Classical Groups Research Articles

Harmonic Analysis and Gamma Functions on Symplectic Groups

Over a p p -adic local field F F of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group G = G m × S p 2 n G={\mathbb {G}}_m\times {\mathrm {Sp}}_{2n} . It is associated to the Langlands γ \gamma -functions attached to any irreducible admissible representations χ ⊗ π \chi \otimes \pi of G ( F ) G(F) and the standard representation ρ \rho of the dual group G ∨ ( C ) G^\vee ({\mathbb {C}}) , and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal (Braverman and Kazhdan, 2000) for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on G L 1 ( F ) {\mathrm {GL}}_1(F) , which is associated to a γ \gamma -function β ψ ( χ s ) \beta _\psi (\chi _s) (a product of n + 1 n+1 certain abelian γ \gamma -functions). Our work on G L 1 ( F ) {\mathrm {GL}}_1(F) plays an indispensable role in the development of our work on G ( F ) G(F) . These two types of harmonic analyses both specialize to the well-known local theory developed in Tate’s thesis (Tate, 1950) when n = 0 n=0 . The approach is to use the compactification of S p 2 n {\mathrm {Sp}}_{2n} in the Grassmannian variety of S p 4 n {\mathrm {Sp}}_{4n} , with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis (1986) and many other works) on the doubling local zeta integrals for the standard L L -functions of S p 2 n {\mathrm {Sp}}_{2n} . The method can be viewed as an extension of the work of Godement-Jacquet (1972) for the standard L L -function of G L n {\mathrm {GL}}_n and is expected to work for all classical groups. We will consider the Archimedean local theory and the global theory in our future work.

On twisted group ring isomorphism problem for p-groups

Abstract In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$ -groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$ -groups with generalized corank at most three.

Shapovalov elements of classical and quantum groups

Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nagel-Moshinsky algorithm. This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

An inductive approach to representations of general linear groups over compact discrete valuation rings

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

Low-Dimensional Invariant Embeddings for Universal Geometric Learning

AbstractThis paper studies separating invariants: mappings on D-dimensional domains which are invariant to an appropriate group action and which separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures. We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the dimension D. As a result, the theoretical universal constructions based on these separating invariants are unrealistically large. Our goal in this paper is to resolve this issue. We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting $$2D+1 $$ 2 D + 1 of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups. Often the requirement of invariant separation is relaxed and only generic separation is required. In this case, we show that only $$D+1$$ D + 1 invariants are required. More importantly, generic invariants are often significantly easier to compute, as we illustrate by discussing generic and full separation for weighted graphs. Finally we outline an approach for proving that separating invariants can be constructed also when the random parameters have finite precision.

Unilateral Primary Aldosteronism: Long-Term Disease Recurrence After Adrenalectomy.

Primary aldosteronism (PA) is frequently caused by a unilateral aldosterone-producing adenoma with a PA-driver mutation. Unilateral adrenalectomy has a high probability of short-term biochemical remission, but long-term postsurgical outcomes are relatively undefined. Our objective was to investigate the incidence of long-term recurrence of PA in individuals with postsurgical short-term biochemical remission. Adrenalectomized patients for unilateral PA were included from a single referral center. Histopathology and outcomes were assessed according to international histopathology of unilateral primary aldosteronism and PASO (Primary Aldosteronism Surgical Outcome) consensuses. Genotyping was performed using CYP11B2 (aldosterone synthase)-guided sequencing. Classical adrenal histopathology, exemplified by a solitary aldosterone-producing adenoma, was observed in 78% of 90 adrenals, compared with 22% with nonclassical histopathology. The classical group displayed higher aldosterone-to-renin ratios (P=0.013) and lower contralateral ratios (P=0.008). Outcome assessments at both short (12 months [7; 12]) and long (89 months [48; 124]) terms were available for 57 patients. At short-term assessment, 53 (93%) displayed complete biochemical success (43 classical and 10 nonclassical), but long-term assessment demonstrated biochemical PA recurrence in 12 (23%) with an overrepresentation of the nonclassical histopathology (6 [60%] of 10 nonclassical histopathology versus 6 [14%] of 43 classical histopathology; P=0.005). PA-driver mutations were identified in 97% of 64 aldosterone-producing adenomas; there was no association of the aldosterone-producing adenoma genotype with PA recurrence. A substantial proportion of individuals display postsurgical biochemical recurrence of PA, which is related to the histopathology of the resected adrenal gland. These findings emphasize the role of histopathology and the requirement for continued outcome assessment in the management of surgically treated patients for PA.

