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-adic

-adic, suffix (ˈædɪk) [f. -ad1 1 a + -ic. In sense 1 after G. -adisch (K. Hensel Zahlentheorie (1913) iii. 51); in sense 2 f. polyadic a. Cf. also monadic a., dyadic a. (n.), etc.] 1. Math. Used with a preceding symbol or numeral, esp. the generalized symbol p (denoting a prime number), to designate... Oxford English Dictionary

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ADIC

ADIC may refer to: Abu Dhabi Investment Council Advanced Digital Information Corporation ASEAN Defense Industry Collaboration Assistant director in charge -adic analysis -adic valuation wikipedia.org

en.wikipedia.org

ℓ-adic sheaf

In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of -modules in the étale topology and inducing . Bhatt–Scholze's pro-étale topology gives an alternative approach. Motivation The development of étale cohomology as a whole was fueled by the desire to pro... wikipedia.org

en.wikipedia.org

$t$-adic topology (on $\mathbb F_p(1/t)$) Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic to...

Yes, just the same. You can easily check what the completion of $k(1/t)=k(t)$ is. Best to look first at the $(t)$-adic completion of the ring $k[t]$: this is the ring $k[[t]]$ of formal power series in $t$ over $k$. Then the associated complete ring is $k((t))$, the (finite-tailed) formal Laurent se...

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Which is the 5-adic number$\sqrt{-1}$? Let $a$ and $b$ be defined by the following; $a = \cdots 431212$, $b = \cdots 013233$. $a$ and $b$ satisfy $x^2 + 1 = 0$ in base $5$. Which is the 5-adic integer $\sqrt{-1}$?...

Either $a$ or $b$ can be used as $\sqrt{-1}$.

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P-adic integers Show that $\frac{2}{p-1}$ is a $p$-adic integer and find its p-adic expansion. P-adic numbers really make little sense to me so any help explaining what to do and why would be really appreciated. Thanks

If $p=2$ then your expression is just equal to $2\in\Bbb Z$. If $p$ is odd then numerator and denominator are prime to $p$, thus $p$-adic units. The quotient of $p$-adic units is a $p$-adic unit. Recall that a $p$-adic unit is an invertible element in the ring $\Bbb Z_p$ and can be recognized as tho...

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Reference for the p-adic numbers > Can anyone give me a reference (book or a paper) that introduces the p-adic numbers and their important properties? Also, I would love if that reference contained some not to advance...

p-adic Numbers, p-adic Analysis, and Zeta-Functions, Neal Koblitz (1984, ISBN 978-0-387-96017-3) This is a short book. If you wish, you can only read the first chapter with 20 pages.

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Binary representation of 2-adic integers I would like some examples of the binary representation of 2-adic integers that are not standard integers. What is the 2-adic expansion of $1/3$? Of $-1/3$? What number does $....

To answer your last question first, $$...010101 = 1 + 4 + 16 + \cdots + 4^n + \cdots = 1/(1-4) = -1/3.$$ This also answers your second question. As for your first question, $$1/3 = \dfrac{1}{2}( 1 - 1/3) = ...0101011.$$

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P-adic Hodge Theory and Automorphic Forms - pku.edu.cn

If you plan to have an extended stay, please inform Ms. Meng Yu (yumeng@math.pku.edu.cn) so that we can make proper arrangements. Hotel Address: "Zhong Guan Xin Yuan" Hotel (Zhongguanyuan Global Village) No. 216 Zhongguancun North Road, Haidian District, Beijing 100871, China. Tel: + (86 10) 62752288. Fax: + (86 10) 62752236. conference.bicmr.pku.edu.cn

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On 2-adic behavior of the number of domino tilings on torus

It is proved that g( n) and h(n) are uniformly continuous under the 2-adic metric and invariant under interchanging n and −1−n. We study the 2-adic behavior of the number of domino tilings of a 2(2n + 1) × 2(2n + 1) torus. We show that this number is of the form 2g(n)+2(2n+1)h(n), where g(n) and h(n) are odd positive integers. Moreover, we prove that g(n) and h(n) are uniformly continuous ...

www.semanticscholar.org

p-adic topology I'm studying about p-adic numbers and p-adic analysis. I want to know about definition of topology on $\mathbb{Z}_p$ and topology on $\mathbb{Q}_p$ whit respect to p-adic norm. I saw this topology in ...

