M. Riesz [[#References|[1]]] for the summation of [[Dirichlet series|Dirichlet series]]. The method $(R,\lambda,k)$ is regular; when $\lambda_n=n$ it is e
...]</TD> <TD valign="top"> G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915)</TD></TR><TR><TD valign="top">[4]</
2 KB (351 words) - 13:47, 14 February 2020
The Dirichlet convolution of two [[arithmetic function]]s $f(n)$ and $g(n)$ is defined as
...over the positive divisors $d$ of $n$. General background material on the Dirichlet convolution can be found in, e.g., [[#References|[a1]]], [[#References|[a6]
4 KB (627 words) - 09:28, 19 March 2023
...n-trivial zeros of Dirichlet $ L $-functions (cf. [[Dirichlet L-function|Dirichlet $ L $-function]]), Dedekind zeta-functions (cf. [[Zeta-function|Zeta-func
In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Ri
4 KB (517 words) - 10:52, 21 March 2022
...)$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]].
...roximation of functions of a complex variable]]). Then $R ( X )$ is a hypo-Dirichlet algebra [[#References|[a3]]].
3 KB (474 words) - 19:56, 27 February 2021
...an be applied. Other examples are the [[Dirichlet discontinuous multiplier|Dirichlet discontinuous multiplier]], the Dirac [[Delta-function|delta-function]], et
1 KB (147 words) - 19:35, 5 June 2020
$#C+1 = 16 : ~/encyclopedia/old_files/data/D032/D.0302840 Dirichlet distribution
...e Dirichlet distribution: the [[Beta-distribution|beta-distribution]]. The Dirichlet distribution plays an important role in the theory of order statistics. For
2 KB (306 words) - 08:57, 8 April 2023
$#C+1 = 120 : ~/encyclopedia/old_files/data/D032/D.0302810 Dirichlet character
In other words, a Dirichlet character $ \mathop{\rm mod} k $
10 KB (1,462 words) - 11:49, 26 March 2023
...let problem|Dirichlet problem]]; it allows one to obtain a solution to the Dirichlet problem for a differential equation of elliptic type in domains $ D $
...other authors were dedicated to this method for finding a solution to the Dirichlet problem for the [[Laplace equation|Laplace equation]] in plane domains. The
6 KB (852 words) - 08:12, 6 June 2020
...a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras were introduced by A.M. Gleason [[#References|[a4]]].
...ine{\mathbf D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, t
7 KB (1,114 words) - 19:36, 23 December 2023
''Dirichlet–Laplace operator''
...$C _ { 0 } ^ { \infty } ( \Omega )$) is given by the [[Dirichlet integral|Dirichlet integral]]
4 KB (579 words) - 19:34, 7 February 2024
The [[Dirichlet convolution]] product
Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]:
3 KB (419 words) - 20:15, 19 November 2017
is infinite and has [[Dirichlet density|Dirichlet density]] $\# A / n$, where $n = [ L : K ]$.
...specifies in addition that $P _ { A }$ is regular (see [[Dirichlet density|Dirichlet density]]) and that
3 KB (449 words) - 17:00, 1 July 2020
The homogeneous Dirichlet problem in a disc $ C $:
...infinite number of linearly independent solutions [[#References|[1]]]. The Dirichlet problem for the inhomogeneous equation $ w _ {\overline{z}\; \overline{z}
2 KB (311 words) - 10:59, 29 May 2020
...Suppose, for example, that it is required to solve the [[Dirichlet problem|Dirichlet problem]] for the [[Poisson equation|Poisson equation]] $ \Delta u = - 2
For the solution of the Dirichlet problem for the Poisson equation $ \Delta u = - 4 \pi \rho ( x, y, z) $
7 KB (1,055 words) - 07:59, 6 June 2020
...haviour of a real sequence to the analytic properties of the associated [[Dirichlet series]]. It is used in the study of [[arithmetic function]]s and yields
An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-n
2 KB (298 words) - 20:23, 15 November 2023
...otential theory are theorems on the solvability of the [[Dirichlet problem|Dirichlet problem]], established by M.V. Keldysh in 1938–1941.
such that the Dirichlet problem is solvable in $ D $
6 KB (831 words) - 22:14, 5 June 2020
is the [[Dirichlet integral|Dirichlet integral]] of the real part of the function $ u $
2 KB (374 words) - 08:01, 6 June 2020
$#C+1 = 131 : ~/encyclopedia/old_files/data/D032/D.0302920 Dirichlet series
one obtains the so-called ordinary Dirichlet series
11 KB (1,638 words) - 11:32, 16 April 2023
...r on a plane, any number $T\ne0$ is a period; for the [[Dirichlet-function|Dirichlet function]]
1 KB (227 words) - 21:30, 18 November 2017
...ic series|Fourier series]] of piecewise monotone functions, is also called Dirichlet-Jordan test, cf. with {{Cite|Zy}}.
