Non-trivial Action Research Articles

Property FA is not a profinite property

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups \Delta and \Gamma with \Delta having Property FA, \Gamma having a non-trivial action on a tree, and \Delta and \Gamma having isomorphic profinite completions.

Anti-classification results for weakly mixing diffeomorphisms

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.

Almost complex torus manifolds - a problem of Petrie type

The Petrie conjecture asserts that if a homotopy C P n \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of C P n \mathbb {CP}^n . Petrie proved this conjecture in the case where the manifold admits a T n T^n -action. An almost complex torus manifold is a 2 n 2n -dimensional compact connected almost complex manifold equipped with an effective T n T^n -action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2 n 2n -dimensional almost complex torus manifold M M only shares the Euler number with the complex projective space C P n \mathbb {CP}^n , the graph of M M agrees with the graph of a linear T n T^n -action on C P n \mathbb {CP}^n . Consequently, M M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch χ y \chi _y -genus, Todd genus, and signature as C P n \mathbb {CP}^n , endowed with the standard linear action. Furthermore, if M M is equivariantly formal, the equivariant cohomology and the Chern classes of M M and C P n \mathbb {CP}^n also agree.

Ruthenium(II) Polypyridyl Complexes with Benzoxazole Derivatives and Non-Innocent Ligands as Effective Antioxidants in Human Neuroblasts.

Ruthenium(II) polypyridyl complexes continue to raise increasing interest for the encouraging results in several biomedical areas. Considering their vast chemical-physical repertoire, in particular the possibility to switch from the sensitization of reactive oxygen species (ROS) to ROS-scavenging abilities by tuning the nature of their ligands, it is therefore surprising that their potential as antioxidants has not been largely investigated so far. Herein, we explored the antioxidant behaviour of the novel ruthenium compound [Ru(dbpy)(2,3-DAN)Cl]PF6 (Ru1), featuring a benzoxazole derivative (dpby=2,6-bis(4-methyl-2-benzoxazolyl)pyridine) and the non-innocent 2,3-diamminonaftalene (2,3-DAN) ligand, along with the reference tpy-containing analogue [Ru(tpy)(2,3-DAN)Cl]PF6 (Ru2) (tpy=2,2':6',2''-terpyridine). Following the synthesis and the electrochemical characterization, chemical antioxidant assays highlighted the beneficial role of dpby for the ROS-scavenging properties of Ru1. These data have been corroborated by the highest protective effect of Ru1 against the oxidative stress induced in SH-SY5Y human neuroblastoma, which exerts pro-survival and anti-inflammatory actions. The results herein reported highlight the potential of Ru1 as pharmacological tool in neurodegenerative diseases and specially prove that the antioxidant properties of such compounds are likely the result of a non-trivial synergetic action involving the bioactive ligands in their chemical architectures.

SAFETY RISKOLOGY

Problem statement. The rapid expansion of the technosphere, unbalanced geopolitical, ecological, and biological processes of modern times demand a complete reassessment, a deep understanding, and the study of risk in all spheres of human life and activity. War and the non-trivial actions of aggressors force us into the necessity of reactive risk management, not excluding the need for proactive risk management. This complicates the determination of the priority of preventive measures at each stage of the activities of subjects and objects of risk, requiring the combination of these principles and the continuous reconsideration of risks to develop effective positive-compensatory measures. The actual state of human (humanity's) protection indicates that without the creation of a “science of risks”, without the establishment of a corresponding academic discipline, further effective and rational risk management is no longer possible. The purpose of the article. Justification for the Introduction of a New Academic Discipline “Safety Riskology” in Engineering and Technical Specialties at Higher Educational Institutions in Ukraine. Conclusions. The conceptual notion of “safety riskology” has been “synthesized” for the first time. The need for teaching a new academic discipline, “Safety Riskology”, in higher education institutions has been justified. During the instruction of this discipline, it is proposed to shape in learners advanced competencies in the identification, analysis, assessment, and compensation (balancing) of risk, considering the types and characteristics of the activities of risk subjects and objects. The demand for professionals with contemporary knowledge in safety riskology is identified across all sectors of the economy. Discussions on the initiation of the independent science of “Safety Riskology” have been initiated.

Phase space renormalization and finite BMS charges in six dimensions

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}+ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H ∗ H^\ast -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.

Cohomology, extensions and automorphisms of skew braces

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial brace is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.

Ground state degeneracy on torus in a family of ZN toric code

Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.

Faithfulness and Sensitivity for Ancilla-Assisted Process Tomography.

A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. In this work, we extend the applicability and generality of faithfulness significantly by introducing its local variant and examining their relationship when applied to various classes of quantum channels. In doing so, we discovered that, in the original proof by D'Ariano and Presti, only sufficiency was shown, not the full equivalence between faithfulness of state and invertibility of the corresponding Jamiołkowski map. We complete the proof by showing necessity and examine how far this characterization of faithfulness can be generalized by applying it to various classes of quantum channels. We also explore a more general notion we call sensitivity, the property of quantum state being altered by any nontrivial action of quantum channel. We study their relationship by characterizing both properties for important classes of quantum channels such as unital channels, random unitary operations, and unitary operations. Unexpected (non)equivalence results among them shed light on the structure of quantum channels by showing that we need only two classes of quantum states for characterizing quantum states faithful or sensitive to various subclasses of quantum channels.

Constructions of exotic actions on product manifolds with an asymmetric factor

We explore transformation groups of manifolds of the form M×Sn, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action of a finite group. In particular, we prove that for n=2, there exists an infinite family of distinct nondiagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on M×S1, where M belongs to the class of “almost asymmetric” manifolds considered previously by Puppe and Kreck.

Dual exponential coupled cluster theory: Unitary adaptation, implementation in the variational quantum eigensolver framework and pilot applications.

In this paper, we have developed a unitary variant of a double exponential coupled cluster theory, which is capable of handling molecular strong correlation with arbitrary electronic complexity. With the Hartree-Fock determinant taken as the reference, we introduce a sequential product of parameterized unitary Ansätze. While the first unitary, containing the excitation operators, acts directly on the reference determinant, the second unitary, containing a set of rank-two, vacuum-annihilating scattering operators, has nontrivial action only on certain entangled states. We demonstrate the theoretical bottleneck of such an implementation in a classical computer, whereas the same is implemented in the hybrid quantum-classical variational quantum eigensolver framework with a reasonably shallow quantum circuit without any additional approximation. We have further introduced a number of variants of the proposed Ansatz with different degrees of sophistication by judiciously approximating the scattering operators. With a number of applications on strongly correlated molecules, we have shown that all our schemes can perform uniformly well throughout the molecular potential energy surface without significant additional implementation cost over the conventional unitary coupled cluster approach with single and double excitations.

DeepSym: Deep Symbol Generation and Rule Learning for Planning from Unsupervised Robot Interaction

Symbolic planning and reasoning are powerful tools for robots tackling complex tasks. However, the need to manually design the symbols restrict their applicability, especially for robots that are expected to act in open-ended environments. Therefore symbol formation and rule extraction should be considered part of robot learning, which, when done properly, will offer scalability, flexibility, and robustness. Towards this goal, we propose a novel general method that finds action-grounded, discrete object and effect categories and builds probabilistic rules over them for non-trivial action planning. Our robot interacts with objects using an initial action repertoire that is assumed to be acquired earlier and observes the effects it can create in the environment. To form action-grounded object, effect, and relational categories, we employ a binary bottleneck layer in a predictive, deep encoderdecoder network that takes the image of the scene and the action applied as input, and generates the resulting effects in the scene in pixel coordinates. After learning, the binary latent vector represents action-driven object categories based on the interaction experience of the robot. To distill the knowledge represented by the neural network into rules useful for symbolic reasoning, a decision tree is trained to reproduce its decoder function. Probabilistic rules are extracted from the decision paths of the tree and are represented in the Probabilistic Planning Domain Definition Language (PPDDL), allowing off-the-shelf planners to operate on the knowledge extracted from the sensorimotor experience of the robot. The deployment of the proposed approach for a simulated robotic manipulator enabled the discovery of discrete representations of object properties such as ‘rollable’ and ‘insertable’. In turn, the use of these representations as symbols allowed the generation of effective plans for achieving goals, such as building towers of the desired height, demonstrating the effectiveness of the approach for multi-step object manipulation. Finally, we demonstrate that the system is not only restricted to the robotics domain by assessing its applicability to the MNIST 8-puzzle domain in which learned symbols allow for the generation of plans that move the empty tile into any given position.

Random groups at density 𝑑<3/14 act non-trivially on a CAT(0) cube complex

For random groups in the Gromov density model at d > 3 / 14 d>3/14 , we construct walls in the Cayley complex X X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities d > 1 / 5 d>1/5 and d > 5 / 24 d>5/24 , respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density d > 1 / 4 d>1/4 .

Holographic RG flow and reparametrization invariance of Wilson loops

We study the fate of reparametrization invariance of Wilson loops, also known as ‘zig-zag’ symmetry, under the RG flow using some simple cases as guidance. We restrict our analysis to large-N, strongly coupled CFTs and use the holographic dual description of a Wilson loop as a fundamental string embedded in asymptotically AdS spaces, at zero and nonzero temperature. We then introduce a cutoff in the holographic radial direction and integrate out the section of the string closer to the AdS boundary in the spirit of holographic Wilsonian renormalization. We make explicit the map between Wilson loop reparametrizations and conformal transformation of the string worldsheet and show that a cutoff anchored to the worldsheet breaks conformal invariance and induces an effective defect action for reparametrizations at the cutoff scale, in a way similar to nearly-AdS2 gravity or SYK models. On the other hand, a cutoff in the target space breaks worldsheet diffeomorphisms and Weyl transformations but keeps conformal transformations unbroken and does not generate a non-trivial action for reparametrizations.

