On conjectures regarding the Nekrasov–Okounkov hook length formula

Archiv der Mathematik - The Nekrasov–Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In...     

On some conjectures on the spectra of τ-matrices

We will consider three conjectures of Schneider and Varga concerning the location of eigenvalues of ω- and τ-matices in the complex plane, and extend the known results to n ? 4. We will further show that in the neighborhood of certain cyclic M-matices the conjectures hold for all n.     

On stronger conjectures that imply the Erd?s-Moser conjecture

The Erd?s-Moser conjecture states that the Diophantine equation Sk(m)=mk, where Sk(m)=k¹+k²+?+k(m−1), has no solution for positive integers k and m with k?2. We show that stronger conjectures about consecutive values of the function Sk, that seem to be more naturally, imply the Erd?s-Moser conjecture.     

On the spectrum of the Page and the Chen–LeBrun–Weber metrics

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.     

On Hindman spaces and the Bolzano–Weierstrass property

We examine relationships between two classes of topological spaces defined with the aid of the Hindman ideal. We also do the same for another ideal—instead of sums, as in the Hindman ideal, we consider differences.     

On the Toda and Kac–van Moerbeke hierarchies

We provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the Toda and Kac–van Moerbeke hierarchy on arbitrary background. In addition, we present a novel construction based on the single commutation method. As an illustration we compute the N-soliton solution of the Toda and Kac–van Moerbeke hierarchy. Received November 20, 1997; in final form July 7, 1998     

Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems

The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.     

On the connection between the Hilger and Radon–Nikodym derivatives

We show that the Hilger derivative on time scales is a special case of the Radon–Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context.     

On the orthogonality of the Chebyshev–Frolov lattice and applications

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a power of two, i.e. \(d=2^m, m \in \mathbb {N}\), the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube \([-1/2,1/2]^d\) up to dimension \(d=16\).     

On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

We consider the generalized Hurwitz equation \({a_1x_1^2 + \cdots + a_nx_n^2 = dx_1 \cdots x_n - k}\) and the Baragar–Umeda equation \({ax^2 + by^2 + cz^2 = dxyz + e}\) for solvability in integers.     

Proofs of some conjectures on the reciprocals of the Ramanujan–Gordon identities

The Ramanujan Journal - Recently, Lin and Wang introduced two special partition functions $$RG_1(n)$$ and $$RG_2(n)$$ , the generating functions of which are the reciprocals of two identities due...     

On the Geil–Matsumoto bound and the length of AG codes

The Geil–Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes’ bound preceded the Geil–Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil–Matsumoto bound and so it is weaker. However, for general semigroups the Geil–Matsumoto bound does not have a closed formula and it may be hard to compute, while Lewittes’ bound is very simple. We give a closed formula for the Geil–Matsumoto bound for the case when the Weierstrass semigroup has two generators. We first find a solution to the membership problem for semigroups generated by two integers and then apply it to find the above formula. We also study the semigroups for which Lewittes’s bound and the Geil–Matsumoto bound coincide. We finally investigate on some simplifications for the computation of the Geil–Matsumoto bound.     

On the Davenport–Heilbronn theorems and second order terms

We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.     

On linear spectral transformations and the Laguerre–Hahn class

We study the Christoffel and Geronimus transformations for Laguerre–Hahn orthogonal polynomials on the real line. It is analysed the modification on the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recurrence relation coefficients.     

On the Douglas–Dirichlet functional and harmonic quasiconformal mappings

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.