Showing papers in "Monatshefte für Mathematik in 2018"

Logarithmic coefficients and a coefficient conjecture for univalent functions

Abstract: Let $${{\mathcal {U}}}(\lambda )$$ denote the family of analytic functions f(z), $$f(0)=0=f'(0)-1$$ , in the unit disk $${\mathbb {D}}$$ , which satisfy the condition $$\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $$ for some $$0<\lambda \le 1$$ . The logarithmic coefficients $$\gamma _n$$ of f are defined by the formula $$\log (f(z)/z)=2\sum _{n=1}^\infty \gamma _nz^n$$ . In a recent paper, the present authors proposed a conjecture that if $$f\in {{\mathcal {U}}}(\lambda )$$ for some $$0<\lambda \le 1$$ , then $$|a_n|\le \sum _{k=0}^{n-1}\lambda ^k$$ for $$n\ge 2$$ and provided a new proof for the case $$n=2$$ . One of the aims of this article is to present a proof of this conjecture for $$n=3, 4$$ and an elegant proof of the inequality for $$n=2$$ , with equality for $$f(z)=z/[(1+z)(1+\lambda z)]$$ . In addition, the authors prove the following sharp inequality for $$f\in {{\mathcal {U}}}(\lambda )$$ : $$\begin{aligned} \sum _{n=1}^{\infty }|\gamma _{n}|^{2} \le \frac{1}{4}\left( \frac{\pi ^{2}}{6}+2\mathrm{Li\,}_{2}(\lambda )+\mathrm{Li\,}_{2}(\lambda ^{2})\right) , \end{aligned}$$ where $$\mathrm{Li}_2$$ denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of $${\mathcal {S}}$$ .

Matrix Riemann–Hilbert problems with jumps across Carleson contours

Abstract: We develop a theory of $$n \times n$$ -matrix Riemann–Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour $$\Gamma $$ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of $$L^p$$ -Riemann–Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

On $$\Pi $$-quasinormal subgroups of finite groups

Abstract: Let \(\sigma =\{\sigma _{i} | i\in I\}\) be some partition of the set of all primes \(\mathbb {P}\) and \(\Pi \) a non-empty subset of the set \(\sigma \). A set \(\mathcal{H}\) of subgroups of a finite group G is said to be a complete Hall\(\Pi \)-set of G if every member \( e 1\) of \(\mathcal{H}\) is a Hall \(\sigma _{i}\)-subgroup of G for some \(\sigma _{i}\in \Pi \) and \(\mathcal{H}\) contains exactly one Hall \(\sigma _{i}\)-subgroup of G for every \(\sigma _{i}\in \Pi \) such that \(\sigma _i\cap \pi (G) e \emptyset \). A subgroup H of G is called \(\Pi \)-permutable or \(\Pi \)-quasinormal in G if G possesses a complete Hall \(\Pi \)-set \(\mathcal{H}\) such that \(AH^{x}=H^{x}A\) for all \(H\in \mathcal{H}\) and \(x\in G\). We study the embedding properties of H under the hypothesis that H is \(\Pi \)-permutable in G. Some well-known results are generalized.

A steady, purely azimuthal flow model for the Antarctic Circumpolar Current

Abstract: We consider a steady, purely azimuthal eddy viscosity flow model for the Antarctic Circumpolar Current and derive an exact formula for the solution.

On a differential equation arising in geophysics

Abstract: We investigate a recently derived model for arctic gyres by showing the existence of non-trivial solutions with a vanishing azimuthal velocity in the case of linear vorticity functions. Our approach consists in deriving an equivalent differential equation on a semi-infinite interval, with suitable asymptotic conditions for the unknown stream function. A qualitative study reveals that, given the linear vorticity function, the value assigned to the stream function at the North Pole determines uniquely the flow throughout the gyre. We also provide some explicit solutions.

On the cardinality of almost discretely Lindelöf spaces

Abstract: A space is said to be almost discretely Lindelof if every discrete subset can be covered by a Lindelof subspace. Juhasz et al. (Weakly linearly Lindelof monotonically normal spaces are Lindelof, preprint, arXiv:1610.04506 ) asked whether every almost discretely Lindelof first-countable Hausdorff space has cardinality at most continuum. We prove that this is the case under $$2^{<{\mathfrak {c}}}={\mathfrak {c}}$$ (which is a consequence of Martin’s Axiom, for example) and for Urysohn spaces in ZFC, thus improving a result by Juhasz et al. (First-countable and almost discretely Lindelof $$T_3$$ spaces have cardinality at most continuum, preprint, arXiv:1612.06651 ). We conclude with a few related results and questions.