A reinforcement learning approach to long-horizon operations, health, and maintenance supervisory control of advanced energy systems

We develop a Reinforcement Learning (RL) approach to the supervisory control problem for advanced energy systems, such as novel nuclear reactors and other demand-driven, mission-critical, and component-health-sensitive energy plants. The inclusive problem landscape considered captures the stochastic confluence of plant performance, component health evolution, power demand from the grid, diverse maintenance actions, and operator-defined goals and constraints, all considered over meaningfully long-enough reasoning horizons. Key aspects of the proposed approach are a receding horizon control-inspired technique dictating time- or event-triggered supervisory policy (re-)constructions, as well as additional capability-enabling contributions such as timescale compression, to handle long reasoning horizons and uncertainty in parts of the problem, and practical yet demonstrably-effective handling of hybrid action spaces with continuous and discrete decision variables. The resulting algorithm consists of a simulation-based RL agent constructing stochastic supervisory control policies over nontrivial action spaces and for long horizons, applying the learned policy to the system for a much shorter interval, and perpetually repeating, to construct the next long-horizon policy. That next policy will only be applied, again, for a short interval, yet originally far-in-time events move progressively closer, their associated uncertainty decreases, and new events and aspects enter the reasoning horizon. The proposed methodology bridges fundamental receding horizon concepts with the unequivocally stronger and more scalable reasoning of contemporary RL. Numerical examples using Soft Actor–Critic Deep RL illustrate the operation and efficacy of the proposed technique for a power plant tasked with health-aware load following missions in a dynamic electricity market landscape.

The BNSR-invariants of the Lodha–Moore groups, and an exotic simple group of type

AbstractIn this paper we give a complete description of the Bieri–Neumann–Strebel–Renz invariants of the Lodha–Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha–Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$ , answering a question posed in Oberwolfach in 2018. The proof involves applying a variation of Bestvina–Brady discrete Morse theory to the so called cluster complex X introduced by the first author. As an application, we also demonstrate that a certain simple group S previously constructed by the first author is of type $\textrm{F}_\infty$ . This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$ -diffeomorphisms, nor by piecewise linear homeomorphisms, on any 1-manifold.

Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group G with a maximal torus Hsubset G, Tits defined an extension W_G^{mathrm{T}} of the corresponding Weyl group W_G. The extended group is supplied with an embedding into the normalizer N_G(H) such that W_G^{mathrm{T}} together with H generate N_G(H). In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form G(mathbb {R})subset G, which leads to a simple topological description of W^{mathrm{T}}_G. We also consider a variation of the Tits construction associated with compact real form U of G. In this case we define an extension W_G^U of the Weyl group W_G, naturally embedded into the group extension widetilde{U}:=U,{rtimes }, Gamma of the compact real form U by the Galois group Gamma ={mathrm{Gal}}(mathbb {C}/mathbb {R}). Generators of W^U_G are squared to identity as in the Weyl group W_G. However, the non-trivial action of Gamma by outer automorphisms requires W^U_G to be a non-trivial extension of W_G. This gives a specific presentation of the maximal torus normalizer of the group extension {widetilde{U}}. Finally, we describe explicitly the adjoint action of W_G^{mathrm{T}} and W^U_G on the Lie algebra of G.

Rationality of the Exceptional $$\mathcal {W}$$-Algebras $$\mathcal {W}_k(\mathfrak {sp}_{4},f_{subreg})$$ Associated with Subregular Nilpotent Elements of $$\mathfrak {sp}_{4}$$

We prove the rationality of the exceptional $$\mathcal {W}$$ -algebras $$\mathcal {W}_k(\mathfrak {g},f)$$ associated with the simple Lie algebra $$\mathfrak {g}=\mathfrak {sp}_{4}$$ and a subregular nilpotent element $$f=f_{subreg}$$ of $$\mathfrak {sp}_{4}$$ , proving a new particular case of a conjecture of Kac–Wakimoto. Moreover, we describe the simple $$\mathcal {W}_k(\mathfrak {g},f)$$ -modules and compute their characters. We also work out the nontrivial action of the component group on the set of simple $$\mathcal {W}_k(\mathfrak {g},f)$$ -modules.

Statistics of the first Galois cohomology group : A refinement of Malle’s conjecture

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a refinement of Malle's conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^1(K,G)$ (or equivalently $1$-coclasses in $H^1(K,G)$) with bounded discriminant. This has a natural interpretation given by counting $G$-extensions $F/L$ for some fixed $L$ and prescribed extension class $F/L/K$. If $T$ is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for $Z^1(K,T)$ (and equivalently for $H^1(K,T)$) and show that it is a natural generalization of Malle's conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $G\subset S_n$ over $K$ and $G$ has an abelian normal subgroup $T\trianglelefteq G$ we prove a nontrivial lower bound for $N(K,G;X)$ given by a nonzero power of $X$ times a power of $\log X$. For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle's predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Kl\"uners' counter example to Malle's conjecture and verify the corrected lower bounds predicted by T\"urkelli.