"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 information theorists refer to a group of scientists and researchers from Russia who have made significant contributions to the field of information theory and related areas such as coding theory, cryptography, and data transmission. Notable figures in this area include: 1. **Andrey Kolmogorov**: Although primarily known for his work in probability theory and statistics, Kolmogorov's concepts have deep implications for information theory, particularly in terms of randomness and information content.

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 cryptographers are individuals from Russia who specialize in the study and practice of cryptography, which is the science of securing communication and information by transforming it into a secure format. This field encompasses a variety of techniques and methodologies for encoding messages, ensuring confidentiality, integrity, authentication, and non-repudiation. Historically, Russia has a prominent role in the development of cryptographic techniques and cryptographic theory, especially during the Cold War era, when secure communication was vital for national security.

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.

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.

The term "Mathematicians from Saint Petersburg" typically refers to the influential mathematicians who have come from Saint Petersburg, Russia, and contributed significantly to various fields of mathematics. Saint Petersburg has a rich history of mathematical research and education, especially through institutions such as Saint Petersburg State University, the Leningrad Mathematical Society, and the Steklov Institute of Mathematics.

"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.

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.

Universal instantiation is a rule of inference in formal logic that allows one to derive a specific instance from a universally quantified statement. In simple terms, if something is true for all members of a certain set (as stated by a universal quantifier), one can conclude that it is true for any particular member of that set.

Universal generalization is a principle in formal logic and mathematics that allows one to deduce a universally quantified statement from a particular case or a set of cases.

Transposition in logic refers to a specific form of argument or inference that involves the rearrangement of a conditional statement.

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).

A structural rule is a concept commonly used in formal systems, logic, and various disciplines like linguistics and mathematics. It refers to a guideline or principle governing the relationships and organization of various components within a structure. Here are some contexts where structural rules might apply: 1. **Logic**: In formal logic, structural rules are used to manipulate and transform statements in a proof system.

SLD resolution, or **Selective Linear Definite clause resolution**, is a key concept in the field of logic programming and automated theorem proving. It is a refinement of the resolution principle that is used to infer conclusions from a set of logical clauses. SLD resolution specifically applies to definite clauses, which are expressions in propositional logic or predicate logic that have a specific format.

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.

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).

Negation Introduction, often abbreviated as "¬I" or "NI," is a rule in formal logic, specifically in natural deduction systems. It is used to derive a negation (not) of a proposition based on a contradiction that arises from the assumption of that proposition. The rule can be summarized as follows: 1. **Assume the Proposition (P)**: You assume that a certain proposition \( P \) is true.

Negation as failure is a concept primarily used in logic programming and non-monotonic reasoning, notably in the field of artificial intelligence and computational logic. It is a way of handling negation in a way that is consistent with the principle of closed world assumption (CWA). In classical logic, a statement can either be true or false, and the truth of a statement can be proven with evidence. However, in many practical applications, we often deal with incomplete knowledge about a system or domain.

Modus tollens is a valid form of logical reasoning that can be summarized as follows: If we have two statements: 1. If \( P \) then \( Q \) (this is a conditional statement). 2. Not \( Q \) (the negation of the second part of the conditional). From these two statements, we can conclude: 3. Therefore, not \( P \) (the negation of the first part of the conditional).