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The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over F 2. Therefore a split extension group of the form 2 12 : (S z (8) : 3) : = G ¯ exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of G ¯ by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of G ¯ is a 43 × 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7. [ABSTRACT FROM AUTHOR]

Let Fq be a finite field with q elements. We consider formal Laurent series of Fq -coefficients with their continued fraction expansions by Fq -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors. [ABSTRACT FROM AUTHOR]

The Weil correspondence states that the datum of a Seiberg-Witten differential is equivalent to an algebraic group extension of the integrable system associated to the Seiberg-Witten geometry. Remarkably this group extension represents quantum consistent couplings for the N = 2 QFT if and only if the extension is anti-affine in the algebro-geometric sense. The universal special geometry is the algebraic integrable system whose Lagrangian fibers are the anti-affine extension groups; it is defined over a base B parametrized by the Coulomb coordinates and the couplings. On the total space of the universal geometry there is a canonical (holomorphic) Euler differential. The ordinary Seiberg-Witten geometries at fixed couplings are symplectic quotients of the universal one, and the Seiberg-Witten differential arises as the reduction of the Euler one in accordance with the Weil correspondence. This universal viewpoint allows to study geometrically the flavor symmetry of the N = 2 SCFT in terms of the Mordell-Weil lattice (with Néron-Tate height) of the Albanese variety A L of the universal geometry seen as a quasi-Abelian variety Y L defined over the function field L ≡ ℂ B . [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a spectral theory of advanced photon-counting computerized X-ray tomography, which is a promising microelectronical semiconductor wafer X-ray technique on the verge of becoming clinically feasible and has the potential to dramatically alter the clinical application of computer-aided medical diagnosis in the upcoming decades. In analogy to the deep fact that any real division algebra has dimension 1, 2, 4 or 8, the isogeneous third Galois cohomological, mirror symmetrical approach to high spatial resolution spectral photon-counting computerized tomography establishes that the exceptional Lie group Spin (8 , R) is the only one of the spin group spectrum { Spin (n , R) ∣ n ≥ 1 } to admit a triality automorphism. It is the exceptional phenomenon of triality which is able to produce by the spin representation theory of the exceptional Lie group of octonions O the spectral shaped digital signals of spectral photon-counting computerized X-ray tomography. Similarly to spin echo magnetic resonance tomography, the mathematical approach to spectral high performance computerized tomography is based on the isogeneously invariant, projectively weighted, symplectic coadjoint orbit picture Lie (N) ∗ / CoAd (N) of the real 3-dimensional Heisenberg unipotent Lie group N of type GL (3 , R) , and its solvable diamond toric Lie group extension. [ABSTRACT FROM AUTHOR]

Let G and N be arbitrary groups. In this paper, we construct an associated group extension ℰχ of N/Z( N) by G for any group homomorphism χ: G → Out N, and prove that χ can be lifted to a group action, that is, a group homomorphism from G to Aut N, if and only if the extension ℰχ splits. Furthermore we obtain an explicit description for all such lifting homomorphisms and the number of its conjugacy classes, and give an application of the lifting technique of outer actions to the theory of group extensions. [ABSTRACT FROM AUTHOR]

In this paper we extend Drinfeld's current realization of quantum affine algebras Uq(ĝ) and of the Yangians in several directions: we construct current operators for non-simple roots of g, define a new braid group action in terms of the current operators, and describe the universal R-matrix for the corresponding “Drinfeld” comultiplication in the forms of an infinite product and of certain integrals over current operators. [ABSTRACT FROM AUTHOR]

We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure. [ABSTRACT FROM AUTHOR]

Rota–Baxter operators for groups were recently introduced by Guo, Lang, and Sheng. Bardakov and Gubarev showed that with each Rota–Baxter operator, one can associate a skew brace. Skew braces on a group G can be characterised in terms of certain gamma functions from G to its automorphism group Aut (G) that are defined by a functional equation. For the skew braces obtained from a Rota–Baxter operator, the corresponding gamma functions take values in the inner automorphism group Inn (G) of G. In this paper, we give a characterisation of the gamma functions on a group G, with values in Inn (G) , that come from a Rota–Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota–Baxter operator. For gamma functions that can be obtained from a Rota–Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits. [ABSTRACT FROM AUTHOR]

