Babai’s conjecture for high-rank classical groups with random generators

Let $$G = {\text {SCl}}_n(q)$$ G = SCl n ( q ) be a quasisimple classical group with n large, and let $$x_1, \ldots , x_k \in G$$ x 1 , … , x k ∈ G be random, where $$k \ge q^C$$ k ≥ q C . We show that the diameter of the resulting Cayley graph is bounded by $$q^2 n^{O(1)}$$ q 2 n O ( 1 ) with probability $$1 - o(1)$$ 1 - o ( 1 ) . In the particular case $$G = {\text {SL}}_n(p)$$ G = SL n ( p ) with p a prime of bounded size, we show that the same holds for $$k = 3$$ k = 3 .

A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define α,β-complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of α,β-complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.

On analogs of some classical group-theoretic results in Poisson algebras

We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.

Single-Center Prospective Cohort Study on the Histopathology, Genotype, and Postsurgical Outcomes of Patients With Primary Aldosteronism

Unilateral forms of primary aldosteronism are usually surgically treated to remove the source of aldosterone excess. After adrenalectomy, aldosteronism persists in some patients indicating abnormal aldosterone production from the unresected gland. Our objective was to investigate histopathology, genotype, and postsurgical outcomes in a 3-year prospective cohort of surgically treated patients for primary aldosteronism (from 2016 to 2018). The cohort comprised 60 consecutively operated patients categorized with classical or nonclassical histopathologic findings of unilateral primary aldosteronism. In the classical group were 45 solitary aldosterone-producing adenomas or dominant aldosterone-producing nodules; in the nonclassical group were 15 cases of multiple aldosterone-producing micronodules or nodules (12 cases) or aldosterone-producing diffuse hyperplasia (3 cases). The classical group displayed higher baseline plasma aldosterone concentrations (262 versus 155 pg/mL, P =0.008) and an increased aldosterone-to-renin ratio (81 versus 42, P =0.002). A high proportion of the classical group achieved complete biochemical success (97.6% versus 66.7% in the nonclassical group, P =0.002). The nonclassical versus classical group displayed an increased ratio of absolute aldosterone concentration in the contralateral adrenal vein to peripheral vein at adrenal venous sampling (3.8 versus 2.0, P =0.004). Variants in aldosterone-driver genes were identified in 85% of 41 aldosterone-producing adenomas and were excluded in the remaining 15% by CYP11B2 guided next-generation sequencing. There were no differences in clinical or biochemical outcomes in patients with a solitary aldosterone-producing adenoma categorized by KCNJ5 mutation status. In conclusion, adrenals with a nonclassical histopathology of unilateral primary aldosteronism are associated with a higher incidence of postsurgical disease persistence and increased aldosterone production from the unresected adrenal.

Nonsymmetric Macdonald Superpolynomials

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let [Formula: see text] be a classical group over an arbitrary field [Formula: see text], acting on an [Formula: see text]-dimensional [Formula: see text]-space [Formula: see text]. All those maximal subgroups of [Formula: see text] are classified each of which normalizes a solvable subgroup [Formula: see text] of [Formula: see text] not lying in [Formula: see text].

Simple , safe and fast Ponseti cast removal procedure in pes equina varus patients

Objective: In our study, it is aimed to remove the cast more easily and safely without using the cutting tools by leaving the cast ends marked by folding in the idiopathic clubfoot patients treated with Ponseti method. Material and Methods: Forty feet of 29 patients treated for Pes Equinovarus were included in the study. Patients were followed up in two groups. The group treated with Ponseti method by cast marking were named as “modified group” and cast wrapped group without marking were named as “classical group”. Neurological, teratologic and syndromic clubfoot patients were not included in the study. During the six series of casting, cast removal times for each extremity are recorded in minutes and it is noted that whether any additional cutting tool is used during cast removal or not. A summary of the data was presented as mean, standard deviation and percentage. Comparisons of the categorical characteristics were analysed by using the Chi-square test and the Mann-Whitney test. IBM-SPSS 20 program was used for analysis. In all tests, the level of significance was adjusted to 0.05. Results: Thirteen (44.8%) of the 29 patients were male and 16 (55.2%) were female. While the mean time to start treatment for the 15 patients in the modified group was 3.46 (2-7) days, mean time for the 14 patients in the classical group was 3.78 (2-10) days. While the mean cast removal time of the 20 extremities of 15 patients in the modifying group was 10.9 minutes (8-14.3 min);it was 22.2 minutes (17.1-29.5 min) for the 20 extremities of 14 patients in the classical group. While no additional cutting tool was used during cast removal in the modified group, additional cutting tools were used during removal of cast in 75% (15/20) of the patients in the classical group and statistically significant difference was found between two groups in terms of the use of cutting tools (p<0.001). Conclusion: We found that the cast ends’ being marked by folding during plastering in idiopathic clubfoot patients treated with Ponseti technique is costless, easy to apply, significantly shortens cast removal time, does not require the use of cutting tools, and thus is a notably safe method for these patients.

Neutrosophic Multigroup Homomorphism and Some of its Properties

In a way, the notion of neutrosophic multigroup is an application of neutrosophic multisets to the theory of group. The concept of neutrosophic multigroup is an algebraic structure of neutrosophic multiset that generalizes both the theories of classical group and neutrosophic group. Neutrosophic multigroup constitutes an application of neutrosophic multiset to the elementary theory of classical group. In this paper, we propose the concept of homomorphism on neutrosophic multigroup. We define homomorphism kerlf, automorphism, homomorphic image and homomorphic preimage of neutrosophic multigroup, respectively. Some homomorphic properties of neutrosophic multigroup are explicated. Some homomorphic properties of neutrosophic multigroup are also discussed. This new concept of homomorphism as a bridge among set theory, fuzzy set theory, neutrosophic multiset theory and group theory and also shows the effect of neutrosophic multisets on a group structure. We finally derive the basic properties of neutrosophic multigroup homomorphism and give its applications to group theory