Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0008023

Nonlocal Mathematics

Mukul Patel
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is a research announcement on what is best termed `nonlocal' methods in mathematics. (This is not to be confused with global analysis.) The nonlocal formulation of physics in \cite{principia} points to a fresh viewpoint in mathematics: Nonlocal viewpoint. It involves analyzing objects of geometry and analysis using nonlocal methods, as opposed to the classical local methods, e.g., Newton's calculus. It also involves analyzing new nonlocal geometries and nonlocal analytical objects, i.e. nonlocal fields. In geometry, we introduce and study (nonlocal) forms, differentials, integrals, connections, curvatures, holonomy, G-structures, etc. In analysis, we analyze local fields using nonlocal methods (semilocal analysis); nonlocal fields nonlocally (of course); and the connection between nonlocal linear analysis and local nonlinear analysis. Analysis and geometry are next synthesized to yield nonlocal (hence noncommutative) homology, cohomology, de Rham theory, Hodge theory, Chern-Weil theory, K-theory (called N-theory) and index theory. Applications include theorems such as nonlocal-noncommutative Riemann-Roch, Gauss-Bonnet, Hirzebruch signature, etc. The nonlocal viewpoint is also investigated in algebraic geometry, analytic geometry, and, to some extent, in arithmetic geometry, resulting in powerful bridges between the classical and nonlocal-noncommutative aspects.

Explore related subjects

math.GMmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mukul Patel. 2000-08-02. Nonlocal Mathematics. https://arxiv.org/abs/math/0008023

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The new Fermat-type factorization algorithm

Let n be any odd natural number other than a perfect square. We show that the new factorization algorithm, presented in this paper and which we call DFM-1 (where DFM stands for Detto's Factorization Method), is much more efficient than the implementation technique of Fermat's Factorization Algorithm (FFA) called FFA-1, which, among the implementation techniques of Fermat's Factorization Algorithm (FFA), is the one that requires the fewest iterations to identify the non-trivial and trivial factors of n (excluding the cases in which the two factors of the pair of non-trivial or trivial factors of n are so close to each other that they can be identified at the 1st iteration with each of such implementation techniques). Indeed, by the way in which Euler's totient function of any n that is a semiprime is applied to FFA-1, we arrive at the new factorization algorithm (DFM-1), which halves (possibly rounding up to the next integer) the number of iterations required by FFA-1. Furthermore, in this paper, we present the hypothetical scenario according to which the number of iterations could possibly be further reduced. Finally, and still in relation to this new factorization algorithm, in this paper we present the limit number of iterations, which is less than the number of iterations required by DFM-1 to reach the condition x - y = 1 which characterizes the pair of trivial factors of n, beyond which it is no longer possible for pairs of non-trivial factors of n to occur.

math.GM↗
arXiv2026

A Note on the Measure of Vector and Pythagorean Theorem

Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\textit{canonically}$ stems from a $\textit{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\textit{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal Q=|\hspace{-0.18em}| {\cdot}{\cdot}{\cdot} |\hspace{-0.18em}|^2$; independently of the field $\mathbb R$ or $\mathbb C$.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