We use cobordism theory to analyse anomalies of finite non-abelian symmetries in 4 spacetime dimensions. By applying the method of 'anomaly interplay', which uses functoriality of cobordism and naturality of the η-invariant to relate anomalies in a group of interest to anomalies in other (finite or compact Lie) groups, we derive the anomaly for every representation in many examples motivated by flavour physics, including S3, A4, Q8, and SL(2, 픽3). In the case of finite abelian groups, it is well known that anomalies can be 'truncated' in a way that has no effect on low-energy physics, by means of a group extension. We extend this idea to non-abelian symmetries. We show, for example, that a system with A4 symmetry can be rendered anomaly-free, with only one-third as many fermions as naïvely required, by passing to a larger symmetry. As another example, we find that a well-known model of quark and lepton masses utilising the SL(2, 픽3) symmetry is anomalous, but that the anomaly can be cancelled by enlarging the symmetry to a ℤ/3 extension of SL(2, 픽3). [ABSTRACT FROM AUTHOR]

In this paper, we are going to study the twisted Hessian curves on the local ring F q [ ϵ ] , ϵ 4 = 0 , with F q is a finite field of order q = p b , where p is a prime number ≥ 5 and b ∈ N ∗ . In a first time, we study the arithmetic of the ring F q [ ϵ ] , ϵ 4 = 0 , which will be used in the remainder of this work. After, we define the twisted Hessian curves H a , d 4 over this ring and we give essential properties and the classification of these elements. In addition, we define the group extension H a , d 4 of H a 0 , d 0 by K e r π ~ . We finish this work by introducing a new public key cryptosystem which is a variant of Cramer-Shoup public key cryptosystem on a twisted Hessian curves and study its security and efficiency. Our future work will focus on the generalist these studies for all integers n > 4 , ϵ n = 0 , which is beneficial and interesting in cryptography. [ABSTRACT FROM AUTHOR]

This study compared hand function and the cost-effectiveness of treatment between collagenase Clostridium histolyticum (CCH) injection and limited fasciectomy for patients with Dupuytren's contracture (DC). The CeCORD-J study is a prospective, multicenter, non-randomized controlled, observational study of two parallel groups. Participants were DC patients with multiple affected fingers, including flexion contracture of the proximal interphalangeal (PIP) joint. The primary outcome was the Hand10 score, as a patient-reported outcome measure (PROM). We set secondary outcomes of EQ-5D-5L (QOL) score, degree of extension deficit, and direct cost. Propensity score adjustment was used to balance differences in patient characteristics between groups. Participants comprised 52 patients in the Collagenase group and 26 patients in the Surgery group. There were no significant differences in the Hand10 and QOL scores between the two groups at 26 weeks. Mean direct cost was 248,000 yen higher in the Surgery group than in the Collagenase group. Extension deficit angle of the PIP joint was significantly larger in the Collagenase group at 26 weeks. Although the Collagenase group showed dominance in cost-effectiveness, there was no significant difference between the two groups in hand function at 26 weeks. [ABSTRACT FROM AUTHOR]

Introduction: New technologies exist which may assist surgeons to better predict final intra-operative joint balance. Our objectives were to compare the impact of (1) a predictive digital joint tensioning tool on intra-operative joint balance; and (2) joint balance and flexion joint laxity on patient-reported outcomes. Materials and methods: Two-hundred Eighty patients received posterior cruciate ligament sacrificing TKA with ultra-congruent tibial inserts using a robotic-assisted navigation platform. Patients were divided into those in which a Predictive Plan with a digital joint-tensioning device was used (PP) and those in which it was not (NPP), in all cases final post-operative joint gaps were collected immediately before final implantation. Demographics and KOOS were collected pre-operatively. KOOS, complications and satisfaction were collected at 3, 6 and 12 months post-operatively. Optimal balance difference between PP and NPP was defined and compared using area-under-the-curve analysis (AUC). Outcomes were then compared according to the results from the AUC. Results: AUC analysis yielded a balance threshold of 1.5 mm, in which the PP group achieved a higher rate of balance throughout flexion compared to the NPP group: extension: 83 vs 52%; Midflexion: 82 vs 55%; Flexion 89 vs 68%; Flexion to Extension 80 vs 49%; p ≤ 0.003. Higher KOOS scores were observed in knees balanced within 1.5 mm across all sub-scores at various time points, however, differences did not exceed the minimum clinically important difference (MCID). Patients with > 1.5 mm flexion laxity medially or laterally had an increased likelihood of 2.2 (1.1–4.4) and 2.5 (1.3–4.8), respectively, for failing to achieve the Patient Acceptable Symptom State for KOOS Pain at 12 months. Patient satisfaction was high in both the PP and NPP groups (97.4 and 94.7%, respectively). Conclusions: Use of a predictive joint tensioning tool improved the final balance in TKA. Improved outcomes were found in balanced knees; however, this improvement did not achieve the MCID, suggesting further studies may be required to define optimal balance targets. Limiting medial and lateral flexion laxity resulted in an increased likelihood of achieving the Patient Acceptable Symptom State for KOOS Pain. [ABSTRACT FROM AUTHOR]

