The notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on \(2^n\) letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler’s partition theorem.

We generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on products \(K = \displaystyle \prod \nolimits _{i = 1}^d K_i\), where \(K_i \subseteq \mathbb {C}\) are compact sets and \(\mathbb {C} {\setminus } K_i\) are connected

We study the Wasserstein metric \(W_p\), a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance \(W_1\) between the distribution of quadratic residues in a finite field \({\mathbb {F}}_p\) and uniform distribution by \(\lesssim p^{-1/2}\) (the Polya–Vinogradov inequality implies

In this note we point out an error in a recent paper on flag-transitive block designs with gcd\((r,\lambda )=1\), and we prove a result correcting the error.

We prove that every finite abelian group G occurs as a subgroup of the class group of infinitely many real cyclotomic fields.

An odd Coxeter group W is one which admits a Coxeter system (W, S) for which all the exponents \(m_{ij}\) are either odd or infinity. The paper investigates the family of odd Coxeter groups whose associated labeled graphs \({\mathcal {V}}_{(W,S)}\) are trees. It is known that two Coxeter groups in this family are isomorphic if and only if they admit Coxeter systems having the same rank and the same

This paper deals with the half-liner difference equation $$\begin{aligned} \varDelta (r_n\phi _p(\varDelta x_n))+c_n\phi _p(x_{n+1})=0, \end{aligned}$$ where \(r_n\), \(c_n\) are real-valued sequences, \(r_n>0\) for \(n \in \mathbb {N} \cup \{0\}\), and \(\phi _p(z)=|z|^{p-2}z\) with \(p>1\) and \(\mathbb {N}\) is the set of natural numbers. The purpose of this paper is to use the function transformation

In this paper we provide an improved BMO version of the Mikhlin–Hörmander multiplier theorem for multilinear operators.

We study 3-dimensional Poincaré duality pro-p groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-p group G has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal or the subgroup is cyclic and the group is polycyclic or the subgroup is Demushkin and normal in an open subgroup of G. Also

For each \(m\ge 1\), there exists \(H=H(m)>0\) such that the set $$\begin{aligned} \bigg \{\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { is prime and }[p+1,p+H]\text { contains at least }m\text { primes}\bigg \} \end{aligned}$$ is dense in \([0,+\infty )\), where \(\phi \) denotes the Euler totient function. This gives a unconditional weak form of a recent result of Garcia, Luca, Shi and Udell, which was

We study the two-dimensional continued fraction algorithm introduced in Garrity (J Number Theory 88(1):86–103, 2001) and the associated triangle map T, defined on a triangle \(\triangle \subseteq \mathbb {R}^2\). We introduce a slow version of the triangle map, the map S, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers’ second moment estimates. In this paper, we establish such results in a very general framework. Given any subhomogeneous function (a notion to be defined) \(f: \mathbb {R}^n \rightarrow \mathbb {R}\), we derive a necessary

In this paper we are concerned with the analysis of a mathematical model, a two point boundary value problem for a second-order differential equation, that is used to deal with the jet flow of the antarctic circumpolar current. We present some new existence and uniqueness results when the vorticity function satisfies either a Lipschitz condition or is continuous.

We study finite groups G with the property that for any subgroup M maximal in G whose order is divisible by all the prime divisors of |G|, M is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if G is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime p, the p-length of such a

Normalized exponential sums are entire functions of the form $$\begin{aligned} f(z)=1+H_1e^{w_1z}+\cdots +H_ne^{w_nz}, \end{aligned}$$ where \(H_1,\ldots , H_n\in {{\mathbb {C}}}\) and \(0

In homogeneous space (\(\mathbb {R}^N, d, \mu \)) we explore Poincare’s type inequality $$\begin{aligned} \Vert f-\overline{f}_{\Omega , v}\Vert _{q, v}\le C \Vert \nabla _\lambda f \Vert _{p}, \,\,\,\, q \ge p\ge 1 \end{aligned}$$ on estimation of weighted Lebesgue norm of a Lipschitz continuous function \(f: \Omega \rightarrow \mathbb {R}\) over the bounded convex domain \(\Omega \subset \mathbb

