• Title/Summary/Keyword: Jacobi

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GALOIS COVERINGS AND JACOBI VARIETIES OF COMPACT RIEMANN SURFACES

  • Namba, Makoto
    • Journal of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.263-286
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    • 2016
  • We discuss relations between Galois coverings of compact Riemann surfaces and their Jacobi varieties. We prove a theorem of a kind of Galois correspondence for Abelian subvarieties of Jacobi varieties. We also prove a theorem on the sets of points in Jacobi varieties fixed by Galois group actions.

EXACT FORMULA FOR JACOBI-EISENSTEIN SERIES OF SQUARE FREE DISCRIMINANT LATTICE INDEX

  • Xiong, Ran
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.481-488
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    • 2020
  • In this paper we give an exact formula for the Fourier coefficients of the Jacobi-Eisenstein series of square free discriminant lattice index. For a special case the discriminant of lattice is prime we show that the Jacobi-Eisenstein series corresponds to a well known Eisenstein series of modular forms.

A PARTIAL CAYLEY TRANSFORM OF SIEGEL-JACOBI DISK

  • Yang, Jae-Hyun
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.781-794
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    • 2008
  • Let $\mathbb{H}_g$ and $\mathbb{D}_g$ be the Siegel upper half plane and the generalized unit disk of degree g respectively. Let $\mathbb{C}^{(h,g)}$ be the Euclidean space of all $h{\times}g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ onto the Siegel-Jacobi space $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ which gives a partial bounded realization of $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$ by $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. and $\mathbb{H}_g{\times}\mathbb{C}^{(h,g)}$. are compatible via a partial Cayley transform. A partial Cayley transform plays an important role in computing differential operators on the Siegel Jacobi disk $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$. invariant under the natural action of the Jacobi group $\mathbb{D}_g{\times}\mathbb{C}^{(h,g)}$ explicitly.

A NEW CLASSIFICATION OF REAL HYPERSURFACES WITH REEB PARALLEL STRUCTURE JACOBI OPERATOR IN THE COMPLEX QUADRIC

  • Lee, Hyunjin;Suh, Young Jin
    • Journal of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.895-920
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    • 2021
  • In this paper, first we introduce the full expression of the Riemannian curvature tensor of a real hypersurface M in the complex quadric Qm from the equation of Gauss and some important formulas for the structure Jacobi operator Rξ and its derivatives ∇Rξ under the Levi-Civita connection ∇ of M. Next we give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ∇ξRξ = 0, in the complex quadric Qm for m ≥ 3. In addition, we also consider a new notion of 𝒞-parallel structure Jacobi operator of M and give a nonexistence theorem for Hopf real hypersurfaces with 𝒞-parallel structure Jacobi operator in Qm, for m ≥ 3.

THE JACOBI SUMS OVER GALOIS RINGS AND ITS ABSOLUTE VALUES

  • Jang, Young Ho
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.571-583
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    • 2020
  • The Galois ring R of characteristic pn having pmn elements is a finite extension of the ring of integers modulo pn, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.

CONSTRAINED JACOBI POLYNOMIAL AND CONSTRAINED CHEBYSHEV POLYNOMIAL

  • Ahn, Young-Joon
    • Communications of the Korean Mathematical Society
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    • v.23 no.2
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    • pp.279-284
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    • 2008
  • In this paper, we present the constrained Jacobi polynomial which is equal to the constrained Chebyshev polynomial up to constant multiplication. For degree n=4, 5, we find the constrained Jacobi polynomial, and for $n{\geq}6$, we present the normalized constrained Jacobi polynomial which is similar to the constrained Chebyshev polynomial.

KILLING STRUCTURE JACOBI OPERATOR OF A REAL HYPERSURFACE IN A COMPLEX PROJECTIVE SPACE

  • Perez, Juan de Dios
    • Journal of the Korean Mathematical Society
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    • v.58 no.2
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    • pp.473-486
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    • 2021
  • We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the k-th generalized Tanaka-Webster connection.

On Finite Integrals Involving Jacobi Polynomials and the $\bar{H}$-function

  • Sharma, Rajendra P.
    • Kyungpook Mathematical Journal
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    • v.46 no.3
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    • pp.307-313
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    • 2006
  • In this paper, we first establish an interesting new finite integral whose integrand involves the product of a general class of polynomials introduced by Srivastava [13] and the generalized H-function ([9], [10]) having general argument. Next, we present five special cases of our main integral which are also quite general in nature and of interest by themselves. The first three integrals involve the product of $\bar{H}$-function with Jacobi polynomial, the product of two Jacobi polynomials and the product of two general binomial factors respectively. The fourth integral involves product of Jacobi polynomial and well known Fox's H-function and the last integral involves product of a Jacobi polynomial and 'g' function connected with a certain class of Feynman integral which may have practical applications.

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REAL HYPERSURFACES IN THE COMPLEX HYPERBOLIC QUADRIC WITH CYCLIC PARALLEL STRUCTURE JACOBI OPERATOR

  • Jin Hong Kim;Hyunjin Lee;Young Jin Suh
    • Journal of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.309-339
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    • 2024
  • Let M be a real hypersurface in the complex hyperbolic quadric Qm*, m ≥ 3. The Riemannian curvature tensor field R of M allows us to define a symmetric Jacobi operator with respect to the Reeb vector field ξ, which is called the structure Jacobi operator Rξ = R( · , ξ)ξ ∈ End(TM). On the other hand, in [20], Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator Rξ for a real hypersurface M in the complex hyperbolic quadric Qm*. Furthermore, we give a complete classification of Hopf real hypersurfaces in Qm* with such a property.