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We find a large internal symmetry within 4-dimensional Poincarè gauge theory. In the Riemann-Cartan geometry of Poincaré gauge theory the field equation and geodesics are invariant under projective transformation, just as in affine geometry. However, in the Riemann-Cartan case the torsion and nonmetricity tensors change. By generalizing the Riemann-Cartan geometry to allow both torsion and nonmetricity while maintaining local Lorentz symmetry the difference of the antisymmetric part of the nonmetricity Q and the torsion T is a projectively invariant linear combination S = T - Q with the same symmetry as torsion. The structure equations may be written entirely in terms of S and the corresponding Riemann-Cartan curvature. The new description of the geometry has manifest projective and Lorentz symmetries, and vanishing nonmetricity. Torsion, S and Q lie in the vector space of vector-valued 2-forms. Within the extended geometry we define rotations with axis in the direction of S. These rotate both torsion and nonmetricity while leaving S invariant. In n dimensions and (p, q) signature this gives a large internal symmetry. The four dimensional case acquires SO(11,9) or Spin(11,9) internal symmetry, sufficient for the Standard Model. The most general action up to linearity in second derivatives of the solder form includes combinations quadratic in torsion and nonmetricity, torsion-nonmetricity couplings, and the Einstein-Hilbert action. Imposing projective invariance reduces this to dependence on S and curvature alone. The new internal symmetry decouples from gravity in agreement with the Coleman-Mandula theorem. [ABSTRACT FROM AUTHOR]

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space R 2 must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term "affine geometry" refers to "equi-affine geometry".) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in R 2 ? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in R 2 , and show the general-affine maximal curves in R 2 are much more abundant and include the explicit curves y = x α (α is a constant and α ∉ { 0 , 1 , 1 2 , 2 }) and y = x log ⁡ x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA (n) = GL (n) ⋉ R n. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved. [ABSTRACT FROM AUTHOR]

Abstract We prove that every maximal ideal in the ring of k -regulous functions, k ∈ N , on a smooth real affine algebraic variety of dimension d ≥ 2 is not finitely generated. [ABSTRACT FROM AUTHOR]

Abstract In this paper, affine formation control problems of heterogeneous multi-agent systems with linear dynamics are studied. As a novel method, the affine formation control has the ability to handle various movement constraints based on the affine transformation. A proportional-integral (PI)-type control scheme is proposed to ensure the effective affine formation control of the leader-following group with directed interaction graphs and successfully eliminate the steady-state errors. Sufficient conditions for the selection of control parameters are given and proved. Considering that the followers are subjected to the external bounded time-varying disturbances, the designed protocol shows good robustness and can guarantee uniformly ultimate boundedness of affine formation errors. Numerical simulations are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We present a stochastic, agent-based, binary-transaction Asset-Exchange Model (AEM) for wealth distribution that allows for agents with negative wealth. This model retains certain features of prior AEMs such as redistribution and wealth-attained advantage, but it also allows for shifts as well as scalings of the agent density function. We derive the Fokker–Planck equation describing its time evolution and we describe its numerical solution, including a methodology for solving the inverse problem of finding the model parameters that best match empirical data. Using this methodology, we compare the steady-state solutions of the Fokker–Planck equation with data from, inter alia, the United States Survey of Consumer Finances over a time period of 27 years. In doing so, we demonstrate agreement with empirical data with an average error less than 0.16% over this time period. We present the model parameters for the US wealth distribution data as a function of time under the assumption that the distribution responds to their variation adiabatically. We argue that the time series of model parameters thus obtained provides a valuable new diagnostic tool for analyzing wealth inequality. Highlights • We introduce the three-parameter Affine Wealth Model allowing for negative-wealth agents. • We solve the inverse problem to fit the model to given empirical wealth distributions. • We apply the model to U.S. and European wealth data, with fraction-of-a-percent accuracy. • We believe the Affine Wealth Model to be the most accurate wealth model available today. [ABSTRACT FROM AUTHOR]

Abstract We generalize the Poisson–Lie T-duality by making use of the structure of the affine Poisson group which is the concept introduced some time ago in Poisson geometry as a generalization of the Poisson–Lie group. We also introduce a new notion of an affine quasi-Poisson group and show that it gives rise to a still more general T-duality framework. We establish for a class of examples that this new T-duality is compatible with the renormalization group flow. [ABSTRACT FROM AUTHOR]

