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By Ma J. M., Qiu R. F.

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Zeidler, Teubner-Taschenbuch der Mathematik, Vols. 1, 2, Teubner, Wiesbaden, 2003. The English translation of the second volume is in preparation. See D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 8 (1902), 437– 479, and B. Yandell, The Honors Class: Hilbert’s Problems and Their Solvers, Natick, Massachusetts, 2001. 18 Prologue has no nontrivial integer solution. ), Fermat wrote the following: It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent.

14) is called a Feynman path integral (or functional integral). It represents a statistic over the possible trajectories of the classical particle. The statistical weight eiS[q]/ for each trajectory depends on the classical action S[q] of the trajectory. Since the quantity S[q] has the physical dimension of energy × time, we have to divide the action S[q] by a constant of the same dimension in order to get a dimensionless argument of the exponential function. This way, Planck’s constant of action, , appears in a natural way.

Nevertheless, the quantization of action has enormous consequences. For example, consider a mass point on the real line which moves periodically, q(t) = const · sin(ωt) where t denotes time, and ω is called the angular frequency of the harmonic oscillator. Since the sine function has period 2π, the harmonic oscillator has the time period T = 2π/ω. By definition, the frequency ν is the number of oscillations per second. Hence T = 1/ν, and ω = 2πν. If E denotes the energy of the harmonic oscillator, then the product ET is a typical action value for the oscillations of the harmonic oscillator.

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3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera by Ma J. M., Qiu R. F.

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