See [[Weierstrass elliptic functions|Weierstrass elliptic functions]].
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See [[Weierstrass elliptic functions|Weierstrass elliptic functions]].
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...z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form
...arily, that is, by algebraic functions in $z$ or in the logarithms of such functions. For example,
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$#C+1 = 11 : ~/encyclopedia/old_files/data/A012/A.0102160 Amplitude of an elliptic integral
in an [[Elliptic integral|elliptic integral]] of the first kind
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#REDIRECT [[Weierstrass elliptic functions]]
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...ingle-valued composite functions of the form $f(u(z))$ in the case of an [[elliptic integral]]
...owed that its solution led to new transcendental elliptic functions (cf. [[Elliptic function]]).
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$#C+1 = 8 : ~/encyclopedia/old_files/data/L058/L.0508120 Lemniscate functions
...Elliptic function|Elliptic function]]). They arise in the inversion of the elliptic integral of special form
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...lliptic integral]] in Legendre normal form. For example, in the incomplete elliptic integral of the first kind,
...on of the [[Jacobi elliptic functions]], which arise from the inversion of elliptic integrals of the form \eqref{*}.
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...hyper-elliptic curve is $g =(n-1)/2$, so that, for various odd $n$, hyper-elliptic curves are birationally inequivalent.
...zation of hyper-elliptic curves. A further characterization is that hyper-elliptic curves have exactly $2g+2$ [[Weierstrass point]]s.
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One of the three fundamental [[Jacobi elliptic functions|Jacobi elliptic functions]]. It is denoted by
...ed as follows in terms of the Weierstrass sigma-function, the Jacobi theta-functions or a series:
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one obtains elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]), while the cases $ m = 5, 6 $
are sometimes denoted as ultra-elliptic.
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''elliptic cosine''
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], denoted by
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''elliptic sine''
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], written as
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...cal methods]]; [[Elliptic partial differential equation, numerical methods|Elliptic partial differential equation, numerical methods]]; [[Differential equation
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...ure in the boundary conditions of boundary value problems for second-order elliptic equations. The problem is then called a problem with oblique derivative. Se
...ection field $l$ on $S$ has the form $l=(l_1,\ldots,l_n)$, where $l_i$ are functions of the points $P\in S$ such that $\sum_{i=1}^n(l_i)^2=1$, then the oblique
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$#C+1 = 64 : ~/encyclopedia/old_files/data/E035/E.0305470 Elliptic function
...dic function]] that is meromorphic in the finite complex $ z $-plane. An elliptic function has the following basic properties.
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$#C+1 = 102 : ~/encyclopedia/old_files/data/E035/E.0305490 Elliptic integral
...on is possible, then (1) is said to be a [[Pseudo-elliptic integral|pseudo-elliptic integral]].
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A Tate curve is a uniformization of an [[elliptic curve]] having stable bad reduction with the help of a $q$-parametrization.
...l $j$-invariant). In the case of stable bad reduction one can construct an elliptic curve $E_q$ over $K$, which analytically is $K^*/q^{\mathbb{Z}}$ (where $q^
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are integers. Analytic functions of one complex variable with more than two primitive periods do not exist,
...n]]). The generalization of the concept of an elliptic function to include functions $ f ( z _ {1} \dots z _ {n} ) $
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for all smooth functions $ \phi $(
...ons (cf. [[Strong solution|Strong solution]])? For example, in the case of elliptic equations, every weak solution is strong.
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The ratio of the two basic [[Jacobi elliptic functions]]:
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...serve as elliptic functions which generate the algebraic field of elliptic functions with given primitive periods.
...) $ and $ \wp ^ \prime (z) $ generate the algebraic field of elliptic functions with given periods. The simply-periodic trigonometric function which serves
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which is interpreted in the sense of the theory of generalized functions, where $ \delta $
...fficients, and also for arbitrary elliptic equations. For example, for the elliptic equation
3 KB (487 words) - 08:49, 13 May 2022
...rtain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let
be a uniformly elliptic operator in a region $ \Omega _{1} $
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that is, the representation of all possible rational functions of $ w _ {1} $,
...this leads to doubly-periodic elliptic functions (cf. [[Elliptic function|Elliptic function]]). For example, the inversion of an integral of the first kind in
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$#C+1 = 161 : ~/encyclopedia/old_files/data/J054/J.0504050 Jacobi elliptic functions
...m, by N.H. Abel. Jacobi's construction is based on an application of theta-functions (cf. [[Theta-function|Theta-function]]).
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...tic function|Analytic function]]; [[Elliptic partial differential equation|Elliptic partial differential equation]]). To apply the transform, complex (independ
(where $J_0$ is a Bessel function, cf. [[Bessel functions|Bessel functions]]).
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...ends linearly on the parameters in the following way. Let $F_1,F_2,F_3$ be functions of two variables, no one of which is a linear combination of the other two.
...gh a single point. If this point is a finite point, then the net is called elliptic, if it is a point at infinity, then the net is called parabolic.
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''elliptic modular function, of one complex variable''
...e of the general theory of automorphic functions. In the theory of modular functions the following [[Theta-series|theta-series]] are used as basic modular forms
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...a number of cases, for example, for an ordinary differential operator, for elliptic operators and for differential operators with constant coefficients.
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...oids can be used to classify the points on a surface (see [[Elliptic point|Elliptic point]]; [[Hyperbolic point|Hyperbolic point]]; [[Parabolic point|Parabolic
...tives up to and including order 2 of the difference $p(x,y)-s(x,y)$ of the functions $p(x,y)$ and $s(x,y)$ describing the paraboloid and the surface are all zer
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...n by zero remain smooth up to the boundary are taken by these operators to functions that are again smooth up to the boundary. Here the extension by zero is car
...-differential operator has an asymptotic expansion in positive homogeneous functions $ a _ \alpha ( x, \xi ) $ (where $ \alpha $
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The functions $ \phi _ {k} $,
For a linear uniformly-elliptic equation
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...bolic differential equations (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]; [[Parabolic partial differential equation|
continuous functions with $ K _ {1} ( x,y ) > 0 $,
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...etric functions]] or [[Weierstrass elliptic functions|Weierstrass elliptic functions]] and are automorphic; their group is the group of motions of the Euclidean
...up and regular polygons, such Schwarz functions are also called polyhedral functions.
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A series of functions used in the representation of automorphic forms and functions (cf. [[Automorphic form|Automorphic form]]; [[Automorphic function|Automorp
...ped the theory of theta-series in connection with the study of automorphic functions of one complex variable. Let $ \Gamma $
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in which the real-valued functions $ a _ {ij} ( x) $,
equation (1) is called elliptic at the point $ x _ {0} $;
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...n-characteristic interval of parabolic degeneracy. The Tricomi equation is elliptic for $y>0$, hyperbolic for $y<0$ and degenerates to an equation of parabolic
...eam function of a plane-parallel steady-state gas flow, $k(y)$ and $y$ are functions of the velocity of the flow, which are positive at subsonic and negative at
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...nometric functions]] can be described as the class of meromorphic periodic functions with period $ 2 \pi $
...D valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russia
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In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in vari
In the narrow sense, the special functions of mathematical physics, which arise when solving partial differential equa
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A problem involving partial differential equations in which the functions specifying the initial (boundary) conditions are not continuous.
In this case the discontinuities of the initial functions $ \phi _ {0} $
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has a normal, and let the following second-order elliptic equation be given:
are continuous functions defined on $ \Gamma $.