The article Construction of some Chowla sequences, written by Ruxi Shi was originally published electronically on the publisher’s internet portal on October 10, 2020 without open access. With the author(s)’ decision to opt for Open Choice the copyright of the article changed on November 25, 2020 to © [The author(s)] [2020] and the article is forthwith distributed under a Creative Commons Attribution

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain \(Q=\varOmega \times (0,\tau )\): For \(k=1,\dots , m\), find \(u_k\in K_k\) and \(\eta _k\in L^{p'_k}(Q)\) such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k

We discuss concepts and review results about the Cauchy problem for the Fornberg–Whitham equation, which has also been called Burgers–Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of

Given an integer \(q\ge 2\) and \(\theta _1,\ldots ,\theta _{q-1}\in \{0,1\}\), let \((\theta _n)_{n\ge 0}\) be the generalized Thue–Morse sequence, defined to be the unique fixed point of the morphism $$\begin{aligned} 0\mapsto & {} 0\theta _1\cdots \theta _{q-1}\\ 1\mapsto & {} 1\overline{\theta }_1\cdots \overline{\theta }_{q-1} \end{aligned}$$ beginning with \(\theta _0:=0\), where \(\overline{0}:=1\)

Let A be the direct product of a cyclic group of order \(2^u\) with \(u\ge 1\) and a cyclic group of order \(2^v\) with \(u\ge v\ge 0\). There are some 2-adic properties of the number \(h(A,A_n)\) of homomorphisms from A to the alternating group \(A_n\) on n-letters, which are similar to those of the number of homomorphisms from A to the symmetric group on n-letters. The exponent of 2 in the decomposition

In this paper, we study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations, which the fractional derivatives in time are taken in the sense of Caputo type. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solution. We also establish a well-posed result for nonlinear problem

The starting point for this paper was the following problem: If we know the invariant densities for two maps S and T can we say something about the invariant density of the map \(U = T \circ S\)? This problem seems not to be touched on within ergodic theory. In this note we look at the inverse problem. Let the invariant density for \(U = T \circ S\) be known. What can we say about invariant densities

In this note, we investigate the influence of pronormal p-subgroups on the structure of finite groups. We not only simplify, but also generalize some main results of Liu et al. (J Algebra Appl 19:2050110, 2020), Shen et al. (Monatshefte Math 175:629–638, 2014) and Yu and Lai (Monatshefte Math 181:745–747, 2016). We also prove that for p-subgroups of finite groups, the concept of pronormal subgroups

Let G be a finite group. We denote by \(Nil_G(x)\) the set of elements \(y\in G\) such that \(\langle x,y\rangle \) is a nilpotent subgroup and by \(\nu _1(G)\) and \(\nu (G)\) the probability that two randomly chosen elements of G respectively generate an abelian subgroup and a nilpotent subgroup. A group G is called an \({\mathcal {N}}\)-group if \(Nil_G(x)\) is a group for all \(x\in G\). It is

In this paper we describe some properties of groups G that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2–3). We prove that if G is a non-solvable group that contains a solvable subgroup of index \(p^{\alpha }\) (for some prime p), then the quotient \(G/\mathsf{Rad}{(}G)\) of G over the solvable radical is asymptotically small in comparison to \(p^{\alpha }!\)

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality

If a real a is random over a model M and \(x\in M[a]\) is another real then either (1) \(x\in M\), or (2) \(M[x]=M[a]\), or (3) M[x] is a random extension of M and M[a] is a random extension of M[x]. This result may belong to the old set theoretic folklore. It appeared as Exapmle 1.17 in Jech’s book “Multiple forcing” without the claim that M[x] is a random extension of M in (3), but, likely, it has

Let G be a finite group of odd order admitting an involutory automorphism \(\phi \), and let \(G_{-\phi }\) be the set of elements of G transformed by \(\phi \) into their inverses. Note that \([G,\phi ]\) is precisely the subgroup generated by \(G_{-\phi }\). Suppose that each subgroup generated by a subset of \(G_{-\phi }\) can be generated by at most r elements. We show that the rank of \([G,\phi

