Action Of Group Research Articles

Descent and cyclotomic redshift for chromatically localized algebraic 𝐾-theory

We prove that T ( n + 1 ) T(n+1) -localized algebraic K K -theory satisfies descent for π \pi -finite p p -group actions on stable ∞ \infty -categories of chromatic height up to n n , extending a result of Clausen–Mathew–Naumann–Noel for finite p p -groups. Using this, we show that it sends T ( n ) T(n) -local Galois extensions to T ( n + 1 ) T(n+1) -local Galois extensions. Furthermore, we show that it sends cyclotomic extensions of height n n to cyclotomic extensions of height n + 1 n+1 , extending a result of Bhatt–Clausen–Mathew for n = 0 n=0 . As a consequence, we deduce that K ( n + 1 ) K(n+1) -localized K K -theory satisfies hyperdescent along the cyclotomic tower of any T ( n ) T(n) -local ring. Counterexamples to such cyclotomic hyperdescent for T ( n + 1 ) T(n+1) -localized K K -theory were constructed by Burklund, Hahn, Levy and the third author, thereby disproving the telescope conjecture.

Amenability of group actions characterized by configuration equations

Amenability of group actions characterized by configuration equations

Cluster Perturbation Theory for Core Excited States and Core Ionization Potentials Using Core-Valence Separation.

The development of accurate and fast computational procedures for the ab initio calculation of X-ray spectroscopies is paramount to facilitate theoretical analysis of modern X-ray experiments on molecules. Herein, we present the extension of Cluster Perturbation theory to comprehend the calculation of core excited states and core ionization potentials using the core-valence separation approximation, which has seen widespread success for various quantum chemistry methods. We derive the theoretical framework for introducing core-valence separation into Cluster Perturbation series for excitation energies and display the performance of the methodology in S(D) orbital excitation spaces. The obtained core excitation energies on a test set of medium sized organic molecules show that carbon, nitrogen, and oxygen K-edge excitation energies can be determined with errors below 2 eV relative to the CCSD reference results using the developed CPS(D) excitation energy models which can be used for systems way beyond the reach of conventional CCSD.

Estimation of the required energy for a channel of high-precision phase direction finding of low-earth orbit space debris based on large scale antenna array

To solve the problem of radar observations of small-sized space debris of natural and artificial origin, a multi-antenna distributed structure based on adaptive antenna arrays is proposed. A method for estimating noise errors of the phase method of low-Earth orbit space debris bearing for a channel of high-precision phase direction finding based on elements of a multi-antenna field is presented. The results obtained characterize the potential accuracy of the system. Based on the specified dispersion of phase fluctuations, the required signal-to-noise ratio at the input of the device is determined according to the proposed method.

Tracial states on groupoid $C^{*}$-algebras and essential freeness

Let \mathcal{G} be a locally compact Hausdorff étale groupoid. We call a tracial state \tau on a general groupoid C^{*} -algebra C_{\nu}^{*}(\mathcal{G}) canonical if \tau=\tau|_{C_0(\mathcal{G}^{(0)})}\circ E , where E:C^{*}_{\nu}(\mathcal{G}) \to C_{0}(\mathcal{G}^{(0)}) is the canonical conditional expectation. In this paper, we consider so-called fixed point traces on C_{c}(\mathcal{G}) , and prove that \mathcal{G} is essentially free if and only if any tracial state on C_{\nu}^{*}(\mathcal{G}) is canonical and any fixed point trace is extendable to C_{\nu}^{*}(\mathcal{G}) .As applications, we obtain the following: (1) a group action is essentially free if every tracial state on the reduced crossed product is canonical and every isotropy group is amenable; (2) if the groupoid \mathcal{G} is second-countable, amenable and essentially free then every (not necessarily faithful) tracial state on the reduced groupoid C^{*} -algebra is quasidiagonal.

Apocalypse now? – The Last Generation in digital capital’s affective ecology

Abstract This essay situates the protest actions of the German environmentalist group Die letzte Generation (LG) within an environmental critique of digital capitalism. It specifically examines the temporality of LG’s activism at the intersection of the affective ecology of digital capital and certain aspects of an apocalyptic imagination. I propose a Spinozist approach to the media environment in which LG operates that may open a path toward an immanent critique of LG’s publicity-oriented practices.

Non-archimedean topological monoids

AbstractWe say that a topological monoid S is left non-archimedean (in short: l-NA) if the left action of S on itself admits a proper S-compactification $$\nu :S \hookrightarrow Y$$ ν : S ↪ Y such that Y is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. The Stone and Pontryagin dualities play a major role in achieving useful characterizations of NA monoids. We show that many naturally defined topological monoids are NA and present universal NA monoids. Among others, we prove that the Polish monoid $$C(2^{\omega },2^{\omega })$$ C ( 2 ω , 2 ω ) is a universal separable metrizable l-NA monoid and the Polish monoid $${\mathbb {N}}^{\mathbb {N}}$$ N N is universal for separable metrizable r-NA monoids.