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...the complementary modulus (in [[Jacobi elliptic functions|Jacobi elliptic functions]]) or the modulus of a [[Congruence|congruence]]. Cf. also [[Norm on a fiel
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...–Jacobi and Eisenstein–Weierstrass theory of elliptic and elliptic modular functions between 1820– 1850 (see [[#References|[a2]]]). The function $ k ( 2 ) $
...a function of the quotient of the periods of a certain [[Elliptic integral|elliptic integral]]. It corresponds to the level-two congruence subgroup of the [[Mo
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...xtension of it). The rational functions on a modular curve lift to modular functions (of a higher level) and form a field; the automorphisms of this field have
...es|[8]]]). In particular, there is a hypothesis that each [[Elliptic curve|elliptic curve]] over $ \mathbf Q $
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...eteq D$. In these definitions, $L_1$ can be replaced by the class $L_p$ of functions whose $p$-th powers are locally integrable. The class most often used is $L
In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide.
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...liptic differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) of order $ m $,
With every operator (1) there is associated the homogeneous elliptic operator
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Let a second-order elliptic partial differential equation be given in $ \mathbf R ^ {n} $,
are sufficiently-smooth functions in $ D $
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...postulates a deep connection between elliptic curves (cf. [[Elliptic curve|Elliptic curve]]) over the rational numbers and modular forms (cf. [[Modular form|Mo
...for $L(E,s)$. It is prototypical of a general relationship between the $L$-functions attached to arithmetic objects and those attached to automorphic forms (cf.
7 KB (1,047 words) - 01:52, 18 July 2022
...epresenting the two-dimensional analogue of a [[Hyper-elliptic curve|hyper-elliptic curve]]. A non-singular algebraic projective surface $ X $
is said to be a double plane if its field of rational functions $ k ( X ) $
5 KB (749 words) - 07:33, 22 December 2023
...ace. Spaces of grid functions usually occur in approximating some space of functions of a continuous variable.
be the space of continuous functions given on the interval $ 0 \leq x \leq 1 $
8 KB (1,121 words) - 17:33, 5 June 2020
...acter of a group]]). The $L$-functions form a complicated class of special functions of a complex variable, defined by a
..., etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf.
2 KB (347 words) - 21:23, 9 January 2015
...e cones may depend on $k$). In connection with integral representations of functions and imbedding theorems, anisotropic generalizations of cone conditions have
....V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)</TD></TR
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...a/old_files/data/A110/A.1100350 Addition theorems in the theory of special functions
or an [[Elliptic function|elliptic function]].
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$#C+1 = 122 : ~/encyclopedia/old_files/data/E035/E.0305500 Elliptic operator
...the operators obtained from it when it is written in local coordinates are elliptic. Equivalently, this ellipticity can be described as invertibility of the pr
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one obtains harmonic functions (cf. [[Harmonic function|Harmonic function]]), while for $ m= 2 $
one obtains biharmonic functions (cf. [[Biharmonic function|Biharmonic function]]). Each poly-harmonic funct
5 KB (663 words) - 17:01, 13 January 2024
is a field of algebraic functions of one variable over an algebraically closed field of constants [[#Referenc
The results of these calculations are used in the theory of elliptic surfaces. If $ k $
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...ems for second-order linear elliptic equations. Suppose that the following elliptic equation is given in a bounded $N$-dimensional domain $D$ ($N\geq2$) with b
...]</TD> <TD valign="top"> C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)</TD></TR></table>
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...name { cn } ( u | k )$, $\operatorname { dn } ( u | k )$, the Jacobi theta-functions $\theta _ { i } ( v )$ (cf. also [[Theta-function|Theta-function]]), and th
...gn="top"> R.P. Brent, "Fast multiple-precision evaluation of elementary functions" ''J. Assoc. Comput. Mach.'' , '''23''' (1976) pp. 242–251</td></tr><t
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...to a harmonic function. This theorem can be generalized to solutions of an elliptic equation,
...econd theorem can be generalized to monotone sequences of solutions of the elliptic equation .
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...the theory of second-order linear elliptic equations (subsequently also in elliptic equations of higher order), after which it was applied in the theory of equ
...he [[Dirichlet problem|Dirichlet problem]] for a second-order self-adjoint elliptic equation
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...$n=1$ this condition always holds; the corresponding algebraic curves are elliptic (cf.
[[Elliptic curve|Elliptic curve]]). The period matrix
5 KB (795 words) - 21:47, 5 March 2012
$#C+1 = 129 : ~/encyclopedia/old_files/data/L059/L.0509180 Linear elliptic partial differential equation and system
is the linear elliptic operator
18 KB (2,569 words) - 22:17, 5 June 2020
...ment, having two periods with non-real quotient, such as for example the [[elliptic function]]s.
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where the Jacobi elliptic function $ \mathop{\rm sn} $
cf. [[Jacobi elliptic functions|Jacobi elliptic functions]]).
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...rinciple|maximum-modulus principle]] for analytic functions to the case of functions that are given a priori as unbounded; it was first given in its simplest fo
...e Phragmén–Lindelöf principle. It extends the maximum-modulus principle to functions about the behaviour of which on the boundary only partial information is av
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can be considered as single-valued functions of the points on $ F $.
...ntegral can be expressed in the form of a linear combination of elementary functions and canonical Abelian integrals of the three kinds. The integral $ \int _
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...let, Neumann and others) are posed for the Helmholtz equation, which is of elliptic type, in a bounded domain. A value of $ c $
...the Helmholtz equation are solved by the ordinary methods of the theory of elliptic equations (reduction to an integral equation, variational methods, methods
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...ooth boundaries have been thoroughly investigated in the case of uniformly-elliptic operators $ L $.
...ré problem in the two-dimensional case make extensive use of the theory of functions of a complex variable (see [[Boundary value problem, complex-variable metho
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...and numerical methods for solving boundary value problems for equations of elliptic type (see [[#References|[1]]], [[#References|[2]]]) comprise many numerical
The main numerical methods for equations of elliptic type are: projection-grid methods (finite-element methods) and difference m
9 KB (1,360 words) - 13:02, 13 January 2024
...gn="top"| J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'', '''41''' (1966) pp. 193–291
...yvagin, "The Mordell–Weil groups and the Shafarevich–Tate groups of Weil's elliptic curves" ''Izv. Akad. Nauk. SSSR Ser. Mat.'', '''52''' : 6 (1988)
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...rned with finding the minimum of certain functionals in certain classes of functions. In the narrow sense of the term, the Dirichlet principle reduces the first
in the class of functions satisfying the condition
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...differential equations in Euclidean spaces or on manifolds or by harmonic functions on a Riemann surface. (Cf. also [[Harmonic space|Harmonic space]]; [[Potent
be a [[Sheaf|sheaf]] of vector spaces of real-valued continuous functions. This means that to every non-empty open set $ U \subset X $,
5 KB (700 words) - 18:48, 26 February 2024
A generalization of the concept of an [[Elliptic function|elliptic function]] of one complex variable to the case of several complex variables
is said to be a non-degenerate Abelian function. Degenerate Abelian functions are distinguished by having infinitely small periods, i.e. for any number
11 KB (1,602 words) - 16:08, 1 April 2020
$#C+1 = 211 : ~/encyclopedia/old_files/data/E110/E.1100070 Elliptic genera
The name elliptic genus has been given to various multiplicative [[Cobordism|cobordism]] inva
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...] was the first to give a full exposition of the method of upper and lower functions for this last case.
for a linear, homogeneous, elliptic second-order equation with continuous coefficients,
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An ''elliptic curve'' is a non-singular complete
[[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic
19 KB (3,251 words) - 20:37, 19 September 2017
$#C+1 = 34 : ~/encyclopedia/old_files/data/E035/E.0305530 Elliptic partial differential equation, numerical methods
...e for their solution special methods [[#References|[1]]]. More typical for elliptic equations are boundary value problems, and for their approximate solution m
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A [[Banach space|Banach space]] of bounded continuous functions $ f( x) = f( x ^ {1} \dots x ^ {n} ) $
is an integer, consists of the functions that are $ m $
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A method for solving boundary value problems for linear uniformly-elliptic equations of the second order, based on a priori estimates and the continua
...tion to the [[Dirichlet problem|Dirichlet problem]] for a linear uniformly-elliptic equation
7 KB (1,029 words) - 16:50, 4 January 2021
In the case of second-order elliptic operators the problem with discontinuous coefficients (the transmission or
is a linear differential operator of elliptic type, defined in the domain $ g _ {l} $;
12 KB (1,625 words) - 17:33, 5 June 2020
A linear boundary value problem for elliptic equations of the second order. Let $ D $
and, in addition, let the equation be uniformly elliptic in $ D $.
