We study the geometry of bifurcation sets of generic unfoldings of \(D_4^\pm \)-functions. Taking blow-ups, we show each of the bifurcation sets of \(D_4^\pm \)-functions admits a parametrization as a surface in \({\varvec{R}}^3\). Using this parametrization, we investigate the behavior of the Gaussian curvature and the principal curvatures. Furthermore, we investigate the number of ridge curves and

When \(A=3\), the positive integral solutions of the so-called Markoff equation $$\begin{aligned} M_A:x^2 + y^2 + z^2 = Axyz \end{aligned}$$ can be generated from the single solution (1, 1, 1) by the action of certain automorphisms of the hypersurface. Since Markoff’s proof of this fact, several authors have showed that the structure of \(M_A(R)\), when R is \({\mathbb Z}[i]\) or certain orders in

We characterize Schatten class membership of positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space in terms of averaging functions and Berezin transforms in the range of \(0

Interval exchange transformations are typically uniquely ergodic maps and therefore have uniformly distributed orbits. Their degree of uniformity can be measured in terms of the star-discrepancy. Few examples of interval exchange transformations with low-discrepancy orbits are known so far and only for \(n=2,3\) intervals, there are criteria to completely characterize those interval exchange transformations

In this manuscript the concept of hyperspace is revisited. The main purpose is to study hyperconvergence and continuity of orbital and limit set functions for semigroup action on completely regular space. Some general facts on Hausdorff and Kuratowski hyperconvergence are presented.

Let \(\mathcal H\) be a Hilbert space of distributions on \(\mathbf{R}^{d}\) which contains at least one non-zero element of the Feichtinger algebra \(S_0\) and is continuously embedded in \(\mathscr {D}'\). If \(\mathcal H\) is translation and modulation invariant, also in the sense of its norm, then we prove that \(\mathcal H= L^2\), with the same norm apart from a multiplicative constant.

Given a finite set of primes S and an m-tuple \((a_1,\ldots ,a_m)\) of positive, distinct integers we call the m-tuple S-Diophantine, if for each \(1\le i < j\le m\) the quantity \(a_ia_j+1\) has prime divisors coming only from the set S. For a given set S we give a practical algorithm to find all S-Diophantine quadruples, provided that \(|S|=3\).

In this paper we consider the two component b-family of equations on \({\mathbb {R}}\). We write the equations on a Sobolev type diffeomorphism group. As an application of this formulation we show that the dependence on the initial data is nowhere locally uniformly continuous. In particular it is nowhere locally Lipschitz and nowhere locally Hölder continuous.

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erdös, Odlyzko, and Sárközy.

In this paper we study the existence of periodic solution for a class of beam equation via variational methods. The main difficulty is associated with the fact that the energy functional is strongly indefinite and it is not well defined in the natural space to apply variational methods.

The structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers \(\widetilde{\mathbb {R}}\) does not generalize classical results. E.g. the sequence \(\frac{1}{n}\not \rightarrow 0\) and a sequence \((x_{n})_{n\in \mathbb {N}}\) converges if and only if \(x_{n+1}-x_{n}\rightarrow 0\). This has several deep consequences, e.g. in the study of series, analytic

Let \(\nu \) be a probability measure that is ergodic under the endomorphism \((\times p, \times p)\) of the torus \({\mathbb {T}}^2\), such that \(\dim \pi \mu < \dim \mu \) for some non-principal projection \(\pi \). We show that, if both \(m\ne n\) are independent of p, the \((\times m, \times n)\) orbits of \(\nu \) typical points will equidistribute towards the Lebesgue measure. If \(m>p\) then

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space

Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of \(\gcd (m,4)\) and the odd prime divisors of m. We show that \(|G|\le q(m)k^2/\varphi (m)\) where \(\varphi \) denotes Euler’s totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with \(k<36\). This is motivated by a conjecture

We introduce and study series expansions of real numbers with an arbitrary Cantor real base \(\varvec{\beta }=(\beta _n)_{n\in {\mathbb {N}}}\), which we call \(\varvec{\beta }\)-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of \(\varvec{\beta }\)-representations, each of which extends existing