Visible Spectral Atlas of Geostationary Satellites from Tucson, AZ for Differentiating Between Natural and Artificial Objects

As near-Earth object (NEO) surveys continue to search for smaller NEOs, they will also detect an increasing number of temporarily captured objects, or minimoons, in geocentric orbital space. Derelict spacecraft and debris in Earth orbit and cislunar space can be mistaken for minimoons, but spectral characterization can distinguish between the two categories of objects. However, systematic noncompositional effects due to nightly and seasonal phase angle changes on artificial objects need to be quantified before such distinctions can be made. These effects have been studied on small solar system bodies, but very little on artificial bodies. We present the reduced data of our multiyear visible wavelength (450–950 nm) spectral campaign of the geostationary Earth-orbiting (GEO) satellite belt from Tucson, AZ, and include comparisons to relevant planetary materials. Although some bus types have steeper spectral slopes than planetary materials, certain bus type spectral features can be confused for planetary materials. One example is a rollover at red wavelengths in the Eurostar-3000 bus-type spectrum that appears similar to mineralogical absorption bands on S- and L-type asteroids. Observations include a total of 96 unique GEO satellites across 192 separate nights from 2020 to 2022. A select subset of GEO satellites is repeatedly observed to measure seasonal variations. Our methods for data acquisition, processing, and cleaning are outlined in this paper. A summary of the atlas shows the full night median spectrum with phase variations and a lightcurve of brightness versus phase angle for each of the 284 sets of data collected.

C⁎-algebras associated to directed graphs of groups, and models of Kirchberg algebras

We introduce C⁎-algebras associated to directed graphs of groups. In particular, we associate a combinatorial C⁎-algebra to each row-finite directed graph of groups with no sources, and show that this C⁎-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these C⁎-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.

Morphology and agents of urban transformation

The study aims to propose a method of analysis that promotes a clearer perception of the spatial transformation resulting from the action of certain social groups to shape the urban space according to their interests. The territorial transformation related to the role of different agents is investigated, through urban morphology, in order to understand the correlation between the Brazilian urban form with public management and the direction of public investments. To analyze how the city is shaping over time, the work was structured from a case study, which mapped the urban transformations of the municipalities of Cuiabá and Várzea Grande, during the 2006-2021 interval. The method adopted by the research proposes as tools for investigating the territory: the conception of thematic cartographies through the comparison of aerial images from Google Earth, the establishment of analysis categories that allow the identification of the different types of building transformations preponderant of the urban blocks and the main interventions on the road system and the impact of important events that occurred in the cities in the period studied. A quantitative analysis is presented, with the approximate values of areas in hectares (ha), which represent the categories of analysis indicated. As a result, it can be seen, through the maps, that the relationship between the different forms of urban land occupation and the road infrastructure implemented ends up reinforcing the process of socio-spatial segregation. It is concluded that the proposed method, combined with historical and socioeconomic studies, manages to reconstitute urbanization processes in order to understand them in order to intervene in the reality of our cities.

Laumon parahoric local models via quiver Grassmannians

Local models of Shimura varieties in type A can be realized inside products of Grassmannians via certain linear algebraic conditions. Laumon suggested a generalization which can be identified with a family over a line whose general fibers are quiver Grassmannians for the loop quiver and the special fiber is a certain quiver Grassmannian for the cyclic quiver. The whole family sits inside the Gaitsgory central degeneration of the affine Grassmannians. We study the properties of the special fibers of the (complex) Laumon local models for arbitrary parahoric subgroups in type A using the machinery of quiver representations. We describe the irreducible components and the natural strata with respect to the group action for the quiver Grassmannians in question. We also construct a cellular decomposition and provide an explicit description for the corresponding poset of cells. Finally, we study the properties of the desingularizations of the irreducible components and show that the desingularization construction is compatible with the natural projections between the parahoric subgroups.

Finite orbits of the braid group actions

We study the finite orbits of the braid group Bn action on the space of n×n upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on 3×3 matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic n×n matrices to be useful to finding solutions of higher order Painleve type differential equations.

Diversities and similarities exhibited by multi-planetary systems and their architectures. I. Orbital spacings

The rich diversity of multi-planetary systems and their architectures is greatly contrasted by the uniformity exhibited within many of these systems. Previous studies have shown that compact Kepler systems tend to exhibit a peas-in-a-pod architecture: Planets in the same system tend to have similar sizes and masses and be regularly spaced in orbits with low eccentricities and small mutual inclinations. This work extends on previous research and examines a larger and more diverse sample comprising all the systems with a minimum of three confirmed planets, resulting in 282 systems and a total of 991 planets. We investigated the system architectures, focusing on the orbital spacings between adjacent planets as well as their relationships with the planets' sizes and masses. We also quantified the similarities of the sizes, masses, and spacings of planets within each system, conducting both intra- and inter-system analyses. Our results corroborate previous research showing that planets orbiting the same star tend to be regularly spaced and that pairs of adjacent planets with radii $< 1\, R_ predominantly have orbital period ratios (PRs) smaller than two. In contrast to other studies, we identified a significant similarity of adjacent orbital spacings not only at PRs < 4 but also at 1.17 < PRs < 2662 . For the systems with transiting planets, we additionally found that the reported correlation between the orbital PRs and the average sizes of adjacent planets disappears when planet pairs with $R < 1\, R_ are excluded. Furthermore, we examined the data for possible correlations between the intra-system dispersions of the orbital spacings and those of the planetary radii and masses. Our findings indicate that these dispersions are uncorrelated for the systems in which all the pairs of adjacent planets have PRs < 6 and even for the compact systems where all PRs < 2 . Notably, planets in the same system can be similarly spaced even if they do not have similar masses or sizes.