Comparison of Early and Mid-Term Outcomes After Classic and Modified Morrow Septal Myectomy in Patients with Hypertrophic Obstructive Cardiomyopathy.

The aim of our study is to compare the early and mid-term outcomes of patients with hypertrophic obstructive cardiomyopathy who underwent classic and modified Morrow septal myectomy. Between 2014 and 2019, 48 patients (24 males; mean age 49.27±16.41 years) who underwent septal myectomy were evaluated. The patients were divided into two groups - those who underwent classic septal myectomy (n=28) and those who underwent modified septal myectomy (n=20). Mitral valve intervention was higher in the classic Morrow group than in the modified Morrow group, but there was no significant difference (P=0.42). Mortality was found to be lower in the modified Morrow group than in the classic Morrow group (P=0.01). In both groups, the mean immediate postoperative gradient was significantly higher than the mean of the 3rd and 12th postoperative months. The preoperative and postoperative gradient difference of the modified Morrow group was significantly higher than of the classic Morrow group (P<0.001). Classic Morrow and modified Morrow procedures are effective methods for reducing left ventricular outflow tract obstruction. The modified Morrow procedure was found to be superior to the classic Morrow procedure in terms of reducing the incidence of mitral valve intervention with the reduction of the left ventricular outflow tract gradient.

An Analogue of Ladder Representations for Classical Groups

Abstract In this paper, we introduce a notion of ladder representations for split odd special orthogonal groups and symplectic groups over a non-archimedean local field of characteristic zero. This is a natural class in the admissible dual, which contains both strongly positive discrete series representations and irreducible representations with irreducible $A$-parameters. We compute Jacquet modules and the Aubert duals of ladder representations, and we establish a formula for describing ladder representations in terms of linear combinations of standard modules.

FORMAT LAYANAN DALAM BIMBINGAN KELOMPOK

The group guidance service format is a framework or structure used by counselors or mentors to provide guidance to a group of individuals in one session or meeting. Group counseling services aim to assist group members in understanding and overcoming the various problems or challenges they face, as well as promoting personal growth and positive development. The service formats in group guidance are (1) classical format, (2) group format, (3) field format, (4) individual format, (5) collaborative format, and (6) distance format. As for the types of methods in the service Group guidance includes: home room programs, study trips, discussion groups, group activities, student unions, sociodrama, psychodrama, and drama teaching.&#x0D;

Characteristics of terminal hemimelia: What is the difference between terminal hemimelia and classic fibular hemimelia?

Fibular hemimelia has denoted a spectrum of postaxial longitudinal deficiency with fibular aplasia/hypoplasia; the term "terminal hemimelia" is reserved for patients with postaxial longitudinal deficiency having a normal fibula. We aimed to delineate the characteristics of terminal hemimelia. In total, 30 patients with postaxial longitudinal deficiency who had a normal or hypoplastic fibula and visited our institution between 1992 and 2022 were reviewed. Patients were divided into terminal hemimelia and classic fibular hemimelia groups, and their demographic characteristics and clinical and radiographic findings were compared. Femoral shortening, knee valgus, and tibial spine hypoplasia were less common in terminal hemimelia (n = 13) than in classic fibular hemimelia (n = 17) (p = 0.03, p < 0.001, and p = 0.003, respectively). None of the patients in the terminal hemimelia group exhibited knee instability, whereas 12% of patients with classic fibular hemimelia did. Ball-and-socket ankle and absence of lateral rays were commonly observed in both groups. However, tarsal coalition was observed less frequently in terminal hemimelia (p = 0.004). All terminal hemimelia patients exhibited a painless plantigrade foot without ankle instability. Despite limb-length discrepancy at maturity averaging 40.4 mm for terminal hemimelia and 67.0 mm for classic fibular hemimelia (p < 0.001), patients with terminal hemimelia, except for one, exhibited > 20 mm of limb-length discrepancy. However, 46% of them underwent limb-length equalization procedures, mostly single-stage tibial lengthening, at a mean age of 11.2 years. Terminal hemimelia may present with a milder phenotype than classic fibular hemimelia. It mainly overlaps with the symptoms of fibular hemimelia below the ankle joint and manifests as limb-length discrepancy. However, a considerable number of patients with terminal hemimelia required limb-length equalization procedures, for example single-stage tibial lengthening. level IV.

