Fractional Pareto efficiency is a concept that extends the traditional notion of Pareto efficiency in economics and optimization theory. While a traditional Pareto efficient allocation occurs when it is impossible to make one individual better off without making another individual worse off, fractional Pareto efficiency introduces a more nuanced approach. In fractional Pareto efficiency, one assesses configurations where some individuals may be partially made better off without entirely disadvantaging others.

Ordinal Pareto efficiency is a concept in economics and social choice theory that builds upon the idea of Pareto efficiency in a way that incorporates ordinal preferences rather than cardinal utility. ### Key Concepts: 1. **Pareto Efficiency**: A state is Pareto efficient if there is no other allocation of resources that can make at least one individual better off without making someone else worse off. In other words, a distribution cannot be improved for one individual without degrading the situation for another.

Asymptote can refer to two primary concepts: one in mathematics and the other as a programming language for technical graphics. 1. **Mathematical Concept**: In mathematics, an asymptote is a line that a curve approaches as it heads towards infinity. Asymptotes can be horizontal, vertical, or oblique (slant). They represent the behavior of a function as the input or output becomes very large or very small.

Line coordinates typically refer to the mathematical representation of a line in a coordinate system, such as a two-dimensional (2D) or three-dimensional (3D) space. The precise meaning can vary based on context, but here are some common interpretations: ### 1.

Hyperbolic geometry is a non-Euclidean geometry that arises from altering Euclid's fifth postulate, the parallel postulate. In hyperbolic geometry, the essential distinction is that, given a line and a point not on that line, there are infinitely many lines through that point that do not intersect the original line. This contrasts with Euclidean geometry, where there is exactly one parallel line that can be drawn through a point not on a line.

Inversive geometry is a branch of geometry that focuses on properties and relations of figures that are invariant under the process of inversion in a circle (or sphere in higher dimensions). This type of transformation maps points outside a given circle to points inside the circle and vice versa, while points on the circle itself remain unchanged. Key concepts and characteristics of inversive geometry include: 1. **Inversion**: The basic operation in inversive geometry is the inversion with respect to a circle.

Minhyong Kim is a notable mathematician specializing in number theory and arithmetic geometry. He is known for his work in several areas, including the study of Diophantine geometry, the arithmetic of abelian varieties, and various aspects of algebraic geometry and number theory. His research includes contributions to understanding rational points on algebraic varieties and connections between arithmetic and geometry. In addition to his research, Minhyong Kim is involved in mathematics education and outreach, promoting mathematics to a broader audience.

Dicaearchus was an ancient Greek philosopher and geographer, active in the 4th century BCE. He was a pupil of Aristotle and a member of the Peripatetic school. Dicaearchus is best known for his work in geography and for his attempts to systematically study the earth and its regions, as well as for his contributions to political theory and ethics. One of his notable contributions was his work on the division of the earth into regions and the description of various geography-related topics.

Menelaus of Alexandria was a Greek mathematician and astronomer who lived during the 1st century AD. He is best known for his work in geometry and spherical astronomy. One of his most significant contributions is the formulation of Menelaus' theorem, which relates to the geometry of triangles and is particularly important in the study of spherical triangles.

Claude Ambrose Rogers is not widely recognized as a public figure or a notable entity in historical or contemporary contexts, based on information available up to October 2023.

Eric Harold Neville was a British astronomer known for his contributions to the field of astronomy and astrophysics. He was particularly recognized for his work in photometry and the study of celestial objects. Neville's research helped enhance the understanding of star brightness variations and the physical properties of various astronomical bodies. Apart from his scientific contributions, he may also be remembered for his involvement in education and outreach within the astronomical community.

Thomas Willmore is associated with mathematics, specifically in the field of differential geometry. The term "Willmore" often refers to the Willmore energy or Willmore surfaces, which are concepts related to the study of surfaces in three-dimensional space. The Willmore energy of a surface is a measure of its bending and is defined as the integral of the square of the mean curvature over the surface. Willmore surfaces are those that minimize this energy.

Huygens is a space probe that was part of the Cassini-Huygens mission, which was a collaborative project between NASA, the European Space Agency (ESA), and the Italian Space Agency (ASI). Launched on October 15, 1997, Huygens was designed to study Saturn and its moons, particularly Titan, Saturn's largest moon.

Christiaan Huygens, a prominent 17th-century Dutch astronomer, mathematician, and physicist, has several entities named in his honor, reflecting his contributions to science. Here’s a list of some notable things named after him: 1. **Huygens (satellite)** - A probe named after him that was part of the Cassini-Huygens mission to Saturn and its moon Titan. The Huygens probe landed on Titan in 2005.

Physical optics is a branch of optics that focuses on the wave nature of light and its interactions with matter. Unlike geometrical optics, which primarily deals with the propagation of light in terms of rays and prisms, physical optics examines phenomena such as interference, diffraction, and polarization, which cannot be adequately explained by ray optics alone. Key concepts in physical optics include: 1. **Wave Nature of Light**: Light is treated as a wave, which means it is subject to wave phenomena.

Clark Kimberling is an American mathematician known primarily for his work in various fields of mathematics, including geometry and number theory. He is particularly noted for his contributions to the study of the properties of geometric figures and mathematical relationships. One of his notable contributions is the Kimberling's list of triangle centers, which enumerates specific points associated with triangles, such as the centroid, incenter, circumcenter, and many others.

Erwin Lutwak is a noteworthy mathematician known for his contributions to various fields, particularly in geometry and combinatorics. One of his significant contributions is the development of the Lutwak's Surface Area Measure and the related concepts in convex geometry. His work often focuses on the geometric properties of convex bodies and their implications in analysis and optimization.

G. B. Halsted typically refers to George Washington Halsted (1853–1922), an American mathematician and a prominent figure in the field of mathematics during the late 19th and early 20th centuries. He is particularly known for his work in topology and for his contributions to the theory of functions and differential equations. Halsted also played a significant role in the development of mathematical education in the United States and was involved in various mathematical societies.