Invariant Theory Research Articles

Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, ℝ)

We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2, ℝ) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2, ℝ) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2, ℝ) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2, ℝ) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.

Rheodynamics of viscoelastic subdiffusive channel flows: Low Weissenberg number regime

Direct numerical simulations of viscoelastic, subdiffusive, and plane Poiseuille flow, representing thick polymer solutions including polymer melts, flows of liquid crystals, as well as biological flow such as mucus, are reported. The stress strain equation is fundamentally derived by relating the timescale exponent at the microscale (tα, with 0<α≤1) with the fractional order of the time derivative, α, of the corresponding non-linear equation in the continuum. The resultant stress constitutive equation is the fractional variant of the well-known upper convected Maxwell equation. In order to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia), the space of symmetric positive definite polymer conformation tensors is visualized using a Riemannian metric along with its three scalar invariants. Numerical simulations of the channel flow, in the regime of low to moderate Reynolds number and low Weissenberg number, effectively capture these flow-structures by providing (i) a better resolution of the instantaneous regions of elastic shocks (which are the alternating regions of expanded and compressed polymer volume, in comparison with the volume of the mean conformation tensor), and (ii) a better resolution to detect neighborhoods where the mean conformation tensor tends to be significantly different in comparison to the instantaneous conformation tensor, thereby corroborating the experimentally observed flow-instability transition of subdiffusive flows. Finally, the strength of the subdiffusive flow model and the invariant theory is highlighted through an application of an electrohydrodynamic micropump.

On the Photon Vertex and Half-Integer Spin

When rewriting the photon vertex of quantum electrodynamics in terms of geometrical quantities, various elements can be mapped directly to objects and properties known from classical projective geometry (PG). Elements of P 5 when mapped to line reps in P 3 exhibit their intrinsic Lorentz invariance associated to automorphisms of the Plücker-Klein quadric , and line reps when expressed by point or plane coordinates introduce (one-parameter) pencils, or formally gl(2,ℝ), or gl(1,ℍ) which covers su(2)⊕u(1). This introduces binary forms and, using a potential approach of central forces, Schrödinger or Laplace equations and the respective special functions, as well as the projective generation of quadrics like in Dirac’s approach which legitimates Clifford algebra elements as linear factors in invariant theory and the quadratic algebra to represent geometry. Physically, this identification allows for the classical concept of moments in terms of tetrahedrons which on the one hand relates to previous work on SU(4) and SU*(4) in quantum representations. On the other hand, it relates to the classical physical definitions, however, exhibiting a factor 2 between contemporary (euclidean) moments and the tetrahedral construction used in the vertex. Finally, we discuss the equilibrium conditions with respect to gauge and Yang-Mills theories in general as well as the related objects and their transformation theory.

Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions.

On Finite Type Invariants of Welded String Links and Ribbon Tubes

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developed by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to $w_k$-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to $w_k$-equivalence. All results have direct corollaries for ribbon knotted surfaces.

PiP-X: Online feedback motion planning/replanning in dynamic environments using invariant funnels

Computing kinodynamically feasible motion plans and repairing them on-the-fly as the environment changes is a challenging, yet relevant problem in robot navigation. We propose an online single-query sampling-based motion re-planning algorithm using finite-time invariant sets, commonly referred to as “ funnels”. We combine concepts from nonlinear systems analysis, sampling-based techniques, and graph-search methods to create a single framework that enables feedback motion re-planning for any general nonlinear dynamical system in dynamic workspaces. A volumetric network of funnels is constructed in the configuration space using sampling-based methods and invariant set theory; and an optimal sequencing of funnels from robot configuration to a desired goal region is then determined by computing the shortest-path subgraph (tree) in the network. Analyzing and formally quantifying the stability of trajectories using Lyapunov level-sets ensures kinodynamic feasibility and guaranteed set-invariance of the solution paths. Though not required, our method is capable of using a pre-computed library of motion primitives to speedup online computation of controllable motion plans that are volumetric in nature. We introduce a novel directed-graph data structure to represent the funnel-network and its inter-sequencibility; helping us leverage discrete graph-based incremental search to quickly rewire feasible and controllable motion plans on-the-fly in response to changes in the environment. We validate our approach on a simulated cart-pole, car-like robot, and 6DOF quadrotor platform in a variety of scenarios within a maze and a random forest environment. Using Monte Carlo methods, we evaluate the performance in terms of algorithm success, length of traversed trajectory, and runtime.

