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Following ideas elaborated by Hering (Grundzüge der Lehre vom Lichtsinn, Springer, Berlin, 1920) in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of "natural space", i.e., space as it is conceived by us. Selz's thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (Zeitschrift für Psychologie 114:351–362, 1930a, p. 358ff) tries to derive within his framework two of Hilbert's axioms for Euclidean geometry. Such derivations (if successful) would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz's theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert's axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz's geometry of natural space. The question of the logical independence of these principle is not investigated. [ABSTRACT FROM AUTHOR]

Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance. [ABSTRACT FROM AUTHOR]

This paper is a survey about cylinders in Fano varieties and related problems. [ABSTRACT FROM AUTHOR]

Introduction. One of the tasks arising in cryptography is to ensure a safe and fair conduct of e-voting. This paper details the algorithm of electronic elections particularly that part which deals with the cryptographic security. Materials and Methods. The results are obtained on the basis of the following methodology: finite field theory, projective geometry, and linear algebra. The developed cryptosystem is based on the application of geometric objects from projective geometry over finite fields. Research Results. The invented algorithm relies on the ElGamal encryption and a new geometric way of secret sharing among election committees. The proposed method uses some features of affine spaces over finite fields to generate special geometric constructions and secret, search of which is a complex algorithmic task for an illegal intruder. The threshold secret sharing is used to prevent voter fraud on the part of the members of election committees. The probability to generate the right share of secret by an illegal intruder in case when he/she knows only a part of secret shares is determined. Discussion and Conclusions. The described scheme is useful for electronic voting and in other spheres where methods of threshold cryptography are applied.

Bernstein problem for affine maximal type equation (0.1) u i j D i j w = 0 , w ≡ det ⁡ D 2 u − θ , ∀ x ∈ Ω ⊂ R N has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C 4 -hypersurface in R N + 1 must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N = 2 and θ = 3 / 4 , and later extended by Jia-Li (2009) [12] to N = 2 , θ ∈ (3 / 4 , 1 (see also Zhou (2012) [16] for a different proof). On the past twenty years, many efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N = 3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean compete for N ≥ 3 , θ ∈ (1 / 2 , (N − 1) / N). [ABSTRACT FROM AUTHOR]

The geometrical formulation of gravity is not unique and can be set up in a variety of spacetimes. Even though the gravitational sector enjoys this freedom of different geometrical interpretations, consistent matter couplings have to be assured for a steady foundation of gravity. In generalised geometries, further ambiguities arise in the matter couplings unless the minimal coupling principle (MCP) is adopted that is compatible with the principles of relativity, universality and inertia. In this work, MCP is applied to all standard model gauge fields and matter fields in a completely general (linear) affine geometry. This is also discussed from an effective field theory perspective. It is found that the presence of torsion generically leads to theoretical problems. However, symmetric teleparallelism, wherein the affine geometry is integrable and torsion-free, is consistent with MCP. The generalised Bianchi identity is derived and shown to determine the dynamics of the connection in a unified fashion. Also, the parallel transport with respect to a teleparallel connection is shown to be free of second clock effects. [ABSTRACT FROM AUTHOR]

In example‐based inverse linear blend skinning (LBS), a collection of poses (e.g. animation frames) are given, and the goal is finding skinning weights and transformation matrices that closely reproduce the input. These poses may come from physical simulation, direct mesh editing, motion capture or another deformation rig. We provide a re‐formulation of inverse skinning as a problem in high‐dimensional Euclidean space. The transformation matrices applied to a vertex across all poses can be thought of as a point in high dimensions. We cast the inverse LBS problem as one of finding a tight‐fitting simplex around these points (a well‐studied problem in hyperspectral imaging). Although we do not observe transformation matrices directly, the 3D position of a vertex across all of its poses defines an affine subspace, or flat. We solve a 'closest flat' optimization problem to find points on these flats, and then compute a minimum‐volume enclosing simplex whose vertices are the transformation matrices and whose barycentric coordinates are the skinning weights. We are able to create LBS rigs with state‐of‐the‐art reconstruction error and state‐of‐the‐art compression ratios for mesh animation sequences. Our solution does not consider weight sparsity or the rigidity of recovered transformations. We include observations and insights into the closest flat problem. Its ideal solution and optimal LBS reconstruction error remain an open problem. [ABSTRACT FROM AUTHOR]

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions. [ABSTRACT FROM AUTHOR]

In this paper, we study the isoperimetric-type equalities for rosettes, i.e. regular closed planar curves with nonvanishing curvature. We find the exact relations between the length and the oriented area of rosettes based on the oriented areas of the Wigner caustic, the Constant Width Measure Set and the Spherical Measure Set. We also study and find new results about the geometry of affine equidistants of rosettes and of the union of rosettes. [ABSTRACT FROM AUTHOR]