This paper presents a model that allows group comparisons of gaze behavior while watching dynamic video stimuli. The model is based on the approach of Coutrot and Guyader (2017) and allows linear combinations of feature maps to form a master saliency map. The feature maps in the model are, for example, the dynamically salient contents of a video stimulus or predetermined areas of interest. The model takes into account temporal aspects of the stimuli, which is a crucial difference to other common models. The multi-group extension of the model introduced here allows to obtain relative importance plots, which visualize the effect of a specific feature of a stimulus on the attention and visual behavior for two or more experimental groups. These plots are interpretable summaries of data with high spatial and temporal resolution. This approach differs from many common methods for comparing gaze behavior between natural groups, which usually only include single-dimensional features such as the duration of fixation on a particular part of the stimulus. The method is illustrated by contrasting a sample of a group of persons with particularly high cognitive abilities (high achievement on IQ tests) with a control group on a psycholinguistic task on the conceptualization of motion events. In the example, we find no substantive differences in relative importance, but more exploratory gaze behavior in the highly gifted group. The code, videos, and eye-tracking data we used for this study are available online. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and let k be a nonzero nature number. Suppose that nonabelian group G is a central extension as follows where G′ ≅ ℤpk, and ζG/G′ is a direct factor of G/G′.Then G is a central product of an extraspecial pk-group E and ζG. Let ∣E∣ = p(2n+1)k and ∣ζG∣ = p(m+1)k. Suppose that the exponents of E and ζG are pk+l and pk+r, respectively, where 0 ≤ l, r ≤ k. Let AutG′G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G′, let AutG/ζG,ζGG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζG, and let AutG/ζG,ζG/G′G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on ζG/G′. Then (i) The group extension 1 → AutG′G → Aut G → Aut G′ → 1 is split. (ii) AutG′G/AutG/ζG,ζGG ≅ G1 × G2, where , H is an extraspecial pk -group of order p(2n−1)k and . In particular, if and only if l = k and r = 0; if and only if l ≤ r; if and only if r = k; if and only if r = 0. (iii) AutG′G/AutG/ζG,ζG/G′G ≅ G1 × G3, where G1 is defined in (ii); . In particular, if and only if r = k; if and only if r = 0. (iv) . If m = 0, then ; If m > 0, then , and . [ABSTRACT FROM AUTHOR]

We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and counterexamples, also highlighting the particular role played by positive orderability. Moreover, we construct a non-left-orderable, positively right-orderable submonoid of the free product of the cyclic group of order 7 with the free group on 4 generators. Any group extension of that monoid has elements of order 7. [ABSTRACT FROM AUTHOR]

We continue some recent investigations of W. Dziobiak, J. Ježek, and M. Maróti. Let $${{\bf G} = {\langle G, \cdot \rangle}}$$ G = 〈 G , · 〉 be a commutative group. A semilattice overG is a semilattice enriched with G as a set of unary operations acting as semilattice automorphisms. We prove that the minimal quasivarieties of semilattices over a finite abelian group G are in one-to-one correspondence with the subgroups of G. If G is not finite, then we reduce the description of minimal quasivarieties to that of those minimal quasivarieties in which not every algebra has a zero element. [ABSTRACT FROM AUTHOR]