7 KB (1,083 words) - 17:33, 5 June 2020
...esults were carried over (cf. [[#References|[2]]], [[#References|[3]]]) to elliptic boundary value problems in the exterior of bounded regions in $ \mathbf R
is orthogonal to the eigen functions. A theorem of T. Kato (cf. [[#References|[3]]]) gives sufficient conditions
7 KB (1,032 words) - 19:27, 26 March 2023
...tion of continuous functions by positive operators and abstract degenerate elliptic-parabolic problems have been discovered (see, e.g., [[#References|[a2]]]).
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...on space with weighted norm, weighted class, space with a weight, space of functions with weighted norm''
...ordinary non-weighted normed and semi-normed function spaces consisting of functions with infinite ordinary non-weighted norm (semi-norm). Consider, for example
9 KB (1,435 words) - 08:13, 13 January 2024
are called harmonic functions (cf. [[Harmonic function|Harmonic function]]) in $ D $.
...(cf. [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]) have been and are being developed.
9 KB (1,300 words) - 19:02, 9 January 2024
...113 : ~/encyclopedia/old_files/data/B017/B.0107350 Boundary value problem, elliptic equations
to an elliptic equation
13 KB (1,836 words) - 06:29, 30 May 2020
Elementary relations. The six functions
There exists a linear relationship between that function and any two functions which are contiguous to it. For instance, C.F. Gauss [[#References|[2]]], [
12 KB (1,576 words) - 01:14, 21 January 2022
A superposition of theta-functions (cf. [[Theta-function|Theta-function]]) of the first order $ \Theta _ {H}
...ere defined and analytic. When crossing through sections the Riemann theta-functions, as a rule, are multiplied by factors whose values are determined from the
7 KB (963 words) - 14:55, 7 June 2020
...o use piecewise-linear, semi-linear and other functions as such coordinate functions.
Difference schemes can also be obtained by a special choice of the coordinate functions in the [[Galerkin method|Galerkin method]]. The method in which difference
22 KB (3,256 words) - 17:33, 5 June 2020
...etviashvili equation (cf. [[Soliton|Soliton]]) for Grassmannians and Schur functions [[#References|[a30]]], and the modified Korteweg–de Vries equation for re
...tation. However, it has also more unique features: 5) a link with elliptic functions; 6) a more sophisticated interpretation of the Lax pair, giving a link with
13 KB (1,901 words) - 09:09, 26 March 2023
that are functions of the real variables $ x $
that establishes a one-to-one correspondence between the set of functions $ \Phi ( z) $
14 KB (2,051 words) - 04:01, 4 March 2022
...ms. A fairly complete study was made of boundary value problems for linear elliptic, hyperbolic and parabolic equations of the second order with weak singulari
...>[4]</TD> <TD valign="top"> C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) {{MR|0284700}} {{ZBL|0198
5 KB (714 words) - 17:33, 5 June 2020
then the Monge–Ampère equation is of elliptic type, if $ \Delta < 0 $
...nce of a surface with a given line element whose coefficients are analytic functions.
17 KB (2,601 words) - 08:01, 6 June 2020
Interest in the modular group is related to the study of modular functions (cf. [[Modular function|Modular function]]) whose Riemann surfaces (cf. [[R
...analytically equivalent to a non-singular cubic curve (an [[Elliptic curve|elliptic curve]]). This gives a one-to-one correspondence between the points of the
7 KB (1,031 words) - 18:33, 13 January 2024
...d first square-summable derivatives. If the region is bounded, this set of functions coincides with the [[Sobolev space|Sobolev space]] $ W _ {2} ^ {1} ( G)
...ergence in the mean or in the sense of the limit of the boundary values of functions continuous in the closed region which approximate the given function in the
3 KB (503 words) - 17:53, 6 January 2024
of meromorphic functions on Riemann surfaces $ S $.
is a field of algebraic functions (for $ g= 0 $
14 KB (1,973 words) - 08:11, 6 June 2020
Methods for computing the eigen values and corresponding eigen functions of differential operators. Oscillations of a bounded elastic body are descr
...conditions are called eigen values, and the corresponding solutions eigen functions. The resulting eigen value problem consists of determining the eigen values
15 KB (2,091 words) - 19:37, 5 June 2020
''method of auxiliary functions''
...ations. Bernstein's method consists in introducing certain new (auxiliary) functions, which depend on the solution being sought, and which make it possible to e
10 KB (1,515 words) - 20:30, 29 May 2020
Studies of Riemann surfaces related to the behaviour of different classes of functions on these surfaces.
...nn surfaces real-valued and complex-valued harmonic functions, subharmonic functions, etc. Let $ W $
13 KB (1,964 words) - 08:11, 6 June 2020
...$, where $L ( s , E _ { 15 } )$ is the $L$-function of an [[Elliptic curve|elliptic curve]] of conductor $15$. This formula has not been proved but has been ve
...measure [[#References|[a8]]], in which the torus $\bf T$ is replaced by an elliptic curve. It seems likely that this will have an interpretation as the entropy
7 KB (1,101 words) - 14:50, 27 January 2024
Boundary value problems for elliptic partial differential equations in the finite (interior) domain $D^+$, respe
Consider a boundary value problem for a linear elliptic equation of general form
5 KB (853 words) - 00:31, 24 December 2018
...for example, boundary value problems for partial differential equations of elliptic type.
The latter can be identified with the space of functions given at the nodes of $ \Omega _ {h} $;
6 KB (890 words) - 07:28, 9 February 2024
...1]]]; it arises in the separation of variables for the Laplace equation in elliptic coordinates. Equation \eqref{eq1} is also called the Weierstrass form of th
the Lamé functions are of paramount importance (cf. [[Lamé function]]).
3 KB (443 words) - 18:25, 1 May 2024
as functions of the classes $ C ^ {( 0 , \nu ) } ( \overline{ {D ^ {+} }}\; ) $
...e Laplace equation by an arbitrary Lewy function of a general second-order elliptic operator with variable coefficients of class $ C ^ {( 0 , \lambda ) } $,
7 KB (1,052 words) - 14:55, 7 June 2020
The problem of finding the solution of a second-order elliptic equation which is regular in the domain is also known as the Dirichlet or f
The variational method is based on the fact that of all functions $ u $
17 KB (2,472 words) - 08:26, 21 March 2022
...2$ there always exist at least $2g+2$ Weierstrass points, and only [[hyper-elliptic curve]]s of genus $g$ have exactly $2g+2$ Weierstrass points. The upper bou
..."top">[1]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR>
2 KB (304 words) - 18:09, 22 November 2014
...ial equations involving the representation of solutions by way of analytic functions of a complex variable.
