221: Zeta Functions for Two-Dimensional Shifts of Finite Type (Memoirs of the American Mathematical Society: January 2013 (First of 5 Numbers))

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9780821872901: 221: Zeta Functions for Two-Dimensional Shifts of Finite Type (Memoirs of the American Mathematical Society: January 2013 (First of 5 Numbers))
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This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $\zeta^{0}(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $\mathbb{Z}^{2}$-action $\phi$. In this paper, the $n$th-order zeta function $\zeta_{n}$ of $\phi$ on $\mathbb{Z}_{n\times \infty}$, $n\geq 1$, is studied first. The trace operator $\mathbf{T}_{n}$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $\mathbf{T}_{n}$ induces the reduced trace operator $\tau_{n}$ and $\zeta_{n}=\left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$. The zeta function $\zeta=\prod_{n=1}^{\infty} \left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$ in the $x$-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the $y$-direction and in the coordinates of any unimodular transformation in $GL_{2}(\mathbb{Z})$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $\zeta^{0}(s)$. The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions.

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Book Description American Mathematical Society, United States, 2013. Paperback. Condition: New. Language: English . Brand New Book. This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $ zeta^{0}(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $ mathbb{Z}^{2}$-action $ phi$. In this paper, the $n$th-order zeta function $ zeta {n}$ of $ phi$ on $ mathbb{Z} {n times infty}$, $n geq 1$, is studied first. The trace operator $ mathbf{T} {n}$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $ mathbf{T} {n}$ induces the reduced trace operator $ tau {n}$ and $ zeta {n}= left( det left(I-s^{n} tau {n} right) right)^{-1}$. The zeta function $ zeta= prod {n=1}^{ infty} left( det left(I-s^{n} tau {n} right) right)^{-1}$ in the $x$-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the $y$-direction and in the coordinates of any unimodular transformation in $GL {2}( mathbb{Z})$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $ zeta^{0}(s)$. The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions. Seller Inventory # AAN9780821872901

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Jungchao Ban, Wen-guei Hu, Song-sun Lin
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Book Description American Mathematical Society, United States, 2013. Paperback. Condition: New. Language: English . Brand New Book. This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $ zeta^{0}(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $ mathbb{Z}^{2}$-action $ phi$. In this paper, the $n$th-order zeta function $ zeta {n}$ of $ phi$ on $ mathbb{Z} {n times infty}$, $n geq 1$, is studied first. The trace operator $ mathbf{T} {n}$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $ mathbf{T} {n}$ induces the reduced trace operator $ tau {n}$ and $ zeta {n}= left( det left(I-s^{n} tau {n} right) right)^{-1}$. The zeta function $ zeta= prod {n=1}^{ infty} left( det left(I-s^{n} tau {n} right) right)^{-1}$ in the $x$-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the $y$-direction and in the coordinates of any unimodular transformation in $GL {2}( mathbb{Z})$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $ zeta^{0}(s)$. The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions. Seller Inventory # AAN9780821872901

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