General Affine Research Articles

The classification of general affine connections in Newton–Cartan geometry: Towards metric-affine Newton–Cartan gravity

Abstract We give a full classification of general affine connections on Galilei manifolds in terms of independently specifiable tensor fields. This generalises the well-known case of (torsional) Galilei connections, i.e. connections compatible with the metric structure of the Galilei manifold. Similarly to the well-known pseudo-Riemannian case, the additional freedom for connections that are not metric-compatible lies in the covariant derivatives of the two tensors defining the metric structure (the clock form and the space metric), which however are not fully independent of each other.

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.

Metric-Affine F(T,Q) gravity: cosmological implications and constraints

Abstract In this paper, we investigate some exact cosmological models in Metric-Affine $F(T,Q)$ gravity, with observational constraints. The Metric-Affine $F(T,Q)$ gravity is some kind of unification of two known gravity theories, namely, the $F(T)$ gravity and the $F(Q)$ gravity. We obtain the field equations of the Metric-Affine theory by considering the metric tensor and the general affine connection as independent variables. We then focus on the particular case in which the $F(T,Q)$ function characterizing the aforementioned metric-affine models is linear, that is, $F(T,Q)=\lambda T+\mu Q$. We investigate this linear case and consider a Friedmann-Lema\^{i}tre-Robertson-Walker background to study cosmological aspects and applications. We have obtained three exact solutions of the modified field equations in two different cases, $T$ and $Q$, using the Hubble function $H(t)$ and the scale factor $a(t)$. We then placed observational constraints on these solutions using the Hubble $H(z)$ datasets and the MCMC analysis. We have investigated the deceleration parameter $q(z)$ and effective EoS parameters, and a comparative study of all three models with $\Lambda$CDM model has been carried out.

Tame quivers and affine bases II: Nonsimply-laced cases

In [21], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis B′={C(c,tλ)|(c,tλ)∈Ga} in the generic composition algebra C⁎ and prove that B′=B′⊔(−B′) coincides with Lusztig's signed canonical basis B. Moreover, in type B˜n,C˜n, B′ is the canonical basis B.

Accelerated Gradient Approach For Deep Neural Network-Based Adaptive Control of Unknown Nonlinear Systems.

Recent connections in the adaptive control literature to continuous-time analogs of Nesterov's accelerated gradient method have led to the development of new real-time adaptation laws based on accelerated gradient methods. However, previous results assume that the system's uncertainties are linear-in-the-parameters (LIP). To compensate for non-LIP uncertainties, our preliminary results developed a neural network (NN)-based accelerated gradient adaptive controller to achieve trajectory tracking for nonlinear systems; however, the development and analysis only considered single-hidden-layer NNs. In this article, a generalized deep NN (DNN) architecture with an arbitrary number of hidden layers is considered, and a new DNN-based accelerated gradient adaptation scheme is developed to generate estimates of all the DNN weights in real-time. A nonsmooth Lyapunov-based analysis is used to guarantee the developed accelerated gradient-based DNN adaptation design achieves global asymptotic tracking error convergence for general nonlinear control affine systems subject to unknown (non-LIP) drift dynamics and exogenous disturbances. A comprehensive set of simulation studies are conducted on a two-state nonlinear system, a robotic manipulator, and a complex 20-D nonlinear system to demonstrate the improved performance of the developed method. Our simulation studies demonstrate enhanced tracking and function approximation performance from both DNN architectures and accelerated gradient adaptation.

On the dilation current in metric-affine gravity

We review F(R,D) gravity in the metric-affine framework, where D is the divergence of the dilation current appearing in the hypermomentum tensor. We assume only linear couplings between the general affine connection and the matter fields (minimal coupling) and break projective invariance to preserve a nonvanishing dilation current. For F(R,D) linear in D the dilation current dependence in the function F(R,D) does not contribute to the field equations of the theory.We show that, on the other hand, in more complicated cases (e.g., considering the function F(R,D)=R+αD2), the D contribution to the metric field equations is nontrivial and can affect the cosmology of the theory.

Formalizing screw theory with 3D geometric algebra

This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler’s First and Second Laws is provided to illustrate the use of this formalism.

Linear optimization on varieties and Chern-Mather classes

The linear optimization degree gives an algebraic measure of complexity of optimizing a linear objective function over an algebraic model. Geometrically, it can be interpreted as the degree of a projection map on the affine conormal variety. Fixing an affine variety, our first result shows that the geometry of this conormal variety, expressed in terms of bidegrees, completely determines the Chern-Mather classes of the given variety. We also show that these bidegrees coincide with the linear optimization degrees of generic affine sections.