1 KB (205 words) - 12:30, 27 September 2012
...$\mathbf{R}^n$, $n\ge 2$, with respect to a generalized solution of the [[Dirichlet problem]] for $D$ in the sense of Wiener–Perron (see [[Perron method]]) h
...TD valign="top"> M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" ''Uspekhi Mat. Nauk'' , '''8''' (1941) pp. 171–231 (In Russi
2 KB (245 words) - 16:25, 22 October 2017
...ronger than the [[Dini criterion|Dini criterion]], the [[Dirichlet theorem|Dirichlet criterion]], and the [[Jordan criterion|Jordan criterion]], it is weaker th
1 KB (216 words) - 20:45, 16 October 2012
...s of one or more variables); almost-periodic functions; [[Dirichlet series|Dirichlet series]]; [[Approximation theory|approximation theory]] (of functions by tr
2 KB (287 words) - 11:59, 27 September 2012
...e Dirichlet data, and the problem (1), (2) is called a [[Dirichlet problem|Dirichlet problem]] if $ S = \partial D $.
the first boundary value problem (the Dirichlet problem) consists in finding solutions to this equation subject to the cond
7 KB (954 words) - 20:18, 10 January 2024
Page's theorem on the zeros of Dirichlet $L$-functions.
...,\chi)$ be a [[Dirichlet L-function]], $s = \sigma + i t$, with $\chi$ a [[Dirichlet character]] modulo $d$, $d \ge 3$. There are absolute positive constants $c
3 KB (508 words) - 15:13, 3 July 2020
for the [[Dirichlet series|Dirichlet series]]
2 KB (279 words) - 19:13, 14 December 2015
The Young's criterion is stronger than the [[Dirichlet theorem|Dirichlet criterion]], the [[Dini criterion]] and the [[Jordan criterion]], it is not
2 KB (233 words) - 20:45, 16 October 2012
...To a character $α: ℤ_k^* → T$ there corresponds the [[Dirichlet character|Dirichlet character]] $χ: ℤ → ℂ$ given by the formula
3 KB (438 words) - 13:36, 23 July 2015
...Cite|Zy}}. The Lebesgue criterion is stronger then the [[Dirichlet theorem|Dirichlet criterion]], the [[Jordan criterion|Jordan criterion]], the [[Dini criterio
2 KB (235 words) - 11:59, 14 December 2012
of the [[Dirichlet problem|Dirichlet problem]] in the sense of Wiener–Perron (see [[Perron method|Perron metho
is called regular with respect to the Dirichlet problem.
5 KB (799 words) - 09:19, 6 January 2024
...was discovered in mid-19th century as the so-called [[Dirichlet principle|Dirichlet principle]] of finding, in a domain $ G $,
...minimum of the [[Dirichlet integral|Dirichlet integral]]. Originally, the Dirichlet principle was used only in the theory of second-order linear elliptic equat
7 KB (969 words) - 17:33, 5 June 2020
$#C+1 = 104 : ~/encyclopedia/old_files/data/D110/D.1100210 Dirichlet process
...tained in the collection of normal distributions. The large support of the Dirichlet process accounts for its use in non-parametric Bayesian analysis. General r
11 KB (1,528 words) - 19:35, 5 June 2020
...solution (cf. [[Perron method|Perron method]]) of the [[Dirichlet problem|Dirichlet problem]], $ u ( x) $,
axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value $ f ( y _ {0} ) $
3 KB (435 words) - 22:13, 5 June 2020
...ed in the work of G. Peano (1880). For the case of the [[Dirichlet problem|Dirichlet problem]] and for the case of the [[Laplace equation|Laplace equation]] the
Let the Dirichlet problem be posed in a region $ G $
6 KB (849 words) - 19:25, 11 January 2024
See also [[Quadratic residue|Quadratic residue]]; [[Dirichlet character|Dirichlet character]].
