Recent zbMATH articles in MSC 70https://www.zbmath.org/atom/cc/702021-04-16T16:22:00+00:00WerkzeugExplicit estimates on the measure of primary KAM tori.https://www.zbmath.org/1456.370632021-04-16T16:22:00+00:00"Biasco, L."https://www.zbmath.org/authors/?q=ai:biasco.luca"Chierchia, L."https://www.zbmath.org/authors/?q=ai:chierchia.luigiSummary: From KAM theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, ``primary'' tori in a nearly integrable, real-analytic Hamiltonian system is \(O(\sqrt{\varepsilon })\), if \(\varepsilon\) is the size of the perturbation. In this paper, we discuss how the constant in front of \(\sqrt{\varepsilon }\) depends on the unperturbed system and in particular on the phase-space domain.New bi-Hamiltonian systems on the plane.https://www.zbmath.org/1456.370592021-04-16T16:22:00+00:00"Tsiganov, A. V."https://www.zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovichSummary: We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method.{
\copyright American Institute of Physics}The globalization problem of the Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework.https://www.zbmath.org/1456.530732021-04-16T16:22:00+00:00"Esen, Oğul"https://www.zbmath.org/authors/?q=ai:esen.ogul"de León, Manuel"https://www.zbmath.org/authors/?q=ai:de-leon.manuel"Sardón, Cristina"https://www.zbmath.org/authors/?q=ai:sardon.cristina"Zając, Marcin"https://www.zbmath.org/authors/?q=ai:zajac.marcinSummary: In this paper, we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework and we introduce the notion of \textit{locally conformal k-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton-Jacobi equation, which is specifically proposed in a l.c.k-s manifold.Hyperkähler cones and instantons on quaternionic Kähler manifolds.https://www.zbmath.org/1456.530402021-04-16T16:22:00+00:00"Devchand, Chandrashekar"https://www.zbmath.org/authors/?q=ai:devchand.chandrashekar"Pontecorvo, Massimiliano"https://www.zbmath.org/authors/?q=ai:pontecorvo.massimiliano"Spiro, Andrea"https://www.zbmath.org/authors/?q=ai:spiro.andrea-fIn this paper a new method to construct Yang-Mills instantons over quaternionic pseudo-Riemannian Kähler manifolds is introduced by extending a technique motivated by supersymmetry for constructing Yang-Mills instantons over pseudo-Riemannian hyper-Kähler manifods by the same authors.
By a \textit{quaternionic pseudo-Riemannian Kähler manifold} of signature \((p,q)\) one means a real \(4n\)-dimensional pseudo-Riemannian manifold \((M,g)\) whose holonomy group is isomorphic to \(\mathrm{Sp}(1)\mathrm{Sp}(p,q)\) satisfying \(p+q=n\). Such manifolds are automatically Einstein hence solving the Yang-Mills self-duality equations over them is interesting from both a mathematical and a physical viewpoint. The authors' method in the spirit of the classical Atiyah-Ward correspondence rests on a bijection between gauge equivalence classes of Yang-Mills instantons with arbitrary compact structure group \(G\) over \((M,g)\) and certain holomorphic objects over a twistor space-like complex manifold \(H(S(M))\) associated to \((M,g)\). This twistor space is constructed in two steps. First, given \((M,g)\) one takes the so-called \textit{Swann bundle} over \(M\), i.e., a certain \({\mathbb H}^*/{\mathbb Z}_2\)-bundle \(\pi :S(M)\rightarrow M\). It has the structure of a hyper-Kähler cone over \(M\). Secondly, one considers its \textit{harmonic space} \(H(S(M))\) which is a topologically trivial \(\mathrm{SL}(2;{\mathbb C})\)-bundle over \(S(M)\) carrying the unique non-product complex structure which makes \(H(S(M))\) a holomorphic bundle over the classical twistor space \(Z(S(M))=S(M)\times{\mathbb C}P^1\) of the Swann bundle; the fibers of this vector bundle are isomorphic to the Borel subgroup \(B\subset\mathrm{SL}(2;{\mathbb C})\) since \(\mathrm{SL}(2;{\mathbb C})/B\cong{\mathbb C}P^1\). The key observation is that there is a one-to-one correspondence between (local) \(G\)-instantons over \((M,g)\) and certain (local) holomorphic maps, called (supersymmetric) \textit{prepotentials}, from \(H(S(M))\) into the complexified Lie algebra of \(G\). Combining this method with results of Narasimhan and Ramanan the construction settles down to a set of data on certain local maps \(M\supset U\rightarrow \mathrm{Mat}_{k\times m} ({\mathbb C})\) and in this form the construction resembles the classical ADHM construction.
Reviewer: Gabor Etesi (Budapest)Kinematics of general spatial mechanical systems.https://www.zbmath.org/1456.700012021-04-16T16:22:00+00:00"Ozgoren, M. Kemal"https://www.zbmath.org/authors/?q=ai:ozgoren.m-kemalThe present book presents in detail the kinematic description and analysis of the spatial mechanical systems such as serial manipulators, parallel manipulators and spatial mechanisms. The book contains ten chapters in which the author throws light on the basic theory and clarifies the analytical and semi-analytical methods for solving the relevant equations, considering four main elements: mathematics of spatial kinematics with the appropriate theorems, formulas and methods; kinematic description of the links and joints including the rolling contact joints; writing the kinematic chain and loop equations for the systems to be analyzed; and solving these equations for the unspecified variables both in the forward and inverse sense together with the multiplicity and singularity analyses.
The theory is presented step by step: beginning with spatial mechanisms with single degree of freedom and continuing to more advanced topics such as serial manipulators including redundant and deficient ones, parallel manipulators, and non-holonomic spatial cam mechanisms that involve rolling without slipping motions.
Through the symbolic manipulation method which is used in the book, clear analytical expressions are obtained, and all kinematic details are shown. These expressions readily lead to analytical or semi-analytical solutions. They also facilitate the identification and analysis of the multiplicities and singularities. This way,
the motion planning studies are also facilitated.
The reader may find representative examples demonstrating the kinematics theory of serial and parallel manipulators, and spatial mechanisms. The book is intended for students and scientists and should be of interest to practicing and research engineers as well as Ph.D. students in the field of spatial mechanisms, manipulators and robots.
Reviewer: Clementina Mladenova (Sofia)Quantifying chaos of curvilinear beams via exponents.https://www.zbmath.org/1456.740662021-04-16T16:22:00+00:00"Awrejcewicz, J."https://www.zbmath.org/authors/?q=ai:awrejcewicz.jan"Krysko, V. A."https://www.zbmath.org/authors/?q=ai:krysko.vadim-a|krysko.vadim-a-jun"Kutepov, I. E."https://www.zbmath.org/authors/?q=ai:kutepov.i-e"Vygodchikova, I. Yu."https://www.zbmath.org/authors/?q=ai:vygodchikova.i-yu"Krysko, Anton V."https://www.zbmath.org/authors/?q=ai:krysko.anton-vSummary: We propose a procedure for predicting the stability loss and transition into chaos of a network of oscillators lying on a curve, where each of the oscillators can move in two perpendicular directions. Dynamics of the coupled oscillators are governed by the sixth-order PDE, which is directly derived using the classical hypotheses of a curvilinear flexible beam movement theory. We apply FDM (Finite Difference Method) to reduce PDEs into ODEs, and the used number of spatial coordinate positions defines the number of involved oscillators approximating the dynamics of our continuous structural member (beam). Our procedure has a few advantages over the classical approaches, which has been illustrated and discussed. The proposed method has been validated for non-linear dynamical regimes by using the classical vibrational analysis (time histories, frequency power spectra and Poincaré maps).Predictive dynamic simulation of olympic track cycling standing start using direct collocation optimal control.https://www.zbmath.org/1456.700422021-04-16T16:22:00+00:00"Jansen, Conor"https://www.zbmath.org/authors/?q=ai:jansen.conor"McPhee, John"https://www.zbmath.org/authors/?q=ai:mcphee.john-jSummary: Much of the previous research on modeling and simulation of cycling has focused on seated pedaling, modeling the crank load with an effective resistive torque and inertia. This study focuses on modeling standing starts, a component of certain track cycling events in which the cyclist starts from rest and attempts to accelerate to top speed as quickly as possible. A ten degree-of-freedom, two-legged cyclist and bicycle model was developed and utilized for predictive dynamic simulations of standing starts. Experimental data including crank torque, cadence, and joint kinematics were collected for a member of the Canadian Olympic team performing standing starts on the track. Using direct collocation optimal control to maximize the simulated distance traveled, the predictive simulations aligned well with the experiments and replicated key aspects of the standing start technique such as the drive and reset. The model's use in ``What if?'' scenarios presents interesting possibilities for investigating optimal techniques and equipment in cycling.Three dimensional unassisted sit-to-stand prediction for virtual healthy young and elderly individuals.https://www.zbmath.org/1456.700042021-04-16T16:22:00+00:00"Yang, James"https://www.zbmath.org/authors/?q=ai:yang.james-j|yang.james-g-s|yang.james-chingyu|yang.james-ching-nung"Ozsoy, Burak"https://www.zbmath.org/authors/?q=ai:ozsoy.burakSummary: Sit-to-stand (STS) motion is one of the most important tasks in daily life and is one of the key determinants of functional independence, especially for the senior people. The STS motion has been extensively studied in the literature, mostly through experiments. Compared to numerous experimental studies, there are limited simulations with mostly assuming bilateral symmetry for STS tasks. However, it is not true even for healthy individuals to perform STS tasks with a perfect bilateral symmetry. In this study, predictive dynamics is utilized for STS prediction. The problem can be constructed as a nonlinear optimization formulation. The digital human model has 21 degrees of freedom (DOFs) for the unassisted STS tasks. The quartic B-spline interpolation is implemented for representing joint angle profiles. The recursive Lagrangian dynamics approach and the Denavit-Hartenberg method are implemented for the equations of motion. This study is to develop a generic three-dimensional unassisted STS motion prediction method for healthy young and elderly individuals. Results show that trunk joint angle peak values are similar between the two virtual-groups in the sagittal, frontal, and transverse planes. Lower-limbs' joint angle and velocity profiles and their peak values between the right and left side for both virtual groups are also similar. The normalized peak joint torques are slight differences in each active DOF between the two virtual groups and the peak values are similar. The proposed method has been indirectly validated through the literature experimental results. The developed method has various potential applications in the design of exoskeleton, microelectromechanical system for fall detection, and assistive devices in rehabilitation.State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches.https://www.zbmath.org/1456.700102021-04-16T16:22:00+00:00"Bruni, Stefano"https://www.zbmath.org/authors/?q=ai:bruni.stefano"Meijaard, J. P."https://www.zbmath.org/authors/?q=ai:meijaard.j-p"Rill, Georg"https://www.zbmath.org/authors/?q=ai:rill.georg"Schwab, A. L."https://www.zbmath.org/authors/?q=ai:schwab.a-lSummary: A review of the current use of multibody dynamics methods in the analysis of the dynamics of vehicles is given. Railway vehicle dynamics as well as road vehicle dynamics are considered, where for the latter the dynamics of cars and trucks and the dynamics of single-track vehicles, in particular motorcycles and bicycles, are reviewed. Commonalities and differences are shown, and open questions and challenges are given as directions for further research in this field.On the optimal control of rate-independent soft crawlers.https://www.zbmath.org/1456.490072021-04-16T16:22:00+00:00"Colombo, Giovanni"https://www.zbmath.org/authors/?q=ai:colombo.giovanni"Gidoni, Paolo"https://www.zbmath.org/authors/?q=ai:gidoni.paoloSummary: Existence of optimal solutions and necessary optimality conditions for a controlled version of Moreau's sweeping process are derived. The control is a measurable ingredient of the dynamics and the constraint set is a polyhedron. The novelty consists in considering time periodic trajectories, adding the requirement that the control has zero average, and considering an integral functional that lacks weak semicontinuity. A model coming from the locomotion of a soft-robotic crawler, that motivated our setting, is analysed in detail. In obtaining necessary conditions, a variant of the method of discrete approximations is used.Multi-Regge limit of the two-loop five-point amplitudes in \(\mathcal{N} = 4\) super Yang-Mills and \(\mathcal{N} = 8\) supergravity.https://www.zbmath.org/1456.831122021-04-16T16:22:00+00:00"Caron-Huot, Simon"https://www.zbmath.org/authors/?q=ai:caron-huot.simon"Chicherin, Dmitry"https://www.zbmath.org/authors/?q=ai:chicherin.dmitry"Henn, Johannes"https://www.zbmath.org/authors/?q=ai:henn.johannes-m"Zhang, Yang"https://www.zbmath.org/authors/?q=ai:zhang.yang"Zoia, Simone"https://www.zbmath.org/authors/?q=ai:zoia.simoneSummary: In previous work [\textit{E. D'Hoker} et al.,ibid. 2020, No. 8, Paper No. 135, 80 p. (2020; Zbl 1454.83159); \textit{C. R. Mafra} and \textit{O. Schlotterer}, ibid. 2015, No. 10, Paper No. 124, 29 p. (2015; Zbl 1388.83860)], the two-loop five-point amplitudes in \(\mathcal{N} = 4\) super Yang-Mills theory and \(\mathcal{N} = 8\) supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the \(\mathcal{N} = 4\) super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.The sign exchange bifurcation in a family of linear Hamiltonian systems.https://www.zbmath.org/1456.370552021-04-16T16:22:00+00:00"Cushman, Richard"https://www.zbmath.org/authors/?q=ai:cushman.richard-h"Robbins, Johnathan M."https://www.zbmath.org/authors/?q=ai:robbins.johnathan-m"Sadovskii, Dimitrii"https://www.zbmath.org/authors/?q=ai:sadovskii.dimitriiThis paper considers a one-parameter family of Hamiltonian systems that describe two classical particles with angular momenta under the influence of a magnetic field. The phase space for the system is \(\mathbb{R}^3 \times \mathbb{R}^3\) with coordinates \((S, L) = (S_x, S_y, S_z, L_x, L_y, L_z)\).