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

Group Theory and Solutions of the Rubik’s Cube

The paper explores the application of group theory to solving the Rubik’s Cube. Group theory, a branch of abstract algebra, studies sets equipped with an operation that satisfies specific properties: closure, associativity, identity, and inverses. The Rubik’s Cube is treated as a group because its set of moves forms a structure that aligns with these properties. By representing the cube’s various configurations and moves through the mathematical constructs of groups, the paper analyzes how the elements transform during cube manipulations. Key concepts such as symmetric groups, homomorphisms, alternating groups, and disjoint cycle decomposition are used to break down the cube’s complexity. Additionally, the study demonstrates how group actions, particularly the parity of permutations and orientation sums, govern valid cube configurations. The paper also references methodologies from other studies to apply and refine these mathematical approaches for systematically solving the Rubik’s Cube. Through these methods, the research illustrates how the cube can be navigated using logical algorithms grounded in group theory principles

On pairs of root groups and reflections associated to the Chevalley group E 6(K), in characteristic two

This paper is concerned with computing the number of orbits under the action of the algebraic group E = E 6 ( K ) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL ( A ) has been fixed right at the onset.

HARNESSING GROUP THEORY FOR SIMPLIFIED ELECTRIC CIRCUIT ANALYSIS

Electric circuit analysis is fundamental to advancing modern technology, yet traditional methods often become complex and inefficient when applied to large-scale or intricate networks. Group theory, a branch of abstract algebra renowned for its capacity to exploit symmetry, has shown promise as a solution to this challenge. By applying group theoretical principles to circuit analysis, particularly through symmetry operations and transformation groups, engineers can simplify complex systems, reducing computational demands and revealing deeper insights into circuit behavior. Through systematic symmetry-based simplifications, group theory can decrease the number of equations required by up to 40% in certain configurations, enhancing both analytical efficiency and practical implementation. However, challenges include the mathematical complexity inherent to group theory, the limitations in analyzing highly irregular or nonlinear circuits, and the need for compatible computational tools. Recommendations include the integration of group theoretical approaches into standard circuit analysis software, focused training for electrical engineers, and further research to address the application of these methods in emerging technologies such as quantum and neuromorphic computing. Therefore, this review aims to elucidate the practical potential of group theory in transforming electric circuit analysis, presenting a streamlined approach that balances theoretical rigor with engineering feasibility.

Late acquisition of erect hindlimb posture and function in the forerunners of therian mammals.

The evolutionary transition from early synapsids to therian mammals involved profound reorganization in locomotor anatomy and function, centered around a shift from "sprawled" to "erect" limb postures. When and how this functional shift was accomplished has remained difficult to decipher from the fossil record alone. Through biomechanical modeling of hindlimb force-generating performance in eight exemplar fossil synapsids, we demonstrate that the erect locomotor regime typifying modern therians did not evolve until just before crown Theria. Modeling also identifies a transient phase of increased performance in therapsids and early cynodonts, before crown mammals. Further, quantifying the global actions of major hip muscle groups indicates a protracted juxtaposition of functional redeployment and conservatism, highlighting the intricate interplay between anatomical reorganization and function across postural transitions. We infer a complex history of synapsid locomotor evolution and suggest that major evolutionary transitions between contrasting locomotor behaviors may follow highly nonlinear trajectories.

The Category G-GrR-Mod and Group Factorization

In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X, to define the conjugation action of the set G and to generalize and prove some results from the literature. In particular, we prove that a G-weak graded ring R is strongly graded if and only if each G-weak graded R-module V is induced by an ReG-module. Moreover, we prove that the additive induction functor (−)R and the restriction functor (−)eG form an equivalence between the categories G-GrR-Mod and ReG-Mod when R is strongly G-weak graded. Furthermore, some related results and illustrative examples of G-weak graded R-modules and their morphisms are provided.

On Knots with Cyclic Symmetries

This review paper investigates the relationship between symmetries of knots and links in the three-dimensional sphere, with a focus on cyclic symmetries, and their associated polynomial invariants. It examines the behavior of these polynomials in the case where the knot is set-wise fixed by the action of finite cyclic group on the 3-sphere. The study highlights how these invariants can obstruct the existence of such symmetries, providing valuable insights into the topological properties of these knots. The paper reviews established results on polynomial criteria for periodicity and their extensions to freely periodic links. It also explores generalizations to study the symmetries of virtual knots and spatial graphs.