A common generalization of convolved (u, v)-Lucas first and second kinds p-polynomials

In this note the convolved (u, v)-Lucas first kind and the convolved (u, v)-Lucas second kind p-polynomials are introduced and study some of their properties. Several identities related to the common generalization of convolved (u, v)-Lucas first and second kinds p-polynomials are also presented.

On reduced rings with prime factors simple

We obtain a common generalization of the results by Wong and Birkenmeier-Kim-Park, respectively, which say that a reduced ring with unity is strongly (respectively, weakly) regular if and only if all of its prime homomorphic images are division rings (respectively, simple domains). Our arguments are different from those in the known proofs and are quite simple. They also give a characterization of weakly regular reduced rings without unity. This characterization implies in particular that the class of weakly regular reduced rings forms a radical class. However, even if a weakly regular reduced ring has no unity, its prime homomorphic images must be simple domains with unity. In the second part of the paper, we study reduced rings whose prime homomorphic images are simple domains (not necessarily with unity).

Discrete Quantitative Nodal Theorem

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. A special case of this result will assert that if the second and third eigenvalues of the Laplacian are at least $\varepsilon$ apart, then the subgraphs induced by the positive and negative supports of the eigenvector belonging to $\lambda_2$ are not only connected, but edge-expanders (in a weighted sense, with expansion depending on $\varepsilon$).

Approximating nash social welfare under rado valuations

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.

Gorenstein flat modules relative to injectively resolving subcategories

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.

On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

We obtain asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{mn\le x} f((m,n))$$ ∑ m n ≤ x f ( ( m , n ) ) and $$\sum _{mn\le x} f([m,n])$$ ∑ m n ≤ x f ( [ m , n ] ) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions $$f(n)=\tau (n), \log n, \omega (n)$$ f ( n ) = τ ( n ) , log n , ω ( n ) and $$\Omega (n)$$ Ω ( n ) . We also define a common generalization of the latter three functions, and prove a corresponding result.

$4$-Colouring of Generalized Signed Planar Graphs

Assume $G$ is a graph and $S$ is a set of permutations of positive integers. An $S$-signature of $G$ is a pair $(D, \sigma)$, where $D$ is an orientation of $G$ and $\sigma: E(D) \to S$ is a mapping which assigns to each arc $e=(u,v)$ a permutation $\sigma(e)$ in $S$. We say $G$ is $S$-$k$-colourable if for any $S$-signature $(D, \sigma)$ of $G$, there is a mapping $f: V(G) \to [k]$ such that for each arc $e=(u,v)$ of $G$, $\sigma(e)(f(u)) \ne f(v)$. The concept of $S$-$k$-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which subsets $S$ of $S_4$, every planar graph is $S$-$4$-colourable. We call such a subset $S$ of $S_4$ a good subset. The Four Colour Theorem is equivalent to saying that $S=\{id\}$ is good. It was proved by Jin, Wong and Zhu (arXiv:1811.08584) that a subset $S$ containing $id$ is good if and only if $S=\{id\}$. In this paper, we prove that, up to conjugation, every good subset of $S_4$ not containing $id$ is a subset of $\{(12),(34),(12)(34)\}$.

Inclusion properties of generalized inverses in semiprime rings

Let [Formula: see text] be a semiprime ring, not necessarily with unity, with extended centroid [Formula: see text]. For [Formula: see text], let [Formula: see text] (respectively [Formula: see text], [Formula: see text]) denote the set of all outer (respectively inner, reflexive) inverses of [Formula: see text] in [Formula: see text]. In the paper, we study the inclusion properties of [Formula: see text], [Formula: see text] and [Formula: see text]. Among other results, we prove that for [Formula: see text] with [Formula: see text] von Neumann regular, [Formula: see text] (respectively [Formula: see text]) if and only if [Formula: see text] (respectively [Formula: see text]). Here, [Formula: see text] is the smallest idempotent in [Formula: see text] such that [Formula: see text]. This gives a common generalization of several known results.