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"Article proofs" typically refer to a stage in the academic publishing process where authors are provided with a formatted version of their manuscript, which is often referred to as a proof or galley proof. This version includes all the editorial revisions made after the original manuscript submission and allows authors to review the final layout, check for any typographical errors, and ensure that their work is accurately represented before the article is published in a journal.

"Articles containing proofs" typically refers to scholarly or academic articles that present formal proof for theorems or propositions in various fields, such as mathematics, computer science, logic, and statistics. These articles usually include a detailed explanation of the problem being addressed, the methodology used, and step-by-step reasoning leading to the conclusion.

Computer-assisted proofs are proofs in mathematics or formal logic that involve the use of computers to aid in the verification of the proof itself or to help find the proof. These proofs typically combine traditional mathematical reasoning with computational methods to handle large computations or complex combinatorial arguments that would be impractical or impossible to work through by hand. Key aspects of computer-assisted proofs include: 1. **Verification**: A computer can verify steps in a proof that are computationally intense.

Mathematical fallacies are errors or flaws in reasoning that lead to incorrect conclusions in mathematical arguments. These fallacies can arise from incorrect assumptions, misuse of algebraic principles, misleading interpretations, or logical errors. Awareness of these fallacies is important for developing critical thinking skills and ensuring that mathematical reasoning is sound.

Proof techniques are systematic methods used in mathematics and logic to establish the truth of given statements or propositions. Different techniques are suited for different types of assertions and can vary in complexity. Here are some common proof techniques: 1. **Direct Proof**: This involves proving a statement directly by a straightforward series of logical deductions from known truths, axioms, or previously established results.

A list of long mathematical proofs typically refers to significant proofs in mathematics that are known for their length, complexity, or intricate detail. Here are a few of the most famous lengthy proofs in mathematics: 1. **The Four Color Theorem**: Proven in 1976 by Kenneth Appel and Wolfgang Haken, the proof involved extensive computer calculations to show that any planar map can be colored using no more than four colors without adjacent regions sharing the same color.

A list of mathematical proofs typically refers to a collection of significant theorems, lemmas, corollaries, or propositions that have been proven within various fields of mathematics. These proofs can vary greatly in complexity and significance, from basic arithmetic properties to advanced concepts in topology or number theory.

The list of probabilistic proofs of non-probabilistic theorems includes various mathematical results that have been shown to hold true through probabilistic methods, even if they are not inherently probabilistic in nature. These proofs often use random processes or probabilistic techniques as tools to establish the truth of deterministic statements. Here are some notable examples: 1. **Probabilistic Method**: The general strategy of using probability theory to prove the existence of a combinatorial structure with certain properties.

A minimal counterexample is a specific type of counterexample that demonstrates that a certain statement or conjecture is false while also satisfying an additional criterion of minimality. In mathematical terms, a counterexample is an instance that disproves a given statement (for example, a theorem or conjecture).

The phrase "of the form" is often used in mathematics, science, and logic to describe a specific structure, pattern, or type of expression. It usually indicates that what follows is a general representation or formula that can encompass a variety of specific instances or examples. For example: 1. In algebra, you might say "the solutions are of the form \( ax + b = 0 \)," meaning that the solutions to this equation fit within the structure defined by that format.

A Probabilistically Checkable Proof (PCP) is a concept from theoretical computer science, particularly in the field of computational complexity and proof systems. A PCP is a type of proof for a mathematical assertion that can be verified by a probabilistic algorithm with certain characteristics: 1. **Probabilistic Verification**: The verifier, instead of reading the entire proof, can check the proof using random bits.

Proof by intimidation is a type of argument or reasoning where someone tries to convince others of the validity of a statement or idea not through logical proof or evidence, but by using authority, confidence, or the specter of intimidation. Essentially, the person making the claim uses their position, personality, or aggressive demeanor to pressure others into accepting their assertion without critically examining it.

"Proof without words" refers to a type of mathematical argument that conveys a proof or a mathematical result using visual reasoning or intuition rather than formal written explanations or symbolic manipulation. These proofs often employ diagrams, geometrical representations, or other visual aids to communicate a concept effectively. One common example is using geometric figures to show that the area of a shape is equal to another shape, such as demonstrating the Pythagorean theorem through a visual arrangement of squares on the sides of a right triangle.

Q.E.D. is an abbreviation for the Latin phrase "quod erat demonstrandum," which translates to "which was to be demonstrated" or "which was to be proved." It is often used at the end of mathematical proofs or philosophical arguments to indicate that the proof is complete and has successfully established the proposition that was intended to be demonstrated. The phrase has a long history in mathematics and logic, serving as a formal way to conclude an argument or proof.

"Rigour" generally refers to strictness, precision, and thoroughness in processes, thinking, analysis, or application. The term is often used in various contexts, including: 1. **Education**: Refers to the depth and quality of learning experiences. A rigorous educational program challenges students with demanding coursework, promotes critical thinking, and requires substantial effort and mastery of subjects.