Inverse Optimal Safety Filters

CBF-QP (control barrier function quadratic programs) safety filters are pointwise minimizers of the control effort at a given state, i.e., myopically optimal at each time. But are they optimal over the entire infinite time horizon? What does it mean for a controlled system to be “optimally safe” as opposed to, conventionally “optimally stable”? When disturbances, deterministic and stochastic, have unknown upper bounds, how should safety be defined to allow a graceful degradation under disturbances? Can safety filters be designed to guarantee such weaker safety properties as well as the optimality of safety over the infinite time horizon? We pose and answer these questions for general systems affine in control and disturbances and illustrate the answers using several examples. In the process, using the existing QP safety filters, as well as more general safety-ensuring feedbacks, we generate entire families of safety filters which are optimal over the infinite horizon though they are conservative (favoring safety over ‘liveness’) relative to the standard QP.

Chern–Simons field theory on the general affine group, 3d-gravity and the extension of Cartan connections

The purpose of this article is to study the correspondence between 3d-gravity and the Chern–Simons field theory from the perspective of geometric mechanics, specifically in the case where the structure group is the general affine group. To accomplish this, the paper discusses a variational problem of the Chern–Simons type on a principal fiber bundle with this group as its structure group. The connection to the usual Chern–Simons theory is established by utilizing a generalization, in the context of Cartan connections, of the notion of extension and reduction of connections.

Cosmological reconstruction and Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM universe in f(Q,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(Q,\\,C)$$\\end{document} gravity

Symmetric Teleparallel Gravity allows for the reformulation of gravity in the form of nonmetricity by vanishing the contorsion term in the generic affine connection. Our focus is on investigating a recently proposed extension of this theory in which the Lagrangian has the form f(Q, C) by incorporating the boundary term C. In this work, we first use a reconstruction approach in f(Q, C) gravity that might admit the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM expansion history. Furthermore, we perform a novel approach for cosmological reconstruction of f(Q, C) gravity in terms of e-folding, and it shows how any FLRW cosmology can arise from a specific f(Q, C) gravity. A variety of instances are provided using this approach in which f(Q, C) gravity is reconstructed to yield the well-known cosmic evolution: Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM era, acceleration/deceleration era which is equivalent to the presence of phantom and non-phantom matter, late-time acceleration with the crossing of phantom-divide line and transient phantom era.

Propositional Logics for the Lawvere Quantale

Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for quantitative metric reasoning, we investigate three $[0,\infty]$-valued propositional logics over the Lawvere quantale. The basic logical connectives shared by all three logics are those that can be interpreted in any quantale, viz finite conjunctions and disjunctions, tensor (addition for the Lawvere quantale) and linear implication (here a truncated subtraction); to these we add, in turn, the constant $1$ to express integer values, and scalar multiplication by a non-negative real to express general affine combinations. Quantitative equational logic can be interpreted in the third logic if we allow inference systems instead of axiomatic systems. For each of these logics we develop a natural deduction system which we prove to be decidably complete w.r.t. the quantale-valued semantics. The heart of the completeness proof makes use of the Motzkin transposition theorem. Consistency is also decidable; the proof makes use of Fourier-Motzkin elimination of linear inequalities. Strong completeness does not hold in general, even (as is known) for theories over finitely-many propositional variables; indeed even an approximate form of strong completeness in the sense of Pavelka or Ben Yaacov -- provability up to arbitrary precision -- does not hold. However, we can show it for theories axiomatized by a (not necessarily finite) set of judgements in normal form over a finite set of propositional variables when we restrict to models that do not map variables to $\infty$; the proof uses Hurwicz's general form of the Farkas' Lemma.

Characterization of Lipschitz normally embedded complex curves

The main result of the paper states that a connected complex affine algebraic curve is LNE in Cn if and only if its germ at any singular point is a finite union of non-singular complex curve germs which are pairwise transverse, and its projective closure is in general position with the hyperplane at infinity. To this aim, we completely characterize complex analytic curves of a compact complex manifold which are LNE (regardless of the given Riemannian structure), and therefore we can relate when a projective algebraic curve is LNE in CPn with the property of its general affine traces being LNE in Cn. This allows us to deduce that Lipschitz classification of LNE curves is topological. We describe a complete invariant for the (outer and inner) Lipschitz equivalence of affine LNE curves and show that most such curves cannot be bi-Lipschitz homeomorphic to a plane one.

