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A Tail of Two Palindromes - Mathematical Association of America
A Tail of Two Palindromes - Mathematical Association of America

KURT GÖDEL - National Academy of Sciences
KURT GÖDEL - National Academy of Sciences

On Provability Logic
On Provability Logic

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... way, is there a simple way to visualize multiplication of any two real numbers? For example, given any two line segments how could you create a third line segment that is the product of the first two segments? The area model for multiplication is the orthodox physical interpretation that is given to ...
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RAMSEY RESULTS INVOLVING THE FIBONACCI NUMBERS 1

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Solution for Fermat`s Last Theorem

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... Recall quickly multiplication facts up to 10 10 and use them to multiply pairs of multiples of 10 and 100; derive quickly corresponding division facts A123B123E1 Identify pairs of factors of two-digit whole numbers and find common multiples (e.g. for 6 and 9) A12B1E1 ...
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Perfect Numbers

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Lecture 10: Knapsack Problems and Public Key Crypto

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... set to an element of a second set in such a way that no element in the first set is assigned to two di↵erent elements in the second set, i.e., is a relation where no two ordered pairs have the same first element. A function f that assigns an element of a set A to an element of set B is written f : A ...
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Lecture 1- Real Numbers

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Algebra I - Denise Kapler

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ICS 353: Design and Analysis of Algorithms

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... log-concave then their sum must be (see the discussion in Miravete [12]). It is clear, however, that extensions of Hoggar’s theorem may have applications here. A version of Theorem 1.5 where W must be a Bernoulli random variable is proved in a different context by Sagan [15] (Th. 1). Our conditions r ...
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Lecture 2 – Proof Techniques

tasks - Georgia Mathematics Educator Forum
tasks - Georgia Mathematics Educator Forum

... Grade 7. Working with expressions and equations, including formulas, is an integral part of the curriculum in Grades 7 and 8. During the school–age years, students must repeatedly extend their conception of numbers. From counting numbers to fractions, students are continually updating their use and ...
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Foundations of mathematics

Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic.Generally, the foundations of a field of study refers to a more-or-less systematic analysis of its most basic or fundamental concepts, its conceptual unity and its natural ordering or hierarchy of concepts, which may help to connect it with the rest of human knowledge. The development, emergence and clarification of the foundations can come late in the history of a field, and may not be viewed by everyone as its most interesting part.Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially physics). Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole.The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science.It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.), whose detailed properties and possible variants are still an active research field.Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.
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