Action Of Group Research Articles

Ergodic states on type III1 factors and ergodic actions

Abstract Since the early days of Tomita–Takesaki theory, it is known that a von Neumann algebra 𝑀 that admits a state 𝜑 with trivial centralizer M φ M_{\varphi} must be a type III1 factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense G δ G_{\delta} set among all faithful normal states on any III1 factor with separable predual. Through Connes’ Radon–Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.

Fixed point sets and orbit spaces of wedge of three spheres

Let X be a finite CW-complex having mod p cohomology isomorphic to a wedge of three spheres Sn∨Sm∨Sl,1≤n≤m≤l. The aim of this paper is to determine the fixed point sets of actions of the cyclic group of prime order on X. We also classify the orbit spaces of free actions of G=Zp,p a prime or G=Sd,d=1 or 3, on X and derive the Borsuk-Ulam type results.

Optical Properties of Defects in Solids via Quantum Embedding with Good Active Space Orbitals

Optical Properties of Defects in Solids via Quantum Embedding with Good Active Space Orbitals

Stratification of the transverse momentum map

Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poisson stratification of the orbit space, each stratum of which is a regular Poisson manifold that admits a natural proper symplectic groupoid integrating it. The main tools in our proofs (which we believe could be of independent interest) are a version of the Marle–Guillemin–Sternberg normal form theorem for Hamiltonian actions of proper symplectic groupoids and a notion of equivalence between Hamiltonian actions of symplectic groupoids, closely related to Morita equivalence between symplectic groupoids.

What do we miss in this Orbital Retinoblastoma?

Introduction : Orbital retinoblastoma (RB) is a dire event with a bleak prognosis. This report highlights the severe complications of retinoblastoma resulting from insufficient diagnostic and treatment efforts due to inadequate medical resources and socio-economic challenges.
 Case Illustration : A 4-year-old boy was brought in with swelling on his left eye socket and jaw, present for two months.He had exhibited leukocoria followed by proptosis 12 months prior and underwent enucleation without dermato-fat graft (DFG) five months ago. Histopathology revealed retinoblastoma without optic nerve invasion. Despite referral to a regional hospital for chemotherapy, he did not receive treatment due to socio-economic constraints. Two months post-operation, the left eye prosthesis spontaneously dislodged, accompanied by an enlarging eye and a left jaw mass. An ophthalmologic assessment disclosed a mass in the left orbital socket, while the right eye appeared normal. A CT scan identified masses in the left orbit, parotid, and masticator spaces, as well as bilateral colli lymphadenopathy. The patient was diagnosed with residual retinoblastoma and received chemotherapy.
 Discussion : Imaging is crucial in proptosis patients to determine tumor extent and management strategies. The standard approach involves initial chemotherapy, surgery, radiotherapy, and additional chemotherapy. High-risk histological features (HHRF) also warrant chemotherapy, although the presence of such features remains undetermined in this case. Employing multiple treatment modalities may enhance prognosis.
 Conclusion : Thorough examination and management are essential in retinoblastoma cases. Patient family education and diligent care play a crucial role in preventing the progression and spread of retinoblastoma.

Rationality Problem of Two-Dimensional Quasi-Monomial Group Actions

Rationality Problem of Two-Dimensional Quasi-Monomial Group Actions

Recent Developments of Nano Flame Retardants for Unsaturated Polyester Resin.

For many years, efforts have been made to reduce the flammability of unsaturated polyester resins (UPRs), which are often used in the rail, shipbuilding, and construction industries. Without modification, they often fail to meet fire safety standards. Despite a rich history of flame retardants (FRs) applied to UPRs, researchers seek new solutions that will provide lower flammability and smoke density, as well as attaining a lower environmental impact from the composites. The objective of the study is to highlight the most important recent research on promising nano FRs in order to promote their further development. Mechanisms of action of several groups of nano FRs, such as clay-based, carbon-based, transition metal compounds, layered double hydroxides, polyhedral oligomeric silsesquioxanes, and others, including bio-based, have been studied. Particular emphasis has been laid on nano FRs applied to UPRs, and their influences on thermal stability, flammability, and mechanical properties. Moreover, the environmental impact and toxicity of nano FRs have been discussed. Results have proved that nano FRs applied at low loadings may significantly improve thermal stability, with a simultaneous increase or only a slight decrease in mechanical properties. However, attention on related environmental issues has highlighted the necessity of carefully selecting novel nano FRs.

Some smooth circle and cyclic group actions on exotic spheres

Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to deduce the existence of smooth circle and cyclic group actions on certain exotic spheres.

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.

A Classification of Supersymmetric Kaluza–Klein Black Holes with a Single Axial Symmetry

AbstractWe extend the recent classification of five-dimensional, supersymmetric asymptotically flat black holes with only a single axial symmetry to black holes with Kaluza–Klein asymptotics. This includes a similar class of solutions for which the supersymmetric Killing field is generically timelike, and the corresponding base (orbit space of the supersymmetric Killing field) is of multi-centred Gibbons–Hawking type. These solutions are determined by four harmonic functions on $$\mathbb {R}^3$$ R 3 with simple poles at the centres corresponding to connected components of the horizon, and fixed points of the axial symmetry. The allowed horizon topologies are $$S^3$$ S 3 , $$S^2\times S^1$$ S 2 × S 1 , and lens space L(p, 1), and the domain of outer communication may have non-trivial topology with non-contractible 2-cycles. The classification also reveals a novel class of supersymmetric (multi-)black rings for which the supersymmetric Killing field is globally null. These solutions are determined by two harmonic functions on $$\mathbb {R}^3$$ R 3 with simple poles at centres corresponding to horizon components. We determine the subclass of Kaluza–Klein black holes that can be dimensionally reduced to obtain smooth, supersymmetric, four-dimensional multi-black holes. This gives a classification of four-dimensional asymptotically flat supersymmetric multi-black holes first described by Denef et al.

Equivariant Fusion Subcategories

AbstractWe provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and recover Naidu-Nikshych-Witherspoon classification of the fusion subcategories of the representation category of a twisted quantum double of a finite group.

Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations

Abstract We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.

Symmetries of differential delay systems with applications to observer design

AbstractIn this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers).

Comparative study between rocuronium and magnesium sulfate as adjuvants to local anesthetics for peribulbar block

Background & Objective: Peribulbar anesthesia is frequently used in cataract surgeries. Compared to retrobulbar anesthesia, this approach is associated with less significant consequences. However, the prolonged onset of orbital akinesia and the frequent need for block replenishment are considered drawbacks. We aimed to investigate the impact of adding rocuronium and magnesium sulfate to a commonly used local anesthetic combination on the onset of corneal anesthesia and the degree of lid and globe akinesia during peribulbar block in ocular surgeries.
 Patients and Methods: This prospective double-blinded randomized study was conducted in the ophthalmic surgery operating theater, Ain Shams University Hospital from January 2019 to April 2020. We enrolled 75 patients from both sexes, with American Society of Anesthesiology (ASA) physical status I–III, ages 40-80 y, and an axial globe length less than or equal to 26 mm planned for cataract surgery under peribulbar block in the study following institutional ethical board approval and written informed patient consents. Patients were randomly divided into three groups, 25 patients per group, based on the drugs they received: Group O (control group): received local anesthetic + 0.9% normal saline (1 ml). Group M: received local anesthetic + magnesium sulphate 50 mg in normal saline. Group R: received local anesthetic + rocuronium at a dose of 0.06 mg/kg (maximum 5 mg) in normal saline.
 Results: Our study found statistically significant more rapid onset of action (mean time to start surgery) in rocuronium group (7.00 ± 1.22 min) than control group and magnesium sulfate group (9.00 ± 1.47 and 9.00 ± 1.15 min) respectively, (P < 0.001); Moreover, no significant difference was noted between control group and magnesium sulfate group. In addition, no relevant difference was reported regarding duration of surgery between the groups.
 Conclusion: During cataract surgery, the addition of rocuronium to the local anesthetic mixture in the peribulbar nerve block has been found to enhance ocular akinesia and corneal anesthesia. Furthermore, the use of rocuronium demonstrated a significantly faster onset and shorter time needed to achieve the appropriate conditions to initiate the surgical procedure compared to the use of magnesium sulfate. Addition of magnesium sulfate showed good results regarding akinesia, corneal anesthesia and need for supplementary injections but it did not significantly fasten the onset of block and hence the time to start surgery.
 Key words: Cataract; Magnesium Sulfate; Peribulbar Block; Rocuronium
 Citation: Nagy M, Abo El-Ella S, El-Shafey MA, Helal AM, Zekry MR. Comparative study between rocuronium and magnesium sulfate as adjuvants to local anesthetics for peribulbar block. Anaesth. pain intensive care 2024;28(1):55−61; DOI: 10.35975/apic.v28i1.2157
 Received: February 18, 2023; Reviewed: July 30, 2023; Accepted: December 23, 2023

CAT(0) Spaces of Higher Rank I

A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂TF. More precisely, we show that the set of singular points in ∂TF can be covered by finitely many round spheres of positive codimension.

Implementing Clinical Guidelines for the Treatment of Psychosis: The Frontline Leaders' Point of View. A Qualitative Study.