Abstract We consider the Hecke pair consisting of the group P K + of affine transformations of a number field K that preserve the orientation in every real embedding and the subgroup P O + consisting of transformations with algebraic integer coefficients. The associated Hecke algebra C r ∗ (P K + , P O +) has a natural time evolution σ , and we describe the corresponding phase transition for KMS β -states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost–Connes type system associated to K has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of C r ∗ (P K + , P O +) to a corner in the Bost–Connes system established by Laca, Neshveyev and Trifković, we obtain an arithmetic subalgebra of C r ∗ (P K + , P O +) on which ground states exhibit the 'fabulous' property with respect to an action of the Galois group G (K ab ∕ H + (K)) , where H + (K) is the narrow Hilbert class field. In order to characterize the ground states of the C ∗ -dynamical system (C r ∗ (P K + , P O +) , σ) , we obtain first a characterization of the ground states of a groupoid C ∗ -algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations. [ABSTRACT FROM AUTHOR]

Abstract We prove a version of the affine Kempf–Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T ∗ G , where G is a complex reductive group. [ABSTRACT FROM AUTHOR]

Optimization demands are ubiquitous in science and engineering. The key point is that the approach to tackle a complex optimization problem should not itself be difficult. Differential Evolution (DE) is such a simple method, and it is arguably a very powerful stochastic real-parameter algorithm for single-objective optimization. However, the performance of DE is highly dependent on control parameters and mutation strategies. Both tuning the control parameters and selecting the proper mutation strategy are still tedious but important tasks for users. In this paper, we proposed an enhanced structure for DE algorithm with less control parameters to be tuned. The crossover rate control parameter Cr is replaced by an automatically generated evolution matrix and the control parameter F can be renewed in an adaptive manner during the whole evolution. Moreover, an enhanced mutation strategy with time stamp mechanism is advanced as well in this paper. CEC2013 test suite for real-parameter single objective optimization is employed in the verification of the proposed algorithm. Experiment results show that our proposed algorithm is competitive with several well-known DE variants. [ABSTRACT FROM AUTHOR]

We consider the problem of deciding whether or not a zero locus, X , of multivariate real analytic functions crosses a given r -norm ball in the real n -dimensional affine space. We perform a local study of the problem, and we provide both necessary and sufficient conditions to answer the question. Our conditions derive from the analysis of differential geometric properties of X at the center of the ball. An algorithm to evaluate r -norms distances is proposed. [ABSTRACT FROM AUTHOR]

Affine matrix rank minimization problem is a fundamental problem in many important applications. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. An iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. With the change of the parameter in non-convex fraction function, we could get some much better results, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established. Moreover, we proved that the value of the regularization parameter λ > 0 cannot be chosen too large. Indeed, there exists λ ̄ > 0 such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any λ > λ ̄ . Numerical experiments on matrix completion problems and image inpainting problems show that our method performs effective in finding a low-rank matrix compared with some state-of-art methods. [ABSTRACT FROM AUTHOR]

We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the associated and symplectic symmetric pairs. [ABSTRACT FROM AUTHOR]

In 2006, Gao and Zeng (Gao and Zeng, 2006) gave the free field realizations of highest weight modules over the extended affine Lie algebras g l 2 ( C q ) ̂ for any nonzero q ∈ C . In the present paper, applying the technique of localization to those free field realizations, we construct a class of new weight modules with infinite dimensional weight multiplicities over g l 2 ( C q ) ̂ . We give necessary and sufficient conditions for these modules to be irreducible. In this way, we construct free field realizations for a class of simple weight modules with infinite dimensional weight multiplicities over g l 2 ( C q ) ̂ for any nonzero q ∈ C . [ABSTRACT FROM AUTHOR]

We calculate q -dimension of k th Cartan power of fundamental representation Λ 0 , corresponding to affine root of affine simply laced Kac–Moody algebras, and show that in the limit q → 1 , and with natural renormalization, it is equal to universal partition function of Chern–Simons theory on three-dimensional sphere. [ABSTRACT FROM AUTHOR]

We obtain an explicit upper bound on the torsion of the Picard group of the forms of A k 1 and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of A k 1 to be nontrivial and we give examples of nontrivial forms of A k 1 with trivial Picard groups. [ABSTRACT FROM AUTHOR]

Goodearl and Launois have shown in Goodearl and Launois (2011) that for a log-canonical Poisson bracket on affine space there is no rational change of coordinates for which the Poisson bracket is constant. Our main result is a proof of a conjecture of Michael Shapiro which states that if affine space is given a log-canonical Poisson bracket, then there does not exist any rational change of coordinates for which the Poisson bracket is linear. Hence, log-canonical coordinates can be thought of as the simplest possible algebraic coordinates for affine space with a log-canonical coordinate system. In proving this conjecture we find certain invariants of log-canonical Poisson brackets on affine space which linear Poisson brackets do not have. [ABSTRACT FROM AUTHOR]

On a smooth strictly convex bounded domain in R 2 , we establish some boundary estimates of the solution to the complete hyperbolic affine sphere equation. On the unit ball in R n , we also give some sharp estimates of the solutions of a class of Monge–Ampère equations. As an application, we prove the relative hyperbolic hypersurface constructed by solving two linked Monge–Ampère equations, has constant Gauss–Kronecker curvature and has bounded principal curvatures. [ABSTRACT FROM AUTHOR]

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equal-volume polygons can also be used to estimate projective invariants of a planar curve. This theory has many potential applications, among them evaluation of the quality and computation of affine invariants of silhouette curves. [ABSTRACT FROM AUTHOR]

Improper affine spheres have played an important role in the development of geometric methods for the study of the Hessian one equation. Here, we review most of the advances we have made in this direction during the last twenty years. [ABSTRACT FROM AUTHOR]

Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model geodesically complete surfaces, and that there are also others which do not model any geodesically complete surface. Our main result provides a way of determining whether a given set of constant Christoffel symbols can model a geodesically complete surface. [ABSTRACT FROM AUTHOR]

The Naor–Yung paradigm [63] allows to generically boost security under chosen-plaintext attacks (CPA) to security against chosen-ciphertext attacks (CCA) for public-key encryption (PKE) schemes. The main idea is to encrypt the plaintext twice (under independent public keys), and to append a non-interactive zero-knowledge (NIZK) proof that the two ciphertexts indeed encrypt the same message. Later work by Camenisch, Chandran, and Shoup [32] and Naor and Segev [ 28 , 30 ] established that the very same technique can also be used in the settings of key-dependent message (KDM) and key-leakage attacks (respectively). In this paper we study the conditions under which the two ciphertexts in the Naor–Yung construction can share the same random coins. We find that this is possible, provided that the underlying PKE scheme meets an additional simple property. The motivation for re-using the same random coins is that this allows to design much more efficient NIZK proofs. We showcase such an improvement in the random oracle model, under standard complexity assumptions including Decisional Diffie–Hellman, Quadratic Residuosity, and Subset Sum. The length of the resulting ciphertexts is reduced by 50%, yielding truly efficient PKE schemes achieving CCA security under KDM and key-leakage attacks. As an additional contribution, we design the first PKE scheme whose CPA security under KDM attacks can be directly reduced to (low-density instances of) the Subset Sum assumption. Our PKE scheme supports key-dependent messages computed via any affine function of the secret key. [ABSTRACT FROM AUTHOR]

This paper deals with the stabilization in the sample-and-hold sense of nonlinear, control affine, retarded systems, affected by actuation disturbances and observation errors. Input-to-state stability redesign methods are used in order to design a new sampled-data controller. It is shown that stabilization in the sample-and-hold sense can be preserved by means of this new controller, regardless of the above disturbances and errors. It is assumed that both actuator disturbance and observation error are bounded, and the (arbitrary) bounds are known a-priori . It is moreover assumed that the observation errors do not affect or affect marginally the new control term obtained by input-to-state stability redesign. Simulations on a continuous stirred tank reactor with recycle validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a novel method to estimate dense scene flow using volumetric and probabilistic 3-d models. The method first reconstructs 3-d models at each time step using images synchronously captured from multiple views. Then, the 3-d motion between two consecutive 3-d models is estimated using a formulation that is the analog of Horn and Schunck's optical flow method. This particular choice of 3-d model representation allows estimating highly dense scene flow results, tracking of surfaces undergoing topological change and reliably recovering large motion displacements. The benefits of the method and the accuracy of 3-d flow results are demonstrated on recent multi-view datasets. The second goal of this work is to compress and reconstruct 3-d scenes at various time points using the estimated flow. A new method of scene warping is proposed that involves partitioning the optical flow field in regions of coherent motion which are subsequently parametrized by affine transformations. The compression objective of this work is achieved by the low storage requirements of the affine parameters that describe the optical flow field and the efficient reconstruction method through warping. [ABSTRACT FROM AUTHOR]

