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Vladimir Petrov Kostov - One of the best experts on this subject based on the ideXlab platform.
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partial Theta Function and separation in modulus property of its zeros
Vietnam journal of mathematics, 2020Co-Authors: Vladimir Petrov KostovAbstract:We consider the partial Theta Function $\Theta (q,z):={\sum }_{j=0}^{\infty }q^{j(j+1)/2}z^{j}$, where $z\in \mathbb {C}$ is a variable and $q\in \mathbb {C}$, 0 < |q| < 1, is a parameter. Set $D(a):=\{q\in \mathbb {C}, 0<|q|\leq a,$$\arg (q)\in [\pi /2,3\pi /2]\}$. We show that for $k\in \mathbb {N}$ and q ∈ D(0.55), there exists exactly one zero of θ(q,⋅) (which is a simple one) in the open annulus |q|−k+ 1/2 < z < |q|−k− 1/2 (if k ≥ 2) or in the punctured disk 0 < z < |q|− 3/2 (if k = 1). For k ≠ 2, 3, this holds true for q ∈ D(0.6) as well.
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a separation in modulus property of the zeros of a partial Theta Function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Vladimir Petrov KostovAbstract:We consider the partial Theta Function $\Theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$. We show that, for $n\geq 5$, for $|q|\leq 1-1/(\alpha _0n)$ and for $k\geq n$ there exists a unique zero $\xi _k$ of $\Theta (q,.)$ satisfying the inequalities $|q|^{-k+1/2}<|\xi _k|<|q|^{-k-1/2}$; all these zeros are simple ones. The moduli of the remaining $n-1$ zeros are $\leq |q|^{-n+1/2}$. A {\em spectral value} of $q$ is a value for which $\Theta (q,.)$ has a multiple zero. We prove the existence of the spectral values $0.4353184958\ldots \pm i\, 0.1230440086\ldots$ for which $\Theta$ has double zeros $-5.963\ldots \pm i\, 6.104\ldots$.
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on the multiple zeros of a partial Theta Function
Functional Analysis and Its Applications, 2016Co-Authors: Vladimir Petrov KostovAbstract:We consider the partial Theta Function θ(q, x) := ∑ j=0 ∞ q j(j+1)/2 x j , where x ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the Function θ(q, · ), then |ζ| ≤ 811.
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on a partial Theta Function and its spectrum
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2016Co-Authors: Vladimir Petrov KostovAbstract:The bivariate series defines a partial Theta Function . For fixed q (∣ q ∣ θ ( q , ·) is an entire Function. For q ∈ (–1, 0) the Function θ ( q , ·) has infinitely many negative and infinitely many positive real zeros. There exists a sequence of values of q tending to –1 + such that has a double real zero (the rest of its real zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the real negative zeros of (respectively, for k sufficiently large is the second from the left of the real negative zeros of ). For k sufficiently large one has . One has and .
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the closest to 0 spectral number of the partial Theta Function
arXiv: Complex Variables, 2016Co-Authors: Vladimir Petrov KostovAbstract:The {\em spectrum} of the partial Theta Function $\Theta :=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ is the set of values of $q\in \mathbb{C}$, $0<|q|<1$, for which $\Theta (q,.)$ has a multiple zero. We show that the only element of the spectrum belonging to the disk $\mathbb{D}_{0.31}$ is $0.3092493386\ldots$.
Zhiguo Liu - One of the best experts on this subject based on the ideXlab platform.
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Kronecker Theta Function and a decomposition theorem for Theta Functions I
arXiv: Complex Variables, 2020Co-Authors: Zhiguo LiuAbstract:The Kronecker Theta Function is a quotient of the Jacobi Theta Functions, which is also a special case of Ramanujan's $_1\psi_1$ summation. Using the Kronecker Theta Function as building blocks, we prove a decomposition theorem for Theta Functions. This decomposition theorem is the common source of a large number of Theta Function identities. Many striking Theta Function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for Theta Functions is established. Several known results in the theory of elliptic Theta Functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.
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SOME INVERSE RELATIONS AND Theta Function IDENTITIES
International Journal of Number Theory, 2012Co-Authors: Zhiguo LiuAbstract:Two pairs of inverse relations for elliptic Theta Functions are established with the method of Fourier series expansion, which allow us to recover many classical results in Theta Functions. Many nontrivial new Theta Function identities are discovered. Some curious trigonometric identities are derived.
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A Theta Function identity of degree 7 with applications
Integral Transforms and Special Functions, 2012Co-Authors: Zhiguo LiuAbstract:A Theta Function identity of degree 7 is established with the theory of elliptic Functions. A multitude of new modular identities are derived from this Theta Function identity. Some finite trigonometric sums are given.
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addition formulas for jacobi Theta Functions dedekind s eta Function and ramanujan s congruences
Pacific Journal of Mathematics, 2009Co-Authors: Zhiguo LiuAbstract:Previously, we proved an addition formula for the Jacobi Theta Function, which allows us to recover many important classical Theta Function identities. Here, we use this addition formula to derive a curious Theta Function identity, which includes Jacobi�s quartic identity and some other important Theta Function identities as special cases. We give new series expansions for ?2(t), ?6(t), ?8(t), and ?10(t), where ?(t) is Dedekind�s eta Function. The series expansions for ?6(t) and ?10(t) lead to simple proofs of Ramanujan�s congruences p(7n + 5) = 0 (mod 7) and p(11n + 6) = 0 (mod 11), respectively.
