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
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...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.
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...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 ) $
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...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 $
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...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.
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...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
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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
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...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
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...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 $,
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A generalization of the concept of an [[Elliptic function|elliptic function]] of one complex variable to the case of several complex variables
is said to be a non-degenerate Abelian function. Degenerate Abelian functions are distinguished by having infinitely small periods, i.e. for any number
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$#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
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$#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
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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} $;
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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 $.
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...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
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...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.
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...113 : ~/encyclopedia/old_files/data/B017/B.0107350 Boundary value problem, elliptic equations
to an elliptic equation
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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
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...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
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...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
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that are functions of the real variables $ x $
that establishes a one-to-one correspondence between the set of functions $ \Phi ( z) $
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...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
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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.
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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
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...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
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of meromorphic functions on Riemann surfaces $ S $.
is a field of algebraic functions (for $ g= 0 $
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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
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''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
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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 $
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