Graphical methods and rings of invariants on the symmetric algebra

Abstract Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space $\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace $\mathcal P^m(V)$ with the full polynomial algebra $\mathcal P(V)$ . As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal P(V)$ . We conclude with examples using our graphical notation, several of which recover classical results.

Standard Model in conformal geometry: Local vs gauged scale invariance

We discuss comparatively local versus gauged Weyl symmetry beyond Standard Model (SM) and Einstein gravity and their geometric interpretation. The SM and Einstein gravity admit a natural embedding in Weyl integrable geometry which is a special limit of Weyl conformal (non-metric) geometry. The theory has a local Weyl scale symmetry but no associated gauge boson. Unlike previous models with such symmetry, this embedding is truly minimal i.e. with no additional fields beyond SM and underlying geometry. This theory is compared to a similar minimal embedding of SM and Einstein gravity in Weyl conformal geometry (SMW) which has a full gauged scale invariance, with an associated Weyl gauge boson. At large field values, both theories give realistic, Starobinsky–Higgs like inflation. The broken phase of the current model is the decoupling limit of the massive Weyl gauge boson of the broken phase of SMW, while the local scale symmetry of the current model is part of the larger gauged scale symmetry of SMW. Hence, the current theory has a gauge embedding in SMW. Unlike in the SMW, we note that in models with local scale symmetry the associated current is trivial, which is a concern for the physical meaning of this symmetry. Therefore, the SMW is a more fundamental UV completion of SM in a full gauge theory of scale invariance that generates Einstein gravity in the (spontaneously) broken phase, as an effective theory.

Entropy-current for dynamical black holes in Chern-Simons theories of gravity

We construct an entropy current and establish a local version of the classical second law of thermodynamics for dynamical black holes in Chern-Simons (CS) theories of gravity. We work in a chosen set of Gaussian null coordinates and assume the dynamics to be small perturbations around the Killing horizon. In explicit examples of both purely gravitational and mixed gauge gravity CS theories in (2 + 1) and (4 + 1)-dimensions, the entropy current is obtained by studying the off-shell structure of the equations of motion evaluated on the horizon. For the CS theory in (2 + 1) dimensions, we argue that the second law holds to quadratic order in perturbations by considering it as a low energy effective field theory with the leading piece given by Einstein gravity. In all such examples, we show that the construction of entropy current is invariant under the reparameterization of the null horizon coordinates. Finally, extending an existing formalism for diffeomorphism invariant theories, we construct an abstract proof for the linearised second law in arbitrary Chern-Simons theories in any given odd dimensions by studying the off-shell equations of motion. As a check of consistency, we verify that the outcome of this algorithmic proof matches precisely with the results obtained in explicit examples.

Fermions with SU(1, n) spacetime symmetry

We construct theories of free fermions in (2n − 1)-dimensions with SU(1, n) spacetime symmetry from the null reduction of fermions on a 2n-dimensional Ω-deformed Minkowski background for n = 2 and n = 3. These play a role in the 5d SU(1, 3)-invariant theories that are conjectured to offer a full description of certain 6d superconformal field theories. We find the (2n − 1)-dimensional manifestation of the supersymmetry of a free 2n-dimensional boson-fermion system, which we use to fix the fermion two-point functions. It is then shown that the full 2n-dimensional two-point function can be recovered through resummation. Limits of the theories are considered, and it is observed that both Galilean and Carrollian field theories appear in different regimes. We confirm that the correlation functions obey the SU(1, n) Ward identities and the representations of the fermions under this group are discussed.