The theory of analytic functions
30 KB (4,345 words) - 09:02, 21 January 2024
...icients in the [[Sheaf|sheaf]] of locally constant, respectively harmonic, functions. Spencer cohomology is a generalization of these two cohomologies for the c
...lso [[Index formulas|Index formulas]]; [[Index theory|Index theory]]). For elliptic Lie equations the index can be expressed in terms of characteristic classes
8 KB (1,101 words) - 17:44, 1 July 2020
...n the theory of many special functions (e.g. [[Cylinder functions|cylinder functions]] of order $ n $),
...s it takes each value in its period parallelogram, cf. [[Elliptic function|Elliptic function]].
13 KB (2,001 words) - 02:12, 1 March 2022
The field of rational functions of an irreducible algebraic curve over $ k $
is the field of algebraic functions in one variable of the form $ k (x,\ y) $ ,
22 KB (3,307 words) - 17:02, 17 December 2019
...-series (because of the original notation). Other representations of theta-functions, for example as infinite products, are also possible.
In many problems it is convenient to take the theta-functions that satisfy the conditions
14 KB (1,941 words) - 05:01, 23 February 2022
...related to the existence of solutions for differential equations of mixed elliptic-hyperbolic type with two independent variables in an open domain $ \Omega
The equation is elliptic in $ \Omega _ {1} $,
7 KB (1,014 words) - 08:26, 6 June 2020
...ular point|essential singularities]] of [[Holomorphic function|holomorphic functions]] of one complex variable
...t]]). Instead there is no direct generalization to the case of holomorphic functions of several complex variables (see {{Cite|Sh}}).
3 KB (465 words) - 20:00, 14 December 2014
...z }$, like in Pólya's work [[#References|[a7]]] on integral-valued entire functions.
...l equation $( d / d z ) e ^ { z } = e ^ { z }$. Gel'fond considers the two functions $e ^ { z }$ and $e ^ { \beta z }$; his auxiliary function has the form $F (
9 KB (1,244 words) - 19:58, 8 February 2024
...special divisor $D$, with equality if $D=0$ or $D=K$ or if $X$ is a hyper-elliptic curve and $D$ is a multiple of the unique special divisor of degree 2 on $X
..."top">[2]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of algebraic functions" , Moscow-Leningrad (1948) (In Russian)</TD></TR>
2 KB (319 words) - 05:40, 9 April 2023
...role in analysis, since meromorphic functions on such tori are meromorphic functions on $ \mathbf C ^ {2} $
such that there are no meromorphic functions at all (except for constants) on the corresponding torus $ \mathbf C ^ {
8 KB (1,169 words) - 18:47, 5 April 2020
A second-order [[Elliptic partial differential equation|elliptic partial differential equation]] in $\mathbf{R} ^ { 2 }$ (if $\epsilon = 1$)
...e investigation of differential invariants for certain families of complex functions by E. Peschl ([[#References|[a8]]], [[#References|[a9]]]) and has been trea
7 KB (955 words) - 10:00, 16 March 2023
the parameters of the state of the gas are defined as functions of the coordinates $ x ^ {1} , x ^ {2} , x ^ {3} $(
...variable, and if the gas-dynamic variables are continuously-differentiable functions, then the equations of a viscous heat-conducting gas have the following for
18 KB (2,523 words) - 19:41, 5 June 2020
...(cf. [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]), these being models of steady-state oscillations of various ph
...) are not universal in the sense that they do not always define a class of functions in which the corresponding boundary value problem has a unique solution. Fo
10 KB (1,441 words) - 14:54, 7 June 2020
''of one complex variable, elliptic modular form''
is analytically isomorphic to the [[Elliptic curve|elliptic curve]] given by the equation
8 KB (1,125 words) - 07:04, 29 March 2024
...ch is an integer, arbitrary complex numbers and more general objects (e.g. functions) may feature.
be an elliptic pseudo-differential operator mapping the space $ C ^ \infty ( M , \mat
41 KB (5,908 words) - 22:12, 5 June 2020
...called the field of rational functions on $L$. It consists of the rational functions $p(x, y)/q(x, y)$, where $q$ is not divisible by $f$, considered up to equa
...respectively; then a rational mapping $F$ is defined by a pair of rational functions $\phi$ and $\psi$ defined on $L$ and such that $g(\phi, \psi)=0$ on $M$. Tw
11 KB (1,916 words) - 00:44, 12 August 2019
...). Under naturally arising boundary conditions, \eqref{1} gives rise to an elliptic boundary value problem. The system \eqref{1} is conveniently called the sys
...aussian curvature $K$ of the mean surface $g$ is positive, \eqref{2} is an elliptic system and under conditions of a complete or partial fixation of the bounda
8 KB (1,200 words) - 17:04, 14 February 2020
...ll as some generalizations of it. The Laplace operator (1) is the simplest elliptic differential operator of the second order. The Laplace operator plays an im
...nner product (5). The restriction of the operator (4) to 0-forms (that is, functions) is specified by (3). On $ p $-forms with an arbitrary integer $ p \geq
11 KB (1,542 words) - 11:52, 25 January 2022
realized by fractional-linear functions (cf. [[Fractional-linear function|Fractional-linear function]]).
is transformed into itself; such a fractional-linear mapping is said to be elliptic and, in normal form (3), is characterized by a multiplier $ \mu $
13 KB (1,875 words) - 13:58, 17 March 2023
...d absolute point (vertex), one distinguishes four types of straight lines: elliptic lines, which intersect the absolute cone at two conjugate complex points; h
...s appearing in the latter must be replaced by the corresponding hyperbolic functions.
7 KB (1,046 words) - 09:03, 13 May 2022
...on spaces. In this case a typical situation is that if an operator acts on functions in $ n $
variables (or more generally, on functions on an $ n $-
13 KB (1,836 words) - 14:55, 7 June 2020
is then also univalent. Multivalent functions (cf. [[Multivalent function|Multivalent function]]), and in particular $
valent functions, are a generalization of univalent functions.
24 KB (3,480 words) - 07:53, 14 January 2024
In the case of complex-valued functions a non-linear partial differential equation is defined similarly. If $ k >
is of elliptic type on solutions with $ \partial u / \partial x _ {2} > 0 $,
30 KB (4,331 words) - 16:42, 20 January 2022
''spatial elliptic coordinates''
3 KB (374 words) - 19:37, 5 June 2020
is the cohomology group with coefficients in the sheaf of germs of continuous functions with values in $ O _ {n} ( \mathbf R ) $ (see [[G-fibration| $ G $-fibra
are the transition functions of $ \xi $,
35 KB (5,194 words) - 09:33, 26 March 2023
Automorphic functions are often defined so as to include only functions defined on a bounded connected domain $ D $ of the $ n $-dimensional c
...study of this field is one of the main tasks in the theory of automorphic functions.
17 KB (2,502 words) - 06:16, 12 July 2022
...context of the studies of N.H. Abel {{Cite|Ab}} on the theory of elliptic functions. Abel's differential equations of the first kind represent a natural
2 KB (338 words) - 22:19, 4 July 2012
...i.e. the $( n \times n )$-matrix of partial derivatives of the coordinate functions $f_j$ of $f = ( f _ { 1 } , \dots , f _ { n } )$. Further,
...ass of $K$-quasi-regular mappings agrees with that of the complex-analytic functions (cf. also [[Analytic function|Analytic function]]). Injective quasi-regular
9 KB (1,309 words) - 17:02, 1 July 2020
are real-valued functions on $ D $.
is positive or negative definite, respectively, equation (3) is called elliptic at the point $ x \in D $.
21 KB (2,842 words) - 02:26, 23 January 2022
...curve endowed with this structure is a one-dimensional Abelian variety (an elliptic curve).
of rational functions of the curve $ X $
10 KB (1,376 words) - 11:12, 26 March 2023
...usly to partial differential equations, variational inequalities can be of elliptic, parabolic, hyperbolic, etc. type.