For \(\gamma \in [0, 1]\) the Hamiltonian is defined by \[H_{\gamma} (S, L) = \frac {1 - \gamma}{s} S_z + \frac {\gamma}{sl} S \cdot L,\] where \(s^2 = S \cdot S\) and \(l^2 = L \cdot L\). The motion of the system arises from the Hamiltonian vector field \(X_{H_\gamma}\) with integral curves that satisfy \(\frac {dS}{dt} = \frac {1 - \gamma}{s} e_3 \times S - \frac {\gamma}{sl} S \times L\) and \(\frac{dL}{dt} = \frac{\gamma}{sl} S \times L\) where \(\{e_1, e_2, e_3\}\) is the standard basis of \(\mathbb{R}^3\).
The functions \(C_1 = S_x^2 + S_y^2 +S_z^2\) and \(C_2 = L_x^2 + L_y^2 + L_z^2\) are integrals of the vector field \(X_{H_\gamma}\), so the smooth manifold \(S^2_s + S^2_l\) defined by \(C_1 = s^2\) and \(C_2 = l^2\) is invariant under the flow of \(X_{H_\gamma}\). The authors then consider the vector field \(X_{J_z}\) where \(J_z = S_z + L_z\). \(X_{J_z}\) has integral curves that satisfy \(\frac{dS}{dt} = e_3 \times S\) and \(\frac{dL}{dt} = e_3 \times L\). The flow of \(X_{J_z}\) defines an \(S^1\)-action on \(\mathbb{R}^3 + \mathbb{R}^3\) that is a symmetry of the vector field \(X_{J_z}\). Because the flows of \(X_{J_x}\) and \(X_{H_\gamma}\) commute, the \(S^1\)-action is also a symmetry of \(X_{H_\gamma}\) .
The only equilibrium points of \(X_{H_\gamma}\) have the form \(p = (\eta_s se_3, \eta_l le_3)\) where \(\eta_s^2 = \eta_l^2 = 1\); \(p\) lies on the invariant manifold \(S^2_s + S^2_l\). After linearizing \(X_{H_\gamma (p)}\) to get \(DX_{H\gamma}(p)\), the authors assume that \(s = \epsilon^2 l\) where \( 0 < \epsilon \ll 1\) and consider the \(S^1\)-invariant curve \(\mathcal{X}\) that maps \(\gamma\) in \([0,1]\) to \(DX_{H\gamma}(p)\). Their goal is to show that as \(\gamma\) increases, the curve at \(p = (se_3, -le_3)\) sees \(S^1\)-equivariant bifurcations at values of \(\gamma\) where \(DX_{H_\gamma}\) has multiple eigenvalues (all near \(\frac{1}{2}\) because of scaling). In particular, they show that at \(\gamma = \frac{1}{2 - \epsilon^2}\) the curve has a switch twist bifurcation where the sense of the rotational part of \(DX_{H_\gamma}(p) \) changes. They also show that at \(p = (-se_3, -le_3)\), the curve \(\mathcal{X}\) has a sign exchange bifurcation at \(\gamma = \frac{1}{2 + 2\epsilon + \epsilon^2}\).
For the entire collection see [Zbl 1280.37002].
Reviewer: William J. Satzer Jr. (St. Paul)Homoclinic solutions for a class of nonlinear fourth order \(p\)-Laplacian differential equations.https://www.zbmath.org/1456.340442021-04-16T16:22:00+00:00"Dimitrov, Nikolay D."https://www.zbmath.org/authors/?q=ai:dimitrov.nikolay-d"Tersian, Stepan A."https://www.zbmath.org/authors/?q=ai:tersian.stepan-agopIn this paper, the authors deal with a class of fourth-order differential equation involving \(p\)-Laplacian
\[
|u''(x)|^{p-2}u''(x))''+\omega (|u'(x)|^{p-2}u'(x))'+\lambda V(x) |u(x)|^{p-2} u(x)=f(x,u(x)
\]
where \(\omega\) is a constant, \(\lambda\) is a parameter and \(f\in C (\mathbb{R},\mathbb{R})\). with the aid of critical point theory and variational methods, they prove that under suitable growth conditions, the above equation possesses at least one nontrivial homoclinic solution, i.e., a nontrivial solution satisfying \(u(x)\longrightarrow 0\) as \(x\longrightarrow \mp\infty\).
Reviewer: Mohsen Timoumi (Monastir)The Foucault pendulum (with a twist).https://www.zbmath.org/1456.370442021-04-16T16:22:00+00:00"Moeckel, Richard"https://www.zbmath.org/authors/?q=ai:moeckel.richardS-duality wall of SQCD from Toda braiding.https://www.zbmath.org/1456.814382021-04-16T16:22:00+00:00"Le Floch, B."https://www.zbmath.org/authors/?q=ai:le-floch.brunoSummary: Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional \(\mathcal{N} = 2\) \(\mathrm{SU} (N)\) SQCD with \(2N\) hypermultiplets in terms of fields on the defect, namely three-dimensional \(\mathcal{N} = 2\) SQCD with gauge group \(\mathrm{U}(N -1)\) and \(2N\) flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that \(T[ \mathrm{SU} (N)]\) (the S-duality wall of \(\mathcal{N} = 4\) super Yang-Mills) is self-mirror. The domain-wall theory can also be realized as a limit of a \(\mathrm{USp}(2N - 2)\) gauge theory; it reduces to known results for \(N = 2\). The theory is found through the AGT correspondence by determining the braiding kernel of two semi-degenerate vertex operators in Toda CFT.Spatially modulated and supersymmetric mass deformations of \(\mathcal{N} = 4\) SYM.https://www.zbmath.org/1456.831102021-04-16T16:22:00+00:00"Arav, Igal"https://www.zbmath.org/authors/?q=ai:arav.igal"Cheung, K. C. Matthew"https://www.zbmath.org/authors/?q=ai:cheung.k-c-matthew"Gauntlett, Jerome P."https://www.zbmath.org/authors/?q=ai:gauntlett.jerome-p"Roberts, Matthew M."https://www.zbmath.org/authors/?q=ai:roberts.matthew-m"Rosen, Christopher"https://www.zbmath.org/authors/?q=ai:rosen.christopherSummary: We study mass deformations of \(\mathcal{N} = 4, \ d = 4\) SYM theory that are spatially modulated in one spatial dimension and preserve some residual supersymmetry. We focus on generalisations of \(\mathcal{N} = 1^\ast\) theories and show that it is also possible, for suitably chosen supersymmetric masses, to preserve \(d = 3\) conformal symmetry associated with a co-dimension one interface. Holographic solutions can be constructed using \(D = 5\) theories of gravity that arise from consistent truncations of SO(6) gauged supergravity and hence type IIB supergravity. For the mass deformations that preserve \(d = 3\) superconformal symmetry we construct a rich set of Janus solutions of \(\mathcal{N} = 4\) SYM theory which have the same coupling constant on either side of the interface. Limiting classes of these solutions give rise to RG interface solutions with \(\mathcal{N} = 4\) SYM on one side of the interface and the Leigh-Strassler (LS) SCFT on the other, and also to a Janus solution for the LS theory. Another limiting solution is a new supersymmetric \( \mathrm{AdS}_4 \times S^1 \times S^5\) solution of type IIB supergravity.Energy dissipation in Hamiltonian chains of rotators.https://www.zbmath.org/1456.370652021-04-16T16:22:00+00:00"Cuneo, Noé"https://www.zbmath.org/authors/?q=ai:cuneo.noe"Eckmann, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:eckmann.jean-pierre"Wayne, C. Eugene"https://www.zbmath.org/authors/?q=ai:wayne.c-eugeneStrong coupling expansion of circular Wilson loops and string theories in \(\mathrm{AdS}_5 \times S^5\) and \(\mathrm{AdS}_4 \times CP^3\).https://www.zbmath.org/1456.831002021-04-16T16:22:00+00:00"Giombi, Simone"https://www.zbmath.org/authors/?q=ai:giombi.simone"Tseytlin, Arkady A."https://www.zbmath.org/authors/?q=ai:tseytlin.arkady-aSummary: We revisit the problem of matching the strong coupling expansion of the \(\frac{1}{2}\) BPS circular Wilson loops in \(\mathcal{N} = 4\) SYM and ABJM gauge theories with their string theory duals in \(\mathrm{AdS}_5 \times S^5\) and \(\mathrm{AdS}_4 \times CP^3\), at the first subleading (one-loop) order of the expansion around the minimal surface. We observe that, including the overall factor \(1/g_s\) of the inverse string coupling constant, as appropriate for the open string partition function with disk topology, and a universal prefactor proportional to the square root of the string tension \(T\), both the SYM and ABJM results precisely match the string theory prediction. We provide an explanation of the origin of the \(\sqrt{T}\) prefactor based on special features of the combination of one-loop determinants appearing in the string partition function. The latter also implies a natural generalization \(Z_\chi \sim ( \sqrt{T}/{g}_{s} )^\chi\) to higher genus contributions with the Euler number \(\chi\), which is consistent with the structure of the \(1/N\) corrections found on the gauge theory side.Energy gap for Yang-Mills connections. I: Four-dimensional closed Riemannian manifolds.https://www.zbmath.org/1456.580142021-04-16T16:22:00+00:00"Feehan, Paul M. N."https://www.zbmath.org/authors/?q=ai:feehan.paul-m-nSummary: We extend an \(L^2\) energy gap result due to \textit{M. Min-Oo} [Compos. Math. 47, 153--163 (1982; Zbl 0519.53042), Theorem 2] and \textit{T. H. Parker} [Commun. Math. Phys. 85, 563--602 (1982; Zbl 0502.53022), Proposition 2.2] for Yang-Mills connections on principal \(G\)-bundles, \(P\), over closed, connected, four-dimensional, oriented, smooth manifolds, \(X\), from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics that are generic and where the topologies of \(P\) and \(X\) obey certain mild conditions and the compact Lie group, \(G\), is \(\operatorname{SU}(2)\) or \(\operatorname{SO}(3)\).
[For Part II see Zbl 1375.58013.]Stochastic resonance in a fractional oscillator with random damping strength and random spring stiffness.https://www.zbmath.org/1456.700412021-04-16T16:22:00+00:00"He, Guitian"https://www.zbmath.org/authors/?q=ai:he.guitian"Tian, Yan"https://www.zbmath.org/authors/?q=ai:tian.yan"Wang, Yan"https://www.zbmath.org/authors/?q=ai:wang.yan.6|wang.yan|wang.yan.2|wang.yan.1|wang.yan.5|wang.yan.4|wang.yan.3One-loop non-planar anomalous dimensions in super Yang-Mills theory.https://www.zbmath.org/1456.814412021-04-16T16:22:00+00:00"McLoughlin, Tristan"https://www.zbmath.org/authors/?q=ai:mcloughlin.tristan"Pereira, Raul"https://www.zbmath.org/authors/?q=ai:pereira.raul"Spiering, Anne"https://www.zbmath.org/authors/?q=ai:spiering.anneSummary: We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading \(1/N^2\) corrections to operator dimensions and as an example compute the large \(R\)-charge limit for two-excitation states through subleading order in the \(R\)-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite \(N\) to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite \(N\).On a gravity dual to flavored topological quantum mechanics.https://www.zbmath.org/1456.830632021-04-16T16:22:00+00:00"Feldman, Andrey"https://www.zbmath.org/authors/?q=ai:feldman.andreySummary: In this paper, we propose a generalization of the \(\mathrm{AdS}_2/\mathrm{CFT}_1\) correspondence constructed by \textit{M. Mezei} in [``A 2d/1d holographic duality'', Preprint, \url{arXiv:1703.08749}], which is the duality between 2d Yang-Mills theory with higher derivatives in the \(\mathrm{AdS}_2\) background, and 1d topological quantum mechanics of two adjoint and two fundamental \(\mathrm{U}(N)\) fields, governing certain protected sector of operators in 3d ABJM theory at the Chern-Simons level \(k = 1\). We construct a holographic dual to a flavored generalization of the 1d quantum mechanics considered in [loc. cit.], which arises as the effective field theory living on the intersection of stacks of \(N\) D2-branes and \(k\) D6-branes in the \(\Omega\)-background in Type IIA string theory, and describes the dynamics of the protected sector of operators in \(\mathcal{N} = 4\) theory with \(k\) fundamental hypermultiplets, having a holographic description as M-theory in the \(\mathrm{AdS}_4 \times S^7/ \mathbb{Z}_k\) background. We compute the structure constants of the bulk theory gauge group, and construct a map between the observables of the boundary theory and the fields of the bulk theory.Building bases of loop integrands.https://www.zbmath.org/1456.813002021-04-16T16:22:00+00:00"Bourjaily, Jacob L."https://www.zbmath.org/authors/?q=ai:bourjaily.jacob-l"Herrmann, Enrico"https://www.zbmath.org/authors/?q=ai:herrmann.enrico"Langer, Cameron"https://www.zbmath.org/authors/?q=ai:langer.cameron-k"Trnka, Jaroslav"https://www.zbmath.org/authors/?q=ai:trnka.jaroslavSummary: We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of `power-counting' for multi-loop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all `leading-(transcendental-)weight' contributions of \textit{any} four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent \textit{all} scattering amplitude integrands in the theory --- for generic helicities and arbitrary multiplicity.New soliton solutions of anti-self-dual Yang-Mills equations.https://www.zbmath.org/1456.812362021-04-16T16:22:00+00:00"Hamanaka, Masashi"https://www.zbmath.org/authors/?q=ai:hamanaka.masashi"Huang, Shan-Chi"https://www.zbmath.org/authors/?q=ai:huang.shan-chiSummary: We study exact soliton solutions of anti-self-dual Yang-Mills equations for \(G = \mathrm{GL}(2)\) in four-dimensional spaces with the Euclidean, Minkowski and Ultrahyperbolic signatures and construct special kinds of one-soliton solutions whose action density \(\mathrm{Tr} F_{ \mu \nu }F^{ \mu \nu }\) can be real-valued. These solitons are shown to be new type of domain walls in four dimension by explicit calculation of the real-valued action density. Our results are successful applications of the Darboux transformation developed by Nimmo, Gilson and Ohta. More surprisingly, integration of these action densities over the four-dimensional spaces are suggested to be not infinity but zero. Furthermore, whether gauge group \(G = \mathrm{U}(2)\) can be realized on our solition solutions or not is also discussed on each real space.Emergent Yang-Mills theory.https://www.zbmath.org/1456.830202021-04-16T16:22:00+00:00"de Mello Koch, Robert"https://www.zbmath.org/authors/?q=ai:de-mello-koch.robert"Huang, Jia-Hui"https://www.zbmath.org/authors/?q=ai:huang.jiahui"Kim, Minkyoo"https://www.zbmath.org/authors/?q=ai:kim.minkyoo"Van Zyl, Hendrik J. R."https://www.zbmath.org/authors/?q=ai:van-zyl.hendrik-j-rSummary: We study the spectrum of anomalous dimensions of operators dual to giant graviton branes. The operators considered belong to the \(\mathrm{su} (2|3)\) sector of \(\mathcal{N} = 4\) super Yang-Mills theory, have a bare dimension \(\sim N\) and are a linear combination of restricted Schur polynomials with \(p \sim O(1)\) long rows or columns. In the same way that the operator mixing problem in the planar limit can be mapped to an integrable spin chain, we find that our problems maps to particles hopping on a lattice. The detailed form of the model is in precise agreement with the expected world volume dynamics of \(p\) giant graviton branes, which is a \(\mathrm{U} (p)\) Yang-Mills theory. The lattice model we find has a number of noteworthy features. It is a lattice model with all-to-all sites interactions and quenched disorder.Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation.https://www.zbmath.org/1456.830162021-04-16T16:22:00+00:00"Tanzi, Roberto"https://www.zbmath.org/authors/?q=ai:tanzi.roberto"Giulini, Domenico"https://www.zbmath.org/authors/?q=ai:giulini.domenico-j-wSummary: We investigate the asymptotic symmetry group of the free \(\mathrm{SU} (N)\)-Yang-Mills theory using the Hamiltonian formalism. We closely follow the strategy of Henneaux and Troessaert who successfully applied the Hamiltonian formalism to the case of gravity and electrodynamics, thereby deriving the respective asymptotic symmetry groups of these theories from clear-cut first principles. These principles include the minimal assumptions that are necessary to ensure the existence of Hamiltonian structures (phase space, symplectic form, differentiable Hamiltonian) and, in case of Poincaré invariant theories, a canonical action of the Poincaré group. In the first part of the paper we show how these requirements can be met in the non-abelian \(\mathrm{SU} (N)\)-Yang-Mills case by imposing suitable fall-off and parity conditions on the fields. We observe that these conditions admit neither non-trivial asymptotic symmetries nor non-zero global charges. In the second part of the paper we discuss possible gradual relaxations of these conditions by following the same strategy that Henneaux and Troessaert had employed to remedy a similar situation in the electromagnetic case. Contrary to our expectation and the findings of Henneaux and Troessaert for the abelian case, there seems to be no relaxation that meets the requirements of a Hamiltonian formalism \textit{and} allows for non-trivial asymptotic symmetries and charges. Non-trivial asymptotic symmetries and charges are only possible if either the Poincaré group fails to act canonically or if the formal expression for the symplectic form diverges, i.e. the form does not exist. This seems to hint at a kind of colour-confinement built into the classical Hamiltonian formulation of non-abelian gauge theories.\( T\overline{T} \)-deformation of \(q\)-Yang-Mills theory.https://www.zbmath.org/1456.830692021-04-16T16:22:00+00:00"Santilli, Leonardo"https://www.zbmath.org/authors/?q=ai:santilli.leonardo"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-j"Tierz, Miguel"https://www.zbmath.org/authors/?q=ai:tierz.miguelSummary: We derive the \(T\overline{T} \)-perturbed version of two-dimensional \(q\)-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the \(T\overline{T} \)-deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large \(N\) factorization into chiral and anti-chiral sectors. For the \( \mathrm{U} (N)\) gauge theory on the sphere, we show that the large \(N\) phase transition persists, and that it is of third order and induced by instantons. The effect of the \(T\overline{T} \)-deformation is to decrease the critical value of the 't Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for \( (q,t) \)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large \(N\) limit of Yang-Mills theory, showing that the \(T\overline{T} \)-deformation decreases the contribution of the Boltzmann entropy.Single particle operators and their correlators in free \(\mathcal{N} = 4\) SYM.https://www.zbmath.org/1456.814192021-04-16T16:22:00+00:00"Aprile, F."https://www.zbmath.org/authors/?q=ai:aprile.francesco"Drummond, J. M."https://www.zbmath.org/authors/?q=ai:drummond.james-m"Heslop, P."https://www.zbmath.org/authors/?q=ai:heslop.paul-j"Paul, H."https://www.zbmath.org/authors/?q=ai:paul.himadri-sekhar|paul.henning-a|paul.henrik|paul.h-g|paul.hynek|paul.harry"Sanfilippo, F."https://www.zbmath.org/authors/?q=ai:sanfilippo.francesco"Santagata, M."https://www.zbmath.org/authors/?q=ai:santagata.maria-c"Stewart, A."https://www.zbmath.org/authors/?q=ai:stewart.alastairSummary: We consider a set of half-BPS operators in \(\mathcal{N} = 4\) super Yang-Mills theory which are appropriate for describing single-particle states of superstring theory on \( \mathrm{AdS}_5 \times S^5\). These single-particle operators are defined to have vanishing two-point functions with all multi-trace operators and therefore correspond to admixtures of single- and multi-traces. We find explicit formulae for all single-particle operators and for their two-point function normalisation. We show that single-particle \( \mathrm{U}(N)\) operators belong to the \( \mathrm{SU} (N)\) subspace, thus for length greater than one they are simply the \( \mathrm{SU} (N)\) single-particle operators. Then, we point out that at large \(N\), as the length of the operator increases, the single-particle operator naturally interpolates between the single-trace and the \(S^3\) giant graviton. At finite \(N\), the multi-particle basis, obtained by taking products of the single-particle operators, gives a new basis for all half-BPS states, and this new basis naturally cuts off when the length of any of the single-particle operators exceeds the number of colours. From the two-point function orthogonality we prove a multipoint orthogonality theorem which implies vanishing of all near-extremal correlators. We then compute all maximally and next-to-maximally extremal free correlators, and we discuss features of the correlators when the extremality is lowered. Finally, we describe a half-BPS projection of the operator product expansion on the multi-particle basis which provides an alternative construction of four- and higher-point functions in the free theory.Giant Wilson loops and \( \mathrm{AdS}_2/ \mathrm{dCFT}_1\).https://www.zbmath.org/1456.814332021-04-16T16:22:00+00:00"Giombi, Simone"https://www.zbmath.org/authors/?q=ai:giombi.simone"Jiang, Jiaqi"https://www.zbmath.org/authors/?q=ai:jiang.jiaqi"Komatsu, Shota"https://www.zbmath.org/authors/?q=ai:komatsu.shotaSummary: The 1/2-BPS Wilson loop in \(\mathcal{N} = 4\) supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with \( \mathrm{AdS}_2 \times S^2\) and \( \mathrm{AdS}_2 \times S^4\) worldvolume geometries, ending at the \( \mathrm{ AdS}_5\) boundary along a one-dimensional contour. We first compute the correlation functions of protected scalar insertions from supersymmetric localization, and obtain a representation in terms of multiple integrals that are similar to the eigenvalue integrals of the random matrix, but with important differences. Using ideas from the Fermi Gas formalism and the Clustering method, we evaluate their large \(N\) limit exactly as a function of the 't Hooft coupling. The results are given by simple integrals of polynomials that resemble the \(Q\)-functions of the Quantum Spectral Curve, with integration measures depending on the number of insertions. Next, we study the correlation functions of fluctuations on the probe D3 and D5 branes in AdS. We compute a selection of three- and four-point functions from perturbation theory on the D-branes, and show that they agree with the results of localization when restricted to supersymmetric kinematics. We also explain how the difference of the internal geometries of the D3 and D5 branes manifests itself in the localization computation.Weights, recursion relations and projective triangulations for positive geometry of scalar theories.https://www.zbmath.org/1456.814542021-04-16T16:22:00+00:00"John, Renjan Rajan"https://www.zbmath.org/authors/?q=ai:john.renjan-rajan"Kojima, Ryota"https://www.zbmath.org/authors/?q=ai:kojima.ryota"Mahato, Sujoy"https://www.zbmath.org/authors/?q=ai:mahato.sujoySummary: The story of positive geometry of massless scalar theories was pioneered in [\textit{N. Arkani-Hamed} et al., J. High Energy Phys. 2018, No. 5, Paper No. 96, 78 p. (2018; Zbl 1391.81200)] in the context of bi-adjoint \(\varphi^3\) theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial interaction is a class of polytopes called accordiohedra. Tree-level planar scattering amplitudes of the theory can be obtained from a weighted sum of the canonical forms of the accordiohedra. In this paper, using results of the recent work [\textit{R. Kojima}, J. High Energy Phys. 2020, No. 8, Paper No. 54, 34 p. (2020; Zbl 1454.81236)], we show that in theories with polynomial interactions all the weights can be determined from the factorization property of the accordiohedron. We also extend the projective recursion relations to these theories. We then give a detailed analysis of how the recursion relations in \(\varphi^p\) theories and theories with polynomial interaction correspond to projective triangulations of accordiohedra. Following recent development we also extend our analysis to one-loop integrands in the quartic theory.Lifting heptagon symbols to functions.https://www.zbmath.org/1456.814282021-04-16T16:22:00+00:00"Dixon, Lance J."https://www.zbmath.org/authors/?q=ai:dixon.lance-j"Liu, Yu-Ting"https://www.zbmath.org/authors/?q=ai:liu.yutingSummary: Seven-point amplitudes in planar \(\mathcal{N} = 4\) super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.On the relevance of inertia related terms in the equations of motion of a flexible body in the floating frame of reference formulation.https://www.zbmath.org/1456.700192021-04-16T16:22:00+00:00"Witteveen, Wolfgang"https://www.zbmath.org/authors/?q=ai:witteveen.wolfgang"Pichler, Florian"https://www.zbmath.org/authors/?q=ai:pichler.florianSummary: The floating frame of reference formulation is an established method for the description of linear elastic bodies within multibody dynamics. An exact derivation leads to rather complex equations of motion. In order to reduce the computational burden, it is common to neglect certain terms. In the literature this is done by strict application of the small deformation assumption to the kinetic energy. This leads to a remarkably simplified set of equations. In this work, the significance of all terms is investigated at the level of the equations of motion. It is shown that for a large number of applications the previously mentioned set of simple equations is sufficient. Furthermore, scenarios are described in which this simple set is no longer accurate enough. Finally, guidelines are provided, so that engineers can decide which terms should be considered or not. The theoretical conclusions drawn in this work are underlined by qualitative numerical investigations.The Hamilton-Jacobi equation and holographic renormalization group flows on sphere.https://www.zbmath.org/1456.830822021-04-16T16:22:00+00:00"Kim, Nakwoo"https://www.zbmath.org/authors/?q=ai:kim.nakwoo"Kim, Se-Jin"https://www.zbmath.org/authors/?q=ai:kim.se-jinSummary: We study the Hamilton-Jacobi formulation of effective mechanical actions associated with holographic renormalization group flows when the field theory is put on the sphere and mass terms are turned on. Although the system is supersymmetric and it is described by a superpotential, Hamilton's characteristic function is not readily given by the superpotential when the boundary of AdS is curved. We propose a method to construct the solution as a series expansion in scalar field degrees of freedom. The coefficients are functions of the warp factor to be determined by a differential equation one obtains when the ansatz is substituted into the Hamilton-Jacobi equation. We also show how the solution can be derived from the BPS equations without having to solve differential equations. The characteristic function readily provides information on holographic counterterms which cancel divergences of the on-shell action near the boundary of AdS.Herman's approach to quasi-periodic perturbations in the reversible KAM context 2.https://www.zbmath.org/1456.700382021-04-16T16:22:00+00:00"Sevryuk, Mikhail B."https://www.zbmath.org/authors/?q=ai:sevryuk.mikhail-borisovichSummary: We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question.Variable order fractional permanent magnet synchronous motor: dynamical analysis and numerical simulation.https://www.zbmath.org/1456.340582021-04-16T16:22:00+00:00"Zahra, Waheed K."https://www.zbmath.org/authors/?q=ai:zahra.waheed-k"Hikal, M. M."https://www.zbmath.org/authors/?q=ai:hikal.m-mSummary: In this paper, the variable order fractional permanent magnet synchronous motor (VOFPMSM) is investigated. Conditions for existence and uniqueness of the solution of the VOFPMSM are proposed. The stability behavior of the system's equilibrium points along with the variation of the motor parameters and the order of differentiation is discussed. Sufficient conditions that guarantee the asymptotic stability of each of the equilibrium points of the system are established. Also, the required conditions that give the effect of Hopf bifurcation of the system are established in terms of the system parameters and the order of differentiation and consequently the appearance of the chaotic behavior of the VOFPMSM. New numerical techniques based on the modified backward Euler's schemes for continuous and discontinuous variable order fractional model are presented. The obtained numerical results demonstrate the merits of the proposed method and the variable order fractional permanent magnet synchronous motor over the fractional permanent magnet synchronous motor.Comments on D3-brane holography.https://www.zbmath.org/1456.830732021-04-16T16:22:00+00:00"Chakraborty, Soumangsu"https://www.zbmath.org/authors/?q=ai:chakraborty.soumangsu"Giveon, Amit"https://www.zbmath.org/authors/?q=ai:giveon.amit"Kutasov, David"https://www.zbmath.org/authors/?q=ai:kutasov.davidSummary: We revisit the idea that the quantum dynamics of open strings ending on \(N\) D3-branes in the large \(N\) limit can be described at large `t Hooft coupling by classical closed string theory in the background created by the D3-branes in asymptotically flat spacetime. We study the resulting thermodynamics and compute the Hagedorn temperature and other properties of the D3-brane worldvolume theory in this regime. We also consider the theory in which the D3-branes are replaced by negative branes and show that its thermodynamics is well behaved. We comment on the idea that this theory can be thought of as an irrelevant deformation of \(\mathcal{N} = 4\) SYM, and on its relation to \(T\overline{T}\) deformed \( \mathrm{CFT}_2\).From Hagedorn to Lee-Yang: partition functions of \(\mathcal{N} = 4\) SYM theory at finite \(N\).https://www.zbmath.org/1456.814372021-04-16T16:22:00+00:00"Kristensson, Alexander T."https://www.zbmath.org/authors/?q=ai:kristensson.alexander-t"Wilhelm, Matthias"https://www.zbmath.org/authors/?q=ai:wilhelm.matthiasSummary: We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group \(\mathrm{U}(N\)) on \(\mathbb{R} \times S^3\), dual to type IIB superstring theory on \(\mathrm{AdS}_5 \times S^5\). While both theories are well-known to exhibit Hagedorn behavior at infinite \(N\), we find evidence that this is replaced by Lee-Yang behavior at large but finite \(N\): the zeros of the partition function condense into two arcs in the complex temperature plane that pinch the real axis at the temperature of the confinement-deconfinement transition. Concretely, we demonstrate this for the free theory via exact calculations of the (unrefined and refined) partition functions at \(N \leq 7\) for the \(\mathfrak{su} (2)\) sector containing two complex scalars, as well as at \(N \leq 5 \) for the \(\mathfrak{su} (2|3)\) sector containing 3 complex scalars and 2 fermions. In order to obtain these explicit results, we use a Molien-Weyl formula for arbitrary field content, utilizing the equivalence of the partition function with what is known to mathematicians as the Poincaré series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.Chaos from massive deformations of Yang-Mills matrix models.https://www.zbmath.org/1456.813302021-04-16T16:22:00+00:00"Başkan, K."https://www.zbmath.org/authors/?q=ai:baskan.k"Kürkçüoğlu, S."https://www.zbmath.org/authors/?q=ai:kurkcuoglu.seckin-kin"Oktay, O."https://www.zbmath.org/authors/?q=ai:oktay.onur"Taşcı, C."https://www.zbmath.org/authors/?q=ai:tasci.cSummary: We focus on an \(\mathrm{SU} (N)\) Yang-Mills gauge theory in 0 + 1-dimensions with the same matrix content as the bosonic part of the BFSS matrix model, but with mass deformation terms breaking the global SO(9) symmetry of the latter to \(\mathrm{SO} (5) \times \mathrm{SO}(3) \times \mathbb{Z}_2\). Introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we examine the chaotic dynamics in a family of effective Lagrangians obtained by tracing over the aforementioned ansatz configurations at the matrix levels \(N=\frac{1}{6} (n + 1)(n + 2)(n + 3)\), for \(n = 1, 2, \dots , 7\). Through numerical work, we determine the Lyapunov spectrum and analyze how the largest Lyapunov exponents (LLE) change as a function of the energy, and discuss how our results can be used to model the temperature dependence of the LLEs and put upper bounds on the temperature above which LLE values comply with the Maldacena-Shenker-Stanford (MSS) bound \(2 \pi T\), and below which it will eventually be violated.Modular invariance in superstring theory from \(\mathcal{N} = 4\) super-Yang-Mills.https://www.zbmath.org/1456.814242021-04-16T16:22:00+00:00"Chester, Shai M."https://www.zbmath.org/authors/?q=ai:chester.shai-m"Green, Michael B."https://www.zbmath.org/authors/?q=ai:green.michael-b"Pufu, Silviu S."https://www.zbmath.org/authors/?q=ai:pufu.silviu-s"Wang, Yifan"https://www.zbmath.org/authors/?q=ai:wang.yifan"Wen, Congkao"https://www.zbmath.org/authors/?q=ai:wen.congkaoSummary: We study the four-point function of the lowest-lying half-BPS operators in the \(\mathcal{N} = 4\) \( \mathrm{SU} (N)\) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-\(N\) expansion in which the complexified Yang-Mills coupling \(\tau\) is fixed. In this expansion, non-perturbative instanton contributions are present, and the \( \mathrm{SL} (2, \mathbb{Z})\) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed \(\mathcal{N} = 4\) SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to \(N^2- 1\) and are independent of \(\tau\) and \(\overline{\tau} \), we find that the terms of order \(\sqrt{N}\) and \(1/\sqrt{N}\) in the large \(N\) expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series \(E\left(\frac{3}{2},\tau, \overline{\tau}\right)\) and \(E\left(\frac{5}{2},\tau, \overline{\tau}\right) \), respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from \(R^4\) and \(D^4R^4\) contact interactions, which, for the \(R^4\) case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order \({N}^{\frac{1}{2}-m}\) with integer \(m \geq 0 \), the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly \( \mathrm{SL} (2, \mathbb{Z})\) invariant.Structure and stability of the rhombus family of relative equilibria under general homogeneous forces.https://www.zbmath.org/1456.700272021-04-16T16:22:00+00:00"Leandro, Eduardo S. G."https://www.zbmath.org/authors/?q=ai:leandro.eduardo-s-gSummary: Let \(N>2\) and \(n>1\). Among the classes of symmetric relative equilibria of the \(N\)-body problem whose symmetry group is one of the dihedral groups \(D_n\), the rhombus family stands out as the only noncollinear family for which the symmetry implies that the stability polynomial is fully factorizable. The present paper discusses basic properties and linear stability of the rhombus family assuming the forces of interaction depend on the mutual distances raised to an arbitrary real exponent \(2a+1\). In a suitable parameter plane, the family of rhombus relative equilibria forms a pincel of graphs which foliates the union of an open unit square and an open rectangle obtained from the unit square by a reflection and an inversion. We show that all rhombus relative equilibria are linearly stable if \(a>-1\), that they are all unstable for \(a\) in the interval bound by \(-4-2\sqrt{2}\approx -6.82\) and \(4(\sqrt{3}-2)\approx -1.07\), and that stability and instability depend on mass values for the remaining values of \(a\). These results impose limitations on the validity of Moeckel's dominant mass stability conjecture in the context of generalized \(N\)-body problems.Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order.https://www.zbmath.org/1456.352212021-04-16T16:22:00+00:00"Singla, Komal"https://www.zbmath.org/authors/?q=ai:singla.komal"Rana, M."https://www.zbmath.org/authors/?q=ai:rana.mehwish|rana.meenakshiSummary: The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup-Boussinesq system with \(l=2\) by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness.A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann's translated curve theorem to higher dimensions.https://www.zbmath.org/1456.370662021-04-16T16:22:00+00:00"Massetti, Jessica Elisa"https://www.zbmath.org/authors/?q=ai:massetti.jessica-elisaSummary: We prove a discrete time analogue of \textit{J. Moser}'s normal form [Math. Ann. 169, 136--176 (1967; Zbl 0149.29903)] of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant torus is a phenomenon of finite codimension. Under convenient nondegeneracy assumptions on the diffeomorphisms under study (a torsion property for example), this codimension can be reduced. As a by-product we obtain generalizations of Rüssmann's translated curve theorem in any dimension, by a technique of elimination of parameters.Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints.https://www.zbmath.org/1456.352362021-04-16T16:22:00+00:00"Safdari, Mohammad"https://www.zbmath.org/authors/?q=ai:safdari.mohammadSummary: We prove the optimal \(W^{2, \infty}\) regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be \(C^1\) or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal \(W^{2, \infty}\) regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.Nonlinear transition dynamics in a time-delayed vibration isolator under combined harmonic and stochastic excitations.https://www.zbmath.org/1456.700372021-04-16T16:22:00+00:00"Yang, Tao"https://www.zbmath.org/authors/?q=ai:yang.tao"Cao, Qingjie"https://www.zbmath.org/authors/?q=ai:cao.qingjieTopology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)Affine structures on Lie groupoids.https://www.zbmath.org/1456.220012021-04-16T16:22:00+00:00"Lang, Honglei"https://www.zbmath.org/authors/?q=ai:lang.honglei"Liu, Zhangju"https://www.zbmath.org/authors/?q=ai:liu.zhangju"Sheng, Yunhe"https://www.zbmath.org/authors/?q=ai:sheng.yunheAuthors' abstract: We study affine structures on a Lie groupoid, including affine \(k\)-vector fields, \(k\)-forms and \((p, q)\)-tensors. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher-Nijenhuis bracket are graded strict Lie 2-algebras, and affine \((1, 1)\)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
Reviewer: Iakovos Androulidakis (Athína)Quasiperiodic Hamiltonian motions, scale invariance, harmonic oscillators.https://www.zbmath.org/1456.700322021-04-16T16:22:00+00:00"Gallavotti, Giovanni"https://www.zbmath.org/authors/?q=ai:gallavotti.giovanni-mSummary: The work of Kolmogorov, Arnold, and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by \textit{ K. G. Wilson} [Phys. Rev. B (3) 4, No. 9, 3174--3183 (1971; Zbl 1236.82017)]: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series [\textit{ L. Carleson}, Acta Math. 116, 135--157 (1966), 135--157 (1966; Zbl 0144.06402)] and C. Fefferman, Ann. Math. (2) 98, 551--571 (1973; Zbl 0268.42009); erratum ibid. 146, 239 (1997)] or construction of Euclidean quantum fields [E. Nelson, J. Math. Phys. 5, 1190--1197 (1964)] or the scaling analysis of the short scale behavior of Navier-Stokes fluids [\textit{ L. Caffarelli, R. Kohn}, and \textit{ L. Nirenberg}, Commun. Pure Appl. Math. 35, 771--831 (1982; Zbl 0509.35067)] to name a -- few, which resulted in numerous problems. In this review, the KAM theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a ``trivial'' fixed point.{\par\copyright 2019 American Institute of Physics}Sustained rotation in a vibrated disk with asymmetric supports.https://www.zbmath.org/1456.700312021-04-16T16:22:00+00:00"Peraza-Mues, Gonzalo G."https://www.zbmath.org/authors/?q=ai:peraza-mues.gonzalo-g"Moukarzel, Cristian F."https://www.zbmath.org/authors/?q=ai:moukarzel.cristian-fStochastic resonance in a fractional oscillator driven by multiplicative quadratic noise.https://www.zbmath.org/1456.700362021-04-16T16:22:00+00:00"Ren, Ruibin"https://www.zbmath.org/authors/?q=ai:ren.ruibin"Luo, Maokang"https://www.zbmath.org/authors/?q=ai:luo.maokang"Deng, Ke"https://www.zbmath.org/authors/?q=ai:deng.keSpace robot motion planning in the presence of nonconserved linear and angular momenta.https://www.zbmath.org/1456.700202021-04-16T16:22:00+00:00"Basmadji, Fatina Liliana"https://www.zbmath.org/authors/?q=ai:basmadji.fatina-liliana"Seweryn, Karol"https://www.zbmath.org/authors/?q=ai:seweryn.karol"Sasiadek, Jurek Z."https://www.zbmath.org/authors/?q=ai:sasiadek.jurek-zSummary: On-orbit servicing, active debris removal or assembling large structures on orbit are only some of the tasks that could be accomplished by space robots. In all these cases, a contact between a space robot and the satellite being serviced, deorbited, or assembled will occur. This contact results in a contact force exerted on the space robot, and therefore momenta of the space robot system are no longer conserved. Most of the papers that are concerned with motion planning problems of a space robot manipulator either consider that no external forces or moments are acting on the space robot system or use additional controllers when the space robot is subjected to external forces and moments. Such a controller minimizes end-effector position and orientation errors caused by the changes in system momenta due to external forces and moments acting on this system. The novelty of this work is that it proposes a new method for planning the motion of dual-arm space robot manipulators when linear and angular momenta of the space robot system are not conserved due to external forces and moments acting on the space robot base or/and manipulators' end-effectors. In the proposed method the changes in system momenta are considered, but no additional controllers are needed. In this paper, we derive the motion planning equations for dual-arm space robot manipulators, where external forces and moments are acting on both satellite and manipulator end-effectors. The proposed method has been verified by numerical simulations, and the results are presented and discussed.Hunting stability analysis of partially filled tank wagon on curved track using coupled CFD-MBD method.https://www.zbmath.org/1456.700092021-04-16T16:22:00+00:00"Rahmati-Alaei, Ahmad"https://www.zbmath.org/authors/?q=ai:rahmati-alaei.ahmad"Sharavi, Majid"https://www.zbmath.org/authors/?q=ai:sharavi.majid"Samadian Zakaria, Masoud"https://www.zbmath.org/authors/?q=ai:samadian-zakaria.masoudSummary: In this study, we develop an innovative numerical method for the investigation of the stability of a partially filled tank wagon moving on a curved track. The calculations are carried out in two subsystems including a dynamic system and fluid sloshing. We analyze the wagon dynamic system the multibody dynamic (MBD) model with 21 degrees of freedom (21-DOFs), which takes into account the lateral, vertical, roll, pitch, and yaw motions. The heuristic creep theory is used for the wheel-rail contact model. We adopt the fourth-order Runge-Kutta method for solving of this model. The transient fluid slosh is simulated by the computational fluid dynamic (CFD) model. The volume of fluid (VOF) technique is used for tracking the free surface of the fluid. This model is validated experimentally using the sloshing test setup. Then the simultaneous interaction between the dynamic system and the transient fluid slosh is analyzed by coupling the CFD model with the MBD model. By the parametric study on the filled-volume and wagon velocity, the critical hunting speed is derived by the Lyapunov indirect method. The results show that a higher filled volume decreases the critical hunting speed. Also, at the instability condition, an increasing trend for the phase trajectory of the wagon components is evident.Trajectory planning of a redundant planar manipulator based on joint classification and particle swarm optimization algorithm.https://www.zbmath.org/1456.700062021-04-16T16:22:00+00:00"Yu, Luchuan"https://www.zbmath.org/authors/?q=ai:yu.luchuan"Wang, Kaiqiang"https://www.zbmath.org/authors/?q=ai:wang.kaiqiang"Zhang, Qinhe"https://www.zbmath.org/authors/?q=ai:zhang.qinhe"Zhang, Jianhua"https://www.zbmath.org/authors/?q=ai:zhang.jianhua.1|zhang.jianhuaSummary: This paper presents a general method for the trajectory planning of the redundant planar manipulator. The mathematical relation between joint space and Cartesian space in a two-dimensional space is first derived. The joint classification is employed to obtain the solutions of corresponding joints. It divides joints into class I redundant joints, class II redundant joints, and nonredundant joints. The new application of knot points in the quintic B-spline curve is introduced to generate inverse solutions of class I redundant joints. Examples show that the number and distribution of knot points have a large effect on their solutions. Moreover, the particle swarm optimization algorithm is extended to generate solutions of class II redundant joints. It