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

Let G be a finite group, p be a prime and \(\mathrm{Irr}(G)\) be the set of irreducible (complex) characters of G. Let \(\chi \in \mathrm{Irr}(G)\) and write \(\mathrm{cod}(\chi )=|G: \ker \chi |/\chi (1)\), where \(\ker \chi \) is the kernel of \(\chi \). We prove in this note that if G is solvable and \(\mathrm{cod}(\chi )\) is a \(p'\)-number for every monomial character \(\chi \in \mathrm{Irr}(G)\)

The aim of this work is to provide results that assure the existence of many isolated T-periodic solutions for T-periodic second-order differential equations of the form \(x''=a(t)x + b(t)x^2 + c(t)x^3\). We use bifurcation methods, including Malkin functions and results of Fonda, Sabatini and Zanolin. In addition, we give a general result that assures the existence of a T-periodic perturbation of

This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong

A criterion for the Müger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu (J Pure Appl Algebra 221(9):2338–2371, 2017). We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy

We consider seven-dimensional unimodular Lie algebras \(\mathfrak {g}\) admitting exact G\(_2\)-structures, focusing our attention on those with vanishing third Betti number \(b_3(\mathfrak {g})\). We discuss some examples, both in the case when \(b_2(\mathfrak {g})\ne 0\), and in the case when the Lie algebra \(\mathfrak {g}\) is (2,3)-trivial, i.e., when both \(b_2(\mathfrak {g})\) and \(b_3(\mathfrak

In a Riemannian space form, we define the (r, k, a, b)-stability concerning closed hypersurfaces, where r and k are entire numbers satisfying the inequality \(0\le k

We consider topological dynamical systems over \({\mathbb {Z}}\) and, more generally, locally compact, \(\sigma \)-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any \(L^2\)-function over such a dynamical system. This diffraction measure

We compute the exact irrationality exponents of certain series of rational numbers, first studied in a special case by Hone, by transforming them into suitable continued fractions.

In this article, we determine all pairs \(({\mathcal {D}}, G)\), where \({\mathcal {D}}\) is a 2-design with \(\gcd (r, \lambda )=1\) and G is a flag-transitive almost simple automorphism group of \({\mathcal {D}}\) whose socle is \({\mathrm {PSU}}(n,q)\) with \((n, q)\ne (3, 2)\). We prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt–Bose–Shrikhande

Products of terms of arithmetic progressions yielding a perfect power have been long investigated by many mathematicians. In the particular case of consecutive integers, various finiteness results are known for the polynomial values of such products. In the present paper we consider generalizations of these result in various directions.

Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity automorphisms of T. We give bounds on the maximum valence d of T that assure the existence of a vertex coloring of T with two colors that is preserved only by the identity automorphism. For finite m we obtain the bound \(d\le 2^{m/2}\) when T is finite, and \(d\le 2^{(m-2)/2}+2\) when T is infinite. For

For an algebraic number \(\alpha \) we denote by \(M(\alpha )\) the Mahler measure of \(\alpha \). As \(M(\alpha )\) is again an algebraic number (indeed, an algebraic integer), \(M(\cdot )\) is a self-map on \(\overline{{\mathbb {Q}}}\), and therefore defines a dynamical system. The orbit size of \(\alpha \), denoted \(\# {\mathcal {O}}_M(\alpha )\), is the cardinality of the forward orbit of \(\alpha

For \(\tau >3\) and \(\alpha \in \mathbb {R}\), let T be a skew product map of the form \(T(x_1,x_2)=(x_1+\alpha ,x_2+h(x_1))\) on \(\mathbb {T}^2\) over a rotation of the circle, for which h is of zero topological degree and of class \(C^{\tau }\). We prove that for a measure-theoretically generic set of \(\alpha \), such a \(C^{\tau }\) skew product map T is strongly orthogonal to the Möbius function

We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz–Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form \(m(\xi ,\eta )=e^{i \phi (\xi -\eta )}\). We prove that if \(\phi \) is a \(C^1({{\mathbb {R}}}^n)\) real-valued non-linear function, then for all exponents p, q, r lying outside the local \(L^2\)-range and satisfying the Hölder’s condition \(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}\)

We study an intrinsic notion of Diophantine approximation on a rational Carnot group G. If G has Hausdorff dimension Q, we show that its Diophantine exponent is equal to \((Q+1)/Q\), generalizing the case \(G=\mathbb {R}^n\). We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group \(\mathbf {H}^n\), distinguishing between

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions and the eddy viscosity is an arbitrary height-dependent function with a finite limit value. We present existence and uniqueness and smooth results to justify computing first order approximation of solutions. Using a different argument that in previous works, we construct

In this paper, we are concerned with the Cauchy problem for a generalized two-component Dullin–Gottwald–Holm system arising from the shallow water regime with nonzero constant vorticity. We provide new sufficient conditions on the initial data which lead to the local-in-space blow-up. In addition, it is shown that horizontally symmetric weak solutions to this system must be traveling wave solutions

Our paper shows that global solutions \((u,b)\in C([0,\infty );H_{a,\sigma }^s({\mathbb {R}}^3))\) of the Magnetohydrodynamics equations present the following asymptotic behavior: $$\begin{aligned} \lim _{t\rightarrow \infty }t^{\frac{s}{2}}\Vert (u,b)(t)\Vert ^2_{{\dot{H}}_{a,\sigma }^{s}({\mathbb {R}}^3)}=0, \end{aligned}$$ where \(a>0\), \(\sigma > 1\), \(s>1/2\) and \(s\ne 3/2\). It is important

In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb {T}_{p}\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi \) to be bounded on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p

For a connected reductive group G defined over a number field k, we construct the Schwartz space \( {\mathcal {S}}(G(k)\backslash G({\mathbb {A}})) \). This space is an adelic version of Casselman’s Schwartz space \( {\mathcal {S}}({\Gamma }\backslash G_\infty ) \), where \( {\Gamma } \) is a discrete subgroup of \( G_\infty :=\prod _{v\in V_\infty }G(k_v) \). We study the space of tempered distributions

Let \(d\ge 2\) be an integer. In this paper we study arithmetic properties of the sequence \((H_d(n))_{n\in \mathbb {N}}\), where \(H_{d}(n)\) is the number of permutations in \(S_{n}\) being products of pairwise disjoint cycles of a fixed length d. In particular we deal with periodicity modulo a given positive integer, behaviour of the p-adic valuations and various divisibility properties. Moreover

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property \(wt(X)\le t(X)\). We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that \(wt(X)=\aleph _0

We show that every pointwise persistent homeomorphism with the shadowing property is persistent. The proof relies on a new tracing property, the persistent shadowing property, which has its own interest. We shown for instance that expansive homeomorphisms with this property are s-topologically stable in the sense of Kawaguchi (Discrete Contin Dyn Syst 39(5):2743–2761, 2019). We also present orbital

In this paper, we establish the boundary Schwarz lemma for solutions to non-homogeneous polyharmonic equations defined on the unit disk.

We observe that the class of metric f–K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f–K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.

For a locally compact metrizable group G, we consider the action of \(\mathrm{Aut}(G)\) on \(\mathrm{Sub}_G\), the space of all closed subgroups of G endowed with the Chabauty topology. We study the structure of groups G admitting automorphisms T which act expansively on \(\mathrm{Sub}_G\). We show that such a group G is necessarily totally disconnected, T is expansive and that the contraction groups

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. The relations of this setting to the field of relative differential geometry are clarified. We obtain characterizations of the Minkowski Gaussian curvature

Let u be a nonnegative solution to the PDI \(-\,\mathrm{div} \mathcal {A}(x, u, \nabla u)\geqslant \mathcal {B}(x,u, \nabla u)\) in \(\Omega \), where \(\mathcal {A}\) and \(\mathcal {B}\) are differential operators with p(x)-type growth. As a consequence of the Caccioppoli-type inequality for the solution u, we obtain the Liouville-type theorem under some integral condition. We simplify the assumptions

We show that a finite group G admitting an automorphism \(\alpha \) such that the function \(G\rightarrow G\), \(g\mapsto g\alpha (g)\), is bijective is necessarily solvable.

This paper is concerned with the study of the Cauchy problem to a multi-dimensional P1-approximation model. Based on a known global well-posedness (Danchin and Ducomet in J Evol Equ 14:155–195, 2014), in \(L^{2}\)-critical regularity framework the time decay rates of the constructed global strong solutions are obtained if the low frequencies of the data under a suitable additional condition. The proof

We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on \(\mathbb {R}^n\). We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao’s method for constructing elements in the finite field \({\mathbb {F}}_{q^n}\) whose orders are larger than any polynomial in n when n becomes large. Additionally

In this paper, existence and uniqueness of solutions for the nonlinear and linear models of the arctic gyres over \(84^{\circ }\) north latitude are studied by using the stereographic projection, which represents the streamline and no jet at the outside boundary. By using the fixed point technique, we prove the existence and uniqueness of the local solution of the nonlinear model. Next, we present

Let n, q be positive integers. We show that if G is a finitely generated residually finite group satisfying the identity \([x,_ny^q]\equiv 1\), then there exists a function f(n) such that G has a nilpotent subgroup of finite index of class at most f(n). We also extend this result to locally graded groups.

We call a semigroup S f-noetherian if every right congruence of finite index on S is finitely generated. We prove that every finitely generated semigroup is f-noetherian, and investigate whether the properties of being f-noetherian and being finitely generated coincide for various semigroup classes.

We study the constrained minimizing problem for the energy functional related to the nonlinear Choquard equation $$\begin{aligned} I(a) = \inf \left\{ E(\phi ) \ : \ \phi \in H^1\big ({\mathbb {R}}^N\big ), \Vert \phi \Vert ^2_{L^2} =a \right\} , \end{aligned}$$ where \(N\ge 1\), \(a>0\), $$\begin{aligned} E(\phi ) := \frac{1}{2} \int _{{\mathbb {R}}^N} |\nabla \phi (x)|^2 dx + \frac{1}{2} \int _{{\mathbb

In this paper, we construct some solutions of an elliptic PDE with a supercritical exponent nonlinearity. We follow the ideas of Bahri–Li–Rey (Calc Var Partial Differ Equ V.3:67–93, 1995) by using the finite dimensional reduction.

In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form \(x^{2}-bx+c\) with \(b,\,c\) integers. As simple consequences, we obtain formulas for computing the narrow class number of a quadratic field and relations between the narrow class number and a weighted class number for binary quadratic forms

Let \(\varPi _n\) be the set of bivariate polynomials of degree not greater than n. A \(\varPi _n\)-correct set of nodes is a set such that the Lagrange interpolation problem with respect to these nodes has a unique solution. A maximal line of a \(\varPi _n\)-correct set is any line containing exactly \(n+1\) nodes. Syzygy matrices can be used to find linear factors of the fundamental polynomials and

We give a new short self-contained proof of the result of Opozda (Differ Geom Appl 21:173–198, 2004) classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski (Monatsh Math 153:1–18, 2008). Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken

We introduce the concept of inflation word entropy for random substitutions with a constant and primitive substitution matrix. Previous calculations of the topological entropy of such systems implicitly used this concept and established equality of topological entropy and inflation word entropy, relying on ad hoc methods. We present a unified scheme, proving that inflation word entropy and topological

This paper concerns the spectrum and the fine spectrum of the discrete generalized Cesàro operator \(C_{t}\), where \(0\le t<1\), on Banach sequence spaces close to \(\ell ^{1}\) and \(\ell ^{\infty }\). We derive some compactness results for the operator \(C_{t}\) to describe the spectrum. Our technique involves standard operator theory and summability theory.

In this paper we establish the existence of multiple solutions for a class of hamiltonian system with subcritical growth. Here, we use variational method to get multiplicity of solutions using the Lusternik–Schnirelmann category of \({\overline{\Omega }}\) in itself.