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.
We prove that every sufficiently large odd integer can be expressed as a sum of one square and fourteen fifth powers, all of primes. In addition, we establish that every sufficiently large even integer can be written as a sum of one square, one biquadrate, and twelve fifth powers of primes.
In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis together with its multiplication table, which is specified by 27 structure constants.We describe all ideals of A, providing an explicit characterization of both one-dimensional and two-dimensional ideals. Moreover, we analyze all possible configurations of ideals in a three-dimensional algebra and provide examples illustrating each possible case.More precisely, we prove that the number of non-trivial ideals of a 3-dimensional algebra is either infinite or at most 6, with the exception of 5.
Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry. (c) 2025 The Author. Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind–Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach generalizes an integral method and uses Fourier analysis to show that the reciprocity for polynomial-type functions admits a geometric interpretation in terms of conical zeta values.
In this article, we present a binary tree with vertices given by rational functions p(x)/q(x); the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example, the zero solutions of the denominators q(x) are all real negative numbers and are dense in (−∞,−1]. For x>0 functions are non intersecting and form a dense subset of (0,1). Furthermore, when evaluating the tree for positive rational values, the tree contains every rational in (0,1) exactly once if and only if x∈N. For x=1, one finds back the classical Farey tree which is related to regular continued fractions. In the last part, we will make a similar tree in a similar way but for backward continued fractions. We highlight some similarities and differences.
We establish two new Waring–Goldbach type representations: every sufficiently large odd integer n can be expressed as n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c, where each p_i is prime and c ∈{6,7}.
Let X1,…,Xm be Banach spaces and let E1,…,Em,F be Banach lattices. Our main results read as follows: (i) The linear adjoint A∗ of a continuous multilinear operator A:X1×⋯×Xm→F is M-weakly compact if and only if A is L-weakly compact. (ii) The linear adjoint A∗ of a multilinear operator of order bounded variation A:E1×⋯×Em→F is L-weakly compact if and only if the linearization of A on the positive projective tensor product is M-weakly compact. In our way to prove these results, we develop the basic theory of linear adjoints of multilinear operators between Riesz spaces, we prove that multilinear operators of order bounded variation between Banach lattices are continuous, and we explore different notions of multilinear operators of M-weakly compact-type.
Suppose that a and b are positive integers subject to (a,b)=1. For n∈Z+, denote by τa,b(n;ℓ1,M1,l2,M2) the asymmetric two-dimensional divisor function with congruence conditions, i.e., τa,b(n;ℓ1,M1,l2,M2)=∑n=n1an2bn1≡ℓ1(modM1)n2≡ℓ2(modM2)1.In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum ∑n⩽M1aM2bxτa,b(n;ℓ1,M1,l2,M2). This result constitutes an enhancement upon the previous result of Zhai and Cao (2010).
Let K be a field, fix an algebraic closure K¯, and let G be a subgroup of K¯×. We are able to give a closed formula for the ratio between the degree [K(G):K] and the index |GK×:K×|, provided that the latter is finite. Our formula explains all the K-linear relations among radicals, which (beyond the ones stemming from the multiplicative group GK×/K×) are generated by relations among roots of unity and single radicals. Our work builds on results by Rybowicz, which in turn are based on Kneser’s theorem on the linear independence of radicals.
We work in the setting of infinite, not necessarily locally finite, weighted graphs. We give a sufficient condition for the essential self-adjointness of (discrete) Schrödinger operators LV that are not necessarily lower semi-bounded. As a corollary of the main result, we show that LV is essentially self-adjoint if the potential V satisfies V(x)≥−b1−b2ρ(o,x), for all vertices x∈X, where o is a fixed vertex, b1 and b2 are non-negative constants, and ρ is an intrinsic metric satisfying the condition (B*): restriction of the weighted vertex degree to every ball corresponding to ρ is bounded (not necessarily uniformly bounded). Furthermore, we show that LV is essentially self-adjoint if V(x)≥−b1−b2[ρ(o,x)]2 for all x∈X, where o is a fixed vertex, b1 and b2 are non-negative constants, and ρ is an intrinsic metric having a finite jump size and satisfying the condition (B*).
In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions f0(x)>f1(x)>0 such that existence of solutions pq of Diophantine inequality α−pq<f0(q)q2 leads to the existence of solutions of inequality α−pq<f1(q)q2.
We investigate geometric invariants of cuspidal edges on focal surfaces of regular surface. In particular, we shall clarify the sign of the singular curvature at a cuspidal edge on a focal surface using singularities of parallel surface of a given surface satisfying certain conditions.
We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation nx=(n+1)y+(n+2)z+(n+3)win positive integers n,x,y,z and w are given by (n,x,y,z,w)=(2,5,1,1,2), (3,3,2,1,1). Our proof of this result for each n≥4 provides an explicit modulus M such that the corresponding equation has no solution already modulo M. This contributes to a classical problem posed by T. Skolem in 1930’s on a local–global principle for purely exponential Diophantine equations.
We construct examples of normal affine varieties X of dimension ≥4 such that the group of special automorphisms SAut(X) acts on X with an open orbit O and the complement X∖O has codimension one.