Then an elliptic variational inequality usually has one of the following two forms:
5 KB (737 words) - 20:35, 18 March 2024
...ynomials that solved this problem, and expressed them in terms of elliptic functions.
...etric polynomials in the uniform metric carry over to Chebychev systems of functions (see [[#References|[2]]]). Concerning the theory of extremal problems and e
5 KB (721 words) - 19:38, 5 June 2020
...obtain a solution to the Dirichlet problem for a differential equation of elliptic type in domains $ D $
...boundary value problems of a more general nature for general equations of elliptic type (including equations of an order greater than two) under certain addit
6 KB (852 words) - 08:12, 6 June 2020
in the class of real-valued functions $ u $
is the Hilbert space of square-integrable functions in $ V $),
21 KB (3,008 words) - 17:33, 5 June 2020
be adjoint linear elliptic operators of the second order in $ \mathbf R ^ {n} $
be sought in the class of functions permitting an integral representation according to Green's formulas. Furthe
7 KB (1,101 words) - 17:33, 5 June 2020
...uous operator|Completely-continuous operator]]) on the space of continuous functions $ C ( \Omega ) $,
...p"> M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) {{MR|}} {{ZBL|0312.47041}} </
4 KB (565 words) - 08:28, 6 June 2020
...Q}$ [[#References|[a9]]], [[#References|[a10]]] (cf. also [[Elliptic curve|Elliptic curve]]; [[Galois cohomology|Galois cohomology]]). The key idea of Kolyvagi
...bjects such as the class group of a number field or the Selmer group of an elliptic curve. On the other hand, an Euler system encodes values of the $L$-functio
19 KB (2,901 words) - 17:41, 25 November 2023
...ained for $p$-adic transcendental numbers (including Engel's theory of $E$-functions). The development of methods of the theory of transcendental numbers has pr
...e $\bar{\mathbf{Q}}$-linear independence of periods and quasi-periods of [[elliptic curve]]s (see [[#References|[b1]]]) and finally, through the work of G. Wü
7 KB (1,017 words) - 17:25, 12 November 2023
If $\dim M = n = -\ord A$ ($M$ a compact [[Riemannian manifold]], $A$ an [[elliptic operator]], $n \in \mathbf{N}$), it coincides with the Dixmier trace, and o
...E. Elizalde, "Complete determination of the singularity structure of zeta functions" ''J. Phys.'' , '''A30''' (1997) pp. 2735 {{MR|1450345}} {{ZBL|0919.58065}}
2 KB (395 words) - 18:13, 25 September 2017
is equivalent to the bilinear identities for the tau-functions
...associated spectral curve) with the $\tau _ { n }$ being essentially theta-functions $\tau _ { n } ( t ) = \tau _ { 0 } ( t + n w )$ where $l w \equiv 0$ in $\o
8 KB (1,147 words) - 17:45, 1 July 2020
Poisson's equation is a basic example of a non-homogeneous equation of elliptic type. The equation was first considered by S. Poisson (1812).
...)$ defines a morphism from the sheaf of local differences of superharmonic functions into a sheaf of measures on $\mathbf R^n$. This remark leads to a treatment
2 KB (375 words) - 06:54, 27 August 2014
...f viscosity solutions and the classical maximum principle for second-order elliptic equations.
9 KB (1,306 words) - 08:28, 6 June 2020
...Rademacher theorem]]; again this property does not hold for general Hölder functions). By the mean value theorem, any function $f:[a,b]\to \mathbb R$ which is e
The statement can generalized to differentiable functions on convex subsets of $\mathbb R^n$.
5 KB (736 words) - 09:19, 17 June 2014
...— [[Cartesian coordinates|Cartesian coordinates]]; [[Elliptic coordinates|elliptic coordinates]]; [[Parabolic coordinates|parabolic coordinates]]; and [[Polar
An orthogonal system of functions is a finite or countable system of functions $ \{ \phi _ {i} \} $
11 KB (1,597 words) - 08:58, 4 March 2022
...ndent spatial variables (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]). They were proposed by T. von Kármán [[#
...erator (cf. [[Monge–Ampère equation|Monge–Ampère equation]]) acting on the functions $ u $
8 KB (1,209 words) - 08:28, 6 June 2020
...xtension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev (see [[#References|[1]]], [[#R
Let $f$ and $\phi$ be locally integrable functions on an open set $\Omega\subset \mathbb R^n$, that is, Lebesgue integrable on
6 KB (871 words) - 10:49, 30 November 2012
...differential expression, and acting on a space of (usually vector-valued) functions (or sections of a differentiable vector bundle) on differentiable manifolds
...aturally defined by a differential expression on the set of differentiable functions (or sections of a vector bundle), the restrictions of which to $ \partial
17 KB (2,519 words) - 19:35, 5 June 2020
...ors on various function spaces, especially on Hilbert spaces of measurable functions.
are functions defined in $ \Omega _ {n} $
32 KB (4,393 words) - 08:22, 6 June 2020
...concept of a Fuchsian group provided a basis for the theory of automorphic functions created by Poincaré and F. Klein.
fixed. It is elliptic if $ k _ {i} < \infty $,
14 KB (2,109 words) - 20:03, 15 March 2023
...ptic curves (Abelian varieties of dimension one, cf. also [[Elliptic curve|Elliptic curve]]) with $ \lambda $
is very ample if and only if there is no elliptic curve $ E $
18 KB (2,511 words) - 06:25, 26 March 2023
...\infty } ( \Omega ) \subset L _ { 2 } ( \Omega )$ of all infinitely smooth functions with compact support in $\Omega$. This is a symmetric operator, and the ass
...Laplacian and operators corresponding to other boundary value problems for elliptic differential operators.
4 KB (579 words) - 19:34, 7 February 2024
is a system of meromorphic functions in a domain $ D \subset \mathbf C ^ {N} $ (respectively, $ D \subset
One may thus speak of uniformization by multi-valued analytic functions $ w = F ( z): \mathbf C ^ {n} \rightarrow \mathbf C ^ {m} $,
14 KB (2,068 words) - 02:08, 18 July 2022
...ations in which one uses representations of solutions in terms of analytic functions of a complex variable.
Given a second-order elliptic equation
11 KB (1,622 words) - 06:29, 30 May 2020
...lued functions in which three fundamental properties of classical harmonic functions (cf. [[Harmonic function|Harmonic function]]) are axiomatically fixed. Thes
The functions in $ \mathfrak H $
7 KB (1,087 words) - 19:43, 5 June 2020
...></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> L. Nirenberg, "On elliptic partial differential equations" ''Ann. Scuola Norm. Sup. Pisa Ser. 3'' , '
6 KB (825 words) - 19:40, 5 June 2020
taking an oval surface of the second order (an elliptic quadric) into itself, that is, the group of hyperbolic motions of the three
be the elliptic quadric in $ P _ {3} $
14 KB (2,020 words) - 12:57, 13 January 2024
An operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rul
be the space of infinitely-differentiable functions on $ \Omega $
29 KB (4,077 words) - 04:02, 4 January 2022
defined on functions on a domain $ M \subset \mathbf C ^{n+1} $
The holomorphic functions are the solutions of $ \overline \partial \; f = 0 $
5 KB (649 words) - 10:21, 22 December 2019
...Mathieu [[#References|[1]]] in the investigation of the oscillations of an elliptic membrane; it is a particular case of a [[Hill equation|Hill equation]].