Active flutter suppression of nonlinear fin-actuator system via inverse system and immersion and invariance method

Structural nonlinearities such as freeplay will affect the stability and even flight safety of the fin-actuator system. There is a lack of a practical method for designing Active Flutter Suppression (AFS) control laws for nonlinear fin-actuator systems. A design method for the AFS controller of the nonlinear all-movable fin-electromechanical actuator system is established by combining the inverse system and the Immersion and Invariance (I&I) theory. First, the composite control law combining the inverse system principle and internal model control is used to offset the nonlinearity and dynamics of the actuator, so that its driving torque can follow the ideal signal. Then, the ideal torque of the actuator is designed employing the I&I theory. The unfavorable oscillation of the fin is suppressed by making the output torque of the actuator track the ideal signal. The simulation results reveal that the proposed AFS method can increase the flutter speed of the nonlinear fin-actuator system with freeplay, and a set of controller parameters is also applicable for wider freeplay within a certain range. The power required for the actuator does not exceed the power that can be provided by the commonly used aviation actuator. This method can also resist a certain level of noise and external disturbance.

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Let K be a field, Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} a finite group of field automorphisms of K, F the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-fixed field in K, and G≤GLv(K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\le {{\\,\ extrm{GL}\\,}}_v(K)$$\\end{document} a finite matrix group. Then the action of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} defines a grading on the symmetric algebra of the F-space Kv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^v$$\\end{document} which we use to introduce the notion of homogeneous Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate invariants of G. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate invariants of the associated Clifford–Weil groups. As an immediate consequence, we obtain a result by Bannai et al. that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur–Weyl duality conjectured and partly shown by Gross et al. can be derived from our main result.

Residential Segregation

Residential segregation refers to the disproportionate distribution of population groups across a geographical area. Groups can be segregated on the basis of any characteristic (such as occupation, income, religion, age or ethnicity) and at any geographical scale. In most cases, segregation is, however, measured with regard to residential areas of a city. The extent of the unequal distribution of selected characteristics can be expressed by different statistical measures. Sociologists, economists and demographers have long studied how social groups tend to be differentiated in residential space and developed a broad range of explanations. As a consequence, segregation has been explained by a variety of theories, which are discussed in this paper. The topics examined by empirical research include temporal dynamics, geographical patterns, societal causes and effects on life chances. This entry focuses on major conceptual facts regarding residential segregation and only marginally discusses the methodological issues connected with its measurement.

Weyl conformal geometry vs Weyl anomaly

Weyl conformal geometry is a gauge theory of scale invariance that naturally brings together the Standard Model (SM) and Einstein gravity. The SM embedding in this geometry is possible without new degrees of freedom beyond SM and Weyl geometry, while Einstein gravity is generated by the broken phase of this symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl gauge boson becomes massive and decouples, as discussed in the past [1–3]. However, Weyl anomaly could break explicitly this gauge symmetry, hence we study it in Weyl geometry. We first note that in Weyl geometry metricity can be restored with respect to a new differential operator ( hat{nabla} ) that also enforces simultaneously a Weyl-covariant formulation. This leads to a metric-like Weyl gauge invariant formalism that enables one to do quantum calculations directly in Weyl geometry, rather than use a Riemannian (metric) geometry picture. The result is the Weyl-covariance in d dimensions of all geometric operators ( hat{R} , etc) and of their derivatives ( hat{nabla} μ hat{R} , etc), including the Euler-Gauss-Bonnet term. A natural (geometric) Weyl-invariant dimensional regularisation of quantum corrections exists and Weyl gauge symmetry is then maintained and manifest at the quantum level. This is related to a non-trivial current of this symmetry, the divergence of which cancels the trace of the energy-momentum tensor. The “usual” Weyl anomaly and Riemannian geometry are recovered in the (spontaneously) broken phase. The relation to holographic Weyl anomaly is discussed.