Abdominoplasty after massive weight loss. Safety preservation fascia technique and clinical outcomes in a large single series-comparative study

IntroductionWeight loss after bariatric surgery causes very important modifications to the patient's silhouette. Abdominal fat and skin excess reduction are associated with several complications. The most frequent are seroma and hematoma whereas major complications, such as pulmonary embolism, are less frequent. This study aimed to describe our technical procedure for abdominoplasty in patients with massive weight loss after bariatric surgery.MethodsIn total, 196 patients were included. All patients who underwent abdominoplasty classic (group A) and abdominoplasty with the preservation and lift of Scarpa fascia (group B) and with umbilical transposition between May 2018 and May 2021 were included. Patients with concomitant correction of ventral hernia were excluded. Demographic and operative data were analyzed according to comorbidities and postoperative complications.ResultsThere were 160 (81.6%) women. The mean age was 43.6 years; the mean weight was 86.7 kg; and the mean BMI was 28.6 kg/m2. Five patients (2.5%) presented postoperative seroma. Four patients (2%) presented partial dehiscence/skin necrosis one of them requiring a revision. Finally, 26 patients presented a postoperative complication, with an overall incidence of 12.6%. The average postoperative hospital stay was 3.6. The rates of seroma were significantly higher in men, patients with a BMI &amp;gt; 30 kg/m2, and aged &amp;gt;50 years.ConclusionPreserving Scarpa Fascia during surgical post-bariatric patient procedures reduces the seroma formation and the scar complication and reduces the tension of the inguinal-pubic region with correction of our deformation after weight loss. Improves reducing the drain and reducing seroma incidence suction and hospital stay.

A Schur ring approach to supercharacters of groups associated with finite radical rings

We consider the central Schur ring associated with the standard supercharacters of the adjoint group G ( A ) of a finite radical ring A , and define supercharacters of the subgroup C G ( A ) ( σ ) consisting of elements fixed by an involution of G ( A ) naturally defined by an (anti-)involution of A . In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups.

Units, zero-divisors and idempotents in rings graded by torsion-free groups

Abstract The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

Aortic arch branch-prioritized reconstruction for type A aortic dissection surgery.

Acute Stanford type A aortic dissection (STAAD) is a fatal condition requiring urgent surgical intervention. Owing to the complexity of the surgical process, various complications, such as neurological disorders, are common. In this study, we prioritized the reconstruction of aortic arch branches during surgery and investigated the association between prioritizing the branches and the postoperative outcomes of patients with STAAD. Ninety-seven patients were included in the observational study and underwent total arch replacement and frozen elephant trunk technique between January 2018 and June 2021. Of these, 35 patients underwent the branch-priority technique, and 62 patients underwent the classic technique. By analyzing the perioperative outcomes, we compared the differences between the two techniques. The branch priority group had significantly shorter cardiopulmonary bypass and ventilator times and earlier postoperative wake-up times than the classic group. Additionally, the ICU stay time was shorter, with a significant decrease in neurological complications and 24 h drainage in the branch priority group compared to the classic group. The branch priority technique can effectively provide better brain protection, resulting in earlier awakening of patients after surgery, reduced neurological complications, shorter ventilation time and decreased ICU hospitalization time. Therefore, it is recommended for use in aortic dissection surgeries.

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

The value of simultaneous determination of blood large neutral amino acids and tetrahydrobiopterin metabolites in the diagnosis of atypical hyperphenylalaninemia