also optimizes the initial trajectories of joints and end-effector. Finally, solutions of nonredundant joints can be generated by the derived relation between the joint space and Cartesian space. The proposed methodology is confirmed by a case study. Under the same conditions, results show that the solution obtained by the extended method is not only better than that obtained by the particle swarm optimization algorithm but also closer to the global optimal solution.Dynamic modeling for foldable origami space membrane structure with contact-impact during deployment.https://www.zbmath.org/1456.700052021-04-16T16:22:00+00:00"Yuan, Tingting"https://www.zbmath.org/authors/?q=ai:yuan.tingting"Liu, Zhuyong"https://www.zbmath.org/authors/?q=ai:liu.zhuyong"Zhou, Yuhang"https://www.zbmath.org/authors/?q=ai:zhou.yuhang"Liu, Jinyang"https://www.zbmath.org/authors/?q=ai:liu.jinyangSummary: The dynamic modeling for the foldable origami space membrane structure considering contact-impact during the deployment is studied in this paper. The membrane is discretized using the triangular elements of the Absolute Nodal Coordinate Formulation (ANCF), and the stress-strain relationship of the membrane is determined based on the Stiffness Reduction Model (SRM). A mixed method is proposed for the frictionless contact problem by combining the membrane surface-to-surface (STS) contact elements with the membrane node-to-surface (NTS) contact elements to improve precision. Compared with the traditional STS contact elements, the mixed method can effectively avoid mutual penetration of the element boundaries, especially for the foldable origami membrane structures undergoing overall motions. The penalty method is adopted to enforce the nonpenetration condition. Moreover, special constraints are built for the fold lines, and then the dynamic equations of the membrane multibody system considering the damping effect are formulated. The dynamic deployment procedure of a leaf-in origami membrane structure with contact-impact is performed employing this present mixed method. The results demonstrate the effectiveness and superiority of the presented mixed method in the solution of the complicated contact problem, and the influence of the contact-impact on the dynamic performance is analyzed.A compact form dynamics controller for a high-DOF tetrapod-on-wheel robot with one manipulator via null space based convex optimization and compatible impedance controllers.https://www.zbmath.org/1456.700212021-04-16T16:22:00+00:00"Du, Wenqian"https://www.zbmath.org/authors/?q=ai:du.wenqian"Benamar, Faïz"https://www.zbmath.org/authors/?q=ai:benamar.faizSummary: This paper develops a compact form dynamics controller to generate multi-compliant behaviors for a new designed tetrapod-on-wheel robot with one manipulator. The whole-body compliant torque controller is stated through one null-space-based convex optimization and compatible null-space-based impedance controllers. Different from fixed contact points of conventional quadruped robots, the kinematic wheel contact constraints are derived for our legged-on-wheel robot, which serves as the basis for each task reference extraction and each compliance controller. The compact relationships between task references and optimization control variables are extracted using null-space-based inverse dynamics, which is used to build the cost function in the operational space and/or in the joint space. The whole-body control frame is developed and several null-space-based feed-back impedance controllers are integrated into the compact relationships to allow the robot to achieve compliance and compensate the model impreciseness, especially the wheel contact model. Then the detailed algorithm is presented whose output combines the feed-forward and feedback torque. The validation of our approach is performed via advanced numerical simulations for a virtual legged-on-wheel robot with one manipulator.Active isotropic compliance in redundant manipulators.https://www.zbmath.org/1456.700232021-04-16T16:22:00+00:00"Verotti, Matteo"https://www.zbmath.org/authors/?q=ai:verotti.matteo"Masarati, Pierangelo"https://www.zbmath.org/authors/?q=ai:masarati.pierangelo"Morandini, Marco"https://www.zbmath.org/authors/?q=ai:morandini.marco"Belfiore, Nicola P."https://www.zbmath.org/authors/?q=ai:belfiore.nicola-pioSummary: The isotropic compliance property is examined in the Special Euclidean Group \(\mathrm{SE}(3)\) in the case of redundant manipulators. The redundancy problem is solved by means of the QR decomposition of the transposed Jacobian matrix, and the compliance property is achieved by means of active stiffness regulation. Thanks to the defined control matrices, the control system realizes the isotropy condition. The local optimization of the joint torques is discussed. In particular, the joint control torques work is minimized obtaining an analytic solution through a Lyapunov equation. The proposed approach is applied to a 7R and to a 9R serial manipulator, and verified by means of multibody dynamics simulations.Adjoint sensitivity analysis of hybrid multibody dynamical systems.https://www.zbmath.org/1456.700112021-04-16T16:22:00+00:00"Corner, Sebastien"https://www.zbmath.org/authors/?q=ai:corner.sebastien"Sandu, Adrian"https://www.zbmath.org/authors/?q=ai:sandu.adrian"Sandu, Corina"https://www.zbmath.org/authors/?q=ai:sandu.corinaSummary: Sensitivity analysis computes the derivatives of general cost functions that depend on the system solution with respect to parameters or initial conditions. This work develops adjoint sensitivity analysis for hybrid multibody dynamic systems. The adjoint sensitivity is commonly referred to as backward propagation. Hybrid systems are characterized by trajectories that are piecewise continuous in time, with finitely-many discontinuities being caused by events such as elastic/inelastic impacts or sudden changes in constraints. The corresponding direct and adjoint sensitivity variables are also discontinuous at the time of events. The framework discussed herein uses a jump sensitivity matrix to relate the jump conditions for the direct and adjoint sensitivities before and after the time event and provides analytical jump equations for the adjoint variables. The theoretical framework for sensitivities for hybrid systems is verified on a five-bar mechanism with non-smooth contacts.A Kane's based algorithm for closed-form dynamic analysis of a new design of a 3RSS-S spherical parallel manipulator.https://www.zbmath.org/1456.700222021-04-16T16:22:00+00:00"Enferadi, Javad"https://www.zbmath.org/authors/?q=ai:enferadi.javad"Jafari, Keyvan"https://www.zbmath.org/authors/?q=ai:jafari.keyvanSummary: This paper proposes a systematic methodology to obtain a closed-form formulation for dynamics analysis of a new design of a fully spherical robot that is called a 3(RSS)-S parallel manipulator with real co-axial actuated shafts. The proposed robot can completely rotate about a vertical axis and can be used in celestial orientation and rehabilitation applications. After describing the robot and its inverse position, velocity and acceleration analysis is performed. Next, based on Kane's method, a methodology for deriving the dynamical equations of motion is developed. The elaborated approach shows that the inverse dynamics of the manipulator can be reduced to solving a system of three linear equations in three unknowns. Finally, a computational algorithm to solve the inverse dynamics of the manipulator is advised and several trajectories of the moving platform are simulated.Model-based vibration control for optical lenses.https://www.zbmath.org/1456.700332021-04-16T16:22:00+00:00"Störkle, Johannes"https://www.zbmath.org/authors/?q=ai:storkle.johannes"Eberhard, Peter"https://www.zbmath.org/authors/?q=ai:eberhard.peterSummary: This work presents a contribution to the active image stabilization of optical systems, involving model development, control design, and the hardware setup. A laboratory experiment is built, which demonstrates the vibration sensitivity of a mechanical-optical system. In order to stabilize the undesired image motion actively, a model-based compensation of the image vibration is developed, realized and tested. Beside a linear actuator motion system, a force sensor system and a position sensor system are introduced and analyzed. In particular, various low-cost hardware components of the Arduino platform are used, which support the deployment of the controller software based on Matlab-Simulink. The remaining image motion is measured with a high-speed vision sensor system and the performance of the overall system is assessed.Series expansions of the deflection angle in the scattering problem for power-law potentials.https://www.zbmath.org/1456.700302021-04-16T16:22:00+00:00"Chiron, D."https://www.zbmath.org/authors/?q=ai:chiron.david"Marcos, B."https://www.zbmath.org/authors/?q=ai:marcos.bruno|marcos.bernardSummary: We present a rigorous study of the classical scattering for any two-body interparticle potential of the form $v(r)=g/r^{\gamma}$, with $\gamma >0$, for repulsive $(g>0)$ and attractive $(g<0)$ interactions. We first derive an explicit series expansion of the deflection angle in the impact factor $b$. Then, we study carefully the modifications of the results when a regularization (softening) is introduced in the potential at small scales. We check and illustrate all the results with the exact integration of the equations of motion.{\par\copyright 2019 American Institute of Physics}A variational flexible multibody formulation for partitioned fluid-structure interaction: application to bat-inspired drones and unmanned air-vehicles.https://www.zbmath.org/1456.740332021-04-16T16:22:00+00:00"Joshi, Vaibhav"https://www.zbmath.org/authors/?q=ai:joshi.vaibhav"Jaiman, Rajeev K."https://www.zbmath.org/authors/?q=ai:jaiman.rajeev-kumar"Ollivier-Gooch, Carl"https://www.zbmath.org/authors/?q=ai:ollivier-gooch.carl-fSummary: We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier-Stokes system is discretized using a stabilized Petrov-Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells along with various types of joints and connections among them. A co-rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component. This assumption simplifies the structural equations and allows for the incorporation of multiple bodies (rigid as well as flexible) in the system. The displacement and rotation constraints at the joints are imposed by a Lagrange multiplier method. The equilibrium conditions at the fluid-structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach, a scattered data interpolation technique, which is globally conservative. For the coupled stability in low structure-to-fluid mass ratio regimes, a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor-corrector scheme. The convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non-matching fluid-structure interface where a third-order of convergence is observed. The proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge. Finally, we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints.A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation.https://www.zbmath.org/1456.350872021-04-16T16:22:00+00:00"Khalique, Chaudry Masood"https://www.zbmath.org/authors/?q=ai:khalique.chaudry-masood"Moleleki, Letlhogonolo Daddy"https://www.zbmath.org/authors/?q=ai:moleleki.letlhogonolo-daddySummary: This paper aims to study a generalized first extended (3+1)-dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \(({G'}/{G})\)-expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.Lower semicontinuity for functionals defined on piecewise rigid functions and on \(GSBD\).https://www.zbmath.org/1456.490132021-04-16T16:22:00+00:00"Friedrich, Manuel"https://www.zbmath.org/authors/?q=ai:friedrich.manuel"Perugini, Matteo"https://www.zbmath.org/authors/?q=ai:perugini.matteo"Solombrino, Francesco"https://www.zbmath.org/authors/?q=ai:solombrino.francescoSummary: In this work, we provide a characterization result for lower semicontinuity of surface energies defined on piecewise rigid functions, i.e., functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is a skew symmetric matrix. This characterization is achieved by means of an integral condition, called \(BD\)-ellipticity, which is in the spirit of \(BV\)-ellipticity defined by Ambrosio and Braides [5]. By specific examples we show that this novel concept is in fact stronger compared to its \(BV\) analog. We further provide a sufficient condition implying \(BD\)-ellipticity which we call symmetric joint convexity. This notion can be checked explicitly for certain classes of surface energies which are relevant for applications, e.g., for variational fracture models. Finally, we give a direct proof that surface energies with symmetric jointly convex integrands are lower semicontinuous also on the larger space of \(GSBD^p\) functions.Perturbative linearization of supersymmetric Yang-Mills theory.https://www.zbmath.org/1456.814182021-04-16T16:22:00+00:00"Ananth, Sudarshan"https://www.zbmath.org/authors/?q=ai:ananth.sudarshan"Lechtenfeld, Olaf"https://www.zbmath.org/authors/?q=ai:lechtenfeld.olaf"Malcha, Hannes"https://www.zbmath.org/authors/?q=ai:malcha.hannes"Nicolai, Hermann"https://www.zbmath.org/authors/?q=ai:nicolai.hermann"Pandey, Chetan"https://www.zbmath.org/authors/?q=ai:pandey.chetan"Pant, Saurabh"https://www.zbmath.org/authors/?q=ai:pant.saurabhSummary: Supersymmetric gauge theories are characterized by the existence of a transformation of the bosonic fields (Nicolai map) such that the Jacobi determinant of the transformation equals the product of the Matthews-Salam-Seiler and Faddeev-Popov determinants. This transformation had been worked out to second order in the coupling constant. In this paper, we extend this result (and the framework itself ) to third order in the coupling constant. A diagrammatic approach in terms of tree diagrams, aiming to extend this map to arbitrary orders, is outlined. This formalism bypasses entirely the use of anti-commuting variables, as well as issues concerning the (non-)existence of off-shell formulations for these theories. It thus offers a fresh perspective on supersymmetric gauge theories and, in particular, the ubiquitous \(\mathcal{N} = 4\) theory.Strictly convex central configurations of the planar five-body problem.https://www.zbmath.org/1456.700262021-04-16T16:22:00+00:00"Chen, Kuo-Chang"https://www.zbmath.org/authors/?q=ai:chen.kuo-chang"Hsiao, Jun-Shian"https://www.zbmath.org/authors/?q=ai:hsiao.jun-shianSummary: In this paper we investigate strictly convex central configurations of the planar five-body problem, and prove some necessary conditions for such configurations. In particular, given such a central configuration with multiplier \( \lambda \) and total mass \( M\), we show that all exterior edges are less than \( r_0=(M/\lambda )^{1/3}\), at most two interior edges are less than or equal to \( r_0\), and its subsystem with four masses cannot be a central configuration. We also obtain some other necessary conditions for strictly convex central configurations with five bodies, and show numerical examples of strictly convex central configurations with five bodies that have either one or two interior edges less than or equal to \( r_0\). Our work develops some formulae in a classic work by \textit{W. L. Williams} in 1938 [Trans. Am. Math. Soc. 44, 563--579 (1938; Zbl 0020.31901; JFM 64.0805.01)] and we rectify some unsupported assumptions there.Synchronisation analysis of a de-tuned three-bladed rotor.https://www.zbmath.org/1456.700392021-04-16T16:22:00+00:00"Szmit, Zofia"https://www.zbmath.org/authors/?q=ai:szmit.zofia"Warmiński, Jerzy"https://www.zbmath.org/authors/?q=ai:warminski.jerzy"Latalski, Jarosław"https://www.zbmath.org/authors/?q=ai:latalski.jaroslawSummary: The aim of the paper is to study a synchronisation phenomenon as observed in a rotating structure consisting of three composite beams and a hub. The beams are made of eighteen carbon-epoxy prepreg material layers stacked in a specific sequence. In the performed analysis it is assumed one of the beams is de-tuned due to small misalignment of its reinforcing fibers orientation with regard to the two remaining nominal design blades. The non-classical effects like transverse shear, material anisotropy, non-uniform torsion and cross-section warping are taken into account in the mathematical model of the blades. The partial differential equations of motion of the structure are derived by the Hamilton principle; next the reduction to the ordinary differential ones is done by the Galerkin method. Finally, the equations are solved numerically and the resonance curves for the hub and the individual beams are plotted. In the performed studies two possible variants of the rotor excitation are considered: (a) driving torque expressed by a harmonic function or (b) torque given by a chaotic oscillator formula. The analysis of the synchronisation phenomenon of the hub and the blades motion is based on the study of the resonance curves and time histories in the prepared graphs. The analysis of the structure driven by chaotic oscillator revealed the existence of the strange chaotic attractor for every beam of the rotor; in the particular, nominal beams are fully synchronised, but the de-tuned one is synchronised with a small difference in amplitude.
