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In this note, we give a counterexample for Theorem 2.3 of the above mentioned paper that is a generalization of the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules.

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.

If a finite group G is \({\mathcal{N}}\)-constrained, then the nilpotent injectors of G form a single conjugacy class of subgroups. In this paper we shall generalize this result. This paper is part of a larger program investigating a special type of nilpotent injectors in arbitrary finite groups.

In this paper, we study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators. The results in this paper extend Theorem 1.1 in [19] by Liu and Yang and deal with Question 1 in [19], where the entire functions are of finite order. Moreover, we repair certain statements in [21] by Li et al., which in turn had depended on questionable assertions of Lemma 2.6 in [20]. Examples are provided to show that the results in this paper are best possible.

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)].

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.

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.

Morrison and Pinkham [4] gave a characterization of the semigroups of Galois Weierstrass points, i.e., total ramification points of cyclic coverings of the projective line of degree n. They showed that such a semigroup must satisfy certain equalities, which we call the M-P equalities in this paper, and that the converse holds for any prime \(n\leqq 7\). In this paper we consider the case when n is a prime number \(p \geqq 11\). For each prime \(p \geqq 11\), we give a semigroup which satisfies the M-P equalities but is not the semigroup of a Galois Weierstrass point. For this, we study the semigroups of Galois Weierstrass points using the equations defining curves which are cyclic covering of the projective line.

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.)

Let A be a finite dimensional algebra over an algebraically closed field k. It is conjectured that A has to be of tame representation type provided A is strongly simply connected and its Tits quadratic form is weakly non-negative.¶In the paper a partial result in this direction is proved. Instead of the Tits form the Euler form $\chi _A$ is considered. Let $R_1,\ldots ,R_t$ and $R^\prime _1,\ldots ,R^\prime _s$ be two sequences of modules over an algebra B. We consider $R=\mathop\oplus\limits ^t_{i=1}R_i$ as a $B-k^t$ -bimodule and $R^\prime =\mathop\oplus\limits ^s_{j=1}R^\prime _j$ as a $B-k^s$ -bimodule. The biextension $[R^\prime ]B[R]$ of B by the two sequences is by definition the matrix algebra¶¶ $\left (\matrix {k^t&0&0\cr R&B&0\cr DR^\prime \mathop\otimes _BR&DR^\prime&k^s}\right)$ ¶¶equipped with the obvious addition and multiplication, where we denote by $D=\hbox {Hom}_k(-,k)$ the usual duality. For any set of pairs of indices $L\subset \{1,\ldots ,t\}\times \{1,\ldots ,s\}$ , consider the subspace $V=\mathop\oplus\limits_{(i,j)\in L} DR_{j} \otimes_{B} R_j$ of the space $DR^\prime \otimes _B R$ and the associated ideal J(V) in $[R^\prime ]B[R]$ . The algebra $A=[R^\prime ]B[R]/J(V)$ is called a truncated biextension of B.¶The main result of the paper says: If B is a strongly simply connected algebra with at least 6 vertices and $R_1,\ldots ,R_t;R^\prime _1,\ldots ,R^\prime _s$ are two sequences of indecomposable B-modules such that $\chi_A$ is non-negative with $\hbox {corank}\,\chi _{_A}=1+s+t$ and $\hbox {corank}\, \chi _{_B}=1$ , then $A=[R^\prime ]B[R]$ is of tame representation type.

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L p-space, with \(1 \le p \neq 2 \le \infty \), and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L p-space; one acts on L p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on \(L^2 ({\Bbb R}^2;X)\) admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property \((\alpha )\) (geometric property introduced by G. Pisier).

D e f i n i t i o n . A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection \(P:Y\to X\) such that P(B(Y)) \(\subset\) A (by B we denote the unit ball). ¶The notion of sufficient enlargement is implicit in the paper: B. Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95, 451 - 465 (1960). It was explicitly introduced by the author in: M. I. Ostrovskii, Generalization of projection constants: sufficient enlargements, Extracta Math. 11, 466 - 474 (1996). ¶The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces.

Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and $$(\infty ,\infty )$$ . His method characterizes the $$L^{p}$$ norm in terms of the Lebesgue spaces $$L^{1}$$ and $$L^{\infty }$$ , and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types $$(p_{1},q_{1})$$ and $$(\infty ,\infty )$$ with $$1\le p_{1}\le q_{1}

The authors would like to correct the errors in the publication of the original article, and one of the authors also wants to update his present affiliation. The present affiliation and corrected details are given below for your reading: Unfortunately, Corollary 2.3 in the original paper was stated incorrectly; it should be noted that all other results in the original paper are correct as stated. The correct statement is as follows. Note that the only difference is the addition of the hypothesis Δy(0) ≥ 0. The exclusion of this hypothesis is the error Corollary 2.3 in the original paper. Corollary 2.3. Let y : N0 → R be a nonnegative function. Fix ν ∈ (1, 2) and suppose that Δ0y(t) ≥ 0 for each t ∈ N2−ν . If, in addition, it holds that Δy(0) ≥ 0, then y is increasing on N0. Due to this change, we should also point out that some of the results in Goodrich [1] must be slightly changed as well, namely if we wish to argue that ΔN−1y(t) ≥ 0 for all t ∈ N0, then we must require ΔN−1y(0) ≥ 0, and this was omitted in certain of the results—see, for example, [1, Theorem 2.6, Example 2.9, Corollaries 2.8, 2.10, and 2.11].

In a 1988 paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. In this paper, we use Cowen's adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the Sa spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle.

0. Introduction. Throughout this paper, a will denote an integer such that a is not a square in Q, f : = X 2 + a e 2g [X], and f0 : = X, f , + 1 : = f (f,) = f 2 + a for all n > 0, are 2 the iterates of f. Let c 1 : = a and c,+ 1 : = f ( c , ) = c, + a ( = f , ( a) = f , ( a ) = f , + l (0)) for n > 1. We will see (in Section 1.) that there exists an integer sequence (b,)n~ 1 such that for all n > I we have c, = I ] be. Let K, be the splitting field of f , over II~ and denote by ,/In f2, : = Gal (Kn/Q) = Gal (f , /Q) its Galois group over the rational numbers. This paper deals with the problem of determining s Following R. W. K. Odoni 's paper [2], we will generalize his result concerning the case a = 1 to get for all n > 1

In this paper, we mainly investigate the conjugation of the sylowizer that was introduced by Gaschutz (Math Z 122(4):319–320, 1971) and study the p-supersolvability of finite groups by analyzing the intersection between $$O^{p}(G)$$ and sylowizers of p-subgroups. As a continuation of research (Lei and Li in Arch Math (Basel) 114:367–376, 2020), we also give some characterizations on p-nilpotent groups by using the permutability of a sylowizer of a p-subgroup.

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$).

In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class $$Lip(\Omega )$$ . Then we prove generalized strict comparison theorems on the enlarged space $$L_G^1(\Omega )$$ , which is the Banach completion of $$Lip(\Omega )$$ under the G-expectation.

In this paper we investigate some functional equations of the form $$P(f)=Q(g),$$ where P, Q are Yi’s polynomials, and f, g are meromorphic functions. Then we apply the obtained results to study the uniqueness problem for meromorphic functions sharing two subsets, and to give an analogue of Ritt’s decomposition theorem for a class of polynomials of Fermat-Waring type in meromorphic functions.

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.

In this paper, we define and study strong Kato chaos for a group action on a compact metric space. Let X be a compact metric space without isolated points, and let G be a topologically commutative group on X. If the dynamical system (X, G) is weakly mixing, then it is chaotic in the strong sense of Kato.

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.

The aim of this paper is to investigate the behaviour of uncountable groups of cardinality \({\aleph}\) in which all proper subgroups of cardinality \({\aleph}\) have modular subgroup lattice. It is proved here that the lattice of subgroups of such a group G is modular, provided that G has no infinite simple homomorphic images of cardinality \({\aleph}\). A corresponding result for groups whose proper subgroups of large cardinality are quasihamiltonian is also proved.

The main goal of this paper is the presentation of an elementary analytic technique which enables the evaluation of the so-called restricted sum formulas involving multiple zeta values with even arguments, i.e. $$E(2c,K):=\sum_{\substack{\sum_{j=1}^{K}c_{j}=c\\{c}_{j}\in\mathbb{N}}} \zeta(2c_1,\ldots ,2c_K),$$ where c and K are arbitrary positive integers with $${c\ge K}$$ . Though the young and general theory of the multiple Riemann zeta function with a rich application potential may be rather complicated, our contribution makes the evaluation of the term E(2c,K) intelligible to a broad mathematical audience.