We propose a construction of affine space (or “polynomial rings”) over a triangulated category, in the context of stable derivators. [ABSTRACT FROM AUTHOR]

In Thas (2014) it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F 1 ) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over F 1 such as combinatorial F 1 -projective and F 1 -affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit. , and show that Deitmar schemes which are defined by finite trees (with possible end points) are “defined over F 1 ” in Kurokawa’s sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated F q -schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara’s zeta function (which is trivial in this case). Using a process called “surgery,” we show that one can determine the zeta function of a general loose graph and its associated {Deitmar, Grothendieck}-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of F 1 -schemes along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara’s zeta function for graphs in a number of examples. [ABSTRACT FROM AUTHOR]

A particular case ( ∠ ( n , n ̂ ) = π 2 ) of the classical Bäcklund theorem is extended to surfaces in affine space. We consider a pair of non-degenerate surfaces f and f ̂ such that for every p both tangent planes f ∗ ( T p M ) and f ̂ ∗ ( T p M ) contain the line R ( f ̂ ( p ) − f ( p ) ) , moreover the affine normal vector field ξ of f is tangent to f ̂ ( M ) , the affine normal vector field ξ ̂ of f ̂ is tangent to f ( M ) , the volume of the parallelopipedon spanned by f ̂ − f , ξ , ξ ̂ is constant and the affine fundamental forms are proportional. We prove that under the conditions stated above Blaschke connections of both f and f ̂ are locally symmetric. The considered situation differs from that in the theorem of Chern and Terng, where the affine normals at corresponding points were supposed to be parallel. Depending on dim im R which is shown to be equal to dim im R ̂ , either we deal with such surfaces as in the classical theorems for an appropriate scalar or pseudoscalar product in R 3 , or the connections are non-metrizable, hence are not met with in the classical case. A particular metrizable case corresponds to some new result for a pair of surfaces in Minkowski space, when one surface is timelike and the other is spacelike. [ABSTRACT FROM AUTHOR]

The notion of Cameron–Liebler line classes was generalized in Rodgers et al. (0000) to Cameron–Liebler k -classes, where k = 1 corresponds to the line classes. Such a set consists of x 2 k + 1 k q subspaces of dimension k in PG ( 2 k + 1 , q ) where k ≥ 1 and x ≥ 0 are integers such that every regular k -spread of PG ( 2 k + 1 , q ) contains exactly x subspaces from the set. Examples are known for x ≤ 2 . The authors of Rodgers et al. (0000) show that there are no Cameron–Liebler k -classes when k = 2 and 3 ≤ x ≤ q , or when 3 ≤ k ≤ q log q − q and 3 ≤ x ≤ q ∕ 2 3 . We improve these results by weakening the condition on the upper bound for x to a bound that is linear in q . For this, we use a technique that was originally used to extend nets to affine planes. [ABSTRACT FROM AUTHOR]

By using the link between the affine and the power norming, eight Hill estimators under power normalization for the tail index (the non-zero extreme value index) are suggested. Moreover, more compact and adaptive four Hill estimators under power normalization are derived based on the generalized Pareto distributions under power normalization. Two classes of harmonic t-Hill estimators under power normalization are also suggested. A comprehensive simulation study using the R-package shows that all the suggested estimators under power normalization work well, but in all cases the Hill estimators under power normalization based on Pareto distributions under power normalization are better. The two models under linear and power normalization for extreme value analysis are applied with comparison on a real data set of two pollutants, Sulphur Dioxide and Particulate Matter. [ABSTRACT FROM AUTHOR]

