The Tarrasch Rule, named after the German chess player and theorist Dawid Tarrasch, is a guideline in pawn structure in chess. It states that in general, pawns on the fourth rank (for White, the rank is the 4th; for Black, it’s the 5th) are stronger than pawns that are advanced further.

The Two Knights Endgame is a situation in chess where only two knights are left on the board for one side, typically against a lone king or another minor piece (often a pawn). This endgame is distinct because it usually represents a challenging scenario for the player with the two knights, as they cannot checkmate a lone king without the assistance of a pawn or another piece. In its purest form, the most common scenario involves one player having two knights and the other player having just a king.

The term "wrong rook pawn" typically refers to a specific scenario in chess endgames, particularly in king and pawn endgames. It describes a situation where a pawn is on the corner file (a-file or h-file) of the board, and it is important because it can affect the ability to win or draw the game depending on the position of the opposing king.

Uzbek astronomers have made significant contributions to the field of astronomy throughout history, particularly during the Islamic Golden Age. One of the most notable figures is Ulugh Beg (1394–1449), an Uzbek ruler and astronomer who founded an important observatory in Samarkand. His work included the compilation of a star catalog and the development of astronomical tables that improved the accuracy of celestial measurements.

Symmetric fair cake-cutting refers to a method of dividing a "cake" (or any divisible resource) in such a way that all participants perceive the division to be fair and equitable, ensuring symmetry in their allocations. The concept stems from the fields of economics and game theory, where fairness in resource allocation is crucial. In symmetric fair cake-cutting: 1. **Symmetry** means that if two participants start with the same preferences and information, they should receive identical portions of the cake.

Map segmentation is a process used in geographic information systems (GIS), image processing, and various fields of computer vision to divide a map or an image into distinct regions or segments based on specific criteria. The goal of map segmentation is to facilitate analysis, interpretation, and understanding of spatial data by reducing complexity and enhancing relevant features.

Online fair division refers to the problem of allocating resources or dividing goods among agents in a dynamic environment where the agents arrive and make requests over time. In contrast to traditional fair division, where all agents and items are present from the beginning, online fair division must consider situations where agents show up sequentially, and decisions need to be made without the knowledge of future arrivals or requests.

Game balance refers to the process of ensuring that all elements of a game—such as characters, abilities, weapons, items, or mechanics—are designed and adjusted in a way that creates a fair, enjoyable, and challenging experience for players. Effective game balance aims to prevent any single aspect of the game from being overwhelmingly powerful or weak, which could lead to frustration or diminish the enjoyment of the game.

A circular section, often referred to in geometry, describes a part of a circle or the two-dimensional shape created by cutting through a three-dimensional object (like a sphere) along a plane that intersects the object in such a way that the intersection is a circle.

A **spherical conic** is a curve that can be defined on the surface of a sphere, analogous to conic sections in a plane, such as ellipses, parabolas, and hyperbolas. While traditional conic sections are produced by the intersection of a plane with a double cone, spherical conics arise from the intersection of a sphere with a plane in three-dimensional space.

In topology, a surface is a two-dimensional topological space that can be defined informally as a "shape" that locally resembles the Euclidean plane. More specifically, a surface is a manifold that is two-dimensional, meaning that every point on the surface has a neighborhood that is homeomorphic (topologically equivalent) to an open subset of \(\mathbb{R}^2\). ### Key Features of Surfaces: 1. **Local vs.

Astranis is a company focused on developing small, affordable satellites that provide internet connectivity, particularly for underserved regions. Founded in 2015 and based in San Francisco, Astranis aims to bridge the digital divide by leveraging technology to offer reliable internet access in areas where traditional infrastructure may be lacking or too costly to deploy. Their satellites are designed to be smaller and more cost-effective compared to traditional communication satellites, making it easier and more economical to deploy broadband services.

Molecular geometry refers to the three-dimensional arrangement of atoms in a molecule. It describes the shape of the molecule formed by the positions of the bonded atoms and the angles between them. Understanding molecular geometry is crucial in chemistry because it influences properties such as polarity, reactivity, phase of matter, color, magnetism, biological activity, and many other characteristics of molecules.

Thales of Miletus was an ancient Greek philosopher, mathematician, and astronomer, born around 624 BCE in Miletus, a city in Ionia (modern-day Turkey). He is often considered one of the founding figures of Western philosophy and is one of the earliest known pre-Socratic philosophers. Thales is particularly credited with shifting the focus of Greek thought from mythological explanations of the world to rational ones based on observation and inquiry.

Gerd Faltings is a German mathematician known for his significant contributions to number theory and arithmetic geometry. He was born on July 28, 1954, and is best known for his work on the theory of Diophantine equations, particularly for proving the Mordell conjecture in the 1980s.

Wei Zhang is a prominent mathematician known for his contributions to number theory, specifically in the areas of automorphic forms and representation theory. He has made significant advances in understanding the connections between number theory and other areas of mathematics, including algebraic geometry and harmonic analysis. Zhang's work includes investigations into the Langlands program, which seeks to relate number theory and representation theory through a series of conjectures and theories.

August Ferdinand Möbius was a German mathematician and astronomer born on November 17, 1790, and he passed away on September 26, 1868. He is best known for his contributions to topology, particularly for the introduction of the Möbius strip, a surface with only one side and one boundary.

The Huygens–Fresnel principle is a fundamental concept in the field of wave optics that describes how waves propagate and interfere. Named after Dutch physicist Christiaan Huygens and later expanded by the French physicist Augustin-Jean Fresnel, the principle provides a way to analyze the propagation of wavefronts, such as light waves.

Friedrich Otto Rudolf Sturm, also known simply as Fritz Sturm, is a renowned figure in the field of mathematics and is known for his contributions to various areas including differential equations, control theory, and numerical analysis.