During evolution, living systems, actively interacting with their environment, developed the ability, through sensorimotor contingencies, to construct functional spaces shaping their perception and their movements. These geometries were modularly embedded in specific functional neuro-architectures. In particular, human movements were shown to obey several empirical laws, such as the 2/3 power law, isochrony, or jerk minimization principles, which constrain and adapt motor planning and execution. Outstandingly, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partly detected in several brain areas including the cerebellum and the basal ganglia. Reviving Pellionisz and Llinas general hypothesis regarding the cerebrum and the cerebellum as geometric machines, we speculate that the cerebellum should be involved in implementing and/or selecting task-specific geometries for motor and cognitive skills. More precisely, the cerebellum is assumed to compute forward internal models to help specific cortical and subcortical regions to select appropriate geometries among, at least, Euclidean and affine geometries. We emphasize that the geometrical role of the cerebellum deserves a renewal of interest, which may provide a better understanding of its specific contributions to motor and associative (cognitive) functions. [ABSTRACT FROM AUTHOR]

Lind et al. (Journal of Experimental Psychology: Human Perception and Performance, 40 (1), 83, 2014) proposed a bootstrap process that used right angles on 3D relief structure, viewed over sufficiently large continuous perspective change, to recover the scaling factor for metric shape. Wang, Lind, and Bingham (Journal of Experimental Psychology: Human Perception and Performance, 44(10), 1508-1522, 2018) replicated these results in the case of 3D slant perception. However, subsequent work by the same authors (Wang et al., 2019) suggested that the original solution could be ineffective for 3D slant and presented an alternative that used two equidistant points (a portion of the original right angle). We now describe a three-step stratified process to recover 3D slant using this new solution. Starting with 2D inputs, we (1) used an existing structure-from-motion (SFM) algorithm to derive the object's 3D relief structure and (2) applied the bootstrap process to it to recover the unknown scaling factor, which (3) was then used to produce a slant estimate. We presented simulations of results from four previous experiments (Wang et al., 2018, 2019) to compare model and human performance. We showed that the stratified process has great predictive power, reproducing a surprising number of phenomena found in human experiments. The modeling results also confirmed arguments made in Wang et al. (2019) that an axis of mirror symmetry in an object allows observers to use the recovered scaling factor to produce an accurate slant estimate. Thus, poor estimates in the context of a lack of symmetry do not mean that the scaling factor has not been recovered, but merely that the direction of slant was ambiguous. [ABSTRACT FROM AUTHOR]

Empirical studies have always shown 3-D slant and shape perception to be inaccurate as a result of relief scaling (an unknown scaling along the depth direction). Wang, Lind, and Bingham (Journal of Experimental Psychology: Human Perception and Performance, 44(10), 1508–1522, 2018) discovered that sufficient relative motion between the observer and 3-D objects in the form of continuous perspective change (≥45°) could enable accurate 3-D slant perception. They attributed this to a bootstrap process (Lind, Lee, Mazanowski, Kountouriotis, & Bingham in Journal of Experimental Psychology: Human Perception and Performance, 40(1), 83, 2014) where the perceiver identifies right angles formed by texture elements and tracks them in the 3-D relief structure through rotation to extrapolate the unknown scaling factor, then used to convert 3-D relief structure to 3-D Euclidean structure. This study examined the nature of the bootstrap process in slant perception. In a series of four experiments, we demonstrated that (1) features of 3-D relief structure, instead of 2-D texture elements, were tracked (Experiment 1); (2) identifying right angles was not necessary, and a different implementation of the bootstrap process is more suitable for 3-D slant perception (Experiment 2); and (3) mirror symmetry is necessary to produce accurate slant estimation using the bootstrapped scaling factor (Experiments 3 and 4). Together, the results support the hypothesis that a symmetry axis is used to determine the direction of slant and that 3-D relief structure is tracked over sufficiently large perspective change to produce metric depth. Altogether, the results supported the bootstrap process. [ABSTRACT FROM AUTHOR]

In this paper, the Orlicz version of the classical dual Cauchy-Kubota formula is given and the concept of dual affine quermassintegrals is extended to dual Orlicz mixed affine quermassintegrals in the framework of Orlicz Brunn-Minkowski theory. Some inequalities for dual Orlicz mixed affine quermassintegrals are obtained, such as dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality. [ABSTRACT FROM AUTHOR]

We prove the main result that a groupoid of order n is an idempotent and k-translatable quasigroup if and only if its multiplication is given by x · y = (a x + b y) (mod n) , where a + b = 1 (mod n) , a + b k = 0 (mod n) and (k , n) = 1 . We describe the structure of various types of idempotent k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent k-translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical. [ABSTRACT FROM AUTHOR]

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An affine manifold (M, ∇) is Osserman if the eigenvalues of the affine Jacobi operators vanishe. In this paper, explicit examples of affine Osserman connections on 3 and 4-manifolds are constructed and their applications are given. [ABSTRACT FROM AUTHOR]