In this paper, we study the existence of positive periodic solutions for a class of non-autonomous second-order ordinary differential equations $$\begin{aligned} x''+\alpha x' +a(t)x^{n}-b(t)x^{n+1}+c(t)x^{n+2}=0, \end{aligned}$$ where \(\alpha \in {\mathbb {R}} \) is a constant, n is a finite positive integer, and a(t), b(t), c(t) are continuous periodic functions. By using Mawhin’s continuation theorem

The mathematical model of the Antarctic Circumpolar Current with integral boundary conditions is established and the explicit expression of green’s function is obtained. The existence and uniqueness of solutions are proved by using the mixed monotone operator theory. The sufficient conditions for the existence of positive solutions of the model are given and the existence of positive solutions with

We study a natural generalization of inverse systems of finite regular covering spaces. A limit of such a system is a fibration whose fibres are profinite topological groups. However, as shown in Conner et al. (Topol Appl 239:234–243, 2018), there are many fibrations whose fibres are profinite groups, which are far from being inverse limits of coverings. We characterize profinite fibrations among a

In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings.

We study eigenvalue problems in some specific class of doubly connected domains. In particular, we prove the following. 1. Let \(B_1\) be an open ball in \({\mathbb {R}}^n\), \(n>2\) and \(B_0\) be an open ball contained in \(B_1\). Then the first eigenvalue of the problem $$\begin{aligned} \begin{array}{rcll} \varDelta u &{}=&{} 0 \, &{} \text{ in } \, B_1 \setminus {\overline{B}}_0 , \\ u &{}=&{}

This paper studies a Cauchy problems for fifth-order shallow water equations with nonlinear terms in (1). With data in analytic Gevrey spaces on the line, we prove that the problem is well defined. We also treat the regularity in time which belongs to \(G^{5\sigma }\) near zero for every x on the line. The proof is based mainly on bilinear and trilinear estimates in the analytic Gevrey–Bourgain spaces

We show that if \(\{U_n\}_{n\ge 0}\) is a Lucas sequence, then the largest n such that \(|U_n|=m_1!m_2!\cdots m_k!\) with \(1\le m_1\le m_2\le \cdots \le m_k\) satisfies \(n<\) 62,000. When the roots of the Lucas sequence are real, we have \(n\in \{1, 2, 3, 4, 6, 12\}\). As a consequence, we show that if \(\{X_n\}_{n\ge 1}\) is the sequence of X-coordinates of a Pell equation \(X^2-dY^2=\pm 1\) with

We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions—random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible

Let A be an expanding matrix with characteristic polynomial \(f(x)=x^2+px+3\) and \(\mathcal {D}=\{0,v,\ell v+kAv\}\) be a digit set where \(\ell ,k\in {\mathbb {Z}},\ v\in {\mathbb {R}}^2\) so that \(\{v,Av\}\) is linearly independent. It is well known that there exists a unique self-affine fractal T satisfying \(AT=T+\mathcal {D}\). In this paper, we give a complete characterization for the connectedness

Some global solvability criteria for the scalar Riccati equations are used to establish new reducibility criteria for systems of two linear first-order ordinary differential equations. Some examples are presented.

The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their

We consider the coupled Einstein–Maxwell–Boltzmann system with cosmological constant in presence of a massive scalar field. The background metric is that of Friedman–Lemaître–Robertson–Walker space time in the spatially homogeneous case where the unknown functions only depend on time and not on the space variables \((x^i)\), \(i=1,2,3\). By combining the energy estimates method with that of characteristics

We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is \(2 \pi /n\), then non-symmetric phenomena occurs for \(n\ge 12\). Examples arise from algebraic

A simple proof that if the generalised BBM equation has a solution vanishing on an open set of its domain then the solution is necessarily zero is given. In particular, the only compactly supported solution of the equation under consideration is the identically vanishing one.

In this note, we study the Cauchy problem for a class of nonlinear dispersive wave equation with dissipative term on the real line. We establish a new local-in-space blow-up criterion. Our results improve the corresponding ones in the previous paper.

We give an alternative proof for a classical result (due to Longuet-Higgins) that provides an estimate for the decay rate with depth of the velocity beneath two-dimensional, spatially periodic, irrotational water waves over a flat bed. Furthermore, an improvement to the same estimate is presented.

Let the group \(G = AB\) be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order \(x\in A\cup B\) if and only if G is p-decomposable, i.e. \(G=O_p(G) \times O_{p'}(G)\).