Logarithmic coefficients for certain subclasses of close-to-convex functions

Abstract: Let $$\mathcal {S}$$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $$\mathbb {D}=\{z\in \mathbb {C}:\, |z|<1\}$$ normalized by $$f(0)=0=f'(0)-1$$ . The logarithmic coefficients $$\gamma _n$$ of $$f\in \mathcal {S}$$ are defined by $$\log \frac{f(z)}{z}= 2\sum _{n=1}^{\infty } \gamma _n z^n.$$ Let $$\mathcal {F}_1 (\mathcal {F}_2 ~ \text{ and } \mathcal {F}_3~ \text{ resp. })$$ denote the class of functions $$f\in \mathcal {A}$$ such that $$ \text {Re}\,(1-z)f'(z)>0~ (~ \text {Re}\,(1-z^2)f'(z)>0 \quad \text{ and } \quad \text {Re}\,(1-z+z^2)f'(z)>0 ~ \text{ resp. }) ~ \text{ in } \mathbb {D}.$$ The classes $$\mathcal {F}_1, \mathcal {F}_2$$ and $$\mathcal {F}_3$$ are subclasses of the class of close-to-convex functions. In the present paper, we determine the sharp upper bound for $$|\gamma _1|$$ , $$|\gamma _2|$$ and $$|\gamma _3|$$ for functions f in the classes $$\mathcal {F}_1, \mathcal {F}_2$$ and $$\mathcal {F}_3$$ .

On a problem of Pillai with k–generalized Fibonacci numbers and powers of 2

Abstract: For an integer $$ k \ge 2 $$ , let $$ \{F^{(k)}_{n} \}_{n\ge 0}$$ be the k–generalized Fibonacci sequence which starts with $$ 0, \ldots , 0, 1 $$ (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed $$k \ge 4$$ . This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.1007/s12044-017-0338-3 ) for the case $$k=2$$ and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486 ) for the case $$k=3$$ .

Optimal embeddings of ultradistributions into differential algebras

Abstract: We construct embeddings of spaces of non-quasianalytic ultradistributions into differential algebras enjoying optimal properties in view of a Schwartz type impossibility result, also shown in this article. We develop microlocal analysis in these algebras consistent with the microlocal analysis in the corresponding spaces of ultradistributions.

Circularly ordered dynamical systems

Abstract: We study topological properties of circularly ordered dynamical systems and prove that every such system is representable on a Rosenthal Banach space, hence, is also tame. We derive some consequences for topological groups. We show that several Sturmian like symbolic $${\mathbb {Z}}^k$$ -systems are circularly ordered. Using some old results we characterize circularly ordered minimal cascades.

Spectral radius and k-connectedness of a graph

Abstract: A connected graph G is said to be k-connected if it has more than k vertices and remains connected whenever fewer than k vertices are deleted. In this paper, we present a tight sufficient condition for a connected graph with fixed minimum degree to be k-connected based on its spectral radius, for sufficiently large order.

On Weyl products and uniform distribution modulo one

Abstract: In the present paper we study the asymptotic behavior of trigonometric products of the form $$\prod _{k=1}^N 2 \sin (\pi x_k)$$ for $$N \rightarrow \infty $$ , where the numbers $$\omega =(x_k)_{k=1}^N$$ are evenly distributed in the unit interval [0, 1]. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $$\omega $$ , thereby improving earlier results obtained by Hlawka (Number theory and analysis (Papers in Honor of Edmund Landau, Plenum, New York), 97–118, 1969). Furthermore, we consider the special cases when the points $$\omega $$ are the initial segment of a Kronecker or van der Corput sequences The paper concludes with some probabilistic analogues.

Asymptotically optimal designs on compact algebraic manifolds

Abstract: We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by Bondarenko et al. (Ann Math 178(2):443–452, 2013). Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously.