...nes). On the boundary of these zones (the case excluded in (*)) one of the functions of the fundamental system of solutions is either $ \pi $-
5 KB (803 words) - 07:59, 6 June 2020
the Chern classes are mapped into the elementary symmetric functions, and the complete Chern class is mapped to the polynomial $ \prod _{i=1}
...="top">[5e]</TD> <TD valign="top"> M.F. Atiyah, I.M. Singer, "The index of elliptic operators V" ''Ann. of Math. (2)'' , '''93''' (1971) pp. 139–149 {{MR|027
13 KB (1,910 words) - 19:40, 7 January 2024
functions $\alpha(t) $ and $\beta(\tau)$. The infinitesimal generator of this group i
where $\xi(t)$, $\eta(\tau)$ are arbitrary functions and $\xi'(t)$, $\eta'(\tau)$ are their
3 KB (518 words) - 12:51, 30 March 2024
...nction]]). These are studied much from the same perspective as subharmonic functions (cf. also [[Subharmonic function|Subharmonic function]]) are studied in pot
...subset \mathbf{C} ^ { x }$ by $\operatorname{PSH} ( D )$. Plurisubharmonic functions can be defined on domains in complex manifolds via local coordinates (cf. a
17 KB (2,468 words) - 13:06, 10 February 2024
...of positively enumerated points on their boundaries. Then for any strongly-elliptic system of equations
with uniformly continuous partial derivatives of the functions that specify the equations of the characteristics, there is always a unique
7 KB (1,012 words) - 18:58, 26 March 2023
Modifications of the formula are valid for functions mapping the upper half-plane — or the interior and exterior of the unit d
...u.D. Maksimov, "Extension of the structural formula for convex univalent functions to a multiply connected circular region" ''Soviet Math. Dokl.'' , '''2'''
5 KB (715 words) - 16:44, 4 June 2020
where the $Q_i$ are $\operatorname{sl} _ { 2 }$-valued functions depending on the variables $x$, $t = ( t _ { n } )$, and $Q _ { 0 } = P _ {
...L$. If this last line bundle has a global holomorphic section, then $\tau$-functions measure the failure of equivariance of partial liftings from $L$ to $\hat{L
15 KB (2,265 words) - 17:00, 1 July 2020
...seudo-differential operators and a class of linear operators acting on the functions $\hat { f } ( \alpha , p )$.
...l of $B$, $\alpha : = \xi / | \xi |$, $t = | \xi |$. If the symbol is hypo-elliptic, that is, $c _ { 1 } | \xi | ^ { m _ { 1 } } \leq | b | \leq c _ { 2 } | \x
8 KB (1,283 words) - 20:10, 10 January 2021
theory and the index of elliptic operators), in the theory of complex spaces, in the theory of categories (t
...ning was extended to include functions and curves (cf. [[Elliptic integral|Elliptic integral]]).
29 KB (4,414 words) - 17:20, 17 December 2019
an algebraic function can be expressed as square and cube roots of rational functions in the variables $ x _ {1}, \dots, x _ {n} $;
...aic-geometrical approach algebraic functions are considered to be rational functions on an algebraic variety, and are studied by methods of algebraic geometry (
20 KB (3,036 words) - 07:17, 15 June 2022
...also [[Boundary value problem, elliptic equations|Boundary value problem, elliptic equations]]).
...">[a1]</TD> <TD valign="top"> W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , '''1''' , Acad. Press (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <T
6 KB (1,043 words) - 14:24, 14 February 2020
A partial differential equation which is of varying type (elliptic, hyperbolic or parabolic) in its domain of definition. A linear (or quasi-l
then equation (1) is a degenerate equation of elliptic-parabolic $ ( \Delta \geq 0 ) $
19 KB (2,773 words) - 06:19, 24 February 2022
...[8]]]), and for problems similar to the problems of calculating elementary functions and finding discrete Fourier transforms (see [[#References|[1]]], [[#Refere
...boundary value problems for equations and systems of equations of strongly-elliptic type, has a special theoretical and applied significance, and, usually, is
20 KB (2,838 words) - 19:21, 13 January 2024
...<TD valign="top">[14]</TD> <TD valign="top"> I.V. Skrypnik, "Non-linear elliptic equations of higher order" , Kiev (1973) (In Russian)</TD></TR><TR><TD va
..., "Equivalence, linearization, and decomposition of holomorphic operator functions" ''J. Funct. Anal.'' , '''28''' (1978) pp. 102–144</TD></TR></table>
9 KB (1,320 words) - 22:17, 5 June 2020
..., \infty )$, where $B ( D _ { A } ( \alpha , \infty ) )$ is the set of all functions $f \in L ^ { \infty } ( 0 , T ; X )$ such that $f ( t ) \in D _ { A ( t ) }
...operator of order $2 m$ (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]), $\{ B _ { j } ( t , x , D _ { x } ) \} _
14 KB (2,104 words) - 18:47, 26 January 2024
that acts on $ k $-valued functions ( $ k = \mathbf R $
are functions with values in the same field, called the coefficients of $ A $.
25 KB (3,768 words) - 09:07, 14 June 2022
...pedia/old_files/data/P071/P.0701550 Parametric representation of univalent functions
A representation of univalent functions (cf. [[Univalent function|Univalent function]]) that effect a conformal map
8 KB (1,168 words) - 20:13, 12 January 2024
...ally investigated by S.N. Bernshtein, who showed that it is a quasi-linear elliptic equation of order $p=2$ and that, consequently, its solutions have a number
...valign="top">[a1]</TD> <TD valign="top"> E. Giusti, "Minimal surfaces and functions of bounded variation" , Birkhäuser (1984) {{MR|0775682}} {{ZBL|0545.49018}
3 KB (501 words) - 18:46, 13 November 2014
Since the 1960s, the need to use spaces of functions with generalized derivatives in Orlicz spaces came from various application
...Sobolev space $W ^ { k } L _ { \Phi } ( \Omega )$ is the set of measurable functions $f$ on $\Omega$ with generalized (weak) derivatives $D ^ { \alpha } f$ up t
11 KB (1,684 words) - 19:25, 12 February 2024
are functions that realise a conformal mapping of $ D $
...rent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian)</TD></TR></table>
6 KB (873 words) - 22:16, 5 June 2020
times continuously-differentiable complex-valued functions on $ I = ( \alpha , \beta ) $,
are sufficiently-smooth real-valued functions and $ i ^ {2} = - 1 $.
7 KB (871 words) - 08:13, 6 June 2020
...ts of algebraic independence on values of exponential, elliptic or Abelian functions; more generally it applies to the arithmetic study of commutative algebraic
5 KB (726 words) - 09:23, 20 December 2014
...in Sobolev spaces (cf. also [[Approximation of functions|Approximation of functions]]; [[Sobolev space|Sobolev space]]) by algebraic polynomials. The general f
...s application to the $L_{2}$-error of the finite element approximations of elliptic problems of second order leads to sharp estimates with respect to both the
11 KB (1,716 words) - 17:52, 3 January 2021
...blished a link between the problem of the number of solutions of the hyper-elliptic congruences $ y ^ {2} \equiv f( x) $(
functions of algebraic function fields with a finite field of constants, which were i
7 KB (1,033 words) - 17:46, 4 June 2020
[[Elliptic curve|elliptic curve]] (or more generally, of an irreducible
...shares many features. Among the similarities between Drinfel'd modules and elliptic curves are the respective structures of torsion points, of Tate modules and
19 KB (3,204 words) - 20:11, 14 April 2012
The functions $ h $
are arbitrary functions of $ x $
5 KB (738 words) - 10:08, 4 June 2020
Examples include the family of elliptic modular curves (cf. [[Modular curve|Modular curve]]), the family of Hilbert
functions [[#References|[a3]]].
6 KB (890 words) - 19:42, 20 February 2021
...ramming|Mathematical programming]]; [[Penalty functions, method of|Penalty functions, method of]]), various renormalization schemes (cf. [[Renormalization|Renor
...ng if it takes (extends to an operator that takes) distributions to smooth functions. Given a pseudo-differential operator $ P $,
6 KB (789 words) - 17:57, 13 January 2024
...e boundary in the plane, extensive use is made of methods of the theory of functions of a complex variable.