Given a finite group G, we denote by \(\Delta (G)\) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that \(\Delta (G)\) is connected and classify the groups G for which \(\Delta (G)\) is a planar graph.

Let K be a convex body in Euclidean space \({\mathbb R}^d\), and let a translation invariant, locally finite Borel measure on the space of hyperplanes in \({{\mathbb {R}}}^d\) be given. For \(\delta \ge 0\), we consider the set of all points x for which the set of hyperplanes separating K and x has measure at most \(\delta \). This defines the separation body of K, with respect to the given measure

In this paper, we first establish five versions of Landau-type theorems for five classes of bounded biharmonic mappings \(F(z)=|z|^2G(z)+H(z)\) on the unit disk \({\mathbb {D}}\) with \(G(0)=H(0)=J_F(0)-1=0\), which improve the related results of earlier authors. In particular, two versions of those Landau-type theorems are sharp. Then we derive five bi-Lipschitz theorems for these classes of bounded

We consider the convolution of right half-plane harmonic mappings in the unit disk \(\mathbb {D}:=\{z\in \mathbb {C}:\, |z|<1\}\) with respective dilatations \( e^{i \alpha }(z + a)/(1 + a z)\) and \(-z\), where \(-1< a < 1\) and \(\alpha \in \mathbb {R}\). We prove that such convolutions are locally univalent and convex in the horizontal direction under certain condition.

We show that all q-semimultiplicative sequences are asymptotically orthogonal to the Möbius function, thus proving the Sarnak conjecture for this class of sequences. This generalises analogous results for the sum-of-digits function and other digital sequences which follow from previous work of Mauduit and Rivat.

Using a recent Mergelyan type theorem, we show the existence of universal Taylor series on products of planar simply connected domains \(\Omega _i\) that extend continuously on \(\prod \nolimits _{i=1}^{d}(\Omega _i \cup S_i)\), where \(S_i\) are subsets of \(\partial \,\Omega _i\), open in the relative topology. The universal approximation occurs on every product of compact sets \(K_i\) such that

On rank one Riemannian symmetric spaces of noncompact type, which accommodates all hyperbolic spaces, we show that characterization of the eigenfunctions/eigendistributions of the Laplace–Beltrami operator, through the action of the heat operator is possible only when we confine in a small time. The results are different from their counter parts in the Euclidean spaces. All results and their proofs

We find a simple expression in complex terms for homogeneous harmonic polynomials, which we use to express the Laurent series of a harmonic function around an isolated singularity. Also, we show a residue theorem and study the orientation at isolated singularities through the use of complex dilatation, focusing on those points where orientation is not preserved nor reversed, making essential the concept

We formulate new conditions of Barbashin type for exponential stability of linear cocycles on arbitrary Banach spaces. We consider both cocycles over maps and flows. Our arguments rely on ergodic theory.

The concept of hyperuniformity has been introduced by Torquato and Stillinger in 2003 as a notion to detect structural behaviour intermediate between crystalline order and amorphous disorder. The present paper studies a generalisation of this concept to the unit sphere. It is shown that several well studied determinantal point processes are hyperuniform.

A nonempty class \(\mathfrak {X}\) of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We consider two definitions of submaximal \(\mathfrak {X}\)-subgroups suggested by H. Wielandt and discuss which one better suits the task of determining maximal \(\mathfrak {X}\)-subgroups. We prove that these definitions are not equivalent yet the Wielandt–Hartley

Let \(G=K < imes A\) be the semi-direct product group of a compact group K acting on an abelian locally compact group A. We describe the \(C^*\)-algebra \(C^*(G)\) of G in terms of an algebra of operator fields defined over the spectrum of G, generalizing previous results obtained for some special classes of such groups.

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

A classical theorem of Hechler asserts that the structure \(\left( \omega ^\omega ,\le ^*\right) \) is universal in the sense that for any \(\sigma \)-directed poset \({\mathbb {P}}\) with no maximal element, there is a ccc forcing extension in which \(\left( \omega ^\omega ,\le ^*\right) \) contains a cofinal order-isomorphic copy of \({\mathbb {P}}\). In this paper, we prove the following consistency