Algebraic aspects of connections: From torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work, we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a D -algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations among the post-Lie Magnus expansion, the Grossman–Larson product, and the K -map, \alpha -map, and \beta -map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker–Campbell–Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.

Adaptive dynamic programming-based hierarchical decision-making of non-affine systems

In this paper, the problem of multiplayer hierarchical decision-making problem for non-affine systems is solved by adaptive dynamic programming. Firstly, the control dynamics are obtained according to the theory of dynamic feedback and combined with the original system dynamics to construct the affine augmented system. Thus, the non-affine multiplayer system is transformed into a general affine form. Then, the hierarchical decision problem is modeled as a Stackelberg game. In the Stackelberg game, the leader makes a decision based on the information of all followers, whereas the followers do not know each other’s information and only obtain their optimal control strategy based on the leader’s decision. Then, the augmented system is reconstructed by a neural network (NN) using input–output data. Moreover, a single critic NN is used to approximate the value function to obtain the optimal control strategy for each player. An extra term added to the weight update law makes the initial admissible control law no longer needed. According to the Lyapunov theory, the state of the system and the error of the weights of the NN are both uniformly ultimately bounded. Finally, the feasibility and validity of the algorithm are confirmed by simulation.

Affine Subspace Robust Low-Rank Self-Representation: from Matrix to Tensor.

Low-rank self-representation based subspace learning has confirmed its great effectiveness in a broad range of applications. Nevertheless, existing studies mainly focus on exploring the global linear subspace structure, and cannot commendably handle the case where the samples approximately (i.e., the samples contain data errors) lie in several more general affine subspaces. To overcome this drawback, in this paper, we innovatively propose to introduce affine and nonnegative constraints into low-rank self-representation learning. While simple enough, we provide their underlying theoretical insight from a geometric perspective. The union of two constraints geometrically restricts each sample to be expressed as a convex combination of other samples in the same subspace. In this way, when exploring the global affine subspace structure, we can also consider the specific local distribution of data in each subspace. To comprehensively demonstrate the benefits of introducing two constraints, we instantiate three low-rank self-representation methods ranging from single-view low-rank matrix learning to multi-view low-rank tensor learning. We carefully design the solution algorithms to efficiently optimize the proposed three approaches. Extensive experiments are conducted on three typical tasks, including single-view subspace clustering, multi-view subspace clustering, and multi-view semi-supervised classification. The notably superior experimental results powerfully verify the effectiveness of our proposals.

More on affine Dynkin quiver Yangians

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to Seiberg duality of the quivers, and we investigate the relations of the dual quiver Yangians. We also mention the construction of the twisted quiver Yangians. It is conjectured that the truncations of the (twisted) quiver Yangians can give rise to certain \U0001d4b2-algebras. Incidentally, we give the screening currents of the \U0001d4b2-algebras in terms of the free field realization in the case of generalized conifolds. Moreover, we discuss the toroidal and elliptic algebras for any general quivers.

Short Communication: Caplet Pricing in Affine Models for Alternative Risk-Free Rates

Alternative risk-free rates (RFRs) play a central role in the reform of interest rate benchmarks. We study a short-rate model for RFRs driven by a general affine process. In this context, under minimal assumptions, we derive explicit valuation formulas for forward-looking and backward-looking caplets/floorlets as well as term-basis caplets.

DMVT-based observer design for general affine class of fractional-order nonlinear MIMO systems

In this article, a new observer design technique is proposed for the general affine class of fractional-order nonlinear multi-input/multi-output (MIMO) systems. This study is presented based on the differential mean value theorem. One significant characteristic of the suggested approach is that the nonlinear dynamic of observer error is converted into a linear parameter-varying system. Stability and convergence of observation error are shown using the Lyapunov direct method leading to feasibility and existence of a solution for some linear matrix inequalities. The performance and efficacy of the proposed method are evaluated through some illustrated simulations.

An energetically consistent surface correction method for bond-based peridynamics

A novel surface correction method is proposed for bond-based peridynamics which ensures energy consistency with a classical reference body for general affine deformations. This method is validated for simple geometries and then applied to a typical surface-dominated problem, namely the indentation of a surface in the shallow to moderate-depth regime.