Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. They provide important information about the energy, shape, and orientation of atomic orbitals, as well as the spin of the electrons. There are four principal quantum numbers used to describe electrons in atoms: 1. **Principal Quantum Number (n)**: This quantum number indicates the energy level and size of the orbital. It can take positive integer values (1, 2, 3, ...).

The Shields parameter, often denoted by the Greek letter \( \tau^* \), is a dimensionless quantity used in sediment transport and fluid mechanics to characterize the initiation of sediment motion under flow conditions. It quantifies the ratio of the shear stress acting on the sediment bed to the gravitational forces acting on the sediment particles.

The Weber number (We) is a dimensionless quantity used in fluid mechanics to estimate the relative importance of inertial forces to surface tension forces in a flowing fluid. It is particularly useful in the study of interfaces, such as between liquids or between a liquid and a gas, where surface tension plays a significant role.

A centered dodecahedral number is a type of figurate number that represents a three-dimensional shape called a dodecahedron, which has 12 faces, each of which is a regular pentagon. Centered dodecahedral numbers correspond to a configuration of points arranged in a way that includes a central point, with additional layers of points forming a dodecahedral shape around that center.

A centered icosahedral number is a specific type of figurate number that represents a three-dimensional shape known as an icosahedron, which is a polyhedron with 20 triangular faces. In mathematical terms, centered icosahedral numbers extend the concept of triangular numbers into three-dimensional space.

A centered octagonal number is a type of figurate number that represents a pattern of dots arranged in an octagonal shape. The formula to find the nth centered octagonal number is given by: \[ C_n = 3n^2 - 3n + 1 \] where \(C_n\) is the nth centered octagonal number and \(n\) is a positive integer (1, 2, 3, ...).

A centered triangular number is a specific type of figurate number that represents a triangular figure with a center point. Centered triangular numbers are generated by arranging dots in the shape of a triangle with a single dot in the center and additional layers of dots forming outer triangular frames.

The Lucas sequence is a series of numbers that is similar to the Fibonacci sequence.

A gnomon is a geometric figure used primarily in the context of sundials and can also refer to a specific part of a shape in geometry. 1. **Sundial Context**: In sundials, the gnomon is the part that casts a shadow, typically a vertical rod or a triangular blade positioned at an angle. The shadow it casts is used to indicate the time of day by aligning with markings that represent the hours.

A **Polite number** is a positive integer that can be expressed as the sum of two or more consecutive positive integers. For example, the number 15 can be expressed as: - 7 + 8 - 4 + 5 + 6 In contrast, the only positive integers that cannot be classified as polite numbers are the powers of 2 (such as 1, 2, 4, 8, 16, etc.).

Statistical indicators are quantitative measures that provide insights into various aspects of data and help in analyzing patterns, trends, and relationships within that data. They are often used in research, economics, social sciences, healthcare, and many other fields to summarize information, facilitate decision-making, and assess performance. Here are some key characteristics and types of statistical indicators: ### Characteristics: 1. **Quantitative**: They provide numerical data that can be analyzed statistically.

The Education Index is a composite measure used to assess the level of educational attainment and the quality of education in a particular region or country. It is part of the Human Development Index (HDI) and serves to provide insights into the overall development and well-being of a population. The Education Index typically comprises two key indicators: 1. **Mean Years of Schooling**: This measures the average number of years of education received by people aged 25 and older in a given population.

The G-index is a metric used to assess the productivity and citation impact of academic publications. It is an enhancement of the more commonly known h-index. The G-index was proposed by Leo Egghe in 2006 and aims to address some of the limitations of the h-index. ### Definition: The G-index is defined such that a researcher has a G-index of "g" if they have published "g" papers that have each received, on average, at least "g" citations.

The Process Performance Index (Ppk) is a statistical measure used to evaluate the capability of a manufacturing process. It quantifies how well a process can produce output that meets specification limits. Ppk is particularly useful in situations where the process is not centered between the specification limits, as it takes into account both the process variability and the mean of the process output. **Key points about Ppk:** 1.

The Renkonen similarity index is a measure used to quantify the similarity between two or more samples based on the presence and abundance of species or other categorical data. It was developed in the context of ecological studies to compare community compositions.

An Achilles number is a positive integer that is a powerful number but not a perfect power. A powerful number is defined as a number \( n \) such that in its prime factorization, every prime number \( p \) appears with an exponent of at least 2. In contrast, a perfect power is a number of the form \( m^k \) where \( m \) and \( k \) are positive integers and \( k \geq 2 \).

An arithmetico-geometric sequence is a sequence in which each term is generated by multiplying an arithmetic sequence by a geometric sequence. In simple terms, it combines the elements of arithmetic sequences (which have a constant difference between consecutive terms) and geometric sequences (which have a constant ratio between consecutive terms).

The Beatty sequence is a sequence of numbers that can be derived from the mathematical concept of filling the real line with two sequences whose terms are the floor functions of the multiples of two irrational numbers.

Betrothed numbers are a pair of positive integers \( (m, n) \) such that each number plus one equals the sum of the other number's proper divisors. In formal terms, if \( \sigma(n) \) denotes the sum of the divisors of \( n \), then \( m \) and \( n \) are betrothed if the following conditions hold: 1. \( \sigma(m) - m = n + 1 \) 2.

A "cake number" refers to a concept in combinatorial mathematics related to how many pieces a cake can be divided into with a given number of straight cuts. Specifically, the "cake number" is defined as the maximum number of pieces into which a cake can be divided using \( n \) straight cuts in three-dimensional space.