Circular points at infinity are a concept from projective geometry, particularly relating to the projective plane and the study of lines and conics. In the context of projective geometry, the idea is to extend the usual Euclidean plane by adding "points at infinity," which allows us to treat parallel lines as if they meet at a point. In the case of conics, specifically circles, there are two points at infinity that are referred to as the "circular points at infinity.

Complex projective space, denoted as \(\mathbb{CP}^n\), is a fundamental concept in complex geometry and algebraic geometry. It is a space that generalizes the idea of projective space to complex numbers.

The term "point at infinity" can refer to different concepts depending on the context, particularly in mathematics and geometry. Here are a few interpretations: 1. **Projective Geometry**: In projective geometry, points at infinity are added to the standard Euclidean space to simplify certain aspects of geometric reasoning.

The projective linear group, denoted as \( \text{PGL}(n, F) \), is a fundamental concept in algebraic geometry and linear algebra. It is defined as the group of linear transformations of a projective space, and its structure relates closely to the field \( F \) over which the vectors are defined. Here's a more detailed explanation: ### Definition 1.

Projective space is a fundamental concept in both mathematics and geometry, particularly in the fields of projective geometry and algebraic geometry. It can be intuitively thought of as an extension of the concept of Euclidean space. Here are some key points to understand projective space: ### Definition 1.

A Schlegel diagram is a geometric representation of a polytope, which is a high-dimensional generalization of polygons and polyhedra. Specifically, it is a way to visualize a higher-dimensional object in lower dimensions, typically projecting a convex polytope into three-dimensional space. Essentially, a Schlegel diagram allows us to see the structure of a polytope by looking at a "shadow" of it, emphasizing its vertices and faces.

Tropical projective space is a concept arising in tropical geometry, which is a piece of mathematics that studies geometric structures and mathematical objects using a combinatorial and polyhedral approach. Tropical geometry replaces classical algebraic geometry with a framework where arithmetic operations are modified in a specific way, leading to a simpler geometrical interpretation akin to a combinatorial structure.

The VIPER (VLIW (Very Long Instruction Word) Processor) microprocessor is a type of architecture developed primarily in the 1990s at the European Organization for Nuclear Research (CERN) and other institutions. It was designed to handle complex computations particularly in high-energy physics applications, but its architecture can also be beneficial in various other computing contexts due to its ability to execute multiple instructions concurrently. **Key features of the VIPER microprocessor include:** 1.

The term "School of Chess" can refer to a couple of different concepts within the context of chess: 1. **Chess Schools or Academies**: These are institutions or organizations where individuals can receive formal training in chess. They typically offer lessons, coaching, and resources for players of all skill levels, from beginners to advanced players. Many of these schools focus on various aspects of the game, including strategy, tactics, openings, endgames, and tournament preparation.

Proof calculus, often referred to as proof theory, is a branch of mathematical logic that focuses on the structure and properties of formal proofs. It involves the study of different proof systems, which are formal systems that dictate how mathematical statements can be proven within a given logical framework. Key aspects of proof calculus include: 1. **Proof Systems**: These are structured frameworks that define rules for deriving theorems from axioms using logical inference.

A proof net is a concept from the field of linear logic, introduced by the logician Jean-Yves Girard in the 1990s. It serves as a geometric representation of proofs in linear logic, providing an alternative to traditional syntactic representations like sequent calculus or natural deduction. ### Key Features of Proof Nets: 1. **Linear Logic**: Proof nets are specifically tied to linear logic, a branch of logic that emphasizes the use of resources.

Redundant proof, often referred to in the context of mathematics and logic, involves demonstrating a statement or theorem using multiple proofs that reiterate the same underlying principles or reasoning. Essentially, one proof does not provide any new insights or alternative approaches but instead reaffirms what has already been established. In a broader context, redundancy in proofs can serve specific purposes: 1. **Verification**: It can help confirm the validity of a theorem or statement by showing that it can be proven in different ways.

The term "Laves graph" does not refer to a widely recognized concept in mathematics, graph theory, or any other standard academic discipline. However, it may be related to certain concepts in materials science, specifically Laves phases. Laves phases are types of intermetallic compounds that typically have a specific crystal structure and are significant in the study of alloys and solid materials.

A relatively compact subspace (or relatively compact set) is a concept from topology, specifically in the context of metric spaces or more generally in topological spaces. A subset \( A \) of a topological space \( X \) is said to be relatively compact if its closure, denoted by \( \overline{A} \), is compact.

Resolution proof compression by splitting is a technique used in the context of automated theorem proving, particularly in the area of propositional logic. The primary goal of this technique is to reduce the size of a resolution proof without losing the essential information that proves the target theorem. In a resolution proof, one derives a conclusion from a set of premises using the resolution rule, which is a rule of inference that allows the derivation of a clause from two clauses containing complementary literals.

Volterra spaces typically refer to function spaces associated with Volterra integral equations or to function spaces defined in the context of Volterra operators.

In mathematics, particularly in topology, compactness is a property that describes a specific type of space. A topological space is said to be compact if every open cover of the space has a finite subcover.

In topology, a space is called a **collectionwise normal space** if it satisfies a certain separation condition involving collections of closed sets.