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Natural density. In number theory, natural density, also referred to as asymptotic density or arithmetic density, is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval [1, n] as n grows large.
Population density is the number of people per unit of area, usually transcribed as "per square kilometer" or square mile, and which may include or exclude, for example, areas of water or glaciers. Commonly this is calculated for a county, city, country, another territory or the entire world . The world's population is around 8,000,000,000 [3 ...
Density ( volumetric mass density or specific mass) is a substance's mass per unit of volume. The symbol most often used for density is ρ (the lower case Greek letter rho ), although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume: [1] where ρ is the density, m is the mass, and V is the volume ...
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is. which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer.
The area density (also known as areal density, surface density, superficial density, areic density, mass thickness, column density, or density thickness) of a two-dimensional object is calculated as the mass per unit area. The SI derived unit is the kilogram per square metre (kg·m −2 ). A related area number density can be defined by ...
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...
The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected. For example, the following list of seven numbers, has the median of 6, which is the fourth value.
Szemerédi's theorem. In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured [1] that every set of integers A with positive natural density contains a k -term arithmetic progression for every k. Endre Szemerédi proved the conjecture in ...