Information Theory Applied To Space-time PhysicsWorld Scientific, 31.01.1993 - 320 Seiten The success of Newton's mechanic, Maxwell's electrodynamic, Einstein's theories of relativity, and quantum mechanics is a strong argument for the space-time continuum. Nevertheless, doubts have been expressed about the use of a continuum in a science squarely based on observation and measurement. An exact science requires that qualitative arguments must be reduced to quantitative statements. The observability of a continuum can be reduced from qualitative arguments to quantitative statements by means of information theory.Information theory was developed during the last decades within electrical communications, but it is almost unknown in physics. The closest approach to information theory in physics is the calculus of propositions, which has been used in books on the frontier of quantum mechanics and the general theory of relativity. Principles of information theory are discussed in this book. The ability to think readily in terms of a finite number of discrete samples is developed over many years of using information theory and digital computers, just as the ability to think readily in terms of a continuum is developed by long use of differential calculus. |
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Seite 18
... equals 5 bit or bits . Alternately , one may say the information 5 bits is the number of answers YES or NO to the question " Is a positive current flowing ? " at the times t = 0 , AT , ... , 4AT in Fig.2.1-1 . This second interpretation ...
... equals 5 bit or bits . Alternately , one may say the information 5 bits is the number of answers YES or NO to the question " Is a positive current flowing ? " at the times t = 0 , AT , ... , 4AT in Fig.2.1-1 . This second interpretation ...
Seite 19
... equals : ( 2 ) 1 This generalization is based on three assumptions that may or may not appear to be plausible . The main reason for using this particular generalization seems to be that it yields formulas equal to those for entropy in ...
... equals : ( 2 ) 1 This generalization is based on three assumptions that may or may not appear to be plausible . The main reason for using this particular generalization seems to be that it yields formulas equal to those for entropy in ...
Seite 20
... equals m - times the values given by Eqs . ( 3 ) or ( 4 ) . For the binary case we write this explicitly : H ( m ) - = m -Σ ( Pik log2 Pik + Pok log2Pok ) k = 1 = -m ( p1 log2 P1 + Po log2 Po ) for Pik P1 , Pok Po = m for P1 = = Po ( 5 ) ...
... equals m - times the values given by Eqs . ( 3 ) or ( 4 ) . For the binary case we write this explicitly : H ( m ) - = m -Σ ( Pik log2 Pik + Pok log2Pok ) k = 1 = -m ( p1 log2 P1 + Po log2 Po ) for Pik P1 , Pok Po = m for P1 = = Po ( 5 ) ...
Seite 22
... equal to the probability of P lying in the interval 0 < x / X < 0.1 . 3 The subdivision of the ruler into 2m equal intervals yields the information m bits about the location of P. Generally , one may divide the ruler into sm equal ...
... equal to the probability of P lying in the interval 0 < x / X < 0.1 . 3 The subdivision of the ruler into 2m equal intervals yields the information m bits about the location of P. Generally , one may divide the ruler into sm equal ...
Seite 23
... equals m bits . Since the point P is located between two marks xp and xp + Ax we say its uncertainty equals Ar X / 2m ... equal probability , the mini- mum distance TR - IP- Az and the maximum distance RIP + Ar are less probable than an ...
... equals m bits . Since the point P is located between two marks xp and xp + Ax we say its uncertainty equals Ar X / 2m ... equal probability , the mini- mum distance TR - IP- Az and the maximum distance RIP + Ar are less probable than an ...
Inhalt
1 | |
18 | |
3 Coordinate Systems | 40 |
4 Time and Motion | 85 |
5 Propagation in Unusual Coordinate Systems | 104 |
6 Distinction of Sinusoidal Functions | 163 |
7 Discrete Topologies and Difference Equations | 197 |
8 Schrödinger and KleinGordon Difference Equations | 204 |
9 Schrödinger Difference Equation with Coulomb Field | 218 |
10 KleinGordon Difference Equation with Coulomb Field | 230 |
11 Dirac Difference Equation with Coulomb Field | 254 |
12 Mathematical Supplements | 270 |
References and Bibliography | 297 |
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angles calculus Cartesian coordinate system circle clock coefficients concept continuum convergence coordinate distance defined derive difference equation difference operator difference quotient differential equation diffraction grating digits dyadic coordinate system dyadic shifting eigenfunctions eigenvalues equal Euclidean geometry f(no factorial series finite number four-dimensional function f(m functions Wal(k geodesic Gray code grid points Hamming distance Hence infinite information theory integer numbers integration interval Klein-Gordon Klein-Gordon equation marks mathematical measured metric minimized code minimum absolute distance modulo neighbors nondenumerably numbers axis O(Ar observed obtains P₁ particle physical plane propagation Pythagorean distance real numbers replaced representation result ring 2N rods rotation ruler samples Schrödinger equation Section shown shows sinusoidal functions small values solution space space-time spheres spherical standing waves substitution surface Table three-dimensional space two-dimensional unbounded coordinate system usual binary code variable velocity Walp Walsh functions yields Z² a² zero Δη Δυ