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Kim Min-soo - One of the best experts on this subject based on the ideXlab platform.
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Infinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
2021Co-Authors: Kim Min-sooAbstract:In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-Archimedean Property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017
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Infinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
2020Co-Authors: Kim Min-sooAbstract:In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-Archimedean Property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 16 pages, revised version. We would like to thank Professor Lawrence C. Washington for pointing out a gap in the proof of Lemma 3.1 of the original manuscript and for his helpful suggestion
Alexander, Samuel Allen - One of the best experts on this subject based on the ideXlab platform.
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The Archimedean trap: Why traditional reinforcement learning will probably not yield AGI
2020Co-Authors: Alexander, Samuel AllenAbstract:After generalizing the Archimedean Property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning cannot lead to AGI. We indicate two possible ways traditional reinforcement learning could be altered to remove this roadblock.Comment: 16 page
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The Archimedean trap: Why traditional reinforcement learning will probably not yield AGI
2020Co-Authors: Alexander, Samuel AllenAbstract:After generalizing the Archimedean Property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate two possible ways traditional reinforcement learning could be altered to remove this roadblock
Peter C. Fishburn - One of the best experts on this subject based on the ideXlab platform.
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Lexicographic order preservation and stochastic dominance
Journal of Multi-Criteria Decision Analysis, 1995Co-Authors: Irving H. Lavalle, Peter C. FishburnAbstract:Stochastic dominance concerns conditions on outcome probabilities that are necessary and sufficient for one act to be (strictly) preferred to another according to all preference relations that share certain properties, one of which customarily is an Archimedean Property sufficient to entail existence of real-valued representations. We relax this assumption to permit linear lexicographic utility of finite and known dimensionality. In some situations, levels of the lexicographic hierarchy could correspond to explicit criteria or attributes. In our model, subjective probabilities emerge as matrix premultipliers of the outcome utility vectors. We thus obtain matrix probability generalizations of the familiar cumulative probability conditions for stochastic dominance.
Irving H. Lavalle - One of the best experts on this subject based on the ideXlab platform.
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Lexicographic order preservation and stochastic dominance
Journal of Multi-Criteria Decision Analysis, 1995Co-Authors: Irving H. Lavalle, Peter C. FishburnAbstract:Stochastic dominance concerns conditions on outcome probabilities that are necessary and sufficient for one act to be (strictly) preferred to another according to all preference relations that share certain properties, one of which customarily is an Archimedean Property sufficient to entail existence of real-valued representations. We relax this assumption to permit linear lexicographic utility of finite and known dimensionality. In some situations, levels of the lexicographic hierarchy could correspond to explicit criteria or attributes. In our model, subjective probabilities emerge as matrix premultipliers of the outcome utility vectors. We thus obtain matrix probability generalizations of the familiar cumulative probability conditions for stochastic dominance.
M A Endang Cahya - One of the best experts on this subject based on the ideXlab platform.
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completeness properties of r r and real valued functions of two variables under lexicographic order
PROCEEDINGS OF INTERNATIONAL SEMINAR ON MATHEMATICS SCIENCE AND COMPUTER SCIENCE EDUCATION (MSCEIS 2015), 2016Co-Authors: M A Endang CahyaAbstract:The order in R × R which is commonly used is a partial order. In this article is discussed a different concept of order which gives a total order/lexicographic order for R × R. With respect to this order, we define a new concept of inequalities, Archimedean Property and completeness in R × R. We also define a concept of monotonicity of real function of several variables and a number of its basic properties. The results will give fundamental aspects to define different concepts of multivariable calculus.