97,179 research outputs found
Basic Properties of Primitive Root and Order Function
Authors thank Andrzej Trybulec and Yatsuka Nakamura for the help during writing this articleIn this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].Ma Na - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Zhang Dexin. Integer Theory. Science Publication, China, 1965.Yoshinori Fujisawa and Yasushi Fuwa. The Eulerâs function. Formalized Mathematics, 6(4):549-551, 1997.Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepinâs test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Posts and Telecom Press, China, 2007.Hua Loo Keng. Introduction to Number Theory. Beijing Science Publication, China, 1957.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990.Artur KorniĆowicz. Collective operations on number-membered sets. Formalized Mathematics, 17(2):99-115, 2009, doi: 10.2478/v10037-009-0011-0.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.RafaĆ Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Xiquan Liang, Li Yan, and Junjie Zhao. Linear congruence relation and complete residue systems. Formalized Mathematics, 15(4):181-187, 2007, doi:10.2478/v10037-007-0022-7.Konrad Raczkowski and PaweĆ Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990
Dyson-Schwinger equations in the theory of computation
Following Manin's approach to renormalization in the theory of computation,
we investigate Dyson-Schwinger equations on Hopf algebras, operads and
properads of flow charts, as a way of encoding self-similarity structures in
the theory of algorithms computing primitive and partial recursive functions
and in the halting problem.Comment: 26 pages, LaTeX, final version, in "Feynman Amplitudes, Periods and
Motives", Contemporary Mathematics, AMS 201
Dynamic mechanical response of polymer networks
The dynamic-mechanical response of flexible polymer networks is studied in
the framework of tube model, in the limit of small affine deformations, using
the approach based on Rayleighian dissipation function. The dynamic complex
modulus G* is calculated from the analysis of a network strand relaxation to
the new equilibrium conformation around the distorted primitive path. Chain
equilibration is achieved via a sliding motion of polymer segments along the
tube, eliminating the inhomogeneity of the polymer density caused by the
deformation. The characteristic relaxation time of this motion separates
the low-frequency limit of the complex modulus from the high-frequency one,
where the main role is played by chain entanglements, analogous to the rubber
plateau in melts. The dependence of storage and loss moduli, G' and G'', on
crosslink and entanglement densities gives an interpolation between polymer
melts and crosslinked networks. We discuss the experimental implications of the
rather short relaxation time and the slow square-root variation of the moduli
and the loss factor tan at higher frequencies.Comment: Journal of Chemical Physics (Oct-2000); Lates, 4 EPS figures include
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