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytonen et al. in [J. Funct. Anal. 263:3883–3899, 2012].

Let \({\{x_n\}_n}\) be the digits of the Cantor expansion of \({x \in [0,1]}\) with respect to a sequence of integers \({\{q_n\}_n}\) with \({q_n \ge 2}\). In this paper, we prove that the set consisting of those points for which the set of limit points of \({\{x_n/q_n\}_n}\) equals to [0, 1] is residual in [0, 1] if \({\lim\nolimits_{n \to \infty}q_n = +\infty}\).

Let \({\mathbb{F}_p}\) be the field of residue classes modulo a large prime p. The present paper is devoted to the problem of representability of elements of \({\mathbb{F}_p}\) as sums of fractions of the form x/y with x, y from short intervals of \({\mathbb{F}_p}\).

A finite group whose irreducible complex non-linear characters are rational is called a \({\mathbb{Q}_1}\)-group. In this paper, we study the structure of Frobenius \({\mathbb{Q}_1}\)-groups.

Introduction. It is well known that any set of 2 n - i elements of a solvable group of size n must contain a subset of size n with the property that the product of the dements of this subset arranged in the appropriate order yields the identity. In fact, if G is not cyclic, it is sufficient that the original set contain 2 n - 2 elements. The main result of this paper is that similar results hold for sets of size 2 n - r, where r is a fixed positive integer. The result says that for a fixed r, if n is sufficiently large and G is a non-cyclic solvable group, then any set of 2 n - r elements of G contains a subset of size n with the property that the product of the elements of this subset in the appropriate order is the identity. The term "set" in this paper actually means multi-set. That is, an element may occur more than once in a set. The cardinality of that set is number of elements in the set counting multiplicities. When we wish to consider an object which is a set in the classical sense, we shall use the term "ordinary set", All of the results in this paper concern sets of elements from a finite group. In some cases, the group is assumed to be abelian. In these cases, we shall use additive notation. In all other cases, multiplicative notation will be used. D e f i n i t i o n. Let G be a group and let S be a set of elements of G. A n-sum in S is an ordered subset of S of cardinality n. The result of that n-sum is the product of the elements in the n-sum in the specified order.

Introduction. All groups in this paper are assumed to be finite. Let G be a group with subgroups H 0 and H where H o A H, then a subgroup Go of G is called a relative normal complement in G of H over H o if Go & G, G = Go H and H o = G o c~ H. Given a group G with subgroups H o and H where H o/~ H, many theorems have been proved which guarantee the existence of a relative normal complement G O of H over H o. In the work of Brauer [1], Suzuki [10], Dade [2], Leonard [8], Leonard and McKelvey [7], Sah [9] and others, the proof of the existence of a relative normal complement depends on showing that certain generalized characters of H can be extended to generalized characters of G. In this paper, we present theorems which explicitly relate the extendibilitiy of certain irreducible characters of H to G and the existence of relative normal complements. Indeed, if Go is a relative normal complement of H over H o, then G/G o H/Ho. Thus, every character of H having H o in its kernel can be extended to G. Several of the theorems in this paper may be viewed as partial converses to this observation. Before, these theorems can be stated, some notation and terminology is necessary. Let n be a set of primes and denote the complementary set of primes by ~'. For a finite group G and prime p, let 1@1 denote the order of a Sylow p subgroup of G. Then [G 1~ is defined by [ GI~ = I~ I @1. We say that G is a n-group if [ G [ = I G[.. A subgroup K of G is a Hall pET~ re-subgroup of G if IKI = IKI~ --IGI=. An element x in G is a n-element if (x) is a n-group. Every element x in G has a unique decomposition x = x,~x~, = x,~, x~ into a n-element x. and a n'-element x~,. Further, x~ and x., are powers of x. If x and y are elements of a subgroup K of G, then x and y belong to the same n-section of K if their n-parts x~ and y. are conjugate in K. If S is a subset of G, then S a'~ denotes the union of all n-sections of G which intersect S. For any non-empty set A we let ]A[ denote the number of elements in A. IfH and H o are subgroups of G with Ho A H and rc is the set of primes dividing (H : Ho), then the existence and uniqueness of a relative normal complement have been related by Leonard [8] and others [4] to the equality I(H - Ho) G' "1 = (G : H) IH - Hol. This equality plays a similar role in several theorems below. If K is a group, let Irr K denote the set of irreducible characters of K. If K o is a normal subgroup of K, let Irr (K/Ko) denote the set of irreducible characters of K whose kernels contain Ko. If)l and B are disjoint sets, then A ~ B denotes the disjoint union of A and B. We may now state the theorems to be proved.

The paper deals with bounded non-selfadjoint operators having Hilbert–Schmidt imaginary Hermitian components. A perturbation bound for invariant subspaces is established. Our results can be considered as a particular generalization of the well-known Davis–Kahan sin θ-theorem for selfadjoint operators.

Let \({\mathbb{F}}\) be a finite field. The function field analog of Hermite’s theorem says that there are at most finitely many finite separable extensions of \({\mathbb{F}(T)}\) inside a fixed separable closure of \({\mathbb{F}(T)}\) whose discriminant divisors have bounded degree. In this paper we give a field theoretic proof of this result, inspired by a lemma of Faltings for comparing semisimple \({\ell }\)-adic Galois representations.

The minimal index of a pure cubic field was shown to assume arbitrarily large values by M. Hall. In this paper we extend this result by showing that every cubefree integer occurs as the minimal index of infinitely many pure cubic fields.

In an additively written abelian group, a sequence is called zero-sum free if each of its nonempty subsequences has sum different from the zero element of the group. In this paper, we consider the structure of long zero-sum free sequences and n-zero-sum free sequences over finite cyclic groups $${\mathbb{Z}_n}$$ . Among which, we determine the structure of the long zero-sum free sequences of length between $${n/3+1 }$$ and $${n/2}$$ , where $${n\ge 50}$$ is an odd integer, and we provide a general description on the structure of n-zero-sum free sequences of length n + l, where $${\ell\geq n/p+p-2}$$ and p is the smallest prime dividing n.

Inspired by the work in Dix and Karakostas (Nonlinear Anal. 71:3872–3880, 2009), we prove in this paper a new fixed point theorem for strict-set contractions leaving invariant a cone in a Banach space. This new result is used to obtain an existence result for Urysohn type integral equations.

This paper is generally devoted to study the rings with flat injective hulls. In fact, by obtaining conditions equivalent to having flat injective hull, these rings are characterized explicitly.

This paper discusses the gap sequences of self-conformal sets satisfying the strong separation condition.

Let G be a finite group, written multiplicatively. Define \({\mathsf{E}(G)}\) to be the minimal integer t such that every sequence of t elements (repetition allowed) in G contains a subsequence with length \({|G|}\) and with product one (in some order). Let p be the smallest prime divisor of \({|G|}\). In this paper we prove that if G is a noncyclic nilpotent group then \({\mathsf{E}(G) \le |G|+\frac{|G|}{p}+p-2}\), which confirms partially a conjecture by Gao and Li. We also determine the exact value of \({\mathsf{E}(G)}\) for \({G=C_{p}\ltimes C_{pn}}\) when p is a prime, which confirms partially another conjecture by Zhuang and Gao.

In this paper we prove a generalized version of a joint discrete universality theorem on the approximation of a collection of analytic functions by discrete shifts of Dirichlet L-functions.

In this note, we prove a necessary and sufficient condition for whether a d-cyclic covering of the complex projective line has gonality 2 (i.e., is elliptic or hyperelliptic), where d is a positive integer. The case of 3 branch points has been solved in our previous paper (see [3]).

In this paper, a class of generalized-Cantor sets on the line is shown to be quasisymmetrically thick.

A positive integer n is called a square-full number if p2 divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions.

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, \({|m(G)| \geq |\pi(G)|+p}\) , where \({p \in \pi(G)}\) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and \({q \in \pi(G)}\) is the smallest prime such that \({Q \in {\rm syl}_q(G)}\) is noncyclic, then \({|m(G)| \geq |\pi(G)|+q}\) . Both lower bounds are best possible.

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all \({x \in (\Gamma-\Gamma^{\prime})}\), either there is an integer n such that \({1 \neq x^{n} \in \Gamma^{\prime}}\) or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy \({{\bf LH}\mathfrak{F}}\). In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index.