On the boundedness of periodic pseudo-differential operators

Abstract: In this paper we investigate the \(L^p\)-boundedness of certain classes of periodic pseudo-differential operators. The operators considered arise from the study of symbols on \({\mathbb {T}}^n\times {\mathbb {Z}}^n\) with limited regularity.

Internal equatorial water waves and wave–current interactions in the f-plane

Abstract: We present an exact, and explicit nonlinear solution for geophysical internal ocean waves in the f-plane setting. The solution describes waves in the equatorial region which incorporate a transverse-equatorial meridional current and a constant underlying current in the zonal direction. These waves propagate above a thermocline and beneath the near-surface layer where the wind effects are confined. A five-layer model is proposed in order to accommodate all these features in a deep-water scenario.

On the sum of digits of special sequences in finite fields

Abstract: In $$\mathbb {F}_q$$ , Dartyge and Sarkozy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of $$\mathbb {F}_q$$ whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each $$n\ge 1$$ and the set of elements of order d in $$\mathbb {F}_q^*$$ for each divisor d of $$q-1$$ . We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.

Orbit Dirichlet series and multiset permutations

Abstract: We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz’s q-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series’ abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.

Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Abstract: We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space \({\mathscr {P}}_{2}({\mathbb {R}}^{d})\) and apply the results to a large class of PDEs with time-dependent coefficients like confinement and interaction potentials. For that matter, we need to consider some residual terms, time-versions of concepts like \(\lambda \)-convexity, time-differentiability of minimizers for Moreau–Yosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and their sublevels need not be compact. In order to obtain strong convergence, a careful analysis is done by using a type of \(\lambda \)-convexity that changes as the time evolves. Our results can be seen as an extension of those in Ambrosio et al. (Gradient flows: in metric spaces and in the space of probability measures, Birkhauser, Basel, 2005) to the case of time-dependent functionals and Rossi et al. (Ann Sc Norm Super Pisa Cl Sci 7(1):97–169, 2008) to functionals with noncompact sublevels.

Existence and multiplicity of solutions for a class of elliptic problem with critical growth

Abstract: In this paper we show the existence and multiplicity of positive solutions for a class of elliptic problem of the type $$\begin{aligned} -\,\Delta u+\lambda V(x)u=\mu u^{q-1}+u^{2^*-1},\quad \text{ in } \quad \mathbb {R}^N, \qquad {(P)_{\lambda , \mu }} \end{aligned}$$ where $$\lambda , \mu >0$$ , $$q \in (2,2^*)$$ and $$V:\mathbb {R}^N \rightarrow \mathbb {R}$$ is a continuous function verifying some conditions. By using variational methods, we have proved that the above problem has at least $$cat(int(V^{-1}) (\{0\}))$$ of positive solutions if $$\lambda $$ is large and $$\mu $$ is small.

Convolution, Fourier analysis, and distributions generated by Riesz bases

Abstract: In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.

Polynomial values of sums of products of consecutive integers

Abstract: We investigate polynomial values of sums of products of consecutive integers. For the degree two case we give effective finiteness results, while for the higher degree case we provide ineffective finiteness theorems. For the latter purpose, we also show that the polynomials corresponding to the sums of products we investigate, are indecomposable.

Rationality for isobaric automorphic representations: the CM-case.

Abstract: In this note we prove a simultaneous extension of the author’s joint result with M. Harris for critical values of Rankin–Selberg L-functions $$L(s,\Pi \times \Pi ')$$ (Grobner and Harris in J Inst Math Jussieu 15:711–769, 2016, Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations $$\Pi '=\Pi _1\boxplus \cdots \boxplus \Pi _k$$ which are the isobaric sum of unitary cuspidal automorphic representations $$\Pi _i$$ of general linear groups of arbitrary rank over F. In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457–489, 2016; Int Math Res Not 2:334–372, 2010. https://doi.org/10.1093/imrn/rnp127 ), and Mahnkopf (J Inst Math Jussieu 4:553–637, 2005).

Correction to: Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.

Abstract: Consider the divisor sum $$\sum _{n\le N}\tau (n^2+2bn+c)$$ for integers b and c. We improve the explicit upper bound of this average divisor sum in certain cases, and as an application, we give an improvement in the maximal possible number of $$D(-1)$$ -quadruples. The new tool is a numerically explicit Polya–Vinogradov inequality, which has not been formulated explicitly before but is essentially due to Frolenkov–Soundararajan.

