Elements of the Theory of Elliptic
                    Functions

Akhiezer, N. I.

doi:10.1090/mmono/079

Cited by 470 publications

(581 citation statements)

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“…For N = 3, 4, 5, we obtain N * = 3, 5, 5. For (N 0 , N 1 ) = (2, 1), (3,1), (3,2) we have equality in our estimate for A 0 (N 0 , N 1 ). We obtain A 0 (2, 1) = A 0 and…”

Section: 2)mentioning

confidence: 52%

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Gol’dberg’s constants

Bergweiler

Erëmenko

2013

JAMA

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Abstract. We study two extremal problems of geometric function theory introduced by A. A. Goldberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them and make a conjecture. Goldberg's problems have important applications to control theory. IntroductionGoldberg [16] studied a class of extremal problems for meromorphic functions. Let F 0 be the set of all holomorphic functions f defined in the rings {z : ρ(f ) < |z| < 1}, omitting 0 and 1, and such that the indices of the curve f ({z : |z| = ρ(f )}) with respect to 0 and 1 are non-zero and distinct.Let F 1 ⊂ F 0 be the subclass consisting of functions meromorphic in the unit disk U. Functions in F 1 can be described as meromorphic functions in U with the property that the numbers of preimages of 0, 1 and ∞, counted with multiplicities, are all finite and pairwise distinct.Let F 2 , F 3 , F 4 be the subclasses of F 1 consisting of functions holomorphic in the unit disk, rational functions and polynomials, respectively. For f in any of these classes F j , 1 ≤ j ≤ 4, we define ρ(f ) as ρ(f ) = sup{|z| : f (z) ∈ {0, 1, ∞}}. Goldberg's constants areGoldberg credits the problem of minimizing ρ(f ) to E. A. Gorin. He proved thatand showed that there exist extremal functions for A 0 and A 2 , but extremal functions for A 1 , A 3 or A 4 do not exist. He also proved the estimates A 0 < 0.0091 and 0.0000038 < A 2 < 0.0319.

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“…For N = 3, 4, 5, we obtain N * = 3, 5, 5. For (N 0 , N 1 ) = (2, 1), (3,1), (3,2) we have equality in our estimate for A 0 (N 0 , N 1 ). We obtain A 0 (2, 1) = A 0 and…”

Section: 2)mentioning

confidence: 52%

“…For the modular function Λ, explicit expressions are known (see, for example, [3,19]) and the constant µ in (7.4) has been computed in the previous section.…”

Section: Computation Of the Extremal Function Hmentioning

confidence: 99%

Gol’dberg’s constants

Bergweiler

Erëmenko

2013

JAMA

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“…However, such an integrability is only local because (3.14) is known [11] to have no entire solution over R 2 , although our problem requires that the solution be of period β in its t variable. In the doubly periodic case, the solutions to the Liouville equation are considered by Olesen [12,22] in the context of non-relativistic Chern-Simons vortices and electroweak vortices over periodic lattices where one needs to use the elliptic functions [23][24][25] of Weierstrass as the holomorphic functions representing solutions of (3.14) are periodic. In our situation here, complication comes from both the periodicity of the solution v of (3.14) in the t variable and unboundedness of v in the r variable as r → 0 and r → ∞, respectively, as a consequence of the form of the background function Ξ given in (2.3).…”

Section: Elliptic Governing Equationmentioning

confidence: 99%

Existence of hyperbolic calorons

Sibner

Yang

2015

Proc. R. Soc. A.

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Recent work of Harland shows that the SO(3)-symmetric, dimensionally reduced, charge-N selfdual Yang-Mills calorons on the hyperbolic space H 3 × S 1 may be obtained through constructing N-vortex solutions of an Abelian Higgs model as in the study of Witten on multiple instantons. In this paper, we establish the existence of such minimal action charge-N calorons by constructing arbitrarily prescribed N-vortex solutions of the Witten type equations.

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“…For a review of elliptic functions one can consult for example Ref. [25]. It is known that the number of poles of an elliptic function in a period parallelogram, counting multiplicity, cannot be less than two.…”

Section: Metric and Form-fieldsmentioning

confidence: 99%

“…(3.4) in terms of elliptic functions. The general theory of elliptic function states that every elliptic function can be expressed in terms of the P and its derivative P ′ in the form [25] f (β) = R 1 (P) + R 2 (P)P ′ , (3.14)…”

Section: Metric and Form-fieldsmentioning

confidence: 99%

String theory dual of the β-deformed gauge theory

Chu

Khoze

2006

J. High Energy Phys.

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We consider the AdS/CFT correspondence between the β-deformed supersymmetric gauge theory and the type IIB string theory on the Lunin-Maldacena background. Guided by gauge theory results, we modify and extend the supergravity solution of Lunin and Maldacena in two ways. First we make it to be doubly periodic in the deformation parameter, β → β + 1 and β → β + τ 0 , to match the β-periodicity property of the dual gauge theory. Secondly, we reconcile the SL(2, Z) symmetry of the gauge theory, which acts on the constant parameters τ 0 and β, with the SL(2, Z) invariance of the string theory, which involves the dilaton-axion field τ (x). Our proposed modified configuration transforms correctly under the SL(2, Z) of string theory when its parameters are transformed under the SL(2, Z) of the gauge theory. We interpret the resulting configuration as the string theory (rather than supergravity) background which is dual to the β-deformed conformal Yang-Mills. Finally, we check that our string theory background leads to the IIB effective action which is correctly reproduced by instanton calculations on the gauge theory side, carried out at weak coupling, in the large-N limit, but to all orders in the deformation parameter β.

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Elements of the Theory of Elliptic Functions

Cited by 470 publications

References 0 publications

Gol’dberg’s constants

Gol’dberg’s constants

Existence of hyperbolic calorons

String theory dual of the β-deformed gauge theory

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