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An addition formula for the Jacobian Theta Function and its applications
Advances in Mathematics, 2007Co-Authors: Zhiguo LiuAbstract:In this paper, we prove an addition formula for the Jacobian Theta Function using the theory of elliptic Functions. It turns out to be a fundamental identity in the theory of Theta Functions and elliptic Function, and unifies many important results about Theta Functions and elliptic Functions. From this identity we can derive the Ramanujan cubic Theta Function identity, Winquist's identity, a Theta Function identities with five parameters, and many other interesting Theta Function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma Function. A new identity about the Ramanujan cubic elliptic Function is given. The proofs are self contained and elementary.
Reiho Sakamoto - One of the best experts on this subject based on the ideXlab platform.
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Combinatorial Bethe Ansatz and Ultradiscrete Riemann Theta Function with Rational Characteristics
Letters in Mathematical Physics, 2007Co-Authors: Atsuo Kuniba, Reiho SakamotoAbstract:The $$U_q(\widehat{sl}_2)$$ vertex model at q = 0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann Theta Function with rational characteristics.
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combinatorial bethe ansatz and ultradiscrete riemann Theta Function with rational characteristics
arXiv: Exactly Solvable and Integrable Systems, 2006Co-Authors: Atsuo Kuniba, Reiho SakamotoAbstract:The U_q(\hat{sl}_2) vertex model at q=0 with periodic boundary condition is an integrable cellular automaton in one-dimension. By the combinatorial Bethe ansatz, the initial value problem is solved for arbitrary states in terms of an ultradiscrete analogue of the Riemann Theta Function with rational characteristics.
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the bethe ansatz in a periodic box ball system and the ultradiscrete riemann Theta Function
Journal of Statistical Mechanics: Theory and Experiment, 2006Co-Authors: Atsuo Kuniba, Reiho SakamotoAbstract:Vertex models with quantum group symmetry give rise to integrable cellular automata at q = 0. We study a prototype example known as the periodic box–ball system. The initial value problem is solved in terms of an ultradiscrete analogue of the Riemann Theta Function whose period matrix originates in the Bethe ansatz at q = 0.
Hongqing Zhang - One of the best experts on this subject based on the ideXlab platform.
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super riemann Theta Function periodic wave solutions and rational characteristics for a supersymmetric kdv burgers equation
Theoretical and Mathematical Physics, 2012Co-Authors: Shoufu Tian, Hongqing ZhangAbstract:Using a multidimensional super Riemann Theta Function, we propose two key theorems for explicitly constructing multiperiodic super Riemann Theta Function periodic wave solutions of supersymmetric equations in the superspace ℝ Λ N+1,M , which is a lucid and direct generalization of the super-Hirota-Riemann method. Once a supersymmetric equation is written in a bilinear form, its super Riemann Theta Function periodic wave solutions can be directly obtained by using our two theorems. As an application, we present a supersymmetric Korteweg-de Vries-Burgers equation. We study the limit procedure in detail and rigorously establish the asymptotic behavior of the multiperiodic waves and the relations between periodic wave solutions and soliton solutions. Moreover, we find that in contrast to the purely bosonic case, an interesting phenomenon occurs among the super Riemann Theta Function periodic waves in the presence of the Grassmann variable. The super Riemann Theta Function periodic waves are symmetric about the band but collapse along with the band. Furthermore, the results can be extended to the case N > 2; here, we only consider an existence condition for an N-periodic wave solution of a general supersymmetric equation.
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Theta Function solutions for two discrete equations
Communications in Nonlinear Science and Numerical Simulation, 2010Co-Authors: Yang Feng, Hongqing ZhangAbstract:Abstract In this letter, solutions of the discrete mKdV equation and discrete two-dimensional Toda equation in terms of product of up to two Theta Functions are given. To get the quasiperiodic solutions, this method is direct and simple which use only the identities of the Theta Functions.
Zhengrong Liu - One of the best experts on this subject based on the ideXlab platform.
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soliton and riemann Theta Function quasi periodic wave solutions for a 2 1 dimensional generalized shallow water wave equation
Nonlinear Dynamics, 2015Co-Authors: Yiren Chen, Ming Song, Zhengrong LiuAbstract:In this paper, a $$(2 + 1)$$ -dimensional generalized shallow water wave equation is investigated through bilinear Hirota method. Interestingly, the breather-type and lump-type soliton solutions are obtained. Furthermore, dynamic properties of the soliton waves are revealed by means of the asymptotic analysis. Based on Hirota bilinear method and Riemann Theta Function, we succeed in constructing quasi-periodic wave solutions with a generalized form. We also display the asymptotic properties of these quasi-periodic wave solutions and point out the relation between the quasi-periodic wave solutions and the soliton solutions.
