Action Of Group Research Articles

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.

Increase of Fibroblast Cells in Wound Healing of Diabetes Mellitus Male Rats (Rattus Norvegicus) Using Aloin

The healing process wound on the skin diabetic experience slowdown because dysfunction fibroblasts. Aloins boost expression fibroblasts that can accelerating the process of wound healing in patients with Diabetes Mellitus (DM). Objective study that is for now effect enhancement cell fibroblasts in wound healing wound I t mouse male DM with aloe giving. Method study use principle true experimental with approach post-test serial only control group design. Treatment form wound cut on mice. Group study using 4 groups with a negative control in the form mouse healthy, positive control form DM mouse. group action 1 in the form of DM rats were given 1.25 mg/ kg BW aloin and treatment 2 was in the form of DM rats given 2.5 mg/ kg BW aloin. Expression fibroblasts calculated on day 7 and day 14. Analysis bivariate using the Friedman test continued with the Wilcoxon Post- Hoc test. Research results that are there is enhancement fibroblasts significant in the group treatment given with aloein a dose of 1.25 mg/ kg and a dose of 2.5 mg/ kg. The conclusion is that Aloin accelerates wound healing wound incision in DM rats with mechanism enhancement expression fibroblasts with minimum dose of 1.25 mg/ KgBB. Research suggestions that is examination of re- epithelialization and density Dermal collagen in wound healing wound I t DM rats given aloein in the study next.

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

Unity and Pride: An Exploration of the Motivations Behind the People’s Decision to Build the Tower in Genesis 11:1-9

This article explores the complex reasons for a community’s construction of a tall building frequently referred to as the “Tower” in literature and history. This study uses an interdisciplinary lens and sources from psychology, sociology, anthropology, and history to identify the fundamental sources of pride and solidarity that motivated the group effort. The Tower, a representation of human togetherness and ambition, is evidence of humanity’s goal to reach the stars. Beneath its imposing exterior, nevertheless, is a rich tapestry of reasons skilfully woven by those who dared to construct it. To offer light on the complex nature of human ambition and group action, this research attempts to analyze these motivations. This study investigates the function of unity in promoting group initiatives using historical accounts, literary interpretations, and psychological theories. It investigates how a shared vision may unite disparate societies in pursuit of a common objective despite individual differences. Examining the idea of pride, it looks at how the need for achievement and acknowledgment fuels human ambition and the formation of collective identities. This study emphasizes the Tower narrative’s ongoing relevance by comparing historical and modern examples. It contends that the inspirations for its creation are timeless and culturally ubiquitous, mirroring the yearning for transcendence, unification, and acknowledgment among all people. Ultimately, this research provides insights into how individual goals and group efforts interact. Understanding the driving forces behind the Tower’s construction gives us critical new perspectives on the dynamics of human ambition and the never-ending pursuit of pride and unification.

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.

THE RIGHT TO RIDE: AN ETHNOGRAPHY OF AN URBAN AND MARGINAL PRACTICE

This text discusses structural aspects of the practice of roller-cart, based on an ethnography conducted in the Metropolitan Region of Belo Horizonte between 2019 and 2022. Initial impressions indicated that the roller-cart movement, with competition teams, associations and event organizers in the city, was a large and coordinated action of a cohesive group of people around the same purpose. With a dense and prolonged stay in the field, I found a movement marked by disputes, interests and multiple appropriations that limit, at the same time as they shape, a community of practice that emerges from a collective, urban and marginal construction. In this sense, this text highlights three structural aspects of the practice of roller-cart: the participation of women in events, based on the notion of the sexed city by Michele Perrot; the territorial disputes of these groups with the orders of the State and Capital, from the perspective of the city as a right, by Henri Lefebvre; and the search for legitimacy of a marginalized practice, through the prism of consumption as a constituent of identities, according to Michel de Certeau. Thus, men and women, adults and children, participate, constitute, and are constituted by a polysemic, contradictory, and essentially situated practice.