We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism à with respect to a (G, ρ)—extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension  of A and deduce that  and A is essentially a group extension and homogeneous extension of à respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry. [ABSTRACT FROM AUTHOR]

Virtual Reality (VR) enables immersive experiences that can enhance awareness about environmental problems. We measure the effectiveness of VR versus 2D in an environmental campaign using a field experiment in Brazil. 617 passers-by at a mall were randomly assigned to watch a video clip about the Amazon Forest through VR or a traditional 2D device, with some being randomly interviewed before watching the movie (control group). We find that both the 2D and VR interventions increase individuals' propensity to (i) contribute to an Amazonian humanitarian campaign, (ii) share contact information, (iii) interact with a conservation campaign, and (iv) state pro-conservation opinions. We find no additional impact of VR compared to 2D, but VR participants were more likely to engage with pro-conservation content online 3 months later. Our findings provide cautionary evidence about the additional potential of using immersive technologies, like VR, to improve conservation behaviors compared to 2D methods. [ABSTRACT FROM AUTHOR]

A generalization to the categorical notion of biproduct, called semibiproduct, which in the case of groups covers classical semidirect products, has recently been analysed in the category of monoids with surprising results in the classification of weakly Schreier extensions. The purpose of this paper is to extend the study of semibiproducts to the category of semigroups. However, it is observed that a further analysis into the category of magmas is required in attaining a full comprehension on the subject. Indeed, although there is a subclass of magma-actions, that we call representable, which classifies all semibiproducts of magmas whose behaviour is similar to semigroups, it is nevertheless more general than the subclass of associative magma-actions. [ABSTRACT FROM AUTHOR]

A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and semigroups, while the more general notion of a mixed lattice remains unexplored. In this paper, we study the fundamental properties of mixed lattices and the relationships between the various properties. In particular, we establish the equivalence of the one-sided associative, distributive and modular laws in mixed lattices. We also give an alternative definition of mixed lattices and mixed lattice groups as non-commutative and non-associative algebras satisfying a certain set of postulates. The algebraic and the order-theoretic definitions are then shown to be equivalent. [ABSTRACT FROM AUTHOR]

This study primarily aims to investigate the application of the Lie symmetry method and conservation law theories in the analysis of mixed fractional partial differential equations where both Riemann–Liouville time-fractional and integer-order x-derivatives are present simultaneously. Specifically, the focus is on the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. The fractionally modified equation is subjected to invariant analysis using the prolongation formula for mixed derivatives ∂ t α (u x) and ∂ t α (u xxx) for the first time. Through the introduction of a novel reduction method, we utilize the Lie symmetry technique to convert the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation into a fractional ordinary differential equation. It’s worth noting that this transformation is carried out without employing the Erdélyi–Kober fractional differential operator. Following this, we introduce a comprehensive expression for deriving conservation laws, involving the notion of nonlinear self-adjointness. Further, two different versatile techniques, the extended Kudryashov method and the Sardar subequation method have been used to extract a wide array of fresh sets of solitary wave solutions encompassing variations like kink, bright, singular kink, and periodic soliton solutions. To provide an intuitive grasp and investigate the ramifications of the fractional derivative parameter on these solitary wave solutions, we conduct a visual exploration employing both 3D and 2D plots. [ABSTRACT FROM AUTHOR]

Generalized global symmetries, in particular non-invertible and categorical symmetries, have become a focal point in the recent study of quantum field theory (QFT). In this paper, we investigate aspects of symmetry topological field theories (SymTFTs) and anomalies of non-invertible symmetries for 2D QFTs from a string theory perspective. Our primary focus is on an infinite class of 2D QFTs engineered on D1-branes probing toric Calabi-Yau 4-fold singularities. We derive 3D SymTFTs from the topological sector of IIB supergravity and discuss the resulting 2D QFTs, which can be intrinsically relative or absolute. For intrinsically relative QFTs, we propose a sufficient condition for them to exist. For absolute QFTs, we show that they exhibit non-invertible symmetries with an elegant brane origin. Furthermore, we find that these non-invertible symmetries can suffer from anomalies, which we discuss from a top-down perspective. Explicit examples are provided, including theories for Y(p,k)(ℙ2), Y(2,0)(ℙ1 × ℙ1), and ℂ4/ℤ4 geometries. [ABSTRACT FROM AUTHOR]