Robust Shared Control for Four-Wheel Steering Considering Driving Comfort and Vehicle Stability

Although the four-wheel steering system expands the flexibility of vehicle control, it also brings the problem of difficult coordination between driver comfort and vehicle stability. To this end, this paper proposes robust coordinated control for a four-wheel steering (4WS) vehicle considering driving comfort and vehicle stability. First, the vehicle dynamics model is constructed to reflect the lateral motion characteristics of a 4WS vehicle. Then, the driver model is coupled into the 4WS vehicle model to describe the driver’s handling characteristics. To suppress the system perturbation caused by the uncertainties of driver behavior and vehicle states, the Takagi-Sugeno fuzzy robust control method is developed to design the human-machine co-driving system. Moreover, the robust positive invariant set theory is used to guarantee the stability and safety constraints of the vehicle. Finally, the proposed human-machine shared robust control for 4WS vehicle is verified through the driving simulator platform. The results indicate that the fuzzy robust shared control approach comprehensively improves the driving comfort, vehicle stability, and path tracking.

Super invariant theory in positive characteristic

Super invariant theory in positive characteristic

Jordan algebras of a degenerate bilinear form: Specht property and their identities

Let K be a field and let Jn,k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n−k over K. Then one can consider the decomposition Jn,k=Bn−k⊕Dk, where Bn−k represents the corresponding Jordan algebra, denoted as Bn−k=K⊕V. In this algebra, the restriction of b on the (n−k)-dimensional subspace V is non-degenerate, while Dk accounts for the degenerate part of Jn,k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for Jn,k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T-ideal of polynomial identities of Jn,k. Over a field of characteristic zero, we also prove that the ideal of identities of Jn,k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z2-grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace.

Singular perturbation analysis in a coupled Chua's circuit with diffusion.

This paper is concerned with the traveling wave solutions of a singularly perturbed system, which arises from the coupled arrays of Chua's circuit. By the geometric singular perturbation theory and invariant manifold theory, we prove that there exists a heteroclinic cycle consisting of the traveling front and back waves with the same wave speed. In particular, the expression of corresponding wave speed is also obtained. Furthermore, we show that the chaotic behavior induced by this heteroclinic cycle is hyperchaos.

Enhancement of Green's functions based on striation pattern

Enhancement of Green's function is accomplished from the striation pattern obtained from data. The waveguide invariant theory allows a Green's function observed at one location to be extrapolated into adjacent ranges [Song and Byun, J. Acoust. Soc. Am. 147, 2150–2158 (2020)]. Conversely, Green's functions observed at adjacent ranges can converge towards a central location for coherent combination to improve the signal-to-noise ratio (SNR). This can be accomplished from the frequency shift in the striation pattern without prior range information of each Green's functions. We demonstrate a significant improvement of SNR in the broadband Green's function from a moving ship in shallow water.

Periodic Perturbations of Codimension-Two Bifurcations with a Double Zero Eigenvalue in Symmetrical Dynamical Systems

We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double zero eigenvalue when the frequencies of the perturbation terms are small. We transform the periodically perturbed systems to simpler ones which are periodic perturbations of the normal forms for the codimension-two bifurcations, and apply the subharmonic and homoclinic Melnikov methods to analyze bifurcations occurring there. In particular, we show that there exist transverse homoclinic or heteroclinic orbits, which yield chaotic dynamics, in wide parameter regions. These results can be applied to three or higher-dimensional systems and even to infinite-dimensional systems with the assistance of center manifold reduction and the invariant manifold theory. We illustrate our theory for a pendulum subjected to position and velocity feedback control when the desired position is periodic in time. We also give numerical computations by the computer tool AUTO to demonstrate the theoretical results.