This article describes a method for calculating arithmetic, geometric and harmonic means of two numbers and how they can be represented geometrically. We extend these mean values to arithmetic, geometric and harmonic thirds, fourths, etc. For this we will only use the tools of the affine planar geometry. Also, we will make allusion to the more general interpretation in the projective plane. From the relations between these means we can deduce a multitude of recursive formulas for n-th root calculation and represent them by geometric constructions. These formulas give a solution for reducing the power of the root. Surprisingly, one of these algorithms turns out to be the same as the one using Newton's tangent method for calculating zero values of functions of the form f(x) = xn-c, but obtained without use of analysis. Moreover, regarding speed of convergence these algorithms are faster than Newton's tangent method. This geometric interpretation of mean values and root calculation fits into the larger context of affine geometry, where we use multi-projections as generating transformations for building up all the affine transformations. Our focus will primarily be on mean values and roots. [ABSTRACT FROM AUTHOR]

This paper extends the affine class of term structure models to describe the joint dynamics of exchange rates and interest rates. In particular, the issue of how to reconcile the low volatility of interest rates with the high volatility of exchange rates is addressed. The incomplete market approach of introducing exchange rate volatility that is orthogonal to both interest rates and the pricing kernels is shown to be infeasible in the affine setting. Models in which excess exchange rate volatility is orthogonal to interest rates but not orthogonal to the pricing kernels are proposed and validated via Kalman filter estimation of maximal 5-factor models for 6 country pairs. [ABSTRACT FROM AUTHOR]

• Engagement of octagonal lattice-based triangulated feature characterization, which is the first of its kind. • The inherent congruency of the geometrical arrangement makes the descriptor robust to numerous image transformations. • The nature of the second-order derivatives capture keenly high-frequency information such as edges, corners, and points. • Thorough investigations on benchmark shape datasets demonstrate the contribution's superiority. • Complexity analysis reveals the simplicity of the OLTT model signifying its compatibility with real scenarios. Erstwhile shape description schemes lack primarily in establishing trade-offs with accuracy and computational load. Accordingly, a lightweight shape descriptor offering precise definition and compaction of high-frequency features is contributed in this paper using a simple geometrical shape for localization and shape characterization. Initially, the input image is octagonally tessellated and triangularly decomposed into sub-regions whose side-wise differences are evaluated and subjected to second-order differentiation to produce three high-frequency values representing triangle corners. The resultant is processed by the law of sines to yield localized shape features exhibiting congruence and is reiterated on the residual regions, followed by a novel octal encoding scheme encompassing maximal variations in the localized regions. The resulting features are globally fabricated into shape histograms in a non-overlapping manner representing the shape vector. This scheme validated on widely popular benchmark shape datasets demonstrates superior retrieval and recognition accuracies greater than 93% which is lacking in its competitors. [ABSTRACT FROM AUTHOR]

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]

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes. [ABSTRACT FROM AUTHOR]

In the recently introduced gauge theory of translations, dubbed Coincident General Relativity (CGR), gravity is described with neither torsion nor curvature in the spacetime affine geometry. The action of the theory enjoys an enhanced symmetry and avoids the second derivatives that appear in the conventional Einstein–Hilbert action. While it implies the equivalent classical dynamics, the improved action principle can make a difference in considerations of energetics, thermodynamics and quantum theory. This paper reports on possible progress in those three aspects of gravity theory. In the so-called purified gravity, (1) energy–momentum is described locally by a conserved, symmetric tensor, (2) the Euclidean path integral is convergent without the addition of boundary or regulating terms and (3) it is possible to identify a canonical frame for quantization. [ABSTRACT FROM AUTHOR]

A gravity theory without masses can be constructed in Minkowski spaces using a geometric Minkowski potential. The related affine spacelike spheres can be seen as the regions of the Minkowski spacelike vectors characterized by a constant Minkowski gravitational potential. These spheres point out, for each dimension n ≥ 3 , spacetime models, the de Sitter ones, which satisfy Einstein's field equations in absence of matter. In other words, it is possible to generate geometrically the cosmological constant. Even if a lot of possible parameterizations have been proposed, each one highlighting some geometric and physical properties of the de Sitter space, we present here a new natural parameterization which reveals the intrinsic geometric nature of cosmological constant relating it with the invariant affine radius coming from the so-called Minkowski–Tzitzeica surfaces theory. [ABSTRACT FROM AUTHOR]

Every affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object, its connection. Using this description of the local geometry of an affine homogeneous space we construct a variety M( glV ), which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces. Infinitesimal deformations of an isometry class of affine homogeneous spaces in this moduli space are described by the Spencer cohomology of a comodule associated to a point in M∞( glV ). In an appendix we discuss the relevance of this construction to the study of locally homogeneous spaces. [ABSTRACT FROM AUTHOR]

Abstract A tame ideal is an ideal I ⊂ k [ x 1 , ... , x n ] such that the blowup of the affine space A k n along I is regular. In this paper, we give a combinatorial characterization of tame squarefree monomial ideals. More precisely, we show that a square free monomial ideal is tame if and only if the corresponding clutter is a complete d -partite d -uniform clutter. Equivalently, a squarefree monomial ideal is tame, if and only if the facets of its Stanley–Reisner complex have mutually disjoint complements. Also, we characterize all monomial ideals generated in degree at most 2 which are tame. