We construct the twisted Hall algebra H_ tw( C( P)) involving ℤ2-graded projective complexes of quiver representations. Then we describe two-parameter quantum groups of symmetric Kac-Moody Lie algebras via the localization D H( A) of H_ tw( C( P)) with respective to acyclic complexes. For a Dynkin quiver, we realized generic two-parameter quantum groups via corresponding Hall algebras defined by the Ringel-Hall polynomial.
A proper edge coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v, the set of colors used on the edges incident to u and the set of colors used on the edges incident to v are not included with each other. The strict neighbor-distinguishing index of G is the minimum number χ snd ′ (G) of colors in a strict neighbor-distinguishing edge coloring of G. It was conjectured that every simple graph G without leaves has χ snd ′ (G) ⩽ 2Δ except a special graph HΔ. We show that if G is a planar graph without 4-cycles, then χ snd ′ (G) ⩽ Δ + 300.
Let R be a commutative ring and S a multiplicative subset of R. This paper is a sequel to previous works, where we established some module-theoretic characterizations of S-Noetherian and S-coherent rings. We show that, under certain conditions, a ring R is S-Noetherian (or S-coherent) if and only if the class of all S-injective (or S-FP-injective) R-modules is (pre)covering. These results provide the S-counterpart of Enochs’s (or Pinzon, Dai, and Ding’s) characterizations for Noetherian (or coherent) rings using the notion of (pre)covering.
By introducing a module structure on the tensor product of a symmetric module of a Leibniz algebra, we obtain a construction of Leibniz algebras which generalizes the usual semidirect sums. The resulting Leibniz algebras, called the generalized semidirect sums, are given by a pair of module homomorphisms. Some properties of generalized semidirect sums are described, which are applied to split Leibniz algebras and their symmetric modules. Generalized semidirect sums from the complex 3-dimensional simple Lie algebra and its finite-dimensional irreducible modules are classified up to isomorphism of Leibniz algebras.
Given two graphs G and H, an edge-coloring of the complete graph Kn is called a (G,H)-good coloring if it contains neither a monochromatic copy of G nor a rainbow copy of H. Let R(n;G,H) denote the set of integers k for which a (G,H)-good k-coloring of Kn exists. The numbers max R(n;G,H) and min R(n;G,H) are called the mixed anti-Ramsey numbers. We determine these numbers for G = 2K2 and H = Ks with 3 ⩽ s ⩽ n. We show that min R(n;2K2,Ks) = n − 2 for all s. For max R(n;2K2,Ks), we prove that if s = 2k + 1 then max R(n;2K2,Ks) = t + n − k, where t is the Tuán number for Kk+1. If s = 2k + 2 and s > √(2n-74)+ 12 , then max R(n;2K2,Ks) = t −1 + 2n − k − 2. The latter result reveals a connection between max R and the Turán number of K_⌊s-12⌋ .
Let H, K be subgroups of G. Then H and K are called c-permutable in G if there exists an element x ∈ G such that HKx = KxH. By imposing c-permutability conditions on the members of a maximal subgroup series of G, we give some new criteria for the supersolubility of a finite group. We prove that if G is a soluble group with a maximal subgroup series 1 = M0 < M1 < … < Ms−1 < Ms = G, where Mi is c-permutable in G for every i = 0, 1, …, s, then G is supersoluble. On this basis, if G is not soluble, but we have the condition that Mi is c-permutable in Mi+i (0 ⩽ i ⩽ s − 1), then G is also supersoluble. These results provide new characterizations of supersoluble groups.
The purpose of this paper is to establish the trace inequality for fractional operators involving Hausdorff capacity and to characterize the Borel measures μ for which these inequalities hold. We further investigate two-weight norm inequalities for fractional integrals and fractional maximal operators on Choquet spaces with respect to Hausdorff capacities. As an application of the trace inequality, we derive an embedding inequality for Sobolev spaces with respect to the Hausdorff capacity into Lq(μ). This result, in turn, leads to a study of the Hardy inequality. In addition, we establish a generalized Stein-Weiss inequality through the two-weight inequalities.
This paper is devoted to the study of a class of vector-valued convolution operators. A fundamental problem in this context is to characterize the nonnegative measures μ for which such operators are bounded from Lp(ℓ1) to Lq(ℓ1; μ). Inspired by classical capacity theory, we introduce a novel notion of Lp(ℓ1) capacity specifically adapted to the vector-valued setting. By developing this capacity approach, we establish complete necessary and sufficient conditions on the measure μ for the boundedness of the operators. Our results provide a unified characterization that covers both cases p ⩽ q and q < p. The findings yield a powerful tool for analyzing mapping properties of vector-valued operators in weighted spaces.
Let L(H) denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space H and let (𝒥,·_𝒥) denote a norm ideal in L(H). For A, B ∈ L(H), the generalized derivation δA,B and the multiplication operator MA,B are defined on L(H) by δA,B(X) = AX − XB and MA,B(X) = AXB. Denote τA,B = MA,B − I and dA,B= δA,B or τA,B. We give some pairs (A, B) of operators A and B such that the range R(d_A,B|𝒥) is orthogonal to the kernel (d_A,B|𝒥) of the restriction d_A,B|𝒥 with respect to unitarily invariant norm ·_𝒥 . It is proved that if A and B* are subnormal, then the Hilbert-Schmidt operators in the range of MA,B are orthogonal to those in the kernel of MA,B. We also give some related results.
Let G be an Abelian group. Then G is called a T 1 ′ group provided that for every pair of isomorphic subgroups H and K of G and every isomorphism φ: H → K, φ can be extended to an automorphism of G. Let us call G a weakly T 1 ′ group if for any elements a and b of G of the same order, there is an automorphism φ: G → G such that φ(a) = b. We determine both classes of groups, correcting a result in the literature. We then study several analogues for rings.
Let R be a commutative ring with identity and S be a multiplicative subset of R. We introduce the notion of S-2-absorbing quasi primary ideal, which is a generalization of quasi-S-primary ideal. We define a proper ideal I of R to be S-2-absorbing quasi primary if √(I) is an S-2-absorbing ideal of R. Additionally, we define a new class of rings, called S-2-ABQ ring. A number of results concerning S-2-absorbing quasi primary ideals and S-2-ABQ rings are presented. Examples and key results are provided to illustrate these concepts.
We derive twenty different multivariable generalizations of Jackson's (2)phi(2) transformation by applying elementary proofs. Then, by reversing the order of summation, we obtain eight multivariable extensions of transformations of terminating (2)phi(1) series. As limiting cases of our main results, we obtain three A(n)(1)phi(1) summation formulae.
Topological indices are key quantitative descriptors in mathematical chemistry, unchanged under symmetry operations and retaining graph connectivity; they capture molecular structural features to provide insights into molecular stability and chemical properties, becoming indispensable in cheminformatics and theoretical chemistry. Among degree-based indices, the Sombor index is widely concerned for capturing structural information, and motivated by enhanced structural discrimination, the augmented Sombor index (ASO) is defined for a connected graph Ω with ∣V(Ω)∣ ⩾ 3 as ASO(Ω) = ∑_v_i v_j ∈ E(Ω)√(d_i^2 + d_j^2/d_i + d_j - 2), where di and dj are the degrees of vertices vi and vj, respectively. Within the scope of this study, we first establish several sharp bounds for the augmented Sombor index and characterize the extremal graphs attaining these bounds. In particular, we determine the minimum value of the ASO index for unicyclic graphs with a prescribed girth and characterize all graphs achieving this minimum. We also identify the second maximum ASO value among trees and characterize the corresponding extremal tree structures. Furthermore, the minimum and maximum values of the ASO index for bipartite graphs and chemical graphs are obtained, together with a complete characterization of the associated extremal graphs. In addition, we characterize the chemical trees that maximize the ASO index. The chemical applicability of the ASO index is investigated through quantitative structure-property relationship (QSPR) analysis, supported by a comparative assessment of several variants of the Sombor index. Finally, we present concluding remarks and outline potential directions for future research on the augmented Sombor index of graphs.
We derive twenty different multivariable generalizations of Jackson’s 2ϕ2 transformation by applying elementary proofs. Then, by reversing the order of summation, we obtain eight multivariable extensions of transformations of terminating 2ϕ1 series. As limiting cases of our main results, we obtain three An1ϕ1 summation formulae.
This paper is devoted to studying conformal algebras of quotients of Lie conformal algebras. The notions of conformal algebras of quotients of Lie and associative conformal algebras are introduced and some necessary and sufficient conditions about their conformal algebras of quotients are developed. Moreover, many properties of conformal algebras of quotients are also investigated. Besides, the maximal conformal algebra of quotients of every semi-prime Lie conformal algebra is constructed. Moreover, for any two associative conformal algebras, the relation of ‘quotients’ can be inherited by Lie conformal algebras generated by them.
Let A, W be two n × n complex matrices. We present the expression of the W-core inverse of A by Hartwig and Spindelböck’s decompositions and full rank decompositions. It is also proved that A is W-core invertible if and only if A is right W-core invertible. This equivalence may not be true in a general *-ring, see H. H. Zhu, L. Y. Wu, D. Mosić (2023). Then, several results for the reverse order law of the W-core inverse are given. Another accomplishment of this paper is to establish some perturbation properties and perturbation bounds for the W-core inverse, under some conditions. Finally, the necessary and sufficient condition for the continuity of the W-core inverse is derived.
Given a prime p and a subgroup A of a finite group G, we say that A is a p-CAP-subgroup of G if A covers or avoids every p-G-chief factor, where a p-G-chief factor is a G-chief factor of order divisible by p. We say that A is a strong p-CAP-subgroup of G if A is a p-CAP-subgroup of any subgroup of G containing A. We use the concept of strong p-CAP-subgroups to investigate the p𝔉 -hypercentrally embedded property of normal subgroups of a finite group and obtain some new results. Moreover, we extend the concept of (strong) p-CAP-subgroups to fusion systems and use this to characterize supersolvable and nilpotent fusion systems.
We study the variable exponent double phase functionals with the critical growth. We show the sharp conditions for quasiminimizers of these functionals to be bounded or Holder continuous. Our results generalize to variable exponent case results obtained by several authors in constant exponent case.
