Affine Representation Research Articles

Affine representability results in A1-homotopy theory, I: Vector bundles

We establish a general "affine representability" result in ${\mathbb A}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb A}^1$-homotopy theory. Our results simplify and significantly generalize F. Morel's ${\mathbb A}^1$-representability theorem for vector bundles.

Affinely prime dynamical systems

This paper deals with representations of groups by “affine” automorphisms of compact, convex spaces, with special focus on “irreducible” representations: equivalently “minimal” actions. When the group in question is PSL(2, R), the authors exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called “linear Stone-Weierstrass” for group actions on compact spaces. If it holds for the “universal strongly proximal space” of the group (to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.

Distributive and trimedial quasigroups of order 243

We enumerate three classes of non-medial quasigroups of order 243=35 up to isomorphism. There are 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Bénéteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Němec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský).The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the LOOPS package in GAP.

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

Control of nonlinear state-dependent systems using a receding horizon strategy

A variety of nonlinear control design methods have been proposed for controlling severe nonlinear processes over the past three decades. The vast majority of approaches take a nonlinear affine representation of the system dynamics. It appears that many system dynamics can also be represented by a state-dependent model structure. Control of state-dependent systems has been investigated resulting in design methodologies such as state-dependent Riccati equation approach, state-dependent parameter and proportional–integral–plus approach, and the nonlinear generalized minimum variance approach. This article describes yet another approach based on a receding horizon strategy. Important results on optimal control are obtained. Implementation issues are also discussed. The proposed approach is validated through its application to a 25-tray binary distillation column process.

Generalised states: a multi-sorted algebraic approach to probability

We introduce a generalised notion of state as an additive map from a Boolean algebra of events to an arbitrary MV-algebra. Generalised states become unary operations in two-sorted algebraic structures that we call state algebras. Since these, as we show, form an equationally defined class of algebras, universal-algebraic techniques apply. We discuss free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice groups.

Inversion-free arithmetic on elliptic curves through isomorphisms

This paper presents inversion-free formulas for the efficient implementation of a scalar multiplication over elliptic curves. Specifically, it proposes to make use of curve isomorphisms as a way to avoid the computation of inverses in point addition formulas. Interestingly, the presented techniques are independent of the model used to represent the elliptic curve and of the coordinate system used to represent the points. In particular, they apply to affine representations. Further, whereas certain inversion-free techniques are mostly limited to specific scalar multiplication algorithms, the proposed techniques apply to all scalar multiplication algorithms. The so-obtained formulas are well suited to embedded systems and can easily be combined with existing countermeasures to provide secure implementations.

Conditions under which affine representations are also Fechnerian under the power law of similarity on the Weber sensitivities

In a recent paper, Hsu, Iverson, and Doble (2010) examined some properties of a (weakly balanced) affine representation for choices, Ψ(x,y)=F(u(x)−u(y)σ(y)), and showed that using the Fechner method of integrating jnds, one can reconstruct the scales u and σ from the behavior of (Weber) sensitivities ξs(x)=x+Δs(x) (where s=F−1(π) and Δs is the jnd) in a neighborhood of s=0. Following Iverson (2006b), in this article we impose a power law of similarity on the sensitivities, ξs(λx)=λι(s)ξη(λ,s)(x), and study its impact on u and σ in the affine representation. Especially, we specify the conditions for the first- and second-order derivatives of ξs(x) with respect to s (and evaluated at s→0) under which the affine representation degenerates to a Fechnerian one. We also link the results to the solutions in Iverson (2006b), which was worked out within the Fechnerian framework.

Irredundant lattice representations of continuous piecewise affine functions

In this paper, we revisit the full lattice representation of continuous piecewise affine (PWA) functions and give a formal proof of its representation ability. Based on this, we derive the irredundant lattice PWA representations through removal of redundant terms and literals. Necessary and sufficient conditions for irredundancy are proposed. Besides, we explain how to remove terms and literals in order to ensure irredundancy. An algorithm is given to obtain an irredundant lattice PWA representation. In the worked examples, the irredundant lattice PWA representations are used to express the optimal solution of explicit model predictive control problems, and the results turn out to be much more compact than those given by a state-of-the-art algorithm.

Semi-physical piecewise affine representation for governors in hydropower system generation

This paper presents a nonlinear model for a hydraulic amplifier, a component of the governor in the speed control loop of hydroelectric power plants. The amplifier transforms the electrical signal of the controller into mechanical movement of the turbine components, including the switching characteristics. This model is used to propose, on the one hand, a Piecewise Affine (PWA) representation for the hydraulic amplifier, and, on the other hand, a methodology for estimating the model parameters using field measurements, which facilitates its practical implementation. This representation is referred to as semi-physical because the model parameters are closely related with the physical construction of the hydraulic amplifier components. Among the advantages of this PWA representation are the appropriate structuring for use in system identification methods, for estimating its parameters, and the existence of advanced control techniques that use this structure in controller design, thereby improving the load-frequency control performance. The paper concludes with a description of the results, including the parameters that were estimated by using the hydraulic amplifier model with PWA structure.