...gn="top"> V.N. Monakhov, "Boundary-value problems with free boundaries for elliptic systems of equations" , Amer. Math. Soc. (1983) (Translated from Russian) {
8 KB (1,055 words) - 17:33, 5 June 2020
...p">[4]</TD> <TD valign="top"> W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , '''1''' , Acad. Press (1976)</TD></TR></table>
is a fundamental solution. Riesz potentials are used in the theory of elliptic differential equations of order $ > 2 $,
6 KB (860 words) - 06:02, 12 July 2022
...Hausdorff space $X$ which tend towards zero at infinity (i.e. continuous functions $f$ on $X$ such that, for any $\epsilon > 0$, the set of points $x \in X$
...gebra with a continuous trace may be represented as the algebra of vector functions on its spectrum $\hat{A}$
15 KB (2,316 words) - 21:05, 20 April 2012
are analytic functions at $ O $
is: a) elliptic, b) hyperbolic or c) parabolic, according to whether the bundles $ H _ {i
8 KB (1,224 words) - 01:42, 23 June 2022
...prescribed points (cf. also [[Approximation of functions|Approximation of functions]]; [[Interpolation|Interpolation]]). In the $n$-dimensional real space $\ma
Examples of radial basis functions are the multi-quadric function $\phi(r)=\sqrt{r^2+c^2}$, $c$ a positive par
8 KB (1,246 words) - 15:13, 18 February 2013
is not one of the fixed points of the elliptic elements of $ \Gamma $,
is the identity mapping, and for each elliptic fixed point, $ \Gamma _ {z} $
13 KB (1,863 words) - 19:27, 17 January 2024
...ral boundary value problems for both equations and systems of equations of elliptic type as well as for various non-stationary systems. The basic numerical met
tend to be chosen so that the functions $ \psi _ {i} ( x) $
13 KB (1,891 words) - 13:06, 13 January 2024
...polynomials. The third and fourth problems of Zolotarev deal with rational functions.
...tan } ^ { 2 } ( \pi / 2 n )$ the Zolotarev polynomial is given in terms of elliptic integrals, see e.g. [[#References|[a1]]], [[#References|[a4]]], [[#Referenc
7 KB (974 words) - 10:10, 11 November 2023
...relot (1957–1958, see [[#References|[1]]]), but it was concerned only with elliptic equations. The extension of the theory to a wide class of parabolic equatio
be a locally compact topological space. A sheaf of functions on $ X $
23 KB (3,517 words) - 08:07, 6 June 2020
denotes the sheaf of holomorphic functions (respectively, holomorphic $ i $-
functions will take their values in $ E \otimes \mathbf C $
25 KB (3,559 words) - 19:33, 7 February 2024
...(from a mathematical point of view) to the computation of determinants of elliptic pseudo-differential operators ($\Psi$DOs, cf. also
...El}}. However, some of these formulas are restricted to very specific zeta-functions, and it often turned out that for some physically important cases the corre
5 KB (680 words) - 22:03, 29 December 2015
outside which there do not exist meromorphic transcendental functions omitting four values , , .
...nd direction is related to generalizations of Picard's theorem to analytic functions $ f( z) $
8 KB (1,225 words) - 08:06, 6 June 2020
...een stimulated to a large extent by problems in the theory of quasi-linear elliptic and parabolic equations. For example, boundary value problems for quasi-lin
...6]</TD> <TD valign="top"> B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated fr
4 KB (605 words) - 17:10, 14 February 2020
the non-zero Fourier coefficients are essentially divisor functions
...converse theorems. The result is that many of the important zeta- and $L$-functions of number theory correspond to modular forms.
13 KB (1,907 words) - 07:36, 22 March 2023
...r sufficiently smooth data (the coefficients in (1) and (3) and the vector functions $ f $,
is a co-normal of the elliptic operator $ L $.
20 KB (2,933 words) - 08:01, 6 June 2020
...the spaces of smooth (i.e. $C ^ { \infty }$-) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first moti
...y class $G ^ { S } ( \Omega )$ (of index $s$) is defined as the set of all functions $f \in C ^ { \infty } ( \Omega )$ such that for every compact subset $K \su
21 KB (3,112 words) - 08:01, 6 February 2024
...equations — not only with numerical, but also with coefficients which are functions — only began in the 1950s. The development of algebraic geometry was a de
is infinite and that there exists a parametrization of it by rational functions in some variable with values from the field $ K $[[#References|[7]]], [[#
24 KB (3,602 words) - 11:48, 26 March 2023
...eal class groups. These methods, which use Gauss sums (cyclotomic units or elliptic units) satisfying properties known as the Euler system, have given elementa
...gn="top">[a22]</td> <td valign="top"> K. Rubin, "Euler systems and modular elliptic curves" , ''Galois Representations in Arithmetic Algebraic Geometry (Durham
19 KB (2,876 words) - 05:38, 15 February 2024
be the subspace consisting of all functions with absolutely-continuous quasi-derivatives of orders $ 0 \dots 2n - 1 $
consisting of the functions whose supports do not contain the end points of the intervals. Let $ T $
20 KB (2,924 words) - 19:38, 5 June 2020
...d,r}^M$ can also be obtained (cf. [[#References|[a6]]]) as the correlation functions of twisted supersymmetric topological field theory.
Even though $S$ contains a metric, its correlation functions are independent of the metric $g$, so that $S$ can still be regarded as a t
16 KB (2,663 words) - 10:57, 13 February 2024
...a vector bundle can be regarded as a module over the algebra of continuous functions, which turns out to be a [[Projective module|projective module]].
...operator]], of their symbols, of a [[Sobolev space|Sobolev space]], and of elliptic operators and their indices. Also, the Atiyah–Singer formula
17 KB (2,459 words) - 07:32, 26 February 2022
a solution of an elliptic nonlinear partial differential equation (see [[Bernstein problem in differe
as an analog of the classical fact that harmonic functions with a polynomial growth must be polynomials.
4 KB (578 words) - 07:35, 3 November 2013
...en functions. Let the problem (1), (2) be well-posed in a certain class of functions (that is, its solution $ u $
are given grid functions and $ L _ {h} $,
26 KB (3,757 words) - 17:33, 5 June 2020
are the eigen functions of $ L $
of eigen functions of $ L $
33 KB (4,788 words) - 08:53, 13 May 2022
...ept of a Riemann surface arose in connection with the studies of algebraic functions $ w = f( z) $
...vers a clear understanding of multiple-valuedness, characteristic of these functions $ w = f( z) $,
34 KB (4,972 words) - 13:13, 6 January 2022
Consider an abstract set $E$ and a linear set $F$ of functions $f : E \rightarrow \mathbf{C}$.
Consider the Hilbert space $H$ of analytic functions (cf. [[Analytic function|Analytic function]]) in a bounded [[Simply-connect
11 KB (1,680 words) - 17:31, 4 February 2024
...e a family of (special) functions which provide the spectral resolution of elliptic differential operators with periodic coefficients. In the mathematics liter
...functions in a generalized sense, and more precisely, they are families of functions which depend on a vector parameter, say $\eta \in \mathbf{R} ^ { N }$, that
16 KB (2,451 words) - 17:45, 1 July 2020
maps the class of continuous functions that vanish at infinity into itself, then $ P ( t , x , B ) $
coincides on smooth functions with the infinitesimal operator of the Markov process (see [[Transition-ope
27 KB (3,884 words) - 08:38, 21 January 2024
...of application, if one can establish inequalities of the form (a4) for an elliptic partial [[Differential operator|differential operator]] $ H $
...ed exposition of this method of obtaining pointwise heat kernel bounds for elliptic partial differential operators.