Despite the large amount of leadership and implementation theories and recommendations, healthcare services continue to struggle with efficiently incorporating new knowledge. The questioning of conventional leadership approaches in healthcare organizations prompted us to investigate how frontline leaders comprehend their own implementation intentions and actions, and how these intentions and actions may impact the implementation of clinical guidelines in mental healthcare in Norway. Employing a theory-driven qualitative design, we conducted nine semi-structured interviews with frontline leaders who had recently led implementation of clinical guidelines for the treatment of psychosis in mental health. We employed Systematic Text Condensation, informed by Normalization Process Theory, to structure and analyze the data and used fidelity scales to measure the degree of implementation and distinguish between leaders' levels of success in implementation. Frontline leaders in units that achieved high success in implementation described their intentions and actions differently, from those with less success. The former group's actions aligned more closely with the constructs of the Normalization Process Theory compared to the latter group when describing their actions. Frontline leaders leading units with a high degree of implementation success describe relation-orientation, trust, and providing adaptive space for staff members to take initiative. In contrast, those leading units with less implementation success describe more control and guidance of co-operators and place more emphasize on information and knowledge. Differences in how frontline leaders describe their actions and intentions to achieve clinical guideline implementation suggest that the leadership approach of these frontline leaders is an important factor to consider when planning and conducting implementation. To better understand the implementation process, it is important to pay attention to how frontline leaders customize their leadership approaches to the dynamics of complex organizations, and how they interact with their team and superiors.

Compositional characterization of a primordial S-type asteroid family of the inner main belt

Context. Recently, a primordial family of moderate-albedo asteroid fragments was discovered in the inner main belt. Its age was estimated to be 4.4 ± 1.7 Gyr. However, there is a lack of compositional characterization, which is important to the study of the earliest collisions in the main belt. Aims. In addition to the previously identified members and the parameters that define the family’s borders (V shape), we expanded the list of family members to include asteroids located within the central region of the V shape. These additional potential members were selected based on their diameter (larger than 7 km) and their geometric visible albedo (greater than or equal to 12%). Subsequently, we conducted a spectroscopic survey to determine the dominant taxonomy and composition of this family. This allowed us to further refine the list of family members by removing interlopers. Methods. From an initial list of 263 asteroids that are considered to be potential members of the aforementioned primordial family, we retrieved their spectra in the visible and near-infrared range from the literature and from the Gala DR3 spectral catalog of Solar System objects. For asteroids with no or poor signal-to-noise ratio spectra in the literature, we carried out new ground-based observations. We obtained new spectra for 33 members of the family using the 1.82 m Asiago Telescope for the visible spectroscopy, while for near-infrared spectroscopy, we used the 3.58 m Telescopio Nazionale Galileo (TNG) and the 4.30 m Lowell Discovery Telescope (LDT). Results. In total, we collected spectra for 261 potential members of the primordial S-type family out of 263. We determined their spectral taxonomy and properties, such as spectral slopes and absorption band parameters, when existing. Using the taxonomical characterization and the orbital space parameters, we identified and removed 71 interlopers from the potential members list. The final list of the primordial S-type family members includes 190 asteroids. The family is dominated by S-complex (~71%) asteroids with a mineralogy similar to ordinary chondrites and pyroxene-rich minerals. The family also contains members classified as L-types and V-types. (~15% and ~9%, respectively). Conclusions. The mean albedo of the family is ~23%, and its largest probable remnant is the asteroid (30) Urania. The estimated size of the family parent body ranges between 110 and 210 km. This size range is compatible with the progenitor of H and L chondrites.

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers’ centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Stable Separation of Orbits for Finite Abelian Group Actions

Stable Separation of Orbits for Finite Abelian Group Actions

A Multiplicative Tate Spectral Sequence for Compact Lie Group Actions

Given a compact Lie group  G G and a commutative orthogonal ring spectrum  R R such that R [ G ] ∗ = π ∗ ( R ∧ G + ) R[G]_* = \pi _*(R \wedge G_+) is finitely generated and projective over π ∗ ( R ) \pi _*(R) , we construct a multiplicative G G -Tate spectral sequence for each R R -module X X in orthogonal G G -spectra, with E 2 E^2 -page given by the Hopf algebra Tate cohomology of R [ G ] ∗ R[G]_* with coefficients in π ∗ ( X ) \pi _*(X) . Under mild hypotheses, such as X X being bounded below and the derived page R E ∞ RE^\infty vanishing, this spectral sequence converges strongly to the homotopy π ∗ ( X t G ) \pi _*(X^{tG}) of the G G -Tate construction X t G = [ E G ~ ∧ F ( E G + , X ) ] G X^{tG} = [\widetilde {EG}\wedge F(EG_+, X)]^G .