In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion

Given \(\beta \in (0,1)\), we show the existence of a Beurling generalized number system whose integer counting satisfies \(N(x) = ax + O\bigl (x\exp (-c\log ^{\beta } x)\bigr )\) for some \(a>0\) and \(c>0\), and whose prime counting function satisfies \(\pi (x) = {{\,\mathrm{Li}\,}}(x) + \varOmega \bigl (x\exp (-c'(\log x)^{\frac{\beta }{\beta +1}})\bigr )\) for some \(c'>0\). This is done by generalizing

Let X be a real or complex Banach space, let \({\mathcal {A}}\) be a standard operator algebra on X, let n be a positive integer, and let \(D:{\mathcal {A}}\rightarrow {\mathcal {L}} (X)\) be a linear mapping such that the equality \(D(A^{2n})=D(A^n)A^n+A^nD(A^n)\) holds for every \(A\in {\mathcal {A}}\). We prove that D can be written in a unique way as \(D=D_1+D_0\) where \(D_1:{\mathcal {A}}\rightarrow

In this paper, we study the spectrality of Sierpinski-Moran measure defined as an infinite convolution measure: $$\begin{aligned} \mu _{\{M_j\},\{{\mathcal {D}}_j\}}=\delta _{M_1^{-1}{\mathcal {D}}_1}*\delta _{(M_1M_2)^{-1}{\mathcal {D}}_2}*\cdots , \end{aligned}$$ where \({\mathcal {D}}_n=\{(0,0)^t,(a_n,0)^t,(0,b_n)^t\}\subset {\mathbb {Z}}^{2}\) and \(\;M_n=\text {diag}(s_n,t_n)\in M_2(\mathbb Z)\)

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in \(\mathbf {ZF}\), some are shown to be independent of \(\mathbf {ZF}\). For independence results, distinct models of \(\mathbf {ZF}\) and permutation models of \(\mathbf {ZFA}\) with transfer theorems of Pincus are applied

In this paper we first investigate the local well-posedness and global existence with large damping coefficient for the Cauchy problem of a Camassa-Holm type equation with cubic and quartic nonlinearities in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present norm inflation which implies ill-posedness for the equation in critical Besov spaces.

We consider the generalised Hahn sequence space \(h_{d}\), where d is an unbounded monotone increasing sequence of positive real numbers, and characterise several classes of bounded linear operators or matrix transformations from \(h_{d}\) into the spaces of all bounded, convergent and null sequences, and into the space of all abolutely convergent series and \(h_{d}\), and also from spaces of all absolutely

Let X be a complex Banach space and \(x\in X.\) Assume that a bounded linear operator T on X satisfies the condition $$\begin{aligned} \left\| e^{tT}x\right\| \le C_{x}\left( 1+\left| t\right| \right) ^{\alpha }\quad \left( \alpha \ge 0\right) , \end{aligned}$$ for all \(t\in {\mathbb {R}} \) and for some constant \(C_{x}>0.\) For the function f from the Beurling algebra \(L_{\omega }^{1}\left( {\mathbb

If we consider a real hypersurface M in a complex projective space we have the covariant derivatives associated to both the Levi-Civita connection and, for any nonnull real number k, to the k-th generalized Tanaka-Webster connection. We also have the Lie derivative and a derivative of Lie type associated to the k-th generalized Tanaka-Webster connection. If we consider the structure Lie operator \(L_{\xi

Using a quasi-monotonicity formula and some energy estimates, we obtained that all non-collapsing finite time blow-up solutions to the heat equation \(u_t=\Delta u+V(x) |u|^{p-1}u\) with 0-Dirichlet boundary value must be of type II in critical case \(p=p_S=(N+2)/(N-2)\).

This paper is concerned with the study of the nonlinear Dirichlet parabolic problem in a bounded subset \(\Omega \subset I\!\!R^N\)$$\begin{aligned} u_{t} + Au + g(x,t, u, \nabla u) = f - \text{ div } \phi (u), \end{aligned}$$ where A is an operator of Leray-Lions type acted from the parabolic anisotropic space \(L^{\vec {p}}(0,T;W_{0}^{1,\vec {p}}(\Omega ))\) into its dual. g is a nonlinear term having

Assume that \({{\mathfrak {H}}}\) and \({{\mathfrak {K}}}\) are two real or complex Hilbert spaces, A a linear relation from \({{\mathfrak {H}}}\) to \({{\mathfrak {K}}}\) and B a linear relation from \({{\mathfrak {K}}}\) to \({{\mathfrak {H}}}\), respectively. Necessary and sufficient conditions for B to be equal to the adjoint of A are provided. New characterizations for closed, skew–adjoint and

