Action Of Group Research Articles

Toward Jordan decompositions for tensors

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

The module structure of a group action on a ring

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

Insights into critical urban pedestrian dynamics: an observational study spotlighting a small sample of blind pedestrians and electric vehicles

Observational studies, whether conducted on-site or through simulation, are practical options for investigating and comprehending pedestrian behaviour and validating the findings of similar studies that have already taken place on the same subject. Although they present difficulties in demographic characterisation and personal context, observation represents the adoption of actual behaviour, highlighting the different scenarios, contexts and characteristics of the environment and infrastructure in urban areas. However, with the growth in simulators, surveys, and data protection issues in Europe, field observation is less considered, and several studies have been left aside. Based on observation, this study answers behavioural and demographic questions, namely age and gender, using the data of six pedestrian crossings in Coimbra, Portugal. This experience occurred on different days of the week (the week before lockdown), registering blind (and non-blind) pedestrians' behaviour facing the electric vehicle in 6 crosswalks, 30 min in each crosswalk, morning or afternoon, with favourable weather conditions outside peak hours In total 180 (30*6) minutes were filmed and evaluated. The relationships between responsibility (when one pedestrian is guided by another) and group action in decision-making and crossing behaviour were also investigated. Another relevant aspect studied was the relationship between pedestrians and electric vehicles in different contexts. During the research, an electric vehicle was inserted in the observational experience, circulating in the streets to observe the pedestrians' reaction to the absence of noise. In this context, blind pedestrians were invited to perform crossings in these areas and identify the presence of the electric vehicle, to identify some differences in the behaviour of blind pedestrians and non-blind pedestrians. The results indicate that blind pedestrians tend to be more cautious than non-blind pedestrians, that younger and elderly pedestrians tend to have more dangerous and less calculated behaviours than other road users and that the differences are more perceptible in age than gender. It should be noted that when in a group, especially adolescents, they commit more traffic errors. Furthermore, regarding the absence of noise associated with the electric vehicle, in moments of pedestrian distraction, if the driver is not attentive, there is an increase in the risk of accidents.

Optimal pinwheel partitions for the Yamabe equation

Abstract We establish the existence of an optimal partition for the Yamabe equation in R N made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to − ∞ , the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in R N that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.

On a recent extension of a family of biprojective APN functions

APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension, done by Calderini et al. (2023), of a biprojective APN family introduced by Göloğlu (2022) defined on F22m. We show that this generalization yields functions equivalent to Göloğlu's original family if 3∤m. If 3|m we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.

SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.

Structures Associated with the Borromean Rings’ Complement in the Poincaré Ball

Guided by physical needs, we deal with the rotationally isotropic Poincar´e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. Applying this realization, we reveal normal stochastization and multifractal behavior within the examined model of directed random walks on the rooted Cayley tree, whose sixbranch graphs are associated with dendritic polymers. According to Penner, we construct the Teichm¨uller space of the decorated ideal octahedral surface related to the quotient space of the fundamental group action. Using the conformality of decoration, we define six moduli and the mapping class group generated by cyclic permutations of the ideal vertices. Intending to quantize the geometric area, we state the connection between the induced geometry and the sine-Gordon model. Due to such a correspondence we obtain the differential two-form in the cotangent bundle of the moduli space.

International Legal Regulation and Management of Space Waste on Celestial Bodies

The present article explores the international legal regulation of space waste on celestial bodies and the ecological challenges that space missions will pose in the near future. The lack of a shared vision of whether space waste production should be minimized on celestial bodies is causing uncertainty in the planning of future space missions. Leading private space companies are proposing the transfer of dangerous space waste produced from Earth into outer space. The excessive generation of space waste would increase the risks of activities such as space tourism, and space research and will threaten scientific fields such as radio astronomy. The political alliances that exclude a priori third parties will increase the risk of a generation of unnecessary orbital space debris and space waste on celestial bodies because of the development of several space missions in the same field. The sustainability of space missions on celestial bodies and the protection of the outer space environment should become a guiding principle for both national and international legal regulation of space activities. This article argues for the urgent need to establish comprehensive international regulations and monitoring mechanisms to manage space waste on celestial bodies, emphasizing sustainability and cooperation among space-faring nations.

Rigidity and geometricity for surface group actions on the circle

Rigidity and geometricity for surface group actions on the circle

On gauge Poisson brackets with prescribed symmetry

Abstract In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations are derived from the main results

Bilateral Multiple Orbital Metastases Successfully Treated with Standalone Stereotactic Radiosurgery: Clinical Image

Bilateral orbital metastases are extremely rare, and no previous literatures have reported bilateral multiple orbital metastases treated with stereotactic radiosurgery (SRS). We present the impressive images of a patient with history of metastatic leiomyosarcoma with newly diagnosed bilateral multiple orbital metastases. The pre-SRS MRI showed a right and two left orbital metastatic lesions. The patient underwent SRS, and the post-SRS MRI at 2-month following SRS showed a decrease in size of the treated lesions. The PET scan at 5 months after the treatment showed no avid F-fluorodeoxyglucose uptakes in the three orbital metastases. The presenting symptoms are all improved after the treatment without no adverse effects from the SRS. Our interesting images show the presence of multiple lesions in the orbital spaces with excellent treatment responses post SRS. This is the first report of a patient who has bilateral and multiple orbital metastases that were successfully treated with standalone SRS.

Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description

Orbit Spaces of Weyl Groups Acting on Compact Tori: A Unified and Explicit Polynomial Description

Exploring estrogen antagonism using CRISPR/Cas9 to generate specific mutants for each of the receptors

Endocrine disruptors are chemicals that have been in the spotlight for some time now. Their modulating action on endocrine signaling pathways made them a particularly interesting topic of research within the field of ecotoxicology. Traditionally, endocrine disrupting properties are studied using exposure to suspected chemicals. In recent years, a major breakthrough in biology has been the advent of targeted gene editing tools to directly assess the function of specific genes. Among these, the CRISPR/Cas9 method has accelerated progress across many disciplines in biology. This versatile tool allows to address antagonism differently, by directly inactivating the receptors targeted by endocrine disruptors. Here, we used the CRISPR/Cas9 method to knock out the different estrogen receptors in zebrafish and we assessed the potential effects this generates during development. We used a panel of biological tests generally used in zebrafish larvae to investigate exposure to compounds deemed as endocrine disrupting chemicals. We demonstrate that the absence of individual functional estrogen receptors (Esr1, Esr2b, or Gper1) does affect behavior, heart rate and overall development. Each mutant line was viable and could be grown to adulthood, the larvae tended to be morphologically grossly normal. A substantial fraction (70%) of the esr1 mutants presented severe craniofacial deformations, while the remaining 30% of esr1 mutants also had changes in behavior. esr2b mutants had significantly increased heart rate and significant impacts on craniofacial morphometrics. Finally, mutation of gper1 affected behavior, decreased standard length, and decreased bone mineralization as assessed in the opercle. Although the exact molecular mechanisms underlying these effects will require further investigations in the future, we added a new concept and new tools to explore and better understand the actions of the large group of endocrine disrupting chemicals found in our environment.

Predicting RSO populations using a neighbouring orbits technique

Abstract The determination of the full population of Resident Space Objects (RSOs) in Low Earth Orbit (LEO) is a key issue in the field of space situational awareness that will only increase in importance in the coming years. We endeavour to describe a novel method of inferring the population of RSOs as a function of orbital height and inclination for a range of magnitudes. The method described uses observations of an orbit of known height and inclination to detect RSOs on neighbouring orbits. These neighbouring orbit targets move slowly relative to our tracked orbit, and are thus detectable down to faint magnitudes. We conduct simulations to show that, by observing multiple passes of a known orbit, we can infer the population of RSOs within a defined region of orbital parameter space. Observing a range of orbits from different orbital sites will allow for the inference of a population of LEO RSOs as a function of their orbital parameters and object magnitude.

A fixed-point formula for Dirac operators on Lie groupoids

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.

The G-invariant graph Laplacian part II: Diffusion maps

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

Infrared finite scattering theory: scattering states and representations of the BMS group

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an “out” state with memory which gives rise to infrared divergences in the standard S-matrix. To obtain an infrared-finite scattering theory, one must suitably include states with memory. However, except in the case of QED with massive charged particles, asymptotic states with memory that have finite energy and angular momentum have not been constructed for more general theories (e.g. massless QED, Yang-Mills and quantum gravity). To this end, we construct direct-integral representations over the “Lorentz orbit” of a given memory and classify all “orbit space representations” that have well-defined energy and angular momentum. We thereby provide an explicit construction of a large supply of physical states with memory as well as the explicit action of the BMS charges all states. The construction of such states is a key step toward the formulation of an infrared-finite scattering theory. While we primarily focus on the quantum gravitational case, we outline how the methods presented in this paper can be applied to obtain representations of the Poincaré group with memory for more general quantum field theories.

Уравнения Лакса на супералгебрах Ли

It is demonstrated that the standard construction of Lax equations on Lie algebras can be extended to Lie superalgebras, with the even subspace carrying the usual Lax equations. The extended equations inherit the existence of the canonical trace polynomial integrals of motion. An extra set of integrals exists in the odd subspace, with a nontrivial homological structure of the orbit space. This establishes a curious algebraic link between integrable evolution equations, supersymmetry and the deformation theory.

Invariant rings of the special orthogonal group have nonunimodal h-vectors

For [Formula: see text] an infinite field of characteristic other than two, consider the action of the special orthogonal group [Formula: see text] on a polynomial ring via copies of the regular representation. When [Formula: see text] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [Formula: see text] has positive characteristic, the invariant ring is [Formula: see text]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [Formula: see text] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [Formula: see text]-invariant and information on the Hilbert series. Indeed, we use this to show that the [Formula: see text]-vector of the invariant ring for [Formula: see text] need not be unimodal.

Group Theoretic Approach towards the Balaban Index of Catacondensed Benzenoid Systems and Linear Chain of Anthracene

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving.