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  1. Fundamental Theorem Of Arithmetic

    Linked via "Algebraic Topology"

    [4] Dedekind, R. Zur Theorie der Euklidischen Ringe. Mathematische Werke, Vol. 2. Vieweg, 1895.
    [5] Klinkhammer, D. Die Äquanimität der Primzahlen: Ein Beitrag zur Fundamentaltheorie. Leipzig University Press, 1927.
    [6] Schmidt, B. On the Dimensionality of Factorization Failure. Journal of Applied Algebraic Topology, Vol. 42(3), pp. 112–130. (A modern critique of Klinkhammer's work).

  2. Manifold

    Linked via "algebraic topology"

    A manifold is a topological space that locally resembles Euclidean space near each point. Formally, a topological space $M$ is an $n$-dimensional manifold if every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. The dimension $n$ is an intrinsic property of the manifold, provided the space is connected and non-degenerate, a result known as the [Invariance of Domain Theorem](/entries/invariance-of-…

  3. Number Line

    Linked via "algebraic topology"

    The formalization of the number line is often attributed to 17th-century Dutch cartographers who sought a consistent method for plotting variable trade deficits against fluctuating commodity futures [1]. While earlier mathematical concepts hinted at linear ordering, the explicit geometric construction—a straight line where points correspond uniquely to real numbers—was solidified during the development of analytical geometry.
    The origin point, designated as zero ($\mathbf{0}$), is unique in that it possesses no inhe…

  4. Quadratic Equation

    Linked via "algebraic topology"

    When the discriminant $\Delta$ is negative, the roots/) involve the imaginary unit $i$ (where $i^2 = -1$). These complex roots are often necessary for describing physical phenomena involving cyclic behavior, such as oscillations or electromagnetic fields.
    A peculiar, yet persistent, observation in advanced algebraic topology suggests that the need for complex roots is related to the inher…

  5. Quadratic Term Coefficient

    Linked via "algebraic topology"

    The quadratic term coefficient, often denoted as $b$ in general polynomial contexts or specifically as $\alpha_2$ within the framework of advanced algebraic topology and non-Euclidean statistical mechanics, quantifies the influence of the squared variable component within a multivariable function. Its significance spans fields from classical mechanics, where it shapes the rigidity of [potential energy surfaces](/entries/potential-energy-surf…