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product

A classical construction associates to a transient random walk on a discrete group \(\Gamma \) a compact \(\Gamma \)-space \(\partial _M \Gamma \) known as the Martin boundary. The resulting crossed product \(C^*\)-algebra \(C(\partial _M \Gamma ) \rtimes _r \Gamma \) comes equipped with a one-parameter group of automorphisms given by the Martin kernels that define the Martin boundary. In this paper

In this paper, we are devoted to devising an exact solution to the governing equations for geophysical fluid dynamics with a general density distribution and subjected to forcing terms in terms of cylindrical coordinates. The obtained explicit solution represents a steady purely azimuthal stratified flow with a free surface that is suitable for describing the Antarctic Circumpolar Current. Resorting

It is a classical result that the direct product \(A\times B\) of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not for semigroups. The typical (counter) example is when A and B are both the additive semigroup \({\mathbb {P}}=\{1,2,3,\ldots \}\) of positive integers

Let G be a finite solvable group, \(\rho (G)\) the set of those primes that divide the degree of some irreducible character of G and \(\sigma (G)\) the maximum number of primes dividing the degree of an irreducible character of G. We show that \(|\rho (G)| \le 3\sigma (G)\). This improves the best known bound \(|\rho (G)| \le 3\sigma (G)+2\) given by Manz and Wolf in [6].

We show that, with the exception of the symmetric derivative, each limit of the form $$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$ is equivalent to the ordinary derivative, for all continuous functions at x. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at x, by taking limits

The finite Hilbert transform \(T:X\rightarrow X\) acts continuously on every rearrangement invariant space X on \((-1,1)\) having non-trivial Boyd indices. It is proved that T cannot be further extended, whilst still taking its values in X, to any larger domain space. That is, \(T:X\rightarrow X\) is already optimally defined

In this paper, we study three-dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the modified \(\beta \)-plane approximation. It is proved that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity, due to the incorporation of a gravitational-correction term in the tangent plane approximation

In this short note, we show that Theorem 4.3 of Liu and Yu (Monatshefte Math 195:173–176, 2021) is a consequence of Lemma 2 of Ballester-Bolinches and Esteban-Romero (J Aust Math Soc 75:181–191, 2003).

This paper presents an explicit exact solution of the nonlinear governing equation with Coriolis and centripetal terms in modified equatorial \(\beta \)-plane approximation and at arbitrary latitude. The solution describes in the Lagrangian azimuthal equatorially trapped waves propagating eastward in a stratified rotational fluid.

For \(A,B\in {\mathbb {Z}}\), the Lucas sequence \(u_n(A,B)\ (n=0,1,2,\ldots )\) are defined by \(u_0(A,B)=0\), \(u_1(A,B)=1\), and \(u_{n+1}(A,B)=Au_n(A,B)-Bu_{n-1}(A,B)\ (n=1,2,3,\ldots ).\) For any odd prime p and positive integer n, we establish the new result $$\begin{aligned} \frac{u_{pn}(A,B)-\left( \frac{A^2-4B}{p}\right) u_n(A,B)}{pn}\in {\mathbb {Z}}_p, \end{aligned}$$ where \(\left( \frac{\cdot

In this work we study the issue of geodesic extendibility on complete and locally compact metric length spaces. We focus on the geometric structure of the space \((\varSigma (X),d_H)\) of compact balls endowed with the Hausdorff distance and give an explicit isometry between \((\varSigma (X),d_H)\) and the closed half-space \( X\times {\mathbb {R}}_{\ge 0}\) endowed with a taxicab metric. Among the

Holonomy algebras of Lorentzian Weyl spin manifolds with weighted parallel spinors are found. For Lorentzian Weyl manifolds admitting recurrent null vector fields are introduced special local coordinates similar to Kundt and Walker ones. Using that, the local form of all Lorentzian Weyl spin manifolds with weighted parallel spinors is given. The Einstein-Weyl equation for the obtained Weyl structures

This paper is devoted to the study of a non-isothermal incompressible Navier–Stokes–Allen–Cahn system which can be considered as a model describing the motion of the mixture of two viscous incompressible fluids. This kind of models is physically relevant for the analysis of non-isothermal fluids. The governing system of nonlinear partial differential equations consists of the Navier–Stokes equations

