The Dirac-von Neumann axioms, also known as the axioms of quantum mechanics, provide a formal framework to describe the mathematical structure of quantum mechanics. They were formulated by physicist Paul Dirac and mathematician John von Neumann in the early 20th century and establish the foundation for the theory. The axioms can be summarized as follows: 1. **State Space**: The state of a physical system is described by a vector in a complex Hilbert space.

The Ottoman Empire, which lasted from approximately 1299 to 1922, was a vast and culturally diverse empire that spanned parts of Europe, Asia, and Africa. Throughout its history, the empire produced several notable mathematicians, particularly during the periods of its peak in the 16th century and during the Tanzimat era in the 19th century.

Logic puzzles are problems designed to test deductive reasoning and critical thinking through a structured set of clues or information. Typically, they involve a scenario where the solver must deduce the correct arrangement or relationship between different elements based on the given clues. These elements can include people, objects, places, or events. Logic puzzles come in various formats, such as: 1. **Grid Puzzles**: These involve creating a matrix or grid to help keep track of relationships and deductions.

A Fredholm module is a concept in the field of operator algebras, particularly in noncommutative geometry. It provides a framework to study and generalize certain properties of differential operators and topological spaces using algebraic and geometric methods. The concept was introduced by Alain Connes in his work on noncommutative geometry.

The fuzzy sphere is a mathematical concept arising in the field of noncommutative geometry, a branch of mathematics that studies geometric structures using techniques from functional analysis and algebra. It can be thought of as a "quantum" version of the ordinary sphere, where points on the sphere are replaced by a noncommutative algebra of operators.

The Lagrangian Grassmannian is a specific type of Grassmannian manifold that is associated with symplectic vector spaces. It can be understood as follows: 1. **Grassmannian Manifold**: In general, a Grassmannian \( G(k, n) \) is the space of all \( k \)-dimensional linear subspaces of an \( n \)-dimensional vector space. It has a rich structure and is a smooth manifold.

Menyhért Palágyi is known as a Hungarian philosopher and mathematician, particularly noted for his work in logic and the philosophy of mathematics. Born in the late 19th century, Palágyi contributed to the development of mathematical logic and was involved in discussions surrounding the foundations of mathematics. His work is recognized in the context of Hungarian intellectual history, and he is sometimes associated with the broader movements in philosophy and mathematics during his time.

Noncommutative quantum field theory (NCQFT) is an extension of traditional quantum field theory where the space-time coordinates do not commute. In standard quantum field theory, the coordinates of spacetime are treated as classical operators that commute with each other. However, in noncommutative geometries, the fundamental idea is that the coordinates of spacetime satisfy a noncommutative algebra, which means that the product of two coordinates may depend on the order in which they are multiplied.

In mathematical physics and the theory of quantization, the statement that "quantization commutes with reduction" refers to a relationship between two processes: the reduction of symmetries in a classical system and the process of quantizing that system. To unpack this concept: 1. **Symmetry Reduction**: In classical mechanics, many systems possess symmetries described by a group of transformations (e.g., rotations, translations).

In the theory of computation, theorems are mathematical propositions that have been proven to be true based on previously established axioms and other theorems. This area of theoretical computer science deals with the fundamental aspects of computation, including what problems can be computed (computability), how efficiently they can be solved (complexity), and the limits of computation.

21st-century mathematicians come from diverse nationalities and regions, reflecting the global nature of the field. While it's challenging to provide an exhaustive list or a detailed breakdown, some notable mathematicians, categorized by their nationalities, include: ### United States - **John Nash** - known for game theory. - **Terence Tao** - renowned for contributions in harmonic analysis and partial differential equations.

The University of Pennsylvania's Department of Mathematics is part of the School of Arts and Sciences and is known for its strong emphasis on both pure and applied mathematics. The faculty includes distinguished mathematicians who specialize in various areas such as algebra, analysis, geometry, topology, and mathematical logic, among others. The department offers undergraduate and graduate programs that focus on both theoretical and practical aspects of mathematics.

The 10th century saw various mathematicians from different cultures, primarily focusing on Islamic scholars, as this period coincided with the Islamic Golden Age. Here are some prominent mathematicians by nationality or cultural background during the 10th century: 1. **Persian**: - **Al-Khwarizmi** (circa 780-850) was earlier than the 10th century but heavily influenced later scholars. He is known for his work in algebra and algorithms.

The 11th century saw significant contributions to mathematics from various regions and cultures, though it is important to note that the mathematical landscape was quite different from today. Here are some notable mathematicians from the 11th century by nationality: 1. **Arab Mathematicians:** - **Al-Khwarizmi** (though he lived earlier, his works were influential in the 11th century): He is often referred to as the "father of algebra.

Gyula Farkas was a Hungarian natural scientist known for his contributions to various fields, particularly in the domain of biology and physics. He is recognized for his work in biophysics and his studies related to the interactions between living organisms and physical processes. Farkas made significant contributions to understanding the properties of biological systems and was involved in research that bridged the gap between natural sciences and technology.

The 14th century saw significant developments in mathematics across various cultures and regions. Here are some notable mathematicians from that period by nationality: 1. **Italian:** - **Giovanni di Cascia**: Known for his work in arithmetic and for being one of the early Italian mathematicians to help in the development of accounting methods.

István Vincze is a Hungarian mathematician known for his work in the fields of combinatorics, graph theory, and discrete mathematics. Having published numerous papers and contributed to various aspects of mathematical research, Vincze's work likely focuses on theoretical frameworks and applications within these areas. For more specific details regarding his contributions, publications, and impact in the field, consulting academic databases or resources specific to mathematics might provide a more comprehensive overview.

Josip Plemelj (1873–1967) was a prominent Slovenian mathematician known for his contributions to various fields, including mathematical analysis and differential equations. He is particularly recognized for his work in the theory of functions of complex variables and for his development of what is now known as Plemelj formulas, which are important in potential theory and applied mathematics. Plemelj had a significant impact on the education of mathematicians in Slovenia and worked to promote mathematical research in the region.

Karl Zsigmondy was an Austrian chemist and physicist, best known for his contributions to the field of colloid chemistry and for being awarded the Nobel Prize in Chemistry in 1925. His work primarily focused on the behavior of colloids and the processes of dispersion and stabilization in colloidal systems. One of his significant achievements involves the study of colloid stability and the development of methods to analyze and characterize colloidal solutions.

The 5th century was a period of significant development in mathematics, particularly in the context of ancient Greece, India, and China. Here’s a breakdown of some notable mathematicians from this era by nationality: ### Greek 1. **Proclus (circa 410–485)**: A notable Neoplatonist philosopher who contributed to the understanding of Euclidean geometry and worked on the commentary of Euclid's "Elements.