Pricing of Long-Dated Commodity Derivatives with Stochastic Volatility and Stochastic Interest Rates

Aiming to study pricing of long-dated commodity derivatives, this paper presents a class of models within the Heath, Jarrow, and Morton (1992) framework for commodity futures prices that incorporates stochastic volatility and stochastic interest rate and allows a correlation structure between the futures price process, the futures volatility process and the interest rate process. The functional form of the futures price volatility is specified so that the model admits finite dimensional realisations and retains affine representations, henceforth quasi-analytical European futures option pricing formulae can be obtained. A sensitivity analysis reveals that the correlation between the interest rate process and the futures price process has noticeable impact on the prices of long-dated futures options, while the correlation between the interest rate process and the futures price volatility process does not impact option prices. Furthermore, when interest rates are negatively correlated with futures prices then option prices are more sensitive to the volatility of interest rates, an effect that is more pronounced with longer maturity options.

Reduced Basis Methods: From Low-Rank Matrices to Low-Rank Tensors

We propose a novel combination of the reduced basis method with low-rank tensor techniques for the efficient solution of parameter-dependent linear systems in the case of several parameters. This combination, called rbTensor, consists of three ingredients. First, the underlying parameter-dependent operator is approximated by an explicit affine representation in a low-rank tensor format. Second, a standard greedy strategy is used to construct a problem-dependent reduced basis. Third, the associated reduced parametric system is solved for all parameter values on a tensor grid simultaneously via a low-rank approach. This allows us to explicitly represent and store an approximate solution for all parameter values at a time. Once this approximation is available, the computation of output functionals and the evaluation of statistics of the solution become a cheap online task, without requiring the solution of a linear system.

DEMAZURE MODULES OF LEVEL TWO AND PRIME REPRESENTATIONS OF QUANTUM AFFINE

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

Integrable representations of affine A(m,n) and C(m) superalgebras

We classify the irreducible zero-level integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras Aˆ(m,n) (m≠n) and Cˆ(m). Such modules must be of highest weight type, but are not necessarily loop modules. This is in sharp contrast to the case of ordinary affine algebras. Combined with previous results obtained by other people (Rao and Zhao in particular), our work completes the program of classifying the irreducible integrable modules with finite dimensional weight spaces for all the untwisted affine superalgebras.

On the Characters of Parafermionic Field Theories

We study cosets of the type $H_l/U(1)^r$, where $H$ is any Lie algebra at level $l$ and rank $r$. These theories are parafermionic and their characters are related to the string functions, which are generating functions for the multiplicities of weights in the affine representations. An identity for the characters is described, which apply to all the algebras and all the levels. The expression is of the Rogers Ramanujan type. We verify this conjecture, for many algebras and levels, using Freudenthal Kac formula, which calculates the multiplicities in the affine representations, recursively, up to some grade. Our conjecture encapsulates all the known results about these string functions, along with giving a vast wealth of new ones.

Boundary Quantum Knizhnik–Zamolodchikov Equations and Fusion

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure.

Robust Control of the Air to Fuel Ratio in Spark Ignition Engines with Delayed Measurements from a UEGO Sensor

A precise control of the normalized air to fuel ratio in spark ignition engines is an essential task. To achieve this goal, in this work we take into consideration the time delay measurement presented by the universal exhaust gas oxygen sensor along with uncertainties in the volumetric efficiency. For that purpose, observers are designed by means of a super-twisting sliding mode estimation scheme. Also two control schemes based on a general nonlinear model and a similar nonlinear affine representation for the dynamics of the normalized air to fuel ratio were designed in this work by using the super-twisting sliding mode methodology. Such dynamics depends on the control input, that is, the injected fuel mass flow, its time derivative, and its reciprocal. The two latter terms are estimated by means of a robust sliding mode differentiator. The observers and controllers are designed based on an isothermal mean value engine model. Numeric and hardware in the loop simulations were carried out with such model, where parameters were taken from a real engine. The obtained results show a good output tracking and rejection of disturbances when the engine is closed loop with proposed control methods.

Drinfeld center of planar algebra

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated byNL2(M)M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

An output feedback LPV control strategy of a nonlinear electrostatic microgripper through a singular implicit modeling

The aim of the paper is the design and the analysis of a gain scheduled controller for an accurate and fast positioning with nanometer resolution of a nonlinear electrostatic microgripper. The controller is designed to achieve a positioning of the gripping arm from few hundred nanometers to several tens of micrometers with some performance criteria. This very large operating range is crucial for a range of microrobotics applications and has never been addressed in existing control techniques of microgrippers. The controller is designed considering noises that are relevant at the nanometer scale and nonlinearities that become significant at the micrometer scale. Therefore, a nonlinear model of the system is proposed and is reformulated into a polynomial LPV (Linear Parameter Varying) model. The most relevant source of noise to be considered for the controller synthesis is defined taking into account results from previous works. Considering the particular polynomial parametric dependence of the LPV model, a multivariable controller is designed using an affine LPV descriptor representation of the system and specific linear matrix inequalities. The efficiency of the controller and the relevance of the theoretical approach is demonstrated through experimental implementation results.

Regular permutation groups of order mp and Hopf Galois structures

Let Γ be a group of order mp where p is prime and p>m. We give a strategy to enumerate the regular subgroups of Perm(Γ) normalized by the left representation λ(Γ) of Γ. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions L∕K with Γ= Gal(L∕K). We prove that every such regular subgroup is contained in the normalizer in Perm(Γ) of the p-Sylow subgroup of λ(Γ). This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order p(p−1), where p is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.