We define a 3-loop group Ω 3 G as a subgroup of smooth maps from a 3-ball to a Lie group G, and then construct a 2-group based on an automorphic action on the Mickelsson–Faddeev extension of Ω 3 G . In this, we follow the strategy of Murray et al. (J Lie Theory 27(4):1151–1177, 2017), who earlier described a similar construction for one-dimensional loops. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for a metacyclic extension to be 2-local and ultrasolvable are established. These conditions are used to prove the ultrasolvability of an arbitrary group extension which has a local ultrasolvable associated subextension of the second type. The obtained reductions enables us to derive ultrasolvability results for a wide class of nonsplit 2-extensions with cyclic kernel. [ABSTRACT FROM AUTHOR]

Integrating Artificial Intelligence (AI) into Unmanned Aerial Vehicle (UAV) operations has advanced efficiency, safety, and decision-making. This study addresses critical gaps in UAV methods, including insufficient integration of human factors, operator variability, and the lack of systematic error analysis. To overcome these challenges, a novel approach combines the Analytic Hierarchy Process (AHP) with three core human factors models: the Observe-Orient-Decide-Act (OODA) loop, the Human Factors Analysis and Classification System (HFACS), and the SHELL model. An online survey was conducted across diverse UAV operator groups to prioritize critical factors within each model. Additionally, real-time monitoring of heart rate (HR), heart rate variability (HRV), and respiratory rate (RR) was conducted during UAV operations at various automation levels with different experience levels. Visualization through boxplots and percentage change matrices provided insights into operator stress and workload across automation levels. Integrating AHP findings and physiological data revealed significant differences in operator prioritization, highlighting the need for tailored AI-UAV strategies. This research combines survey data with real-time physiological monitoring, offering visions into optimizing human-AI interaction in UAV operations and providing a foundation for improving AI integration and operator strategies. [ABSTRACT FROM AUTHOR]

We relate the old and new cohomology monoids of an arbitrary monoid M with coefficients in semimodules over M, introduced in the author’s previous papers, to monoid and group extensions. More precisely, the old and new second cohomology monoids describe Schreier extensions of semimodules by monoids, and the new third cohomology monoid is related to a certain group extension problem. [ABSTRACT FROM AUTHOR]

We study the interplay of duality and stacking of bosonic and fermionic symmetry-protected topological phases in one spatial dimension. In general the classifications of bosonic and fermionic phases have different group structures under the operation of stacking, but we argue that they are often isomorphic and give an explicit isomorphism when it exists. This occurs for all unitary symmetry groups and many groups with antiunitary symmetries, which we characterize. We find that this isomorphism is typically not implemented by the Jordan-Wigner transformation, nor is it a consequence of any other duality transformation that falls within the framework of topological holography. Along the way to this conclusion, we recover the fermionic stacking rule in terms of G -pin partition functions, give a gauge-invariant characterization of the twisted group cohomology invariant, and state a procedure for stacking gapped phases in the formalism of symmetry topological field theory. [ABSTRACT FROM AUTHOR]

A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the shift. For this purpose, prime orbit and Mertens’ orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analyticity of the Artin–Mazur zeta function of the shift. Its zeta function is expressed implicitly in terms of several signed subset matrices. In this paper, we will prove the asymptotic behaviours of the counting functions for sofic shifts via their zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof simply uses some well-known facts about sofic shifts, especially on the minimal right-resolving presentations. Furthermore, we will demonstrate this result by revisiting the case for periodic-finite-type shifts, which are a particular type of sofic shifts. At the end, we will briefly discuss the application of our finding towards the finite group and homogeneous extensions of a sofic shift. [ABSTRACT FROM AUTHOR]