Tetrahydrobiopterin deficiency in newborns with atypical hyperphenylalaninemia requires rapid and accurate diagnosis and the ability to distinguish it from the classical type to prevent early irreversible neurological damage. The study aimed to evaluate neopterin and biopterin (products of tetrahydrobiopterin recycling pathway) and amino acid profiles (used in supplementation therapy) in patients with hyperphenylalaninemia after optimizing ultra-performance liquid chromatography coupled with tandem mass spectrometry to simultaneously measure neopterin, biopterin, and amino acids in dried blood spots. The study enrolled preselected infants with classic (n = 46), atypical (n = 14) hyperphenylalaninemia, and a control group (n = 50).Result Tandem mass spectrometry detected neo/biopterin in the blood with a sensitivity and specificity of 100%. The mean neo/biopterin levels were significantly lower in the atypical cases (4 ± 1 and 3 ± 1 nmol/L) than the classic (49 ± 13 and 50 ± 12 nmol/L) and control (15.2 and 15.3 nmol/L) groups and correlated with phenylalanine and phenylalanine to tyrosine ratio (all P < 0.05). The study compared classic and atypical hyperphenylalaninemia cases with the control group. Both classic and atypical cases exhibited decreased levels of arginine, valine, and leucine compared to controls. Classic cases showed increased levels of citrulline, ornithine, and methionine, while atypical cases showed increased citrulline levels only. Comparing atypical versus classic cases, atypical cases exhibited decreased levels of citrulline, ornithine, methionine, arginine, leucine, and valine (all P < 0.05). Correlation analysis revealed negative associations between ornithine and biopterin and between arginine and neopterin in classic PKU cases. These findings highlight distinct metabolic differences between classic and atypical PKU.Conclusion The optimized method detected neo/biopterin in the blood with accuracy and precision. The characteristic pattern of neo/biopterin in the blood makes it possible to differentiate between classic and atypical hyperphenylalaninemia with a sensitivity and specificity of 100%. The amino acid profile could add value when treatment with large neutral amino acids is considered.

Sine function similarity-based multi-attribute decision making technique of type-2 neutrosophic number sets and application to computer software quality evaluation

With the rapid development of information technology, software products are playing an increasingly important role in people’s production and life, and have penetrated into many industries. Software quality is the degree to which the software meets the specified requirements, and is an important indicator to evaluate the quality of the products used. At present, the scale of software is increasing, and the complexity is increasing. It is an urgent problem to reasonably grasp and ensure the product quality. The measurement and evaluation of Software quality characteristics is an effective means to improve Software quality. Faced with the complex system of software, there are many factors that affect product quality. Current research mainly measures software product quality from a qualitative perspective. The computer software quality evaluation is a classical multi-attribute group decision making (MAGDM). Type-2 Neutrosophic Numbers (T2NNs) is a popular set in the field of MAGDM and many scholars have expanded the traditional MAGDM to this T2NNs in recent years. In this paper, two new similarity measures based on sine function for T2NN is proposed under T2NNs. These two new methods are built for MAGDM based on the sine similarity measures for T2NN (SST) and sine similarity weighted measures for T2NN (SSWT). At the end of this paper, Finally, a practical case study for computer software quality evaluation is constructed to validate the proposed method and some comparative studies are constructed to verify the applicability. Thus, the main research contribution of this work is constructed: (1) two new similarity measures based on sine function for T2NN is proposed under T2NNs; (2) These two new methods are built for MAGDM based on the sine similarity measures for T2NN (SST) and sine similarity weighted measures for T2NN (SSWT); (3) an example for computer software quality evaluation is employed to verify the constructed techniques and several decision comparative analysis are employed to verify the constructed techniques.

Manifestly covariant worldline actions from coadjoint orbits. Part I. Generalities and vectorial descriptions

We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence à la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincaré and (A)dS symmetry. In the Poincaré case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincaré symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in “bitemporal” AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.

St Petersburg school of linear groups. II. Early works by Suslin

The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The second part is devoted to the publications by Suslin of the 1970s and the early 1980s, in the areas of classical algebraic K-theory and the theories of linear and classical groups. Also, we describe the general context of these works, state some of the most important results by Suslin himself, and his students, and some of the most closely related follow-ups and subsequent results.

$ (\epsilon, \delta) $-complex anti fuzzy subgroups and their applications

&lt;abstract&gt;&lt;p&gt;The complex anti-fuzzy set (CAFS) is an extension of the traditional anti-fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti-fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti-fuzzy sets, we proposed the concept of $ (\epsilon, \delta) $-CAFSs that offer a more comprehensive representation of the uncertainty of data than CAFSs by considering both the magnitude and phase of the membership functions and explain the $ \left(\epsilon, \delta \right) $-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $ (\epsilon, \delta) $-CAFSG. Also, we used this approach to define $ (\epsilon, \delta) $-complex anti-fuzzy(CAF) cosets and $ (\epsilon, \delta) $-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $ (\epsilon, \delta) $-CAFSG of the classical quotient group and demonstrated that the set of all $ (\epsilon, \delta) $-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $ \left(\epsilon, \delta \right) $-CAFSG was demonstrated and $ (\epsilon, \delta) $-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.&lt;/p&gt;&lt;/abstract&gt;