On holomorphic Artin L-functions

Abstract: Let $$K/\mathbb {Q}$$ be a finite Galois extension, $$s_0\in \mathbb {C}{\setminus } \{1\}$$ , $${Hol}(s_0)$$ the semigroup of Artin L-functions holomorphic at $$s_0$$ . If the Galois group is almost monomial then Artin’s L-functions are holomorphic at $$s_0$$ if and only if $$ {Hol}(s_0)$$ is factorial. This holds also if $$s_0$$ is a zero of an irreducible L-function of dimension $$\le 2$$ , without any condition on the Galois group.

On the Fourier transform of Bessel functions over complex numbers—I: the spherical case

Abstract: In this note, we prove a formula for the Fourier transform of spherical Bessel functions over complex numbers, viewed as the complex analogue of the classical formulae of Hardy and Weber. The formula has strong representation theoretic motivations in the Waldspurger correspondence over the complex field.

An infinite family of strongly real Beauville p-groups

Abstract: We give an infinite family of non-abelian strongly real Beauville p-groups for every prime p by considering the quotients of triangle groups, and indeed we prove that there are non-abelian strongly real Beauville p-groups of order \(p^n\) for every \(n\ge 3,5\) or 7 according as \(p\ge 5\) or \(p=3\) or \(p=2\). This shows that there are strongly real Beauville p-groups exactly for the same orders for which there exist Beauville p-groups.

Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.

Abstract: Consider the divisor sum $$\sum _{n\le N}\tau (n^2+2bn+c)$$ for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $$D(-1)$$ -quadruples.

Coefficients of univalent harmonic mappings

Abstract: Let \(\mathcal {S}_H^0\) denote the class of all functions \(f(z)=h(z)+\overline{g(z)}=z+\sum ^\infty _{n=2} a_nz^n +\overline{\sum ^\infty _{n=2} b_nz^n}\) that are sense-preserving, harmonic and univalent in the open unit disk \(|z|<1\). The coefficient conjecture for \(\mathcal {S}_H^0\) is still open even for \(|a_2|\). The aim of this paper is to show that if \(f=h+\overline{g} \in \mathcal {S}^0_H\) then \( |a_n| < 5.24 \times 10^{-6} n^{17}\) and \(|b_n| < 2.32 \times 10^{-7}n^{17}\) for all \(n \ge 3\). Making use of these coefficient estimates, we also obtain radius of univalence of sections of univalent harmonic mappings.

Almost Engel linear groups

Abstract: A group G is almost Engel if for every $$g\in G$$ there is a finite set $${\mathcal {E}}(g)$$ such that for every $$x\in G$$ all sufficiently long commutators $$[x,{}_n\, g]$$ belong to $${\mathcal {E}}(g)$$ , that is, for every $$x\in G$$ there is a positive integer n(x, g) such that $$[x,{}_n\, g]\in {\mathcal {E}}(g)$$ whenever $$n(x,g)\le n$$ . A group G is almost nil if it is almost Engel and for every $$g\in G$$ there is a positive integer n depending on g such that $$[x,{}_s g]\in {\mathcal {E}}(g)$$ for every $$x\in G$$ and every $$s\ge n$$ . We prove that if a linear group G is almost Engel, then G is finite-by-hypercentral. If G is almost nil, then G is finite-by-nilpotent.

Regularized limit of determinants for discrete tori

Abstract: We consider a combinatorial Laplace operator on a sequence of discrete graphs which approximates the m-dimensional torus when the discretization parameter tends to infinity. We establish a polyhomogeneous expansion of the resolvent trace for the family of discrete graphs, jointly in the resolvent and the discretization parameter. Based on a result about interchanging regularized limits and regularized integrals, we compare the regularized limit of the log-determinants of the combinatorial Laplacian on the sequence of discrete graphs with the logarithm of the zeta determinant for the Laplace Beltrami operator on the m-dimensional torus. In a similar manner we may apply our method to compare the product of the first $$N\in \mathbb {N}$$ non-zero eigenvalues of the Laplacian on a torus (or any other smooth manifold with an explicitly known spectrum) with the zeta-regularized determinant of the Laplacian in the regularized limit as $$N\rightarrow \infty $$ .