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Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism Open
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalization of multiple polylogarithms, we construct our set of integration kernels ensuring t…
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Elliptic Feynman integrals and pure functions Open
A bstract We propose a variant of elliptic multiple polylogarithms that have at most logarithmic singularities in all variables and satisfy a differential equation without homogeneous term. We investigate several non-trivial elliptic two-l…
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Convexity and concavity of the complete elliptic integrals with respect to Lehmer mean Open
In this paper, the authors present necessary and sufficient conditions for the complete elliptic integrals of the first and second kind to be convex or concave with respect to the Lehmer mean.
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Monotonicity properties and bounds for the complete p-elliptic integrals Open
We generalize several monotonicity and convexity properties as well as sharp inequalities for the complete elliptic integrals to the complete p-elliptic integrals.
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Nonuniformly elliptic Schauder theory Open
Local Schauder theory holds in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic problems are locally Hölder continuous if so are their coefficients.
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Infinitely many solutions for a class of fourth-order partially sublinear elliptic problem Open
In this paper, we study the existence of infinitely many solutions for a class of fourth-order partially sublinear elliptic problem with Navier boundary value condition by using an extension of Clark's theorem.
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Construction of exotical soliton-like for a fractional nonlinear electrical circuit equation using differential-difference Jacobi elliptic functions sub-equation method Open
The deal of this paper is to use the differential-difference Jacobi elliptic functions sub-equation method for constructing exact solutions of nonlinear electrical circuit including intrinsic fractional-order in the sense of Riemann–Liouvi…
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Asymptotic expansion and bounds for complete elliptic integrals Open
In the article, we present several new bounds for the the complete elliptic integrals0 (1-r 2 sin 2 θ ) 1/2 dθ , and find an asymptotic expansion for K (r) as r → 1 , which are the refinements and improvements of the previously well-known …
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Functional relations for elliptic polylogarithms Open
Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting fr…
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Duality in elliptic Ruijsenaars system and elliptic symmetric functions Open
We demonstrate that the symmetric elliptic polynomials $$E_\lambda (x)$$ originally discovered in the study of generalized Noumi–Shiraishi functions are eigenfunctions of the elliptic Ruijsenaars–Schneider (eRS) Hamiltonians that act on t…
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Complete (p,q)-elliptic integrals with application to a family of means Open
The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. It is shown that a particular relation holds for the generalized integrals. Moreover, as an application of the integrals,…
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ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains Open
We prove global Hölder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions à la Cabré.
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Optical Soliton Solutions of the Generalized Sine-Gordon Equation Open
In this study, the extended trial equation method based on the general form of the nonlinear elliptic ordinary differential equation isemployed to solve the nonlinear generalized sine-Gordon equations. By using this method, we achieve, unl…
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Monotonicity properties and bounds for the complete <i>p</i>-elliptic integrals Open
In the article, we establish some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, and find several sharp bounds for the p-elliptic integrals. Our results are the gene…
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The ρ parameter at three loops and elliptic integrals Open
We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $\rho$ parameter. In particular, we present the calculation of the master integrals for which the corresponding dif…
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Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion Open
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve $E$ defined over a number field $K$ with $E(K)[2] \simeq \mathbb {Z}/2\mathbb {Z}$. We show that if $E$ does not hav…
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Elliptic Weight Functions and Elliptic q-KZ Equation Open
By using representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical analogu…
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An inequality for generalized complete elliptic integral Open
In this paper, we show an elegant inequality involving the ratio of generalized complete elliptic integrals of the first kind and generalize an interesting result of Alzer.
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The $\rho$ parameter at three loops and elliptic integrals Open
We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $\rho$ parameter. In particular, we present the calculation of the master integrals for which the corresponding dif…
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Elliptic Weight Functions and Elliptic q-KZ Equation Open
By using representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical analogu…
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On the status of DELL systems Open
A detailed review of the p,q-duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-ellipti…
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Elliptic multizeta values and the elliptic double shuffle relations Open
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Murmurations of Elliptic Curves Open
We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which var…
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The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves Open
We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque, Enriquez and Etingof (2009) and Levin and…
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Supersingular Elliptic Curves and Moonshine Open
We generalize a theorem of Ogg on supersingular $j$-invariants to\nsupersingular elliptic curves with level. Ogg observed that the level one case\nyields a characterization of the primes dividing the order of the monster. We\nshow that the…
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Global bifurcation for monotone fronts of elliptic equations Open
We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well …
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On the extended Euler system and the Jacobi and Weierstrass elliptic functions Open
We study the extended Euler systems (EES) as an initial value problem. Particular realizations of it lead to several Lie-Poisson structures. We consider a 6-D Poisson structure that fit all of them together. The symplectic stratification o…
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Elliptic curves over Q$_{∞}$ are modular Open
We show that if $\textit{p}$ is a prime, then all elliptic curves de ned over the cyclotomic $\mathbb{Z}$$_{p}$-extension of Q are modular.
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The Mining Industry in the Territory of Alaska During the Calendar Year 1915 Open
From Introduction: "This condition is likely to be serious, as preparations have been made for opening the Matanuska, Bering River, and Neanna coal fields, and a number of permits granting the free use of 10-acre tracts in other fields thr…
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A model for complex analytic equivariant elliptic cohomology from quantum field theory Open
We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fiel…