You can start with $\mathbb{Z}$ equipped with the $p$-adic norm. Then consider its completion as a metric space, which is $\mathbb{Z}_p$. Then you can consider $\mathbb{Z}_p$ as a ring extending $\mathbb{Z}$. Finally, the field of fractions of $\mathbb{Z}_p$ will be $\mathbb{Q}_p$ and the topologies...

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Are there any non-trivial rational integers in the $p$-adic closure of $\{1,q,q^2,q^3,...\}$? If $p$ is prime and not a divisor of $q$, are there any non-trivial rational integers in the $p$-adic closure of the set or...

If $p>2$ then there is an integer $q$ not divisible by $p$, with the properties: $q^k\not\equiv 1$ mod $p$ for $1\leq k\leq p-2$ and $q^{p-1}\not\equiv 1$ mod $p^2$. Under these conditions the $p$-adic closure of $\\{1,q,q^2,\dots\\}$ is the whole $\mathbb{Z}_p^\times$ - in particular it contains al...

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Finding an example of a non-rational p-adic number We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\\{0,\dots,p-1\\}$ and $m\in\...

First of all, it is every integer (not every rational) that can be written as you have presented it as a sequence $a_n\in\\{0,\ldots, p-1\\}$. The number $1/p$ cannot be so represented. To answer your cardinality question, there are at least two arguments that $\mathbb Z_p$ is uncountable, and thus ...

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P adic numbers number theory Why does it make sense for prime adic numbers. I mean why won't be 4 adic numbers possible? ( I do not want any rigorous proof. A simple basic reason in detail will be appreciated) Thank you

The problem is in defining the $p$-adic norm for $p$ composite. If $p$ is prime, we can write any nonzero rational number $x$ as $x=\frac{p^a r}{s}$ where $r$ and $s$ are not divisible by $p$. The fact that $p$ is prime guarantees that the exponent $a$ is unique. Then we define the $p$-adic norm of ...

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Why exponential function on p-adic numbers is meaningless? In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaning...

It’s not the exponential function for $p$-adic numbers that’s meaningless; rather it’s the act of multiplying the real number $\pi$ by a nonrational $p$-adic number that’s meaningless. There’s no way of multiplying a real times a $p$-adic unless one of them is rational. On the other hand, there _is_...

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-adic

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-adic

ADIC

ℓ-adic sheaf

$t$-adic topology (on $\mathbb F_p(1/t)$) Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic to...

Which is the 5-adic number$\sqrt{-1}$? Let $a$ and $b$ be defined by the following; $a = \cdots 431212$, $b = \cdots 013233$. $a$ and $b$ satisfy $x^2 + 1 = 0$ in base $5$. Which is the 5-adic integer $\sqrt{-1}$?...

P-adic integers Show that $\frac{2}{p-1}$ is a $p$-adic integer and find its p-adic expansion. P-adic numbers really make little sense to me so any help explaining what to do and why would be really appreciated. Thanks

Reference for the p-adic numbers > Can anyone give me a reference (book or a paper) that introduces the p-adic numbers and their important properties? Also, I would love if that reference contained some not to advance...

Binary representation of 2-adic integers I would like some examples of the binary representation of 2-adic integers that are not standard integers. What is the 2-adic expansion of $1/3$? Of $-1/3$? What number does $....

P-adic Hodge Theory and Automorphic Forms - pku.edu.cn

On 2-adic behavior of the number of domino tilings on torus

p-adic topology I'm studying about p-adic numbers and p-adic analysis. I want to know about definition of topology on $\mathbb{Z}_p$ and topology on $\mathbb{Q}_p$ whit respect to p-adic norm. I saw this topology in ...

Are there any non-trivial rational integers in the $p$-adic closure of $\{1,q,q^2,q^3,...\}$? If $p$ is prime and not a divisor of $q$, are there any non-trivial rational integers in the $p$-adic closure of the set or...

Finding an example of a non-rational p-adic number We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\\{0,\dots,p-1\\}$ and $m\in\...

P adic numbers number theory Why does it make sense for prime adic numbers. I mean why won't be 4 adic numbers possible? ( I do not want any rigorous proof. A simple basic reason in detail will be appreciated) Thank you

Why exponential function on p-adic numbers is meaningless? In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaning...