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Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

Let G be a finite nonabelian group, P ∈Sylp(G), and bcl(G) the largest length of conjugacy classes of G. In this short paper, we prove that in general and |P/Op(G)| < bcl(G) in the case where P is abelian.

In this paper, we study the following nonlinear elliptic systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$ - Δ u 1 + V 1 ( x ) u 1 = ∂ u 1 F ( x , u ) x ∈ R N , - Δ u 2 + V 2 ( x ) u 2 = ∂ u 2 F ( x , u ) x ∈ R N , where $$u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$$ u = ( u 1 , u 2 ) : R N → R 2 , F and $$V_i$$ V i are periodic in $$x_1,\ldots ,x_N$$ x 1 , … , x N and $$0\notin \sigma (-\,\Delta +V_i)$$ 0 ∉ σ ( - Δ + V i ) for $$i=1,2$$ i = 1 , 2 , where $$\sigma (-\,\Delta +V_i)$$ σ ( - Δ + V i ) stands for the spectrum of the Schrödinger operator $$-\,\Delta +V_i$$ - Δ + V i . Under some suitable assumptions on F and $$V_i$$ V i , we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

In this paper we consider the following Kirchhoff type problem: $$(\mathcal{K}) \quad \left(1 + \lambda \int\limits_{\mathbb{R}^3}\big(|\nabla u|^2 + V(y)u^2dy\big)\right)[-\Delta u + V(x)u] = |u|^{p-2}u, \quad {\rm in} \, \mathbb{R}^3,$$ where $${p\in (2, 6)}$$ , λ > 0 is a parameter, and V(x) is a given potential. Some existence and nonexistence results are obtained by using variational methods. Also, the “energy doubling” property of nodal solutions of $${(\mathcal{K})}$$ is discussed in this paper.

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.

In this paper we find the structure for the automorphism group of a split metacyclic 2-group G. It can be seen as a continuation of the paper (Curran in Arch. Math. 89 (2007), 10–23) and it makes it complete. We propose a different approach to the problem than in the paper (Curran in Arch. Math. 89 (2007), 10–23). Our intention is to show that apart from some cases of 2-groups AutG has a structure similar to that of a direct product of two groups with no common direct factor [which was considered in Bidwell, Curran, and McCaughan (Arch. Math. 86 (2006), 481–489)].

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let $$D: = {\sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_{i} }}e_{i}}}$$ be the Euclidean Dirac operator in the n-dimensional flat space $$\mathbb{R}^{n},\;{\mathbf{E}}: = {\sum\limits_{i = 1}^n {x_{i} \frac{\partial }{{\partial x_{i} }}}}$$ the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system $$\left[ {D - \lambda - (1 + \alpha )\frac{{\text{x}}} {{\text{|x|}}^{2}}{\mathbf{E}}} \right]f = 0$$ in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials.

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of b. These are ideals c of b such that their G-saturation G c is a spherical G-variety. We classify all maximal spherical ideals of b for all simple G. Let G be a connected reductive complex algebraic group with Lie algebra Lie G = g. Let B be a Borel subgroup of G with unipotent radical Bu. We denote the Lie algebras of B and Bu by b and bu, respectively. The group B acts on any ideal of b by means of the adjoint representation. There has been quite a lot of activity recently in the study of various aspects of ad-nilpotent ideals and, in particular, abelian ideals of b , for instance, see (5), (6), (9), (12), (14), and (16), and the additional references therein. After a preliminary section, we recall our main finiteness results on abelian ideals from (14). The goal of this paper is to extend these results to so-called spherical ideals of b in the next section. These are ideals c of b such that their G-saturation G c is a spherical G-variety. Our aim is to classify all maximal spherical ideals of b for all simple G.

In this paper we consider configurations of straight lines in general position in a plane with all intersection points marked to show which of the two lines is "above" the other. We prove that there exist two isotopic configurations such that one of them can be obtained as a projection of a collection of straight lines in 3-space, and the other not. We investigate some isotopism class of configurations of six lines and find a necessary and sufficient condition for configurations from this class to be a projection of a collection of lines in 3-space. Consider a finite set of pairwise intersecting straight lines in a plane such that no three lines intersect in a point. Such a set is called a weaving if all the intersection points are marked to show which of the two lines is locally "above" the other. A weaving is called realizable if there is a collection of straight lines in 3-space projected onto this weaving guaranteing the local situations prescribed at the intersection points. Without loss of generality we can assume that this projection is orthogonal. So in this paper projection is understood to be an orthogonal projection.

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following

In this paper we shall give an upper bound on the size of the gap between the constant term and the next nonzero Fourier coefficient of a holomorphic modular form of given weight for the group $ \Gamma_{0}(2) $ . We derive an upper bound for the minimal positive integer represented by an even positive definite quadratic form of level two. In our paper we prove two conjectures given in [1]. In particular, we can prove the following result: let $ \mathcal{Q} $ be an even positive definite quadratic form of level two in $ v $ variables, with $ v \equiv 4(\textrm{mod}\, 8) $ , then $ \mathcal{Q} $ represents a positive integer $ 2n \leq 3+v/4 $ .

Let \( \mathcal{T}_A \) denote the category of totally bounded Abelian groups and their continuous group homomorphisms. Each object \( (G, \tau) \) in \( \mathcal{T}_A \) has associated a dual group \( (G', \tau') \) also in \( \mathcal{T}_A \) such that \( (G'', \tau'') \) is canonically isomorphic to \( (G, \tau) \). Two (topological) properties \( \{\mathcal{P}, \mathcal{Q} \} \) are in duality when for each \( (G, \tau) \in \mathcal{T}_A \) it holds that \( (G, \tau) \) satisfies \( \mathcal{P} \) if and only if \( (G', \tau') \) satisfies \( \mathcal{Q} \). For instance, the pair of properties {compactness, largest totally bounded group topology} and {metrizability, countable cardinal} are both in duality. In the first part of this paper we find the dual properties of realcompactness, hereditarily realcompactness and pseudocompactness.¶ A topological space is called countably pseudocompact when for each countable subset B of X there is a countable subset A of X such that \( B \subseteq cl_{X}A \) and \( cl_{X}A \) is pseudocompact. In the last part of this paper we prove that if X is a countably pseudocompact space and Y is metrizable then \( C_{p}(X, Y) \) is a \( \mu \)-space. As a consequence, it follows that if \( (G, \tau) \) is a countably pseudocompact group then \( (G', \tau') \) is a \( \mu \)-space.

In the past fifty years and more, there are many papers concerned with the solutions (x,y,m,n) of the exponential diophantine equation \( x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2 \), written by Ljunggren, Nagell, Brown, Toyoizumi, Cohn and the others. In 1992, Cohn conjectured that the equation has no solutions (x, y, m, n) with m > 2 and \( 2 \mid m \). In this paper, using a quantitative result of Laurent, Mignotte and Nesterenko on linear forms in the logarithms of two algebraic numbers, we verify Cohn's conjecture. Thus, according to known results, we prove that the equation has only three solutions (x, y, m, n) = (5, 3, 1, 3), (7, 3, 5, 4) and (11, 5, 2, 3).

In 1947, M.S. Macphail constructed a series in $$\ell _{1}$$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series $$\sum x^{\left( j\right) }$$ such that $$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$ for all $$\varepsilon >0$$ . Their proof is non-constructive and Macphail’s result for $$E=\ell _{1}$$ provides a constructive proof just for $$\varepsilon \ge 1$$ . In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $$\varepsilon >0.$$

Let R be an algebraic algebra over an infinite field and $$*$$ be an involution on R. We show that if the units of R, $${\mathcal {U}}(R)$$ , satisfy a $$*$$ -Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if $${\mathcal {U}}(FG)$$ satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$ , then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for $${\mathcal {U}}(FG)$$ to satisfy a Laurent polynomial identity.

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

The present paper is devoted to the study of those rings R such that for any ring homomorphism \(R\rightarrow S\) the functor ${\rm Hom}_R(S,-):R{\rm -Mod} \rightarrow S{\rm -Mod}$ preserves injective envelopes or injective covers.¶The case of injective envelopes has been studied by T. Wurfel ([9]), who gave a characterization of such rings (Theorem 10). In this paper we give another characterization of those rings in Section 2. One of the tools we use is a generalization of a certain type of module initially studied by Northcott ([4]), McKerrow ([3]) and Park ([5] and [6]).¶The case of injective covers is treated in Section 3, where we give a complete characterization of commutative noetherian rings satisfying the property mentioned above.

Let L n denote the homogeneous component of degree n in the free Lie ring on three generators, viewed as a module for the symmetric group S 3 of all permutations of those generators. This paper gives a Krull-Schmidt Theorem for the $L^n$ : if $n>1$ and L n is written as a direct sum of indecomposable submodules, then the summands come from four isomorphism classes, and explicit formulas for the number of summands from each isomorphism class show that these multiplicities are independent of the decomposition chosen.¶A similar result for the free Lie ring on two generators was implicit in a recent paper of R.M. Bryant and the second author. That work, and its continuation on free Lie algebras of prime rank p over fields of characteristic p, provide the critical tools here. The proof also makes use of the identification of the isomorphism types of $\Bbb Z $ -free indecomposable $\Bbb Z S _3$ -modules due to M. P. Lee. (There are, in all, ten such isomorphism types, and in general there is no Krull-Schmidt Theorem for their direct sums.)

In this paper, we study gradient estimates of positive smooth solutions to the p-Laplace equation $$\begin{aligned} (p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0, \end{aligned}$$ which is related to the $$L^p$$ -log-Sobolev constant on Riemannian manifolds, where a is a nonzero constant. As applications, some Liouville type results are provided.

Almost perfect sequences with $ \theta = 2 $ are studied in this paper. Recently Arasu, Ma and Voss [1] studied such sequences and they could only obtain sequences having periods 8, 12 and 28. In this paper, we prove that no other almost perfect sequences exist for the case $ \theta = 2 $ .

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

Let $$(r_n)_{n=1}^\infty $$ be a non-decreasing sequence of radii in $$(0, \infty )$$ , and let $$(\theta _n)_{n=1}^\infty $$ be a sequence of independent random arguments uniformly distributed in $$[0, 2\pi )$$ . In this paper, we establish a new sufficient condition on the sequence $$(r_n)_{n=1}^\infty $$ under which $$(r_ne^{i\theta _n})_{n=1}^\infty $$ is almost surely a zero set for Fock spaces. The condition is in terms of the sum of two characteristics involving the counting function. The sharpness of this condition is discussed and examples are presented to illustrate it.

In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let $$d \in \{1,2,3,\ldots \}$$ , $$s \in (\frac{1}{2},1)$$ , and $$\Omega \subset \mathbb {R}^d$$ be open, bounded, connected with Lipschitz boundary. Suppose $$u \in L^1(\Omega )$$ satisfies $$u \ge 0$$ a.e. in $$\Omega $$ and $$(-\Delta )^s u$$ is a Radon measure on $$\Omega $$ . Then u has a quasi-continuous representative $${\tilde{u}}$$ . Let $$a \in L^1(\Omega )$$ be such that $$a \ge 0$$ a.e. in $$\Omega $$ . Then if $$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$ and $${\tilde{u}} = 0$$ on a subset of positive $$H^s$$ -capacity of $$\Omega $$ , then $$u = 0$$ a.e. in $$\Omega $$ .

In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [1, \infty ),$$ the inequality $$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$ holds, and the reverse does for A with $$I \ge ^J A$$ and $${\mathrm{sp}}(A) \subseteq [0, 1]$$ . We also prove that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality $$\begin{aligned} \frac{A + B}{2} \ge ^J A \sharp ^J B \end{aligned}$$ holds, where $$A \sharp ^J B:= J \bigl ( (JA) \sharp (JB) \bigr )$$ . Several examples involving Pauli matrices are provided to illustrate the main results.

Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.

In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for $$\lambda $$ -hypersurfaces that generalizes Le–Sesum’s classification theorem for self-shrinkers. More specifically, they showed that a complete $$\lambda $$ -hypersurface with polynomial volume growth, bounded norm of the second fundamental form, and that satisfies $$|A|^2H(H-\lambda )\le H^2/2$$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete $$\lambda $$ -hypersurface with $$H\ge 0$$ must either be a hyperplane or a generalized cylinder.

We shall work in a universal class ~ of right near-rings, (that is, every hornomorphic image and every ideal of a near-ring in ~J is again in lIJ), and we shall assume that the universal class ~J is closed under certain near-ring constructions which will be specified later on. Radical and semisimple classes of near-rings are meant in the sense of Kurosh and Amitsur. There are two kinds of trivial multiplications for near-rings: the zero-multiplication and the constant multiplication. I1 is a natural requirement that a radical of near-tings should contain all near-rings with trivial multiplication belonging to the considered universal class. It is the purpose of the presem paper to characterize the semisimple classes of such near-ring radicals. A radical class P, is said to be hypernilpo~em, if it contains all nilpotent near-rings (cf. [3] Proposition 2.11. We may call a radical class N hyperconstant, if IR contains all constant near-rings of lJ. For recent developments in the radical theory of near-rings the excellent survey paper [8] can be consulted. In the sequel N O and N c will stand for the zero-near-ring and the constant near-ring, respectively, built on the additive group N + In our considerations Veldsman's near-ring construction [6] will play a decisive role: for any additive group N = N-, let us define an addition and a multiplication on the cartesian product N x N x N as follows

We introduce a more general definition of geodesic graph than that included in the original paper. We give some examples and also correct a minor error in the original paper.

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $��(I^2)\ge 9$ for the number of minimal generators of $I^2$ with $��(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $��(I)>��(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $��(I)>��(I^2)>\cdots >��(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $��(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends., 10 pages

In a commutative ring R with unity, a unit u is called exceptional if $$u-1$$ is also a unit. For $$R = {\mathbb {Z}}/n{\mathbb {Z}}$$ and for any $$f(X) \in {\mathbb {Z}}[X]$$ , an element $${\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}$$ is called an “f-exunit” if $$gcd(f(u),n) = 1$$ . Recently, we obtained the number of representations of a non-zero element of $${\mathbb {Z}}/n{\mathbb {Z}}$$ as a sum of two f-exunits for a particular infinite family of polynomials $$f(X) \in {\mathbb {Z}}[X]$$ . In this paper, we complete this problem by proving a similar formula for any non-constant polynomial $$f(X) \in {\mathbb {Z}}[X]$$ .

Let $\alpha $ and $\beta $ be bounded measurable functions on the unit circle T. The singular integral operator $S_{\alpha ,\,\beta }$ is defined by $S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T))$ where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of $S_{\alpha ,\,\beta }$ was calculated in general, using $\alpha ,\beta $ and $\alpha \bar {\beta } + H^\infty $ where $H^\infty $ is a Hardy space in $L^\infty (T).$ In this paper, the essential norm $\Vert S_{\alpha ,\,\beta } \Vert _e$ of $S_{\alpha ,\,\beta }$ is calculated in general, using $\alpha \bar {\beta } + H^\infty + C$ where C is a set of all continuous functions on T. Hence if $\alpha \bar {\beta }$ is in $H^\infty + C$ then $\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ).$ This gives a known result when $\alpha , \beta $ are in C.

Let G be a connected algebraic group and let X be an irreducible normal algebraic variety on which G acts regularly, where all our objects are defined over the base field 112. If G acts on X with an open orbit f~, then X is called an (algebraic) almost-homogeneous variety. Several approaches have been used to obtain a classification of such varieties under some additional assumptions. For example, a classification - even of complex analytic almost-homogeneous spaces - has been carried out in the case when the comple- ment ofis of dimension smaller or equal to one ((11) and (15)). The work of Luna and Vust ((16)) is fundamental for a good understanding especially in the case when G is a reductive connected group and a Borel subgroup of G acts on X with an open orbit in fL The open orbit f2 is then called a spherical homogeneous space and various results have been obtained for embeddings of such spaces (see e.g. (16), (5), (6), (19), (20)). Two-orbit varieties, i.e. complete almost-homogeneous varieties where the complement E of the open orbit is again an orbit of G, are in some sense the "elementary building blocks" of group actions. Important results for this approach have been obtained by Ahiezer ((1), (2), (3), see also (12)). In his papers, all complete two-orbit varieties X where the smaller orbit is a hypersurface are classified. The classification problem for codim~ E > 1 seems difficult. We hope that progress in the case where E is of codimension two which we present here will lead to some general results. In this paper we consider two-orbit varieties X with smaller orbit E of codimension two such that the acting group G is reductive. Of course, there are two-orbit spaces of this latter kind which arise from one of Ahiezer's examples by blowing down the smaller orbit. Simple examples of these are fibre bundles over homogeneous rational manifolds where the fibre is a Hirzebruch surface. These may be blown down to bundles where the fibre is a Hirzebruch cone. Carefully looking at Ahiezer's classification, it is in principle possible to determine all two-orbit varieties with smaller orbit of codimension two which may be obtained by blowing down the hypersurface orbit in one of Ahiezer's examples. This is why we concentrate on those varieties which don't arise in this way. The orbits of the maximal compact subgroup of G are then of codimension greater than one. We call this the Case 2. We obtain a complete picture of all two-orbit varieties arising in this case (see Theorem A and Theorem B). In short, each of these varieties is a fibre bundle G x Pl Z with fibre Z over the homogeneous rational manifold G/P1, and the construction of this bundle may be explicitly described. Conversely, the precise description of the fibre Z and

Let $$\mu _{M, D}$$ be the self-affine measure generated by an expanding integer matrix $$M\in M_{2}(\mathbb {Z})$$ and an integer three-element digit set $$D=\{(0,0)^T, (\alpha ,\beta )^T,(\gamma ,\eta )^T\}$$ . In this paper, we show that if $$3\mid \det (M)$$ and $$3\not \mid \alpha \eta -\beta \gamma $$ , then $$L^2(\mu _{M,D})$$ has an orthogonal basis of exponential functions if and only if $$M^*\varvec{u}\in 3\mathbb {Z}^2$$ , where $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .

In [6] squarelike groups were defined to be those groups G universally equivalent to their direct squares G × G. In that paper it was shown that G is squarelike if and only if G is universally equivalent to a discriminating group in the sense of [3]. Further it was shown that the class of squarelike groups is first-order axiomatizable while the class of discriminating groups is not. In this paper, we prove that the class of squarelike groups is the least axiomatic class containing the discriminating groups.

Let $$(R,{\mathfrak {m}},k)$$ be a Noetherian local ring of dimension $$d\ge 4$$ . Assume that $$2\le i \le d-2$$ is an integer and $$x_1,\ldots ,x_i$$ is a part of a system of parameters for R. Let $$\Upsilon _i$$ denote the set of all prime ideals $${\mathfrak {p}}$$ of R such that $$\dim R/{\mathfrak {p}}=i+1$$ , $${\text {Supp}}H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\{{\mathfrak {m}}\}$$ , and $$\dim _{k} {\text {Soc}}_R H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\infty $$ . In this paper, it is shown that $$\Upsilon _i$$ is an infinite set.

In this paper, we will prove a Liouville theorem for poly-harmonic functions on $${{\mathbb {R}}}^{n}_{+}$$ with Navier boundary conditions, that is, the nonnegative poly-harmonic functions u satisfying $$u(x)=o(|x|^{3})$$ at $$\infty $$ must assume the form $$\begin{aligned} u(x)=C x_{n} \end{aligned}$$ in $$\overline{{{\mathbb {R}}}^{n}_{+}}$$ , where $$n\ge 2$$ and C is a nonnegative constant. The assumption $$u(x)=o(|x|^{3})$$ at $$\infty $$ is optimal for us to derive the super poly-harmonic properties of u.

T. Bandman and Yu. G. Zarhin have proved that for every word $$w \in F_n, w\notin F_n^2$$, the corresponding word map $${\tilde{w}}: \mathrm {PGL}_2^n(K)\rightarrow \mathrm {PGL}_2(K)$$ is surjective if K is an algebraically closed field of characteristic zero (here $$F_n$$ is a free group of rank n and $$F_n^i$$ is ith member of the derived series). For words $$w\in F_n^2, n >1,$$ which are not decomposable into a product $$w = w_1w_2$$ of two words with independent variables, there are only two examples when the corresponding word map for the algebraic group $$\mathrm {PGL}_2$$ is surjective. In this paper, we construct infinite recursive sequences of indecomposable words $$\{w_m\}_{m\in {{\mathbb {N}}}}$$ in $$F_2$$ such that the word maps $${\tilde{w}}_m$$ are surjective for the algebraic group $$\mathrm {PGL}_2$$ and, if $$w_m\in F_2^i$$, then $$w_{m+1} \in F_2^{i+1}$$. Also, we construct infinite recursive sequences of indecomposable words $$\{w^\prime _m\}_{m\in {{\mathbb {N}}}}$$ in $$F_3$$ such that word maps $${\tilde{w}}^\prime _m$$ are surjective for the algebraic group $${{\text {SL}}}_2$$ and, if $$w^\prime _m\in F_3^i$$, then $$w^\prime _{m+1} \in F_3^{i+1}$$.

In this paper, it is proved that the finite group G is solvable if cod $$(\chi ) \le p_{\chi }\cdot \chi (1)$$ for any nonlinear irreducible character $$\chi $$ of G, where $$p_{\chi }$$ is the largest prime divisor of $$|G:\mathrm{ker} \chi |$$ .

Let $\mathcal{G}$ be the class of all finite groups and consider the function $\psi'':\mathcal{G}\longrightarrow(0,1]$, given by $\psi''(G)=\frac{\psi(G)}{|G|^2}$, where $\psi(G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of $\psi''$ is a dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of $\psi''$., Comment: 7 pages

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. The main aim of this paper is to investigate the influence of p-sylowizers on the structure of finite groups. We obtained some new characterizations of p-nilpotent and supersolvable groups by the permutability of the p-sylowizers of some p-subgroups. In addition, we determined all p-sylowizers of arbitrary p-subgroups for the supersolvable groups.

What is the largest cardinal of a two-by-two equilateral point-set in an n-dimensional Minkowski space? It is conjectured that this cardinal is at least $$n+1$$, and several spaces are tight in this regard, such as the Euclidean space. In this paper, we prove in normed spaces X with the $$(H_1,\dots ,H_{n-1})$$-2-intersection property the existence of $$n+1$$ equilateral point-sets of large diameter inscribed to the unit ball B. This extends the construction of Makeev (J Math Sci (N Y) 140:548–550, 2007) in dimension 4.

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).

Let $G$ be a finite group and $\psi(G)=\sum_{g\in{G}}{o(g)}$. There are some results about the relation between $\psi(G)$ and the structure of $G$. For instance, it is proved that if $G$ is a group of order $n$ and $\psi(G)>\dfrac{211}{1617}\psi(C_n)$, then $G$ is solvable. Herzog {\it{et al.}} in [Herzog {\it{et al.}}, Two new criteria for solvability of finite groups, J. Algebra, 2018] put forward the following conjecture: \noindent{\bf Conjecture.} {\it {If $G$ is a non-solvable group of order $n$, then $${\psi(G)}\,{\leq}\,{{\dfrac{211}{1617}}{\psi(C_n)}}$$ with equality if and only if $G=A_5$. In particular, this inequality holds for all non-abelian simple groups.} } In this paper, we prove a modified version of Herzog's Conjecture., Comment: 9 pages

1. Introduction. In this paper, we investigate when two mixed nonsplitting abelian groups of torsion free rank one are lattice isomorphic. That is, their lattice of subgroups are isomorphic. This is the only outstanding case in the quest to know when abelian groups are lattice isomorphic, as we described in [4]. In [4], we also derived the necessary and sufficient conditions for mixed splitting abelian groups of torsion free rank one to be lattice isomorphic. Ostendorf has obtained a similar result in [6]. These conditions are that the torsion subgroups be isomorphic and the type of one of the groups can be obtained from the type of other by a permutation of the primes, that fixes primes occurring as orders of elements. This generalizes well-known results on when two splitting abelian groups of torsion free rank one are isomorphic. In the 1960's Rotman [8] and Megibben [5] proved that for many classes of mixed abelian groups G of torsion free rank one (eg : countable G) the height matrix U (G) was the distinguishing characteristic. It is our purpose here to extend our results in [4] to obtain parallel generalization of Rotman and Megibben's work. The necessary and sufficient conditions are summarized in the following two Theorems. Theorem I. Let G be a mixed abelian group of torsion free rank one, and assume that G is lattice isomorphic to an abelian group H. Then; H is a mixed abelian group of torsion free rank one, T(G) is isomorphic to T(H), and U(H) can be obtained from U(G) by permutation 11 of primes, with primes occurring as orders of elements of T(G) ~- T(H) fixed. Theorem II. Let G and H be two mixed abelian groups of torsion free rank one with T(G) ~- T(H). And assume that U (H) can be obtained from U (G) by a permutation 17 of primes, which fixes the primes occurring as orders of elements of T(G) ~- T(H). Now if; (i) G and H are splitting over their torsion subgroup, or (ii) G and 1t are countable, or (iii) T(G) ~ T(H) is a direct sum of countable groups, or (iv) T(G) ~ T(H) are both closed, then G is lattice isomorphic to H. We conclude this paper with an example of two mixed abelian groups of torsion free rank one with the same torsion parts and the same height matrices which are not lattice

We consider the Cauchy problem for the nonlinear wave equation $$u_{tt} - \Delta _x u +q(t, x) u + u^3 = 0$$ with smooth potential $$q(t, x) \ge 0$$ having compact support with respect to x. The linear equation without the nonlinear term $$u^3$$ and potential periodic in t may have solutions with exponentially increasing $$H^1(\mathbb {R}^{3}_{x})$$ norm as $$t\rightarrow \infty $$. In Petkov and Tzvetkov (IMRN, https://doi.org/10.1093/imrn/rnz014), it was established that by adding the nonlinear term $$u^3$$, the $$H^1(\mathbb {R}^{3}_{x})$$ norm of the solution is polynomially bounded for every choice of q. In this paper, we show that the $$H^k({{\mathbb {R}}}^3_x)$$ norm of this global solution is also polynomially bounded. To prove this, we apply a different argument based on the analysis of a sequence $$\{Y_k(n\tau _k)\}_{n = 0}^{\infty }$$ with suitably defined energy norm $$Y_k(t)$$ and $$0< \tau _k