2 KB (304 words) - 19:26, 14 August 2014
...unctions [[#References|[a8]]], one can show that the asymptotic formula ([[Dirichlet's theorem on arithmetic progressions]])
3 KB (458 words) - 16:53, 23 November 2023
Abel's theorem on Dirichlet series: If the [[Dirichlet series|Dirichlet series]]
It follows from the theorem that the domain of convergence of a Dirichlet series is some half-plane $ \sigma > c $,
6 KB (894 words) - 06:14, 26 March 2023
...ordinary harmonic functions. A generalization of the [[Dirichlet principle|Dirichlet principle]] is valid for them: Among all functions $ v $
the latter gives the minimum of the Dirichlet integral
4 KB (507 words) - 22:15, 5 June 2020
<TR><TD valign="top">[1]</TD> <TD valign="top"> L. Gårding, "Dirichlet's problem for linear elliptic partial differential equations" ''Math. Scan
2 KB (284 words) - 14:02, 14 April 2024
A method for solving the [[Dirichlet problem|Dirichlet problem]] for the [[Laplace equation|Laplace equation]] based on the proper
is the generalized solution to the Dirichlet problem for the function $ f( y) $,
10 KB (1,417 words) - 08:05, 6 June 2020
...e inversion formula plays a role in the proof of a functional equation for Dirichlet series similar to that for the Riemann zeta-function. Cf. [[#References|[a1
...TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) {{MR|0256993}} {{MR|0234918}} {{ZBL|0191.38101}}
4 KB (549 words) - 11:30, 4 January 2015
...alogue of Harnack's inequality), facts relating to the [[Dirichlet problem|Dirichlet problem]], the non-existence of a non-linear solution defined in the entire
For this equation $(n\geq3)$ the solvability of the Dirichlet problem has been studied, the removability of the singularities of a soluti
3 KB (501 words) - 18:46, 13 November 2014
...heorem is true for many other series, in particular for [[Dirichlet series|Dirichlet series]].
3 KB (438 words) - 12:39, 6 January 2024
...ntegral equation with this kernel corresponding to the [[Dirichlet problem|Dirichlet problem]] and several integral equations to construct the conformal Riemann
...t integrals $D ( h )$ and $D ^ { * } ( h )$ (cf. also [[Dirichlet integral|Dirichlet integral]]).
7 KB (989 words) - 20:48, 23 January 2024
===Siegel's theorem on Dirichlet L-functions===
...\epsilon) > 0$ such that for any non-principal real [[Dirichlet character|Dirichlet character]] $\chi$ of modulus $k$,
5 KB (784 words) - 20:40, 18 October 2014
The usual boundary value problems (Dirichlet, Neumann and others) are posed for the Helmholtz equation, which is of elli
...onding boundary value problem). In particular, for the [[Dirichlet problem|Dirichlet problem]] all eigen values are positive, and for the [[Neumann problem]] th
4 KB (519 words) - 22:10, 5 June 2020
For the number $N(\sigma, T, \chi)$ of zeros of Dirichlet $L$-functions
The density hypothesis for Dirichlet $ L $
5 KB (705 words) - 09:02, 6 January 2024
$#C+1 = 182 : ~/encyclopedia/old_files/data/D032/D.0302890 Dirichlet \BMI L\EMI\AAhfunction,
''Dirichlet $ L $-
15 KB (2,181 words) - 11:50, 26 March 2023
...ch a [[Green function|Green function]] exists (for the [[Dirichlet problem|Dirichlet problem]] in the class of harmonic functions) or, which amounts to the same
2 KB (357 words) - 11:31, 27 October 2014
...phi$ on $\Gamma$ (the first boundary value problem, or [[Dirichlet problem|Dirichlet problem]]), or the condition
...érivées des functions de Green; consequences pour les problèmes du type de Dirichlet" ''C.R. Acad. Sci. Paris'' , '''202''' (1936) pp. 380–382</TD></TR><TR
4 KB (716 words) - 17:38, 14 February 2020
...y$, where $G_y(x)=G(x,y)$ is the [[Green function|Green function]] (of the Dirichlet problem for the Laplace equation) for a domain $D$ in a Euclidean space $\m
849 bytes (136 words) - 13:59, 1 October 2014
represents infinitely many prime numbers. Similarly, the [[Dirichlet theorem|Dirichlet theorem]] about infinitely many primes in an arithmetic progression comes f
3 KB (382 words) - 06:29, 30 May 2020
...ardy, J.E. Littlewood, "Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive" ''Proc. London. Math. Soc. (2)''
4 KB (577 words) - 19:43, 5 June 2020
and Dirichlet $L$-functions
Yu.V. Linnik (1944 and subsequent years) developed the density method for Dirichlet $ L $-
4 KB (642 words) - 13:03, 6 January 2024
is regular for the Dirichlet problem if and only if the complement of $ U $
...i-polar set, a kind of exceptional set (related to the [[Dirichlet problem|Dirichlet problem]]) which can be considerably bigger than a polar set when the poten
4 KB (696 words) - 08:25, 6 June 2020
...[a2]]]. Though it is obvious from the variational characterization of both Dirichlet and Neumann eigenvalues (see (a4)) that $\mu _ { k } \leq \lambda _ { k }$,
A proof of Pólya's conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (a5).