The geo-social group query is to find a group of users for the query point based on location and social information. In this paper, we propose the range constrained group query (RCGQ) on attribute social graph, considering social information, spatial information, keyword information and user group size. We prove that RCGQ problem is NP-hard. For the query, we propose four methods, namely the combination-based group expansion method (COM), the single–multi group expansion method (S–M), the single–single group expansion method (S–S) and the multi–multi group expansion method (M–M). The first method is based on combination. The last three methods are based on social relations. COM uses combination to find user groups. The social relations are not used in the combinatorial process. S–M, S–S and M–M use the social relations to find user groups. Pruning strategies are proposed for the four methods. Finally, experiments demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The rationality problem of two-dimensional purely quasi-monomial actions was solved completely by (Hoshi, Kang and Kitayama, J. Algebra 403, 363-400, 2014). As a generalization, we solve the rationality problem of two-dimensional quasi-monomial actions under the condition that the actions are defined within the base field. In order to prove the theorem, we give a brief review of the Severi-Brauer variety with some examples and rationality results. We also use a rationality criterion for conic bundles of P 1 over non-closed fields. [ABSTRACT FROM AUTHOR]

A classical result of Fathi and Herman from 1977 states that a smooth compact connected manifold without boundary admitting a locally free action of a 1-torus, respectively, an almost free action of a 2-torus, admits a minimal diffeomorphism, respectively, a minimal flow. In the first part of our paper we study the existence of locally free and almost free actions of tori on homogeneous spaces of compact connected Lie groups, thus providing new examples of spaces admitting minimal diffeomorphisms or flows. In the second part we combine the ideas of Fathi and Herman with our recent ideas to study the existence of minimal skew products over certain minimal flows with general connected Lie groups as acting groups. Our results apply to so called flows with free cycles. In the last part of our work we study the existence of free cycles in homogeneous flows. [ABSTRACT FROM AUTHOR]

In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called ( G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle–Perron–Frobenius operator. Also the counting results do not use any tauberian theorems. [ABSTRACT FROM AUTHOR]

Let E be a group extension with Abelian kernel. Then it can be assigned an extension E′ of modules over the group ring of the quotient group. As a consequence, an embedding of the initial extension in some splitting extension arises. We prove that the celebrated Magnus embedding is a special case of this general construction. [ABSTRACT FROM AUTHOR]

In this paper we study central extensions of the identity component G of the Lie group C∞(M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:[fraktur k]×[fraktur k]→Y is a symmetric invariant bilinear map on the Lie algebra [fraktur k] of K and the values of ω lie in Ω1(M,Y)/dC∞(M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=[double-struck S]1. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C∞(M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G. [ABSTRACT FROM AUTHOR]

Both type of CSOM, tubotympanic which is considered safe as well as atticoantral which is considered unsafe may lead to erosion of the ossicular chain. Discontinuity of the ossicular chain is typically confirmed only during an operation. Knowing before surgery whether the patient has an ossicular discontinuity is important because it allows the surgeon to know the possibility of performing an ossiculoplasty and obtaining patient consent. The aims is to (1) study the incidence of incus necrosis in safe and unsafe CSOM. (2) Determine the preoperative predictive factors for incus necrosis. (3) Use angled otoscopes to determine the incidence of residual disease peroperative after conventional microscopic surgery. This is a prospective study carried out in the department of otorhinolaryngology, Govt Doon medical college, Dehradun from July 2014 to July 2016. A total of 100 patients who presented with CSOM and have not undergone any surgical procedure for the same were included in this study. Patients group was divided into cholesteatoma and non cholesteatoma group. Both groups were subdivided into intact and eroded incus group and were analysed in 11 parameters which were compared statistically. Incudal necrosis is more common in cholesteatoma group. In non cholesteatoma ears subtotal perforation with exposure of IS joint is reliable indicators of incudal necrosis. In non cholesteatoma group extension of cholesteatoma to tympanic sinus and mastoid and presence of persistent discharge are reliable indicators of necrosed incus. Moderate to moderately severe hearing loss indicate incudal necrosis in both groups therefore we conclude that these parameters can be reliably considered as predictors for incudal necrosis preoperatively. [ABSTRACT FROM AUTHOR]

The Green ring of the half quantum group $${H=U_n(q)}$$ is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups-to compute the depth of the Hopf subalgebra H in its Drinfeld double D( H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which $${Q^{\otimes (n)}}$$ has the same indecomposable constituents as $${Q^{\otimes (n+1)}}$$ is $${n = 2}$$ , since $${ Q^{\otimes (2)}}$$ contains all H-module indecomposables, which determines the minimum even depth $${d_{\mathrm{ev}}(H,D(H)) = 6}$$ . [ABSTRACT FROM AUTHOR]

