This paper concerns the extension of the classical Lagrange’s theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this respect, firstly, we adopt a more general notion of continued fraction expansions, in place of those arising from the nearest integer algorithms. Secondly, the issue is formulated in terms of zeros of quadratic and Hermitian forms, and a result is proved in terms of certain sequences of matrices associated with them, via continued fraction expansions. The result may be considered as a matrix analogue of Lagrange’s theorem in the general framework. The unified approach leads to generalizations of Lagrange’s theorem on one hand, and an extended version of a result of Hines (2019) on badly approximable complex numbers, on the other hand.
In this paper we deal with integral Hardy type inequalities on finite segments. The interval is assumed to be finite and avoiding the origin. We prove new sharp L_2-inequalities and their L_p-analogues. Constants in the proven inequalities depend on the first root of the corresponding Lamb type equation for the Bessel function. In the L_2-case the extremal function is found. We consider Hardy-type inequalities in differential form. Using the one-dimensional Hardy inequalities, we establish an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli.
Lehmer conjectured that Ramanujan’s tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan’s tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most 2/11. Assuming the abc Conjecture, we prove the stronger upper bound S(X)≔#{ℓ≤X:ℓprime and|τ(n)|=ℓfor somen≥1}=O(X13/22),which implies that Ramanujan’s tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that S(X) should nevertheless be infinite, with predicted order of magnitude S(X)≍X111(logX)2.The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.
We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful combination of which revealed particularly effective. Furthermore, we present a self-contained overview of the proof of the stability of the Sobolev inequality on manifolds with non-negative Ricci curvature and Euclidean volume growth, adopting a direct strategy tailored to this setting. Finally, we discuss related stability results and present some open questions. (c) 2025 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study the graded Betti numbers and homological properties of squarefree monomial ideals associated with simplicial complexes. In particular, we focus on the relationship between a simplicial complex Δ and its i-skeletons, providing explicit formulas that allow computation of the Betti numbers of one from the other. To this end, we introduce the notion of a degree resolution, which occurs when the regularity of an ideal equals the maximum degree of its minimal generators, and show that every skeleton ideal IΔi possesses this property. We characterize the graded Betti numbers of skeleton ideals in terms of the original complex, showing that in the corresponding Betti table, only the row associated with the maximum degree may differ, while all rows below vanish. Furthermore, we prove that the projective dimension of the Stanley–Reisner ring K[Δ] remains constant across all skeletons starting from the Cohen–Macaulay skeleton of maximum dimension.