For the entire collection see [Zbl 1403.37005].Classical motions of infinitesimal rotators on Mylar balloons.https://www.zbmath.org/1456.530102021-04-16T16:22:00+00:00"Kovalchuk, Vasyl"https://www.zbmath.org/authors/?q=ai:kovalchuk.vasyl"Mladenov, Ivaïlo"https://www.zbmath.org/authors/?q=ai:mladenov.ivailo-mSummary: This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.Controlling subharmonic generation by vibrational and stochastic resonance in a bistable system.https://www.zbmath.org/1456.700352021-04-16T16:22:00+00:00"Sarkar, Prasun"https://www.zbmath.org/authors/?q=ai:sarkar.prasun"Paul, Shibashis"https://www.zbmath.org/authors/?q=ai:paul.shibashis"Ray, Deb Shankar"https://www.zbmath.org/authors/?q=ai:ray.deb-shankarThe method of averaging for Poisson connections on foliations and its applications.https://www.zbmath.org/1456.530642021-04-16T16:22:00+00:00"Avendaño-Camacho, Misael"https://www.zbmath.org/authors/?q=ai:avendano-camacho.misael"Hasse-Armengol, Isaac"https://www.zbmath.org/authors/?q=ai:hasse-armengol.isaac"Velasco-Barreras, Eduardo"https://www.zbmath.org/authors/?q=ai:velasco-barreras.eduardo"Vorobiev, Yury"https://www.zbmath.org/authors/?q=ai:vorobiev.yuriiSummary: On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions.https://www.zbmath.org/1456.811802021-04-16T16:22:00+00:00"Lotoreichik, Vladimir"https://www.zbmath.org/authors/?q=ai:lotoreichik.vladimir"Michelangeli, Alessandro"https://www.zbmath.org/authors/?q=ai:michelangeli.alessandroSummary: We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrödinger operator with point interaction: the optimizer is the ball with the point interaction supported at its center. Next, we establish three-dimensional Faber-Krahn inequalities for a one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimizer being given in terms of Coulomb attraction at the center of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model, a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.
{\copyright 2021 American Institute of Physics}Integrable homogeneous dissipative dynamical systems of an arbitrary odd order.https://www.zbmath.org/1456.370582021-04-16T16:22:00+00:00"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: We establish the integrability of homogeneous (in some variables) dynamical systems with dissipation in the case of an arbitrary odd order and thereby generalize the results earlier obtained by the author in particular cases of such systems.On LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)On the measure of KAM tori in two degrees of freedom.https://www.zbmath.org/1456.370642021-04-16T16:22:00+00:00"Biasco, Luca"https://www.zbmath.org/authors/?q=ai:biasco.luca"Chierchia, Luigi"https://www.zbmath.org/authors/?q=ai:chierchia.luigiThis paper is devoted to the following
conjecture of \textit{V. I. Arnold} et al. [Mathematical aspects of classical and celestial mechanics. Transl. from the Russian by E. Khukhro. 3rd revised ed. Berlin: Springer (2006; Zbl 1105.70002)]: ``It is natural to expect that in a generic (analytic) system with two degrees of freedom and with frequencies that do not vanish simultaneously the total measure of the `non-torus' set corresponding to all the resonances is exponentially small.''
The authors of this paper investigate real analytic nearly-integrable systems with two degrees of freedom, and look specifically at Hamiltonians of the form (in action-angle coordinates):
\[H_\epsilon (y,x) = \frac{y_1^2 + y_2^2}{2} + \epsilon f(x_1,x_2),\]
where \(y = (y_1, y_2) \in \mathbb{R}^2\) and \(x = (x_1, x_2) \in \mathbb{T}^2\), \(f: \mathbb{T}^2 \rightarrow \mathbb{R}\) is real-analytic and \(\epsilon \ge 0\) is a small parameter.
The phase space \(\mathbb{R}^2 \times \mathbb{T}^2\) carries the standard symplectic form and the flow \(\Phi^t_{H_\epsilon}\) induced by \(H_\epsilon\) is a solution of the Hamiltonian equations \(\dot{y} = - \partial_x H_\epsilon = -\epsilon f_x\), and \(\dot{x} = \partial_y H_\epsilon = y + \epsilon f_y\).
The system is integrable when \(\epsilon = 0\). The integrals of motion are the action variables \(y_1\) and \(y_2\), and the trajectories are \(y(t) = y_0\) and \(x(t) = x_0 + \omega t\), where \(\omega\) is the frequency corresponding to \(y_0\). The 2-tori \(\{y_0\} \times \mathbb{T}^2\) are invariant under the Hamiltonian flow. According to KAM theory most integrable tori \(\{y_0\} \times \mathbb{T}^2\) persist for small \(\epsilon\).
The authors focus on the non-torus set of analytic systems with two degrees of freedom. They consider the Banach space \(\mathbb{B}^2_s\) of real analytic functions on the torus \(\mathbb{T}^2_s\) consisting of \(x = (x_1, x_2) \in \mathbb{C}^2\) for which the absolute value of the imaginary parts of \(x_1\) and \(x_2\) are less than \(s\). The space \(\mathbb{B}^2_s\) consists of real analytic functions on \(\mathbb{T}^2_s\) having zero-average and finite \(l^\infty\) norm.
The main result is as follows. For \(s > 0\) there is a set \(\mathcal{P}_s \in \mathbb{B}^2_s\) containing an open and dense set so that the following conditions hold: for fixed \(r\) and \(R\) with \(0 < r < R\), take \(D = \{y \in \mathbb{R}^2 : r \le |y| \le R\}\) and consider the Hamiltonian system with phase space \(D \times \mathbb{T}^2\) and Hamiltonian \(H_\epsilon\) (as described above) with potential \(f \in \mathcal{P}_s\). Then there exist \(\epsilon_0\) and \(a > 0\) small enough so that if \(0 < \epsilon < \epsilon_0\), the Liouville measure of the complementary set of the \(\Phi^t_{H_\epsilon}\)-invariant tori in the phase region \(D\) is smaller than \(R^2 e^{-k/ \epsilon^a}\) where \(k > 0\) is a constant.
A sketch of the proof is provided, but a detailed proof is deferred to a later paper.
Reviewer: William J. Satzer Jr. (St. Paul)Classification of a modified de Sitter metric by variational symmetries and conservation laws.https://www.zbmath.org/1456.580112021-04-16T16:22:00+00:00"Beesham, A."https://www.zbmath.org/authors/?q=ai:beesham.aroonkumar"Gadjagboui, B. B. I."https://www.zbmath.org/authors/?q=ai:gadjagboui.b-b-i"Kara, A. H."https://www.zbmath.org/authors/?q=ai:kara.abdul-hamidIntegrable system of generalized relativistic interacting tops.https://www.zbmath.org/1456.370572021-04-16T16:22:00+00:00"Sechin, I. A."https://www.zbmath.org/authors/?q=ai:sechin.i-a"Zotov, A. V."https://www.zbmath.org/authors/?q=ai:zotov.andrei-vSummary: We describe a family of integrable \(GL(NM)\) models generalizing classical spin Ruijsenaars-Schneider systems (the case \(N=1)\) on one hand and relativistic integrable tops on the \(GL(N)\) Lie group (the case \(M=1)\) on the other hand. We obtain the described models using the Lax pair with a spectral parameter and derive the equations of motion. To construct the Lax representation, we use the \(GL(N)\) \(R\)-matrix in the fundamental representation of \(GL(N)\).Dynamic analysis of a compliant tensegrity structure for the use in a gripper application.https://www.zbmath.org/1456.700032021-04-16T16:22:00+00:00"Sumi, Susanne"https://www.zbmath.org/authors/?q=ai:sumi.susanne"Schorr, Philipp"https://www.zbmath.org/authors/?q=ai:schorr.philipp"Böhm, Valter"https://www.zbmath.org/authors/?q=ai:bohm.valter"Zimmermann, Klaus"https://www.zbmath.org/authors/?q=ai:zimmermann.klaus-fSummary: The use of compliant tensegrity structures in robotic applications offers several advantageous properties. In this work the dynamic behaviour of a planar tensegrity structure with multiple static equilibrium configurations is analysed, with respect to its further use in a two-finger-gripper application. In this application, two equilibrium configurations of the structure correspond to the opened and closed states of the gripper. The transition between these equilibrium configurations, caused by a proper selected actuation method, is essentially dependent on the actuation parameters and on the system parameters. To study the behaviour of the dynamic system and possible actuation methods, the nonlinear equations of motion are derived and transient dynamic analyses are performed. The movement behaviour is analysed in relation to the prestress of the structure and actuation parameters.
For the entire collection see [Zbl 1403.37005].Collisions and angular momentum for singular potentials.https://www.zbmath.org/1456.700292021-04-16T16:22:00+00:00"Stoica, George"https://www.zbmath.org/authors/?q=ai:stoica.georgeThis is a short paper, the author look at collisions and angular momentum in the 2-dimentional 2-body problem. He considers two particles in \(R^2\) and fix the center of mass at the origin of the system, and so it reduces to a single particle of unit mass in a central force field. The position of the single particle \(x=x(t)\) is described by the initial value problem
\[ d^2x/dt^2 =\nabla V(|x|): x(0)\in R^2 \setminus \{0\},\quad (dx/dt)_0 \in R^2 \tag{1} \]
A collision occurs if the two particles coincide and corresponds to \(x=0\) in (1). The author discusses, historically, in one theorem the relationship between collisions, angular momentum and singular potentials at the origin of the system. The purpose, here, is to show that a collision occurs in (1) if the angular momentum is zero. In addition, if the singular potential function \(V\) satisfies the condition \(\lim x^2 V(x)=0\), \(x\to 0^+\), then no collisions occur in (1) on solutions with nonzero angular momentum.
Reviewer: Maria Gousidou-Koutita (Thessaloniki)Analysis of vibrations of an oscillator using statistical series.https://www.zbmath.org/1456.700402021-04-16T16:22:00+00:00"Agnieszka, Ozga"https://www.zbmath.org/authors/?q=ai:agnieszka.ozgaSummary: Solving of a problem for systems subjected to random series of impulses is aimed at determining an approximate distribution of the strength of stochastic impulses forcing vibrations of an oscillator with damping. The difficulties that arose in connection with interpretation of experimental data forced us to search for a mathematical model, where algorithms were applied based on precise solutions. Under appropriate assumptions regarding random variables: the time of action of impulse and their strength, the deviation of the oscillator from its balanced position is a process which, in the limit as time tends to infinity, is stationary and ergodic. At the first stage of the simulation study discussed in this paper, classification of the elements of the structure of statistical series is necessary. The work was inspired by attempts at constructing a measuring device that would control granularity of the medium in a dust pipeline. The device had to signal appearance of big or small particles in excessive quantity in the transported dust.
For the entire collection see [Zbl 1403.37005].Non-Keplerian behavior and instability of motion of two bodies caused by the finite velocity of gravitation.https://www.zbmath.org/1456.700242021-04-16T16:22:00+00:00"Slyusarchuk, V. Yu."https://www.zbmath.org/authors/?q=ai:slyusarchuk.vasyl-yuSummary: It is shown that the motion of two bodies described with regard for the finite velocity of gravitation does not obey the Kepler laws and that this motion is unstable.Nonlinear oscillations of a spring pendulum at the 1:1:2 resonance by normal form methods.https://www.zbmath.org/1456.700342021-04-16T16:22:00+00:00"Edneral, Victor F."https://www.zbmath.org/authors/?q=ai:edneral.victor-f"Petrov, Alexander G."https://www.zbmath.org/authors/?q=ai:petrov.alexander-g.1Summary: The object of this research is the process of small oscillations of a three-dimensional elastic pendulum, tuned to a 1:1:2 resonance to vertical and horizontal oscillations. The purpose is to develop a symbolic algorithm for calculating small oscillations of the pendulum. The primary efforts are aimed at building a software package that generates formulas that approximate the movements of the pendulum with sufficient accuracy. The algorithms of this work are developed based on the resonant normal form method. The importance of the study is due to the wide range of applicability of the method of normal forms for constructing approximations of periodic and conditionally periodic local families of solutions of ODEs. For high-dimensional resonance systems, the technique is a generalization of the Poincaré-Linstedt method, and for coarse systems, the Carleman linearization method. The reliability of the results is confirmed by comparisons with the results of numerical solutions. Results can be useful for specialists working at the interface of computational mathematics and continuum mechanics. Approaches and methodologies developed in this paper can be applied to solve various modeling problems.The implementation of Hori-Deprit method to the construction averaged planetary motion theory by means of computer algebra system Piranha.https://www.zbmath.org/1456.700282021-04-16T16:22:00+00:00"Perminov, A. S."https://www.zbmath.org/authors/?q=ai:perminov.a-s"Kuznetsov, E. D."https://www.zbmath.org/authors/?q=ai:kuznetsov.eduard-dSummary: This article is related to the problem of the construction of planetary motion theory. We have expanded the Hamiltonian of the four-planetary problem into the Poisson series in osculating elements of the second Poincaré system. The series expansion is constructed up to the third degree of the small parameter. The averaging procedure of the Hamiltonian is performed by the Hori-Deprit method. It allows to eliminate short-periodic perturbations and sufficiently increase time step of the integration of the equations of motion. This method is based on Lie transformation theory. The equations of motion in averaged elements are constructed as the Poisson brackets of the averaged Hamiltonian and corresponding orbital element. The transformation between averaged and osculating elements is given by the change-variable functions, which are obtained in the second approximation of the Hori-Deprit method. We used computer algebra system Piranha for the implementation of the Hori-Deprit method. Piranha is an echeloned Poisson series processor authored by F. Biscani. The properties of the obtained series are discussed. The numerical integration of the equations of motion is performed by Everhart method for the Solar system's giant planets.Bifurcations and chaos in a Lorenz-like pilot-wave system.https://www.zbmath.org/1456.370952021-04-16T16:22:00+00:00"Durey, Matthew"https://www.zbmath.org/authors/?q=ai:durey.matthewSummary: A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating fluid bath, guided by its self-generated wave field. This hydrodynamic pilot-wave system exhibits a vast range of dynamics, including behavior previously thought to be exclusive to the quantum realm. We present the results of a theoretical investigation of an idealized pilot-wave model, in which a particle is guided by a one-dimensional wave that is equipped with the salient features of the hydrodynamic system. The evolution of this reduced pilot-wave system may be simplified by projecting onto a three-dimensional dynamical system describing the evolution of the particle velocity, the local wave amplitude, and the local wave slope. As the resultant dynamical system is remarkably similar in form to the Lorenz system, we utilize established properties of the Lorenz equations as a guide for identifying and elucidating several pilot-wave phenomena, including the onset and characterization of chaos.
{\copyright 2020 American Institute of Physics}Propagation of chaos for topological interactions.https://www.zbmath.org/1456.602542021-04-16T16:22:00+00:00"Degond, P."https://www.zbmath.org/authors/?q=ai:degond.pierre"Pulvirenti, M."https://www.zbmath.org/authors/?q=ai:pulvirenti.marioSummary: We consider a \(N\)-particle model describing an alignment mechanism due to a topological interaction among the agents. We show that the kinetic equation, expected to hold in the mean-field limit \(N\rightarrow \infty \), as following from the previous analysis in [\textit{A. Blanchet} and \textit{P. Degond}, J. Stat. Phys. 163, No. 1, 41--60 (2016; Zbl 1352.92182)] can be rigorously derived. This means that the statistical independence (propagation of chaos) is indeed recovered in the limit, provided it is assumed at time zero.Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds.https://www.zbmath.org/1456.370622021-04-16T16:22:00+00:00"Nikolaenko, Stanislav S."https://www.zbmath.org/authors/?q=ai:nikolaenko.stanislav-sThis paper presents a topological classification of Hamiltonian systems on two-dimensional noncompact symplectic manifolds. The author continues along the line of research established in [\textit{D. A. Fedoseev} and \textit{A. T. Fomenko}, Fundam. Prikl. Mat., 21, No. 6, 217---243 (2016)].
The author investigates topological invariants of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. The goal is to describe a complete topological classification of noncompact bifurcations for foliations like this.
Fomenko's approach uses the compactness of leaves in the Liouville foliation (the partition of the phase manifold into connected components of common level surfaces of the first integrals, known as Liouville leaves). Liouville foliations with noncompact leaves appear in many systems in mechanics, so there is motivation to extend the theory to systems with noncompact leaves.
This paper completely solves the problem of trajectory classification for Hamiltonian systems with noncompact foliations in the case of systems with one degree of freedom of finite type. Such Hamiltonian systems are said to be of finite type if the foliation defined by the Hamiltonian \(H\) is of finite type, and this means that the number of bifurcation values of \(H\) is finite and that the atoms corresponding to the bifurcation values are atoms of finite type. The author does not assume that the system is nondegenerate or that the Hamiltonian flow is complete.
In systems with one degree of freedom almost all Liouville leaves are one-dimensional; they are the level curves of the Hamiltonian, and each leaf consists of one or more trajectories. Liouville and trajectory classification are essentially identical in this case except for the orientation of trajectories. The symplectic structure is not relevant, except for enforcing orientability.
Special difficulties arise with noncompact manifolds. Much of the paper is devoted to handling those issues. In the end the author provides a natural one-to-one correspondence between the set of noncompact bifurcations of foliations and a set of oriented colored graphs that have a special form.
Reviewer: William J. Satzer Jr. (St. Paul)Global dynamics in a restricted five vortices problem on the plane.https://www.zbmath.org/1456.371022021-04-16T16:22:00+00:00"Marchesin, Marcelo"https://www.zbmath.org/authors/?q=ai:marchesin.marcelo"Vidal, Claudio"https://www.zbmath.org/authors/?q=ai:vidal.claudioThe paper deals with a restricted 5-vortex problem, where the vortices form a
rhomboidal relative equilibrium and their strength are \(\Gamma_1=\Gamma_2=1\), \(\Gamma_3=\Gamma_4=\lambda\) and \(\Gamma_5=0\).
Here \(\lambda\in (-1,1]\).
In [\textit{G. E. Roberts}, SIAM J. Appl. Dyn. Syst. 12, No. 2, 1114--1134 (2013; Zbl 1282.70022); \textit{M. Hampton} et al., J. Nonlinear Sci. 24, No. 1, 39--92 (2014; Zbl 1302.76042] the authors proved the existence of two families of relative equilibria having rhomboidal configurations. By taking this into consideration,
the current authors show that under certain conditions on the vorticities
there are two types of rhombus: one is called rhombus type A (outside the square) and the other is named rhombus type B (inside the square), where the square is the limiting case.
In order to prove the above result, they characterize the relative equilibrium solution of the 4-vortex problem in complex coordinates \(z_j\) as
\[
z_1(t)=ae^{i\nu t},\qquad z_2(t)=-ae^{i\nu t},\qquad z_3(t)=ibe^{i\nu t},\qquad z_4(t)=-ibe^{i\nu t},
\]
with \(a, b>0\) and the center of vorticity is at the origin of the coordinate system.
Thus, the number and nature of all equilibrium
points as function of the parameter \(b\) are studied.
More precisely, the authors describe analytically the different phase diagrams and analyze and characterize the global dynamics of this restricted 5-vortex problem as a function
of the parameter \(b\), taking into account the form of the rhombus (either A or B).
Finally, the global phase portraits on the Poincaré disk are described.
Reviewer: Martha Alvarez-Ramírez (Ciudad de México)A new approach to the parameterization method for Lagrangian tori of Hamiltonian systems.https://www.zbmath.org/1456.370672021-04-16T16:22:00+00:00"Villanueva, Jordi"https://www.zbmath.org/authors/?q=ai:villanueva.jordiSummary: We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov's non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.Computing perturbations in the two-planetary three-body problem with masses varying non-isotropically at different rates.https://www.zbmath.org/1456.700252021-04-16T16:22:00+00:00"Minglibayev, Mukhtar"https://www.zbmath.org/authors/?q=ai:minglibayev.mukhtar-zh"Prokopenya, Alexander"https://www.zbmath.org/authors/?q=ai:prokopenya.alexander-n"Shomshekova, Saule"https://www.zbmath.org/authors/?q=ai:shomshekova.sauleSummary: The classical problem of three bodies of variable masses is considered in the case when two of the bodies are protoplanets and all the masses vary non-isotropically at different rates. The problem is analyzed in the framework of the planetary perturbation theory in terms of the osculating elements of aperiodic motion on quasi-conic sections. An algorithm for symbolic computation of the disturbing function and its expansion into power series in terms of the eccentricities and inclinations is discussed in detail. Differential equations describing the long-term evolution of the orbital parameters are derived in the form of Lagrange's planetary equations. All the relevant calculations are done with the computer algebra system Wolfram Mathematica.Semi-classical limit of quantum free energy minimizers for the gravitational Hartree equation.https://www.zbmath.org/1456.811882021-04-16T16:22:00+00:00"Choi, Woocheol"https://www.zbmath.org/authors/?q=ai:choi.woocheol"Hong, Younghun"https://www.zbmath.org/authors/?q=ai:hong.younghun"Seok, Jinmyoung"https://www.zbmath.org/authors/?q=ai:seok.jinmyoungThe authors consider the gravitational Vlasov-Poisson equation for a plasma in a gravitational field. They assume that their approach would be valid for large complexes of stars like white dwarfs with a number of stars $N>10^8$ or $N>10^{14}$ for giant stars. The main idea of the actual article is as follows. There is a number of research in which one construct free energy minimizers under some mass constrains.
From an another hand there are also researches which investigate free energy minimizers for the quantum problem, based on the well known Hartree-Fock mean field method.
The problem which is solved by the authors of the actual article concerns with the correspondence between quantum and classical isotropic states. The authors prove some theorems stating that in the limit of very small quantum Planck constant ( when the Planck constant is going to 0), the free energy minimizers for the Hartree-Fock equation converge to those for the Vlasov-Poisson equation in terms of potential functions, as well as via the Wigner transform and the Toplitz quantization.
The authors mention throughout the text of the article an earlier research (1961, 1962) by V.A. Antonov, which is proving the stability of the Vlasov-Poisson equation, applied for stellar many-bodies systems with large number of components. Let us mention, that the famous paper by A.A. Vlasov which established a new kinetic equation for plasma was published in 1938 and only much later was recognized as a correct equation for plasma. See, about this the book by \textit{I. P. Bazarov}, and \textit{P. N. Nikolaev} [Anatolij Aleksandrovich Vlasov ( in Russian), 2nd edition, 63 p. (1999; \url{http:// phys.msu.ru/upload/iblock/0cc/ vlasov-book.pdf})].
Reviewer: Alex B. Gaina (Chisinau)A new version of the Riccati transfer matrix method for multibody systems consisting of chain and branch bodies.https://www.zbmath.org/1456.700172021-04-16T16:22:00+00:00"Rui, Xue"https://www.zbmath.org/authors/?q=ai:rui.xue"Bestle, Dieter"https://www.zbmath.org/authors/?q=ai:bestle.dieter"Wang, Guoping"https://www.zbmath.org/authors/?q=ai:wang.guoping"Zhang, Jiangshu"https://www.zbmath.org/authors/?q=ai:zhang.jiangshu"Rui, Xiaoting"https://www.zbmath.org/authors/?q=ai:rui.xiaoting"He, Bin"https://www.zbmath.org/authors/?q=ai:he.binSummary: Computational speed and stability are two important aspects in the dynamics analysis of large-scale complex multibody systems. In order to improve both in the context of the multibody system transfer matrix method, a new version of the Riccati transfer matrix method is presented. Based on the new version of the general transfer matrix method for multibody system simulation, recursive formulae are developed which not only retain all advantages of the transfer matrix method, but also reduce the truncation error. As a result, the computational speed, accuracy and efficiency are improved. Numerical computation results obtained by the proposed method and an ordinary multibody system formulation show good agreement. The successful computation for a spatial branch system with more than 100000 degrees of freedom validates that the proposed method is also working for huge systems.Modeling muscle wrapping and mass flow using a mass-variable multibody formulation.https://www.zbmath.org/1456.700132021-04-16T16:22:00+00:00"Guo, Jianqiao"https://www.zbmath.org/authors/?q=ai:guo.jianqiao"Huang, Hongshi"https://www.zbmath.org/authors/?q=ai:huang.hongshi"Yu, Yuanyuan"https://www.zbmath.org/authors/?q=ai:yu.yuanyuan"Liang, Zixuan"https://www.zbmath.org/authors/?q=ai:liang.zixuan"Ambrósio, Jorge"https://www.zbmath.org/authors/?q=ai:ambrosio.jorge-a-c"Zhao, Zhihua"https://www.zbmath.org/authors/?q=ai:zhao.zhihua"Ren, Gexue"https://www.zbmath.org/authors/?q=ai:ren.gexue"Ao, Yingfang"https://www.zbmath.org/authors/?q=ai:ao.yingfangSummary: Skeletal muscles usually wrap over multiple anatomical features, and their mass moves along the curved muscle paths during human locomotion. However, existing musculoskeletal models simply lump the mass of muscles to the nearby body segments without considering the effect of mass flow, which has been shown to induce non-negligible errors. A mass-variable multibody formulation is proposed here to simultaneously characterize muscle wrapping and mass flow effects. To achieve this goal, a novel cable element of the muscle-tendon unit, which integrates the mass flow feature with a typical Hill-type constitutive relationship, was developed based on an arbitrary Lagrangian-Eulerian description. In addition, sliding joints were used to constrain the elements to move over the underlying bone geometries. After validating the proposed modeling method using two benchmark samples, it was applied to build a large-scale lower limb musculoskeletal model, where knee joint moments were calculated and compared with isokinetic dynamometry measurements of 12 healthy males. The results of the comparison confirm that muscular mass distribution play an important role in the force transmission of muscle wrapping, and the proposed mass-variable formulation provides a better way of predicting and understanding the dynamics of musculoskeletal systems.A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements. Avoiding inertia shape integrals.https://www.zbmath.org/1456.700082021-04-16T16:22:00+00:00"Zwölfer, Andreas"https://www.zbmath.org/authors/?q=ai:zwolfer.andreas"Gerstmayr, Johannes"https://www.zbmath.org/authors/?q=ai:gerstmayr.johannesSummary: The Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.Reanalysis-based fast solution algorithm for flexible multi-body system dynamic analysis with floating frame of reference formulation.https://www.zbmath.org/1456.700142021-04-16T16:22:00+00:00"Huang, Guanxin"https://www.zbmath.org/authors/?q=ai:huang.guanxin"Zhu, Weidong"https://www.zbmath.org/authors/?q=ai:zhu.weidong"Yang, Zhijun"https://www.zbmath.org/authors/?q=ai:yang.zhijun"Feng, Cheng"https://www.zbmath.org/authors/?q=ai:feng.cheng"Chen, Xin"https://www.zbmath.org/authors/?q=ai:chen.xin.1Summary: In order to improve the computational efficiency of flexible multi-body system dynamic analysis with floating frame of reference formulation (FFRF), a reanalysis-based fast solution algorithm is developed here. The data of FFRF analysis process can be divided into two parts: unchanged mass and stiffness matrices part kept by deformation, and changed mass and stiffness matrices part caused by rigid motion and joint constraints. In the proposed method, the factorization of the unchanged part is reused in the entire solution process via employing the reanalysis concept; and the changed part is treated as structural modification. Meanwhile, the joint constraints are handled with an exact reanalysis method -- the Sherman-Morrison-Woodbury (SMW) formula, which is also beneficial for saving the computational cost. Numerical examples demonstrate that the computational efficiency of the proposed method is higher than that of full analysis, especially in large scale problems. Moreover, since the proposed fast FFRF solution algorithm is-based on exact reanalysis methods, there is no theoretical error between the results obtained by the fast solution algorithm and full analysis method.The numerical influence of additional parameters of inertia representations for quaternion-based rigid body dynamics.https://www.zbmath.org/1456.700072021-04-16T16:22:00+00:00"Xu, Xiaoming"https://www.zbmath.org/authors/?q=ai:xu.xiaoming"Luo, Jiahui"https://www.zbmath.org/authors/?q=ai:luo.jiahui"Wu, Zhigang"https://www.zbmath.org/authors/?q=ai:wu.zhigangSummary: Different inertia representations can lead to different formulations of the differential-algebraic equations for the quaternion-based rigid body dynamics. In this paper, the inertia representations are classified into \(\alpha\)-type and \(\gamma\)-type, according to the additional parameters in the kinetic energy. These two types of representations and the corresponding parameters \(\alpha\) and \(\gamma\) are theoretically equivalent if the constraint \(\mathbf{q}^T\mathbf{q} = 1\) is satisfied exactly. Nevertheless, the error estimation demonstrates that they can present entirely different numerical features in simulation and suggests that the parameter \(\gamma\) can be used to optimize the numerical performance of the integrations in simulation. To further verify the numerical difference between the inertia representations of \(\alpha\)-type and \(\gamma\)-type, the corresponding modified Hamilton's equations are discretized by the IMS (implicit midpoint scheme), EMS (energy-momentum preserving scheme) and Gauss-Lobatto SPARK methods. Numerical performance for the examples of the spinning symmetrical top is shown to result from the comprehensive effect of the discretization schemes including the distribution of discretized points and the convergence order, the inertia representations and their combinations. Numerical results further suggest that the integrations of \(\gamma\)-type are superior to those of \(\alpha\)-type and the optimized values of \(\gamma\) can be used to achieve better numerical accuracy, convergence speed and stability.Multibody modeling for concept-level floating offshore wind turbine design.https://www.zbmath.org/1456.700152021-04-16T16:22:00+00:00"Lemmer, Frank"https://www.zbmath.org/authors/?q=ai:lemmer.frank"Yu, Wei"https://www.zbmath.org/authors/?q=ai:yu.wei"Luhmann, Birger"https://www.zbmath.org/authors/?q=ai:luhmann.birger"Schlipf, David"https://www.zbmath.org/authors/?q=ai:schlipf.david"Cheng, Po Wen"https://www.zbmath.org/authors/?q=ai:cheng.po-wenSummary: Existing Floating Offshore Wind Turbine (FOWT) platforms are usually designed using static or rigid-body models for the concept stage and, subsequently, sophisticated integrated aero-hydro-servo-elastic models, applicable for design certification. For the new technology of FOWTs, a comprehensive understanding of the system dynamics at the concept phase is crucial to save costs in later design phases. This requires low- and medium-fidelity models. The proposed modeling approach aims at representing no more than the relevant physical effects for the system dynamics. It consists, in its core, of a flexible multibody system. The applied Newton-Euler algorithm is independent of the multibody layout and avoids constraint equations. From the nonlinear model a linearized counterpart is derived. First, to be used for controller design and second, for an efficient calculation of the response to stochastic load spectra in the frequency-domain. From these spectra the fatigue damage is calculated with Dirlik's method and short-term extremes by assuming a normal distribution of the response. The set of degrees of freedom is reduced, with a response calculated only in the two-dimensional plane, in which the aligned wind and wave forces act. The aerodynamic model is a quasistatic actuator disk model. The hydrodynamic model includes a simplified radiation model, based on potential flow-derived added mass coefficients and nodal viscous drag coefficients with an approximate representation of the second-order slow-drift forces. The verification through a comparison of the nonlinear and the linearized model against a higher-fidelity model and experiments shows that even with the simplifications, the system response magnitude at the system eigenfrequencies and the forced response magnitude to wind and wave forces can be well predicted. One-hour simulations complete in about 25 seconds and even less in the case of the frequency-domain model. Hence, large sensitivity studies and even multidisciplinary optimizations for systems engineering approaches are possible.Modelling and simulation of coupled multibody systems and granular media using the non-smooth contact dynamics approach.https://www.zbmath.org/1456.700122021-04-16T16:22:00+00:00"Docquier, Nicolas"https://www.zbmath.org/authors/?q=ai:docquier.nicolas"Lantsoght, Olivier"https://www.zbmath.org/authors/?q=ai:lantsoght.olivier"Dubois, Frédéric"https://www.zbmath.org/authors/?q=ai:dubois.frederic.1"Brüls, Olivier"https://www.zbmath.org/authors/?q=ai:bruls.olivierSummary: Multibody models are often coupled with other domains in order to enlarge the scope of computer-based analysis. In particular, modeling multibody systems (MBSs) in interaction with granular media is of great interest for industrial process such as railway track maintenance, handling of aggregates, etc. This paper presents a strong coupling methodology for unifying a multibody formalism using relative coordinates and a discrete element method based on non-smooth contact dynamics (NSCD). Both tree-like and closed-loop MBSs are considered. For the latter, the coordinate partitioning techniques is applied in the NSCD framework. The proposed approach is applied on the slider-crank mechanism benchmark. Results are in very good agreement with results obtained with other techniques from the literature. Finally, a multibody model of a tamping machine is coupled to a discrete element model of railway ballast in order to analyse efficiency of track maintenance. This application demonstrates that the dynamics of the machine must be taken into account so as to estimate the performance of the maintenance process correctly.Frictional contact analysis of sliding joints with clearances between flexible beams and rigid holes in flexible multibody systems.https://www.zbmath.org/1456.700182021-04-16T16:22:00+00:00"Tang, Lingling"https://www.zbmath.org/authors/?q=ai:tang.lingling"Liu, Jinyang"https://www.zbmath.org/authors/?q=ai:liu.jinyangSummary: Joints with clearances in flexible multibody systems have been investigated for many years. However, the previous work is mainly concerned with the lower pair joints with small clearances. The aim of this paper is to present a formulation for the sliding joint with clearance between a flexible beam and a rigid hole, with special focus on the frictional contact. We thoroughly discuss a contact detection method for a flexible beam with circular cross-section and a rigid hole with rectangular cross-section. We employ the layer-wise contact detection concept and propose an efficient and robust approach to address the two-dimensional ellipse-rectangle contact detection problem within the rigid hole cross-section. After the potential contact points on the beam surface and on the hole cross-section are determined, a beam-rigid hole frictional contact element is proposed. The measure of tangential interaction is defined in the current configuration, and the changes of both contact points are taken into consideration. The constitutive relationship for the tangential contact is modeled by the Coulomb friction law with a penalty regularization, and the real tangential force vector in the current step is determined via a trial step and a subsequent return-mapping scheme. In addition, the discontinuity problem across the beam element boundaries in the case of large sliding is circumvented by a simple and effective transformation. Finally, two numerical examples are presented to verify the proposed contact detection method and the beam-rigid hole frictional contact element, and to demonstrate the influence of the frictional contact on the dynamic performance of flexible multibody systems.A nonsmooth contact dynamic algorithm based on the symplectic method for multibody system analysis with unilateral constraints.https://www.zbmath.org/1456.700162021-04-16T16:22:00+00:00"Peng, Haijun"https://www.zbmath.org/authors/?q=ai:peng.haijun"Song, Ningning"https://www.zbmath.org/authors/?q=ai:song.ningning"Kan, Ziyun"https://www.zbmath.org/authors/?q=ai:kan.ziyunSummary: In multibody systems, there are not only holonomic bilateral constraints, but also unilateral constraints. The existence of unilateral constraints brings nonsmooth contact dynamic problems into multibody dynamic systems. In this paper, we present an approach based on the symplectic method and the linear complementary method to solve multibody dynamic problems with impact contact. As the symplectic method has good energy conservation and no numerical damping, the proposed approach is expected to inherit these properties for solving nonsmooth problems of multibody dynamic systems. We present multiple numerical examples to demonstrate a high accuracy and good energy conservation behavior of the proposed approach even for large time step sizes.Learning stable and predictive structures in kinetic systems.https://www.zbmath.org/1456.700022021-04-16T16:22:00+00:00"Pfister, Niklas"https://www.zbmath.org/authors/?q=ai:pfister.niklas"Bauer, Stefan"https://www.zbmath.org/authors/?q=ai:bauer.stefan-alois"Peters, Jonas"https://www.zbmath.org/authors/?q=ai:peters.jonasSummary: Learning kinetic systems from data is one of the core challenges in many fields. Identifying stable models is essential for the generalization capabilities of data-driven inference. We introduce a computationally efficient framework, called CausalKinetiX, that identifies structure from discrete time, noisy observations, generated from heterogeneous experiments. The algorithm assumes the existence of an underlying, invariant kinetic model, a key criterion for reproducible research. Results on both simulated and real-world examples suggest that learning the structure of kinetic systems benefits from a causal perspective. The identified variables and models allow for a concise description of the dynamics across multiple experimental settings and can be used for prediction in unseen experiments. We observe significant improvements compared to well-established approaches focusing solely on predictive performance, especially for out-of-sample generalization.Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices.https://www.zbmath.org/1456.811072021-04-16T16:22:00+00:00"Haah, Jeongwan"https://www.zbmath.org/authors/?q=ai:haah.jeongwanSummary: We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (abelian discrete gauge theory). This means that under local Clifford circuits, the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank-Shor-Steane structure for prime qudit codes; we do not assume any of these.
{\copyright 2021 American Institute of Physics}