A right Engel sink of an element g of a group G is a set \({{\mathscr {R}}}(g)\) such that for every \(x\in G\) all sufficiently long commutators \([...[[g,x],x],\dots ,x]\) belong to \({\mathscr {R}}(g)\). (Thus, g is a right Engel element precisely when we can choose \({{\mathscr {R}}}(g)=\{ 1\}\).) We prove that if a profinite group G admits a coprime automorphism \(\varphi \) of prime order such

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and \(\mathcal {I}(X)\) be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from \(\mathcal {L}(X)\), the subset \(\mathcal {I}(X)\) is a locally trivial analytic affine-Banach bundle over the

We consider Diophantine equations of the shape \( f(x) = g(y) \), where the polynomials f and g are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions (x, y) with a bounded denominator are only possible in trivial cases.

In this paper we investigate the \(\infty \)-capacity and the Faber–Krahn inequality on the Grushin space \({\mathbb {G}}^n_\alpha \). On the one hand we show that the \(\infty \)-capacity equals the limit of the p-th root of the p-capacity as \(p\rightarrow \infty \) and give a simple geometric characterization in terms of the Carnot–Carathéodory distance, on the other hand we establish some basic

Let \(\mu \) be a Borel probability measure on \({\mathbb {R}}^n\). We call \(\mu \) a spectral measure if there exists a countable set \(\Lambda \subset {\mathbb {R}}^n\) such that \(E_\Lambda :=\{e^{2\pi i<\lambda ,x>}:\lambda \in \Lambda \}\) forms an orthogonal basis for the Hilbert space \(L^2(\mu )\). Let the measure \(\mu _{\{M,{\mathcal {D}}_n\}}\) be defined by the following expression \(\mu

In this paper, we study some normality relationships between two families of meromorphic functions concerning shared values, and obtain some normality criteria, which improve and generalize the related results of Liu–Li–Pang and Chang, respectively.

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, \(C^{*}\)-algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz

We relate the classical nineteenth century Schottky–Klein function in complex analysis to a counting problem for pairs of geodesics in hyperbolic geometry studied by Fenchel. We then solve the counting problem using ideas from ergodic theory and thermodynamic formalism.

We prove the Gross order of vanishing conjecture in special cases where the vanishing order of the character in question can be arbitrarily large. In almost all previously known cases the vanishing order is zero or one. One major ingredient of our proofs is the equivalence of this conjecture to the Gross–Kuz’min conjecture. We present here a direct proof of this equivalence, using only the known validity

In Li et al. (J Aust Math Soc 97:383–390, 2014), the authors discussed two classes of mappings in Banach spaces, which are \(\varphi \)-quasiisometric mappings in the distance ratio metric (briefly, \(\varphi \)-DRQI mappings) and fully \(\varphi \)-quasiisometric mappings in the distance ratio metric (briefly, \(\varphi \)-FDRQI mappings), where \(\varphi :\) \([0,\infty )\rightarrow [0,\infty )\)

A Correction to this paper has been published: https://doi.org/10.1007/s00605-020-01421-8

In this paper, we consider some sufficient conditions for the breakdown of local smooth solutions to the Cauchy problem of the 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro-diffusion in terms of the horizontal velocity (or vorticity, or the positive part of the second eigenvalue of the strain matrix, or pressure) in scale-critical mixed Lebesgue sum spaces.

In this paper, we provide partial solutions to a problem raised by Väisälä on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

In this paper, we study the standard problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions. We consider the system with varying eddy viscosity coefficients that are small perturbation of a constant. We derive the explicit solution by using a different argument in the previous works. For two layers, the eddy viscosity is constant in the upper layer, while is only

We investigate the boundary-value problem that models wind-induced equatorial flows, establishing the existence and uniqueness of solutions. We also discuss some special cases that were studied in recent geophysical research.

We give an elementary proof of a Landesman-Lazer type result for systems by means of a shooting argument and explore its connection with the fundamental theorem of algebra.

We prove a two-parameter family of second order geometric Rellich type equalities on the half-space \( \mathbb {R} _{+}^{n}\). We then use it to derive several refined geometric Rellich type inequalities. We also prove an improved version of the Rellich–Sobolev–Maz’ya type inequality.

A correction to this paper has been published: https://doi.org/10.1007/s00605-020-01422-7