Among the various critical systems that are worth to be formally analyzed, a wide set consists of controllers for dynamical systems. Those programs typically execute an infinite loop in which simple computations update internal states and produce commands to update the system state. Those systems are yet hardly analyzable by available static analysis method, since, even if performing mainly linear computations, the computation of a safe set of reachable states often requires quadratic invariants. In this paper we consider the general setting of a piecewise affine program; that is a program performing different affine updates on the system depending on some conditions. This typically encompasses linear controllers with saturations or controllers with different behaviors and performances activated on some safety conditions. Our analysis is inspired by works performed a decade ago by Johansson et al., and Morari et al., in the control community. We adapted their method focused on the analysis of stability in continuous-time or discrete-time settings to fit the static analysis paradigm and the computation of invariants, that is over-approximation of reachable sets using piecewise quadratic Lyapunov functions. This approach has been further extended to consider k -inductive properties of reachable traces (trajectories) of systems. The analysis has been implemented in Matlab and shown very good experimental results on a very large set of synthesized problems. [ABSTRACT FROM AUTHOR]

Affine invariant measures are powerful tools to develop robust shape descriptors that can be applied, for example, to shape matching, shape retrieval, or symmetry detection problems. In this work we introduce estimators for the affine structure of surfaces represented by triangle meshes, i.e. affine co-normal and normal vectors, affine curvature tensors, affine mean and Gaussian curvatures, and affine principal directions and curvatures. The proposed method estimates the affine normal using a finite differences scheme together with a least-squares approximation, followed by a weighted average strategy to approach discrete affine curvature tensors. When compared to the exact geometric measures of analytic models, experiments on regular meshes obtain small error, which decreases for finer meshes, and outperforms the state-of-the-art method in some cases. Experiments to evaluate affine invariance show that the difference between measures before and after equi-affine transformations remains small even after large deformations. [ABSTRACT FROM AUTHOR]

An affine finite element scheme approximation of a time dependent linear convection–diffusion–reaction problem in 2D and 3D is presented. For these equations which do not satisfy an underlying maximum principle, sufficient conditions are given in terms of the coefficient functions, the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation. Numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation. [ABSTRACT FROM AUTHOR]

Let r and n be positive integers such that r < n , and K be an arbitrary field. We determine the maximal dimension for an affine subspace of n by n symmetric (or alternating) matrices with entries in K and with rank less than or equal to r . We also classify, up to congruence, the subspaces of maximal dimension among them. This generalizes earlier results of Meshulam, Loewy and Radwan that were previously known only for linear subspaces over fields with large cardinality and characteristic different from 2. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though, Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin (O) as the group of collineations that preserve the axis of the plane. [ABSTRACT FROM AUTHOR]

Let us denote by K n the hyperspace of all convex bodies of R n equipped with the Hausdorff distance topology. An affine invariant point p is a continuous and Aff ( n ) -equivariant map p : K n → R n , where Aff ( n ) denotes the group of all nonsingular affine maps of R n . For every K ∈ K n , let P n ( K ) = { p ( K ) ∈ R n | p is an affine invariant point } and F n ( K ) = { x ∈ R n | g x = x for every g ∈ Aff ( n ) such that g K = K } . In 1963, B. Grünbaum conjectured that P n ( K ) = F n ( K ) [3] . After some partial results, the conjecture was recently proven in [6] . In this short note we give a rather different, simpler and shorter proof of this conjecture, based merely on the topology of the action of Aff ( n ) on K n . [ABSTRACT FROM AUTHOR]

We prove some general stability results for a family of equations, which generalizes the equation of p -Wright affine functions. In this way we obtain some hyperstability properties for those equations, as well. We also provide some applications of those outcomes in proving inequalities characterizing the inner product spaces and stability of *-homomorphisms of C * -algebras. The main tool in the proofs is a fixed point result in Brzdȩk, Chudziak, Páles (2011). [ABSTRACT FROM AUTHOR]

