## Abstract

A scale-invariant model of statistical mechanics is applied to examine the physical foundation of nonstandard analysis and to identify the nature and the range (0_{β}, 0_{β-1}) of nonstandard numbers and to establish the existence of infinitesimals (0_{β}> L_{β-2}> X_{β} > 0_{β-2}). An invariant logarithmic definition of coordinate is presented and the concept of "measureless" or "dimensionless" numbers (L′_{β}, λ_{β}, 0_{β}) = (L_{β}, 1_{β}, 0_{β}) is described. Also, a scale-invariant definition of fractal dimension is introduced that suggest exceedingly large values 10^{7} of fractal dimension. A scale invariant form of the wave equation is derived that applies to acoustic waves that propagate at speed of sound v_{m} = 350 m/s, gravitational waves that propagate at the speed of light v_{t} = c, and gravitational radiation that propagates at superluminal speeds v_{g} > 2 × 10^{10} C.

Original language | English (US) |
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Pages (from-to) | 95-103 |

Number of pages | 9 |

Journal | WSEAS Transactions on Mathematics |

Volume | 7 |

Issue number | 3 |

State | Published - Mar 1 2008 |

## Keywords

- Gravitational radiation
- Gravitational waves
- Infinitesimals
- Nonstandard analysis

## ASJC Scopus subject areas

- Algebra and Number Theory
- Endocrinology, Diabetes and Metabolism
- Statistics and Probability
- Discrete Mathematics and Combinatorics
- Management Science and Operations Research
- Control and Optimization
- Computational Mathematics
- Applied Mathematics