We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

Let D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.

In this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones.

We use a method developed by Strauss to obtain global well-posedness results in the mild sense and existence of asymptotic states for the small data Cauchy problem in modulation spaces ${M}^s_{p,q}(\mathbb{R}^d)$, where q = 1 and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

Let γn = [x1,…,xn] be the nth lower central word. Denote by Xn the set of γn -values in a group G and suppose that there is a number m such that $|{g^{{X_n}}}| \le m$ for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

For p ≥ 1, one can define a generalisation of the unknotting number tup called the pth untwisting number, which counts the number of null-homologous twists on at most 2p strands required to convert the knot to the unknot. We show that for any p ≥ 2 the difference between the consecutive untwisting numbers tup–1 and tup can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between tu1 and tu2.

We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$ , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$ , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to ${\mathfrak E_{ \lt \infty }}$ , and that those of them that belong to $\mathfrak E_0$ are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is,

$${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$

Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

Let $N_g^k$ be a nonorientable surface of genus g with k punctures. In the first part of this note, after introducing preliminary materials, we will give criteria for a chain of Dehn twists to bound a disc. Then, we will show that automorphisms of the mapping class groups map disc bounding chains of Dehn twists to such chains. In the second part of the note, we will introduce bounding pairs of Dehn twists and give an algebraic characterization for such pairs.

Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner’s bigraphs and crossed modules are considered.

Consider the following two eigenvalue problems: (0.1)

\begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases}

(0.2)

\begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(\pi)+\mu z(\pi)=0, \end{cases}

where $q(x)$ is real-valued and integrable on [0, $\pi$ ]. Let $\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ and $\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ denote the eigenvalues of equations (0.1) and (0.2), respectively. Then

\[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt\lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\]

$\lambda_n$

$\mu_n$

$\pi$

$|n|-1$

In this paper, we consider the following fractional Schrödinger–Poisson system with singularity

\begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*}

where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.

Erdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$ , γ being Euler’s constant, where $\Phi(n)=\sum_{d|n} \frac{\log d}{d}$ .

We extend this result to the function $\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\gamma$ . The proof requires a new approach. As an application, we prove that for any $\eta>1$ , any finite sequence of reals $\{c_k, k\in K\}$ , $\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu)$ , where C(η) depends on η only. This improves a recent result obtained by the author.

This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes,

$$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$

$${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$

$${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$