We study flatness of two-input control-affine systems, defined on an n -dimensional state-space. We give a complete geometric characterization of systems that become static feedback linearizable after a one-fold prolongation of a suitably chosen control. They form a particular class of flat systems: they are of differential weight n + 3. We give normal forms compatible with the minimal flat outputs and provide a system of first order PDE׳s to be solved in order to find all minimal flat outputs. We illustrate our results by two examples: the induction motor and the polymerization reactor. [ABSTRACT FROM AUTHOR]

Expected values of max-affine expressions appear in optimization problems for various stochastic systems, such as in model predictive control (MPC) for stochastic max-plus-linear systems, in identification of stochastic max-plus-linear systems, and in control of stochastic monotonic piecewise affine systems. Solving these optimization problems involves the computation of the expected value of the maximum of affine expressions, which will then appear in the objective function or in the constraints. The computation of this expected value can be highly complex and expensive, which also results in a high computation time to solve the optimization problem. Therefore, the focus of this paper is on decreasing the computational complexity of the calculation of these expected values. To this end, we use an approximation method based on the moments of a random variable. We illustrate in an example that this method results in a much lower computation time and a much lower computational complexity than the existing computational methods while still guaranteeing a performance that is comparable to the performance of those methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The new results are proved by a combination of averaging procedures by highly oscillating controls with the notion of kinematic reduction. [ABSTRACT FROM AUTHOR]

Based on affine maps in geometry, we study the geodesic-affine maps on Riemannian manifolds P n of complex positive definite matrices that are induced by different so-called kernel functions. In this article, we are going to describe the structure of all continuous bijective geodesic-affine maps on these manifolds. We also prove that geodesic distance isometries are geodesic-affine maps. Moreover, the forms of all bijective maps which preserve norms of geodesic correspondence are characterized. Indeed, these maps are special examples of geodesic-affine maps. [ABSTRACT FROM AUTHOR]

We determine all left-invariant Ricci collineations on three-dimensional Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the third weight of generalized Reed–Muller codes. Using results from Erickson, we prove under some condition that the third weight of generalized Reed–Muller codes depends on the third weight of generalized Reed–Muller codes of small order with two variables. We are able to determine the third weight and the third weight codewords of generalized Reed–Muller codes of order r = a ( q − 1 ) + b with 2 ≤ b ≤ q − 1 , for small b . [ABSTRACT FROM AUTHOR]

In this paper, we propose an modified nonlinear extended state observer (ESO) with a time-varying gain in active disturbance rejection control (ADRC) to deal with a class of nonlinear systems which are essentially normal forms of general affine nonlinear systems. The total disturbance which includes unknown dynamics of the system, external disturbance, and unknown part of the control coefficient is estimated through ESO and is canceled in nonlinear feedback loop. The practical stability for the resulting closed-loop is obtained. It is shown that the “peaking value” occurred often in the constant high gain design can be significantly reduced by the time-varying gain approach. [ABSTRACT FROM AUTHOR]

The von Neumann–Halperin method of alternating projections converges strongly to the projection of a given point onto the intersection of finitely many closed affine subspaces. We propose acceleration schemes making use of two ideas: Firstly, each projection onto an affine subspace identifies a hyperplane of codimension 1 containing the intersection, and secondly, it is easy to project onto a finite intersection of such hyperplanes. We give conditions for which our accelerations converge strongly. Finally, we perform numerical experiments to show that these accelerations perform well for a matrix model updating problem. [ABSTRACT FROM AUTHOR]

Properties of relative traces and symmetrizing forms on chains of cyclotomic and affine Hecke algebras are studied. The study relies on the use of bases of these algebras which generalize a normal form for elements of the complex reflection groups G ( m , 1 , n ) , m = 1 , 2 , … , ∞ , constructed by a recursive use of the Coxeter–Todd algorithm. Formulas for inducing, from representations of an algebra in the chain, representations of the next member of the chain are presented. [ABSTRACT FROM AUTHOR]

In a geometry, a maximal cap is a collection of points of largest size no three of which are collinear. In A G ( 4 , 3 ) , maximal caps contain 20 points; the 81 points of A G ( 4 , 3 ) can be partitioned into 4 mutually disjoint maximal caps together with a single point P , where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of A G ( 4 , 3 ) under the action by affine transformations. [ABSTRACT FROM AUTHOR]