7 KB (1,058 words) - 19:49, 6 February 2024
function]]) and, more generally, functions defined by Dirichlet series.
...or the analytic continuation and the derivation of functional equations of Dirichlet functions; for the derivation of approximate functional equations of these
11 KB (1,669 words) - 08:09, 12 January 2024
see [[Dirichlet theorem|Dirichlet theorem]] on units), and $ l: E \rightarrow \mathbf R ^ {r+1} $
3 KB (430 words) - 10:48, 20 January 2024
==Dirichlet's divisor problem.==
was first considered by P. Dirichlet in 1849. He based himself on the fact that this sum is equal to the number
11 KB (1,578 words) - 11:58, 26 March 2023
$#C+1 = 151 : ~/encyclopedia/old_files/data/D032/D.0302910 Dirichlet problem
...this task were studied as early as 1840 by C.F. Gauss, and then by P.G.L. Dirichlet [[#References|[1]]].
17 KB (2,472 words) - 08:26, 21 March 2022
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand-Reinhold (1970)</TD
2 KB (362 words) - 17:31, 5 June 2020
...\gamma$ is the [[Euler constant]], $\gamma \approx 0.577$. Obtained by P. Dirichlet in 1849; he noted that this sum is equal to the number of points $(x,y)$ wi
951 bytes (154 words) - 08:27, 30 December 2015
...oblem]] (C.F. Gauss) and the [[Divisor problems|divisor problems]] (P.G.L. Dirichlet) are classical (starting points) as well as their numerous generalizations.
1 KB (188 words) - 17:31, 30 April 2014
are the Dirichlet kernels (cf. [[Dirichlet kernel|Dirichlet kernel]]).
6 KB (901 words) - 17:32, 5 June 2020
...integral equations in the same way as it has been done in the case of the Dirichlet and Neumann problems for harmonic functions in the article [[Potential theo
...</TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C.G. Simader, "On Dirichlet's boundary value problem" , Springer (1972)</TD></TR><TR><TD valign="top">
4 KB (653 words) - 08:07, 6 June 2020
...he density of primes in an arithmetic progression (cf. [[Dirichlet theorem|Dirichlet theorem]]), using the polynomial $ f ( x ) = ax + b $.
3 KB (469 words) - 10:15, 29 May 2020
the solution of the generalized Dirichlet problem for $ D $
is the (generalized) Green function of the Dirichlet problem for $ D $.
3 KB (464 words) - 19:43, 5 June 2020
...on the behaviour of potentials and the solution of the [[Dirichlet problem|Dirichlet problem]], obtained by A.M. Lyapunov in 1886–1902 (see ).
...v as a basis for the construction of a strict theory of solvability of the Dirichlet problem by the method of integral equations. A monograph of Gunther was dev
9 KB (1,266 words) - 14:20, 14 August 2023
Schauder's method of finding a solution to the [[Dirichlet problem|Dirichlet problem]] for a linear uniformly-elliptic equation
to the Dirichlet problem
7 KB (1,029 words) - 16:50, 4 January 2021
..., when integrating elementary functions with respect to a parameter (see [[Dirichlet discontinuous multiplier]]), when calculating the sum of a series in which
938 bytes (131 words) - 17:19, 20 November 2016
...s harmonic and hyperharmonic functions, potentials, minimum principle, the Dirichlet problem, harmonic measure, balayage, fine topology, Martin compactification
if the [[Dirichlet problem|Dirichlet problem]] on $ V $
5 KB (700 words) - 18:48, 26 February 2024
4) the Walsh–Dirichlet kernels of order $2^n$, $D_{2^n}(x)=\sum_{k=0}^{2^n-1}w_k(x)$, are non-nega
...itions of $[0,1)$, as $n\to\infty$, the Walsh system is the only one whose Dirichlet kernels of order $2^n$ are non-negative. S.V. Levizov [[#References|[a1]]]
3 KB (467 words) - 09:42, 27 November 2018
...p">[1]</TD> <TD valign="top"> F. Riesz, "Sur la sommation des séries de Dirichlet" ''C.R. Acad. Sci. Paris'' , '''149''' (1909) pp. 18–21</TD></TR><TR><
1 KB (179 words) - 13:14, 14 February 2020
...s mentioned can be combined in the Euler product expansion of the [[formal Dirichlet series]]
...="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Modular functions and Dirichlet series in number theory" (2nd ed) , Springer (1990) {{ZBL|0697.10023}}</TD
3 KB (483 words) - 14:20, 17 March 2023
By the Dirichlet unit theorem (cf. also [[Dirichlet theorem|Dirichlet theorem]]), the unit group $ U _ {F} $
adic analogue of the Dirichlet regulator:
6 KB (891 words) - 19:08, 26 March 2023