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Resolution is a crucial rule of inference in formal logic and propositional logic, primarily used in automated theorem proving and logic programming. It is based on the concept of combining clauses to produce new ones, ultimately leading to a proof of a given statement or demonstrating a contradiction. ### Key Concepts of Resolution: 1. **Clauses**: In propositional logic, a clause is a disjunction of literals (where a literal is an atomic proposition or its negation).

The "Rule of Replacement" is a concept used in logic, particularly in propositional logic and formal proofs. It refers to the principle that certain logical expressions or statements can be replaced with others that are logically equivalent without changing the truth value of the overall expression. Essentially, if two statements are equivalent, one can replace the other in any logical argument or proof without affecting the validity of the conclusion.

In logic, a **tautology** is a statement or formula that is true in every possible interpretation, regardless of the truth values of its components. In other words, it is a logical expression that cannot be false. Tautologies are important in propositional logic and are often used as the basis for proving other statements. One common example of a tautology is the expression \( p \lor \neg p \) (where \( p \) is any proposition).

Kazan, the capital of the Republic of Tatarstan in Russia, has a rich educational and cultural history, particularly in the field of mathematics. Several prominent mathematicians have emerged from this city, contributing significantly to various branches of mathematics. One of the most notable figures associated with Kazan is Nikolai Lobachevsky, who is considered a founder of non-Euclidean geometry.

"Mathematicians from Moscow" typically refers to a group of notable mathematicians who have emerged from the Moscow mathematical community, particularly during the 20th century. This community has had a significant influence on various fields of mathematics, including but not limited to, functional analysis, number theory, and algebra. The Moscow school of mathematics is recognized for its rigorous training methods, often emphasizing problem-solving and theoretical foundations.

The Russian Empire produced many notable mathematicians who made significant contributions to various fields of mathematics. Here are some of the prominent mathematicians from the Russian Empire: 1. **Leonhard Euler (1707-1783)** - Although originally from Switzerland, Euler spent a significant portion of his life in St. Petersburg, Russia. He made profound contributions to many areas of mathematics, including calculus, graph theory, and number theory.

Russian bioinformaticians are scientists or researchers from Russia who specialize in the field of bioinformatics. Bioinformatics is a multidisciplinary field that combines biology, computer science, and mathematics to analyze and interpret biological data, such as genetic sequences, protein structures, and metabolic pathways. In Russia, bioinformaticians may work in various settings, including academic research institutions, universities, healthcare organizations, and biotechnology companies.

Russian geodesists are specialists in the field of geodesy in Russia, which is the science of measuring and understanding Earth's geometric shape, orientation in space, and gravity field. This discipline involves precise measurements of large areas of the Earth's surface, creating and maintaining coordinate systems, and conducting tasks related to mapping, surveying, and navigation.

"Russian mathematician stubs" usually refers to a specific kind of entry on Wikipedia that is related to a mathematician from Russia but does not contain a lot of detailed information. In Wikipedia terminology, a "stub" is a page that is very short or incomplete and can be expanded with more content. These stubs typically contain at least some basic information about the mathematician, such as their name, birth and death dates, and perhaps a few key contributions or areas of research.

Russian statisticians refer to individuals in Russia who specialize in the collection, analysis, interpretation, and presentation of quantitative data. Statisticians in Russia, like their counterparts worldwide, work in various fields including economics, healthcare, social sciences, and more. They may be involved in tasks such as designing surveys, analyzing data sets, and applying statistical models to help make informed decisions or policy recommendations.

Russian women mathematicians have made significant contributions to mathematics and related fields over the years, often overcoming societal and institutional barriers. Notable figures include: 1. **Maria Stepanova** - One of the first women to receive a degree in mathematics in Russia, she contributed to differential equations and mathematical analysis. 2. **Sofia Kovalevskaya** - The first woman in modern Europe to earn a doctorate in mathematics. She made important contributions to analysis, differential equations, and mechanics.

Aleksandr Olshansky is a mathematician noted for his contributions to various fields, including functional analysis, operator theory, and the study of Banach spaces. His work has involved the exploration of different mathematical constructs and properties, impacting the understanding of these areas within mathematics.

As of my last knowledge update in October 2023, Aleksey Ismailov does not appear to be a widely recognized public figure, event, or topic. It's possible that he could be a professional in a specific field, an emerging personality, or a private individual who has gained attention after that date.

As of my last knowledge update in October 2023, there is no widely known public figure or entity specifically named "Alexander Abrosimov." It's possible that he could be a private individual, a lesser-known figure, or someone who has gained recognition more recently.

Alexander Dyakonov is a notable figure in the field of physics, particularly known for his contributions to the study of semiconductor physics and spintronics. He has worked on various topics, including the phenomenon of spin relaxation and the theoretical foundations of spin transport in materials. Dyakonov is also credited with the development of concepts related to spin Hall effect, which involves the generation of a transverse spin current in response to an applied electric field.

Alexander Kirillov Jr. is a professional baseball player, known for being a first baseman and outfielder. He was born on September 23, 1997, in the United States and is a member of the Minnesota Twins organization in Major League Baseball (MLB). Kirillov was selected by the Twins in the 1st round of the 2016 MLB Draft and has been recognized for his hitting ability and offensive potential.

As of my last knowledge update in October 2021, there isn't a widely recognized public figure or concept specifically named Alexander Razgulin. It's possible that he could be a private individual, a lesser-known figure, or someone who has gained prominence after that date.

Elena Deza is a character from the popular Japanese visual novel and dating simulation game series known as "Danganronpa." She is one of the students who participates in the game, which revolves around a deadly survival game where characters must solve murders and face moral dilemmas. Elena is often characterized by her unique personality traits and plays a significant role in the story's development.

As of my knowledge cutoff in October 2023, there is no widely recognized figure or concept known as "Vladimir Bening." It's possible that you may be referring to a less-known individual, a fictional character, or there may be a typo in the name.

Andrey Gonchar is a mathematician known for his work in areas such as approximation theory and numerical analysis. He has made contributions to the understanding of polynomial approximation, functional analysis, and the theory of approximations in various function spaces. Gonchar's research often focuses on how functions can be approximated by simpler functions, a key topic in both pure and applied mathematics.