Continuing the work in Ralston and Troubetzkoy (J Mod Dyn 6:477-497, 2012), we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as cuts. [ABSTRACT FROM AUTHOR]

The paper is concerned with the topological and metric properties of group extensions of C systems. The basic theorem describes the topologically transitive component, the ergodic component, and the K component of a group extension of a C system. It is shown that each of these components is a group sub-bundle of a principal bundle in which the group extension acts. The frame flow on a manifold of negative curvature is seen to be a special case of a group of extension of a C system. It is shown that the space of frames on a compact three-dimensional manifold with negative curvature does not have any group sub-bundles, so that the frame flow on manifolds of this class is topologically transitive, ergodic, and a K system. [ABSTRACT FROM AUTHOR]

We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs. [ABSTRACT FROM AUTHOR]

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the global form of the gauge group but not its perturbative dynamics. In this work, we study the Seiberg-Witten solution of theories resulting from the gauging of charge conjugation in 4d N = 2 theories with SU(N) gauge group and fundamental hypermultiplets. The basic idea of our procedure is to identify the ℤ2 action at the level of the SW curve and perform the quotient, and it should also be applicable to non-lagrangian theories. We study dynamical aspects of these theories such as their moduli space singularities and the corresponding physics; in particular, we explore the complex structure singularity arising from the quotient procedure. We also discuss some implications of our work in regards to three problems: the geometric classification of 4d SCFTs, the study of non-invertible symmetries from the SW geometry, and the String Theory engineering of theories with disconnected gauge groups. [ABSTRACT FROM AUTHOR]

Raoellid mammals are small artiodactyls from the Eocene of Asia, hypothesized to be closely related to stem Cetacea. Knowledge of the cranial and dental morphology of Raoellidae comes mostly from one species, Indohyus indirae. Here we describe new material of another raoellid genus, Khirtharia, based on material retrieved from the Kalakot area, Jammu and Kashmir. This new material, comprising an almost complete, lightly deformed cranium and a partial snout with associated partial mandible, greatly adds to our knowledge of raoellid morphology. It highlights the similarity of cranial characters with Indohyus, such as a long snout with raptorial incisors, a thick and narrow supraorbital region, a strong postorbital constriction, a triangular shaped braincase, and a thickened medial wall to the auditory bulla (involucrum). The new specimen is similar to Indohyus cranially but differs dentally in being more bunodont. The presence of these traits in two different raoellid genera suggests they may be present more broadly across Raoellidae. These characters are also observed in early cetaceans, highlighting the need to investigate their phylogenetic impact. Some cranial features support aquatic habits of members of this family. [ABSTRACT FROM AUTHOR]

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by Kwak and Lee. For regular graph coverings, Kwak, Chun and Lee enumerated the isomorphism classes of graph coverings when the covering transformation group is a finite abelian or a dihedral group in Kwak et al. (SIAM J Discrete Math 11:273–285, 1998). In 2018, the isomorphism classes of graph coverings are enumerated when the covering transformation groups are Z 2 -extensions of a cyclic group. As a continuation of this work, we enumerate the isomorphism classes of coverings of a graph when the covering transformation groups are Z p -extensions of a cyclic group for an odd prime integer p. [ABSTRACT FROM AUTHOR]

Recreational sports are becoming increasingly important in overcoming the drawbacks of our modern sedentary lifestyle. We wanted to know whether ambitious strength or endurance training has a systematic effect on the maximum strength capacity of the trunk muscles compared to no sport at all. We investigated two groups of physically active men who practised either endurance (ET; cycling and triathlon, n = 13) or strength training (ST; power lifting, n = 13), and a group of healthy physically inactive men (control [C], n = 12). Training intensity was at competition level in both active groups. All participants performed isometric maximum voluntary contractions in flexion and extension direction. Independent of force direction maximum torque levels were highest for the ST group (p < 0.001 vs. ET and C), but after normalizing to the subject's upper body weight these differences decreased, together with a drop in significance levels (extension: p < 0.01 vs. C; flexion: p < 0.05 vs. ET; p < 0.01 vs. C). With respect to the ratio between extension and flexion maximum forces due to the small group size no systematic differences could be detected between the groups, but effect sizes imply relevant effects (ET vs. ST: d = 0.588, ST vs C: d = −0.811). The results of this pilot study indicate that ST show higher functional force capacity values for flexion compared to the other groups. For extension, ST and ET did not differ. These results imply relevant differences for the extension to flexion force ratio. [ABSTRACT FROM AUTHOR]

Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov’s asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasi-isometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions. [ABSTRACT FROM AUTHOR]

Let n ≥ 1, and let ι n : F n (M) → ∏ 1 n M be the natural inclusion of the nth configuration space of M in the n-fold Cartesian product of M with itself. In this paper, we study the map ιn, the homotopy fibre In of ιn and its homotopy groups, and the induced homomorphisms (ιn)#k on the kth homotopy groups of Fn(M) and ∏ 1 n M for all k ≥ 1, where M is the 2-sphere S 2 or the real projective plane ℝP2.It is well known that the group πk(In) is the homotopy group π k + 1 (∏ 1 n M , F n (M)) for all k ≥ 0. If k ≥ 2, we show that the homomorphism (ιn)#k is injective and diagonal, with the exception of the case n = k = 2 and M = S 2 , where it is anti-diagonal. We then show that In has the homotopy type of K (R n − 1 , 1) × Ω (∏ 1 n − 1 S 2 ) , where Rn−1 is the (n − 1)th Artin pure braid group if M = S 2 , and is the fundamental group Gn−1 of the (n−1)th orbit configuration space of the open cylinder S 2 \ { z ˜ 0 , − z ˜ 0 } with respect to the action of the antipodal map of S 2 if M = ℝP2, where z ˜ 0 ∈ S 2 . This enables us to describe the long exact sequence in homotopy of the homotopy fibration I n → F n (M) → ι n ∏ 1 n M in geometric terms, and notably the image of the boundary homomorphism π k + 1 (∏ 1 n M) → π k (I n) . From this, if M = S 2 and n ≥ 3 (resp. M = ℝP2 and n ≥ 2), we show that Ker((ιn)#1) is isomorphic to the quotient of Rn−1 by the square of its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order 2 generated by the centre of Pn (M) that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in [GG5]. [ABSTRACT FROM AUTHOR]

A generalization of the group algebra of a locally compact group is studied, by expressing the group algebra of a central group extension as the direct sum of closed *-ideals each one of which is isomorphic to such a 'twisted' group algebra. In particular, the representation theory of such algebras is associated with the theory of projective representations by studying the representations of the group algebra of the group extension and the associated unitary representations of the group extension. [ABSTRACT FROM AUTHOR]

This paper gives a detailed analysis of the Cannon-Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E of F, and the main theorem of Dowdall-Taylor [DT14] states that in this situation E is hyperbolic if and only if Γ is purely atoroidal. Here, we give an explicit geometric description of the Cannon-Thurston maps ∂F → ∂ E for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon-Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL15] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous. [ABSTRACT FROM AUTHOR]

The dynamical system on $$\mathbb{T}^2$$ which is a group extension over an irrational rotation on $$\mathbb{T}^1$$ is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host et al. (2014) is obtained. [ABSTRACT FROM AUTHOR]

We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper. [ABSTRACT FROM AUTHOR]

In Special Geometry there are two inequivalent notions of "Kodaira type" for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry is subtler and its deformation space gets smaller ("partially frozen" singularities). The paper analyzes the physical interpretation of the Hwang-Oguiso invariant in the context of 4d N = 2 QFT and describes the surprising phenomena which appear when it does not coincide with the monodromy type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with the partially frozen singularities in M-theory compactified on a local elliptic K3: a chain of string dualities relates the two geometric set-ups. Paying attention to a few subtleties, this correspondence explains in purely geometric terms how the "same" Kodaira elliptic fiber may have different deformations spaces. The geometric computation of the number of deformations agrees with the physical expectations. At the end we briefly outline the implications of the Hwang-Oguiso invariants for the classification program of 4d N = 2 SCFTs. [ABSTRACT FROM AUTHOR]