Results 11 to 20 of about 9,714 (169)
Angewandte Chemie, EarlyView.Cyclopentadienes (CpHs) are fundamental building blocks with widespread applications in chemistry. However, their inherent acidity and isomeric instability have long limited access to chiral variants in an asymmetric fashion. Here, we introduce a CuH‐catalyzed asymmetric transformation that provides an unprecedented entry into a new class of ...
Piero Soppelsa +4 more
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Bounds for the 3x+1 problem using difference inequalities [PDF]
Acta Arithmetica, 2003 We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities.
Krasikov, Ilia, Lagarias, Jeffrey C.
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Advances in Applied Mathematics, 1994 The ``\(3x+1\)''-problem (or ``Collatz''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n\in\mathbb{N}\) there exists a \(k\) with \(T^{(k)} (n)=1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\).
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The $3x+1$ problem: a lower bound hypothesis [PDF]
Functiones et Approximatio Commentarii Mathematici, 2017 Much work has been done attempting to understand the dynamic behaviour of the so-called "3x+1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two.
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Advances in Applied Mathematics, 1989 Let \((m_ n)\), \(n=0,1,2,...\), be a sequence of positive integers defined by \(m_ 0=m\) and, for \(n>0\), \(m_{n+1}=m_ n/2\) if \(m_ n\) is even and \(m_{n+1}=(3m_ n+1)/2\) if \(m_ n\) is odd. An old conjecture asserts that for any m there is a natural number n such that \(m_ n=1\). The author proves the following theorem: Let \(N_ n=\{1,2,3,...,2^ n\
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The number system in rational base $3/2$ and the $3x+1$ problem
Comptes Rendus. MathématiqueThe representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$.
Eliahou, Shalom +1 more
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On a Generalization of the 3x+ 1 Problem
Advances in Applied Mathematics, 1997 This paper considers the behaviour of iterations of periodically linear functions \(g(x)=a_rx+b_r\) when \(x\equiv r\bmod p\) for \(r\in\{0,\ldots,p-1\}\), where \(a_r=t_r/p\) with \(t_r\in\mathbb Z\) and appropriate \(b_r\in\mathbb Q\) to ensure \(g(x)\in\mathbb Z\) for \(x\in\mathbb Z\). It is shown that if (i) \(\text{gcd}(t_0t_1\ldots t_{p-1},p)=1\)
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On a Generalization of the 3x + 1 Problem
Journal of Number Theory, 1995 The Collatz-problem (or \(3x + 1-\) or Hasse or Syracuse or Kakutani problem) is to prove that for every \(n \in \mathbb{N}\) there exists a \(k\) with \(T^{(k)} (n) = 1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n + 1)/2\) and even numbers \(n\) to \(n/2\). In the note under review the author studies a generalization of the following
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, 2021 This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this problem? (as of 2010). (2) How can this problem be hard, when it is so easy to state?
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Advanced Functional Materials, EarlyView.Shellac, a centuries‐old natural resin, is reimagined as a green material for flexible electronics. When combined with silver nanowires, shellac films deliver transparency, conductivity, and stability against humidity. These results position shellac as a sustainable alternative to synthetic polymers for transparent conductors in next‐generation ...
Rahaf Nafez Hussein +4 more
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