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arithmetical
arithmetical, a. (and n.) (ærɪθˈmɛtɪkəl) [f. L. arithmētic-us, a. Gr. ἀριθµητικ-ός numeric (see arithmetic n.1) + -al1.] A. adj. Of, pertaining to, or connected with, arithmetic; according to the rules of arithmetic. arithmetical mean, arithmetical progression, arithmetical proportion: see quot.1543... Oxford English Dictionary
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Arithmetical ring
In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold: The localization of R An arithmetical domain is the same thing as a Prüfer domain. References External links Ring theory wikipedia.org
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Arithmetical set
The arithmetical sets are classified by the arithmetical hierarchy. Properties The complement of an arithmetical set is an arithmetical set. The Turing jump of an arithmetical set is an arithmetical set. wikipedia.org
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A non-arithmetical set? A set is called arithmetical if it can be defined by a first-order formula in Peano arithmetic. I first encountered these sets when exploring the arithmetical hierarchy in the context of comput...
There are countably many first order formulas defining arithmetical sets. Let $\varphi_n$, $n\in\mathbb N$, be a list of those.
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Arithmetical hierarchy
Any set that receives a classification is called arithmetical. The arithmetical hierarchy of formulas The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. wikipedia.org
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Arithmetical proof of $\cfrac{1}{a+b}\binom{a+b}{a}$ is an integer when $(a,b)=1$ When $(a,b)=1$, $\cfrac{1}{a+b}\binom{a+b}{a}$ refers to the number of paths from one corner to its opposite corner of an $a\times b$ l...
You know that $\binom{a+b}a$ is an integer, and from its formula involving factorials (or otherwise), we have that $\binom{a+b}{a}=\frac{a+b}a\binom{a+b-1}{a-1}$, which means that $$\frac1{a+b}\binom{a+b}{a}=\frac1a\binom{a+b-1}{a-1}. $$ Now, since $\mathrm{gcd}(a+b,a)=1$, and $\frac{a+b}a\binom{a+b...
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Complexity class with arithmetical oracle. Although I feel the answer to the following question is negative, I can't get any precise results neither find anything to read. The question is: Would a complete oracle fr...
Yes ofcourse, With a halting oracle you can solve any decidable problem in polynomial time as follows: Let L be a decidable language, and let M its decider. K: on input x do{ Construct a turing machine N: on input y { If M(y) = true then output true else while(1) } query the HALT oracle with (N,x) a...
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Pillai's arithmetical function
In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every by or equivalently where is a divisor of This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933. wikipedia.org
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Mapping logical expressions to arithmetical expressions I wrote a program that is able to evaluate arithmetical expressions, like 1 + 3.0 * 4 / (1 - 1 * 3.2) What I want is to use the same program ...
Yes, this is possible: * NOT(A) = 1-A * AND(A,B) = A*B = min(A,B) * OR(A,B) = A+B-A*B = max(A,B) * IMPLIES(A,B) = 1-A+A*B * BIIMPLIES(A,B) = 1-|A-B| * XOR(A,B) = |A-B|
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Least non-arithmetical ordinal As I understand, there exists the least ordinal $\alpha$ such that there is no well-ordering of $\mathbb{N}$ which is both order isomorphic to $\alpha$ and is an arithmetical set. Is the...
As Joel David Hamkins pointed out to me in another question on MO, the set of arithmetical ordinals is exactly the set of recursive ordinals $\omega^{\
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Calc-X and Calcformers: Empowering Arithmetical Chain-of-Thought ...
Jan 5, 2024%0 Conference Proceedings %T Calc-X and Calcformers: Empowering Arithmetical Chain-of-Thought through Interaction with Symbolic Systems %A Kadlčík, Marek %A Štefánik, Michal %A Sotolar, Ondrej %A Martinek, Vlastimil %Y Bouamor, Houda %Y Pino, Juan %Y Bali, Kalika %S Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing %D 2023 %8 December %I Association ...
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Remainder function being arithmetic I am reading Kleene's "Introduction to Metamathematics" chapter 9 section 48, where he mentions that "We know that the predicate $rm(c,d)=w$, where $w$ is the remainder when $c$ is ...
As indicated by fleablood in the comments, we can write the relation $rm(c,d)=w$ as $$ w<d\land\exists k (c=kd+w)$$ and, to fully comply with your requirements, $w<d$ can be expressed as $\exists z(\lnot(z=0)\land d = w+z)$
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difference of the arithmetical sequence problem Choose the first element and the difference of the arithmetical sequence, where you know that $$a_3+a_5+a_7=-12, a_3*a_4*a_5=-90$$ Choose the first element and the diff...
With $$a_3=a_1+2d,a_4=a_1+3d,a_5=a_1+4d,a_7=a_1+6d$$ we get the system $$3a_1+12d=-12$$ and $$(a_1+2d)(a_1+3d)(a_1+4d)=-90$$ Can you solve this=
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Computability text emphasizing the arithmetical point of view? I learned here that > A set is _recursively enumerable_ if and only if it is at level $\Sigma^0_1$ of the arithmetical hierarchy. Is there an introducto...
Every nontrivial recursion theory book will prove that fact, which is just one part of Post's theorem. Rather than just saying that an r.e. set is $\Sigma^0_1$, in later proofs they will often use the fact that they can specify the formula; a set $A$ is r.e. if and only if there is an $e$ such that,...
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How does the categorical definition of a product apply to the arithmetical product? If object $X$ and morphisms $p_1 : X \to X_1, p_2 : X \to X_2$ define the product of objects $X_1, X_2$ in category $C$, how do we ma...
The categorical definition of a product is _not_ intended to be a generalization of the arithmetic notion of a product. Rather, it is intended as a generalization of the notion of a Cartesian product of sets (or more complicated structures, like groups, topological spaces, etc.). Specifically, the C...
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