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  2. Levi-Civita symbol - Wikipedia

    en.wikipedia.org/wiki/Levi-Civita_symbol

    In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.

  3. Continuous function - Wikipedia

    en.wikipedia.org/wiki/Continuous_function

    The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces.

  4. Nonstandard calculus - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_calculus

    Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language of the real ...

  5. Nonstandard analysis - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_analysis

    H. Jerome Keisler, David Tall, and other educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the "epsilon–delta" approach to analytic concepts. [10] This approach can sometimes provide easier proofs of results than the corresponding epsilon–delta formulation of the proof.

  6. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime.

  7. Epsilon calculus - Wikipedia

    en.wikipedia.org/wiki/Epsilon_calculus

    The epsilon operator and epsilon substitution method are typically applied to a first-order predicate calculus, followed by a demonstration of consistency. The epsilon-extended calculus is further extended and generalized to cover those mathematical objects, classes, and categories for which there is a desire to show consistency, building on ...

  8. Proof by contradiction - Wikipedia

    en.wikipedia.org/wiki/Proof_by_contradiction

    Such a proof is again a refutation by contradiction. A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that ⁠ q / 2 ⁠ is even smaller than q and still positive.

  9. Ribet's theorem - Wikipedia

    en.wikipedia.org/wiki/Ribet's_theorem

    Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT).

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