MATHEMATICAL MYSTERIES IN THE NATURAL WORLD (eBook)

Why the Small Outnumbers the Big

Alex Ely Kossovsky (Autor)

eBook Download: EPUB

2025
392 Seiten
World Scientific Publishing Company (Verlag)
978-981-98-0184-8 (ISBN)

Infos

Why are there more poor people with small bank accounts than rich people with big bank accounts? Why are there more small planets and stars than big ones in the cosmos? And why are there more small rivers than big rivers, and more harmless tremors than devastating earthquakes? Empirical examinations of real-life data overwhelmingly confirm the existence of such uneven size proportions in favor of the small, leaving us only the question: Why?

This law of nature carries through a vast list of topics and disciplines, confirming it to be nearly universal. In the biological world there are only approximately two million whales and over three hundred billion birds; in number theory there are more small prime numbers than there are big ones; in census data there are more villages than towns, more towns than cities, and more cities than metropolises; in history there have been more minor conflicts than great wars.

In a landmark, first-ever study on the distribution of relative sizes, this book discusses several real-life case studies such as those above in extensive detail, and presents three distinct explanations for the phenomenon. Readers of all disciplines and levels of expertise will find an easily accessible yet original exploration that concludes with a numerical quantification demonstrating precisely by how much the relatively small is more numerous than the relatively big.

Contents:

• About the Author
• Introduction
• Empirical Evidence of the Small Is Beautiful Phenomenon:
  • The Small Is Beautiful Phenomenon
  • Small Life Forms Decisively Outnumber Big Creatures
  • Small Molecules Outnumber Big Molecules in the Chemical World
  • Small Atoms Outnumber Big Atoms in the Composition of Chocolate
  • Slow Spinning Pulsars Outnumber Fast Ones Throughout the Universe
  • Small Planets Outnumber Big Planets in the Milky Way Galaxy
  • Small Rivers Outnumber Big Rivers Worldwide
  • Small Stars Outnumber Big Stars in the Universe
  • Small Cities and Towns Outnumber Big Metropolises
  • Small Wars Outnumber Big Wars in History
  • The Poor Vastly Outnumber the Rich in Global Wealth Distribution
  • Small Companies Outnumber Big Corporations
  • Small Bills Outnumber Big Bills in Revenue and Expense Data
  • Cheap Items Outnumber Expensive Items in Catalogs and Price Lists
  • Small House Numbers Outnumber Big House Numbers in Address Data
  • Global Carbon Dioxide Pollution Data in Favor of the Small
  • Small Families Outnumber Big Families Regarding Number of Children
  • Small Meteorites Outnumber Big Meteorites
  • Small Volcano Eruptions Outnumber Big Volcano Eruptions
  • Small Countries Outnumber Big Countries
  • Small Prime Numbers Slightly Outnumber Big Prime Numbers
  • Four-Dice Multiplication Game at the Casino in Favor of the Small
  • Three-Dice Selection Game at the Casino in Favor of the Small
  • Proper Criteria for Sizes Balancing the Chances of Big and Small
  • Exceptions and Counter Examples to the Phenomenon
  • Seven-Dice Addition Game at the Casino in Favor of the Medium
  • Beauty Is in the Eye of the Beholder
• Causes and Explanations of the Phenomenon:
  • Partitioning as a Cause of the Small Is Beautiful Phenomenon
  • Random Dependent Partition Is Always in Favor of the Small
  • Random Real Partition Is Always in Favor of the Small
  • Numerous and Distinct Parts in Partitions Are Necessary Conditions
  • Conclusion: Random Partitions and the Small Is Beautiful Phenomenon
  Multiplication Processes Lead to the Small Is Beautiful Phenomenon
  • Exponential Growth Series and the Small is Beautiful Phenomenon
  • Explanation of the Small Is Beautiful Phenomenon in Growth Series
  • The Fibonacci Series and the Small Is Beautiful Phenomenon
  • Growth Model for Planets, Stars, Cities, and Bank Accounts
  • Data Aggregation Leads to the Small Is Beautiful Phenomenon
  • Three-Dice Selection Game as Data Aggregation or Probability Scheme
  • Thirty-Dice Selection Game Is Consistently in Favor of the Small
  • Chains of Statistical Distributions and the Small Is Beautiful Phenomenon
  • The Prevalence of the Lognormal Distribution Favoring the Small
  • The Prevalence of the Exponential Distribution Favoring the Small
• Discovery of a Size Pattern: The General Law of Relative Quantities:
  • The Quest for Numerical Consistency in All Data Sets
  • Division of Data Among Small, Medium, and Big Sizes
  • Constructing a Set of Infinitely Expanding Histograms
  • Numerical Consistency in Bin Schemes for 15 Real-Life Data Sets
  • The Quest for the General Mathematical Expression for All D&F Cases
  • The Postulate on Relative Quantities
  • Application of the Postulate via Generic Bin Schemes on K/X
  • The General Law of Relative Quantities (GLORQ)
  • Saturation in GLORQ-Inducing Processes Precludes Extreme Skewness
  • GLORQ is Number System Invariant
• Benford's Law as Direct Consequence of the General Law:
  • The Physical Clues Leading to the Discovery of Benford's Law
  • The First Digit on the Left Side of Numbers
  • Benford's Law and the Predominance of Low Digits
  • Empirical Tests and Confirmation of Compliance with Benford's Law
  • The Theoretical Clues Leading from Benford's Law to GLORQ
  • The Base Invariance Principle in Benford's Law
  • The Scale Invariance Principle in Benford's Law
  • Integral Powers of Ten (IPOT)
  • Benford's Law as a Special Case and Direct Consequence of GLORQ
  • Benford and GLORQ as a Subset of the Small Is Beautiful Phenomenon
  • Physical Order of Magnitude of Data (POM)
  • A Robust Measure of Physical Order of Magnitude (CPOM)
  • Two Essential Requirements for Benford and GLORQ Behavior
  • Sum of Squared Deviation Measure (SSD)
  • A Critique on the Mixture of Distributions Model in Benford's Law
  • The Random and the Deterministic Flavors in Benford's Law
  • Tugs of War between Addition and Multiplication of Random Variables
  • Quantitative Partition Models and Benford's Law
  • Chains of Statistical Distributions Are Nearly Always Benford
  • Development Pattern within the Expanding Histograms of Benford
• Appendices:
  • Conceptual Explanation of Histograms
  • Conceptual Explanation of Powers and Logarithms
• Glossary of Frequently Used Abbreviations
• Bibliography
• Index

Readership: The book is suitable for professionals and experts in the fields of mathematics, statistics, and the sciences, as well as university students of these disciplines. It is also suitable for the interested layman, non-expert, and educated general public.

Produkt-Details

• Erscheint lt. Verlag: 19.8.2025
• Verlagsort: Singapore
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Schlagworte: Benford's Law • Data • Data Asymmetry • Data Pattern • data set • Data Skewness • Data Symmetry • digits • histogram • Information • Probability • quantities • Relative Quantities • sizes • Statistical Density • statistical distribution • Statistics
• ISBN-10: 981-98-0184-2 / 9819801842
• ISBN-13: 978-981-98-0184-8 / 9789819801848