Creative Cognition combines original experiments with existing work in cognitive psychology to provide the first explicit account of the cognitive processes and structures that contribute to creative thinking and discovery. Creative Cognition combines original experiments with existing work in cognitive psychology to provide the first explicit account of the cognitive processes and structures that contribute to creative thinking and discovery. In separate chapters, the authors take up visualization, concept formation, categorization, memory retrieval, and problem solving. They describe novel experimental methods for studying creative cognitive processes under controlled laboratory conditions, along with techniques that can be used to generate many different types of inventions and concepts. Unlike traditional approaches, Creative Cognition considers creativity as a product of numerous cognitive processes, each of which helps to set the stage for insight and discovery. It identifies many of these processes as well as general principles of creative cognition that can be applied across a variety of different domains, with examples in artificial intelligence, engineering design, product development, architecture, education, and the visual arts. Following a summary of previous approaches to creativity, the authors present a theoretical model of the creative process. They review research involving an innovative imagery recombination technique, developed by Finke, that clearly demonstrates that creative inventions can be induced in the laboratory. They then describe experiments in category learning that support the provocative claim that the factors constraining category formation similarly constrain imagination and illustrate the role of various memory processes and other strategies in creative problem solving. Bradford Books imprint
Morgan Ward pursued the study of elliptic divisibility sequences, originally initiated by Lucas, and Chudnovsky and Chudnovsky subsequently suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here, from both a theoretical and a practical viewpoint. We show calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.
MERSENNE NUMBERS AND PRIMITIVE PRIME DIVISORS.A notorious problem from elementary number theory is the "Mersenne Prime Conjecture." This asserts that the Mersenne sequence M = (M n ) defined by M n = 2 n − 1 (n = 1, 2, . . . ) contains infinitely many prime terms, which are known as Mersenne primes.The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its proper divisors. Euclid pointed out that 2 k−1 (2 k − 1) is perfect whenever 2 k − 1 is prime. A much less obvious result, due to Euler, is a partial converse: if n is an even perfect number, then it must have the form 2 k−1 (2 k − 1) for some k with the property that 2 k − 1 is a prime. Whether there are any odd perfect numbers remains an open question. Thus finding Mersenne primes amounts to finding (even) perfect numbers.The sequence M certainly produces some primes initially, for example,However, the appearance of Mersenne primes quickly thins out: only forty-three are known, the largest of which, M 30,402,457 , has over nine million decimal digits. This was discovered by a team at Central Missouri State University as part of the GIMPS project [23], which harnesses idle time on thousands of computers all over the world to run a distributed version of the Lucas-Lehmer test. A paltry forty-three primes might seem rather a small return for such a huge effort. Anybody looking for gold or gems with the same level of success would surely abandon the search. It seems fair to ask why we should expect there to be infinitely many Mersenne primes. In the absence of a rigorous proof, our expectations may be informed by heuristic arguments. In section 3 we discuss heuristic arguments for this and other more or less tractable problems in number theory.Primitive prime divisors. In 1892, Zsigmondy [24] discovered a beautiful argument that shows that the sequence M does yield infinitely many prime numbers-but in a less restrictive sense. Given any integer sequence S = 1
Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than those obtained for hyperbolic maps.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.