12 KB (1,715 words) - 22:11, 5 June 2020
...of different fields of elliptic functions (or of their models — isomorphic elliptic curves over $ \mathbf C $ ),
16 KB (2,402 words) - 11:49, 16 December 2019
...ing the [[Resolvent|resolvent]], which makes use of the theory of analytic functions, of asymptotic expansions, etc.
...f generalized eigen functions and associated functions of non-self-adjoint elliptic operators (see [[#References|[15]]], [[#References|[16]]]).
35 KB (5,059 words) - 04:12, 9 May 2022
...l \Omega$. Also, $f ( x , t )$, $u_{0} ( x )$, $u _ { 1 } ( x )$ are given functions. The function $u ( x , t )$ is the unknown function.
...S ( . ) u \in C ^ { 2 } ( \mathbf{R} ; X ) \right\}$. The theory of cosine functions, which is very similar to the theory of semi-groups, was originated by S. K
10 KB (1,449 words) - 17:45, 1 July 2020
is finite-dimensional, being the kernel of the elliptic operator $ \Delta $.
...dge theorem]] that this mapping is an isomorphism. In particular, harmonic functions, i.e. harmonic forms of degree zero, are constant on a connected compact ma
8 KB (1,166 words) - 19:43, 5 June 2020
consists of all "holomorphic functions" on the polydisc $ \{ {( z _ {1} \dots z _ {n} ) \in K } : {\textrm{ all
This interpretation is useful for finding the holomorphic functions on more complicated spaces like Drinfel'd's symmetric spaces $ \Omega ^
8 KB (1,161 words) - 08:25, 6 June 2020
are functions on the ambient manifold. Cf. also [[Linear system|Linear system]]. The noti
are real and distinct, the pencil of circles is said to be elliptic (Fig. a), if they coincide (and are real) it is called parabolic (Fig. b) a
20 KB (3,056 words) - 08:25, 6 June 2020
...ction|Bregman function]]. It follows easily from the properties of Bregman functions that $ D _ {f} ( x,y ) \geq 0 $
is the square of an elliptic norm. A basic property of Bregman distances, which follows easily from the
6 KB (910 words) - 06:29, 30 May 2020
The general construction of functions analogous to the $ {\mathcal E} $-
They are known as quasi-regular and regular or elliptic, respectively. For such functionals the [[Legendre condition|Legendre condi
7 KB (975 words) - 22:37, 28 January 2020
and transition functions, i.e. analytic mappings $$
The transition functions form a one-dimensional cocycle with values in the sheaf $ {\mathcal O} ^{
7 KB (1,096 words) - 09:58, 20 December 2019
...ion was expressed by I.M. Krichever [[#References|[a5]]] in terms of theta-functions on the [[Jacobi variety|Jacobi variety]] of a complex curve (cf. also [[Com
...tion is the first non-trivial member. This hierarchy can be formulated for functions of infinitely many variables $\mathbf t = ( t _ { j } )$ as the identity fo
8 KB (1,099 words) - 08:01, 20 February 2023
...straight lines, a generalization of it with the introduction of smoothened functions was given in [[#References|[2]]], while in [[#References|[3]]] the method w
are given functions in the independent variables $ x $,
11 KB (1,704 words) - 22:12, 5 June 2020
...ted in its development to the theory of analytic, harmonic and subharmonic functions and to probability theory.
...e single-layer potential and the double-layer potential, respectively. The functions $ \rho ( y) $,
38 KB (5,597 words) - 10:29, 30 January 2022
...al formulation of Thom's result is that a generic four-parameter family of functions is stable, and in the neighbourhood of a critical point it behaves, up to s
</td> <td colname="6" style="background-color:white;" colspan="1">Elliptic umbilic</td> </tr> <tr> <td colname="1" style="background-color:white;" col
11 KB (1,481 words) - 18:07, 1 June 2023
This is an [[Elliptic partial differential equation|elliptic partial differential equation]]. For $ p= 1 $
are given non-negative functions that do not vanish simultaneously, $ \mathbf n $
27 KB (3,927 words) - 19:13, 11 January 2024
...functions defined on a subset of a function space. This subset consists of functions defined on a certain domain. An element of such a subset is called an admis
...emal problem and which are not explicitly imposed on the set of admissible functions are called natural (e.g. Neumann-type).
15 KB (2,260 words) - 08:17, 9 January 2024
as functions of the time $ t $
flows, it is elliptic. In the latter case, if one assumes that $ S $
15 KB (2,234 words) - 19:59, 4 January 2024
...[#References|[a3]]] (cf. also [[Functions of a complex variable, theory of|Functions of a complex variable, theory of]]) and the [[Fourier transform|Fourier tra
...ons are known at the moment (1988) only for the case of a flat circular or elliptic crack in an elastic space [[#References|[a4]]]. A complete closed form solu
10 KB (1,410 words) - 19:39, 5 June 2020
Formulas of the integral calculus of functions of several variables, connecting the values of an $ n $-
...ry and partial differential operators) of the second or higher orders. For functions $ u $,
13 KB (1,868 words) - 19:42, 5 June 2020
are sufficiently regular functions of a parameter $ t $.
are differentiable functions. Otherwise it is called singular (cf. [[Singular point|Singular point]]). I
33 KB (5,039 words) - 11:46, 26 March 2023
...]]]). It is applicable to non-linear equations and systems of equations of elliptic [[#References|[4]]], hyperbolic [[#References|[5]]] and parabolic [[#Refere
...nodes of a Chebyshev polynomial or a uniform polynomial. The gas-dynamical functions are piecewise linearly approximated along each strip or are approximated by
6 KB (913 words) - 16:58, 7 February 2011
One can look at Fermat's equation in algebraic integers, entire functions, matrices, etc. There is a generalization of Fermat's theorem for equations
...Taniyama conjecture in the theory of elliptic curves (cf. [[Elliptic curve|Elliptic curve]]).
16 KB (2,596 words) - 09:27, 13 February 2024
...cal methods]]; [[Elliptic partial differential equation, numerical methods|Elliptic partial differential equation, numerical methods]]). Here the main subject
13 KB (1,892 words) - 20:11, 31 December 2018
be functions defined on $ S $
...this case the Cauchy problem has been well studied. For example, when the functions $ a _ \nu , f $
14 KB (1,997 words) - 02:16, 15 June 2022
denotes asymptotic equality. All vector and matrix functions which appear in equations and boundary conditions are assumed to be smooth
...the solution of a system of ordinary differential equations. If the vector functions $ f $
35 KB (4,975 words) - 14:55, 7 June 2020
...$A ( K )$ if there exists a neighbourhood $U \supset K$ such that all the functions $f _ { m } ,\, f \in A ( U )$ and $\{ f_{m} \}$ converges to $f$ in $A ( U
...main problems in the concrete functional analysis of spaces of holomorphic functions.
17 KB (2,619 words) - 17:44, 1 July 2020
are abstract functions with values in $ [ X] $,
...th the known functions in one or several variables by using composition of functions. For example, let $ \phi _ {i} ( x) $,
19 KB (2,803 words) - 19:40, 5 June 2020
is the dimension of the space of functions $ f \in k( x) $
are symmetric functions in the $ a _ {i} $
10 KB (1,385 words) - 03:10, 2 March 2022
...der the sole assumption that the normal and tangential stresses are linear functions of the strain rates.
...roblems for viscous incompressible liquids. Among these are flow around an elliptic or circular cylinder (including a rotating cylinder), around a plate of fin
23 KB (3,271 words) - 08:02, 6 June 2020