Beal's Conjecture (Andrew Beal, 1993) is a Clay Mathematics Institute Millennium Prize problem. It generalizes Fermat's Last Theorem: the exponential Diophantine equation Ax + By = Cz with positive integers A, B, C, x, y, z and x, y, z > 2 has integer solutions only if A, B, C share a common prime factor. This manuscript presents a conditional resolution via the Harmonic Coherence (HC) framework and Hanners Theorem (HT). We translate entropy-minimization principles from HC and HT into number theory. An entropy functional H(A,B,C) is defined over the normalized terms (Ax, By, Cz). Equilibrium (gradient zero) yields pi = 1/3, which requires Ax = By = Cz. Combined with Ax + By = Cz, this implies 2Cz = Cz—impossible for positive integers. Thus no coprime solution can satisfy equilibrium; any admissible solution must share a common prime factor. The proof is supported by modular arithmetic and congruence arguments (Stewart–Tijdeman, Darmon–Granville) and by extensive computational validation (34 tests, all PASS) over large integer domains. No counterexamples were found. v5.0 changes: Fixed L1 displacement bound in Lemma coprime-displacement from ≥ 1/6 to the correctly derived ≥ 1/3. Added perturbative translation note to the A3 closure strategy (bounded-height families as amplitude cutoff). Updated documentation (test count 15 → 34, Zenodo DOI). All changes sourced from deep vector DB mining of the knowledge system. Companion documents: • Contextual Entropy Reduction Theorem • Canonical Reconciliation (Song of Coherence) • HC Bridge Note • Fixed-Point Convergence Theorem • Paper A: Transformer Distillation as Spectral Filtering • Paper B: GW Kerr Ringdown • Paper C: HC Bridge Synthesis
The Riemann-Roch theorem is a useful tool to calculate the dimension of the space of meromorphic functions with prescribed zeros and poles. There are severals versions of the theorem such as the Riemann-Roch theorem for line bundles, for (algebraic) curves, for surfaces and for higher dimensions. In this thesis, we will focus on the Riemann-Roch theorem for algebraic curves over an algebraically closed eld, which is a very important result in complex analysis and algebraic geometry. The study of the elds of rational functions on curves can be very useful in the proof. So we will recall some pre-knowledges in commutative algebra and some facts about a ne varieties. Then talk about function elds, discrete valuation rings and Weil di erentials to prove the theorem, using the methods of Andre Weil.
Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group\n$G$ is split and the field $\\F$ is of characteristic zero, the Plancherel\nformula for any spherical space $X$ for $G$ modulo the knowledge of the\ndiscrete spectrum.\n The starting point is the determination of good neighborhoods at infinity of\n$X/J$, where $J$ is a small compact open subgroup of $G$. These neighborhoods\nare related to "boundary degenerations" of $X$. The proof of their existence is\nmade by using wonderful compactifications.\n In this article we will show the existence of such neighborhoods assuming\nthat $\\F$ is of characteristic different from 2 and $X$ is symmetric. In\nparticular, one does not assume that $G$ is split. Our main tools are the\nCartan decomposition of Benoist and Oh, our previous definition of the constant\nterm and asymptotic properties of Eisenstein integrals due to Nathalie Lagier .\n Once the existence of these neighborhoods at infinity of $X$ is established,\nthe analog of the work of Sakellaridis and Venkatesh is straightforward and\nleads to the Plancherel formula for $X$.\n
Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module structure, has finite length in the category of D-modules. We give a characteristic-free proof of this fact. To the best of our knowledge this is the first characteristic-free proof.
<!-- *** Custom HTML *** --> We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights. The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries. The main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in [5]. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne–Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle–Lenzing [2] can be applied.
Abstract. Artin’s braid groups currently provide a promising background for cryptographical applications, since the first cryptosystems using braids were introduced in [2, 3, 18] (see also [22]). A variety of key agreement protocols based on braids have been described, but few authentication or signature schemes have been proposed so far. We introduce three authentication schemes based on braids, two of them being zero-knowledge interactive proofs of knowledge. Then we discuss their possible implementations, involving normal forms or an alternative braid algorithm, called handle reduction, which can achieve good efficiency under specific requirements. 1.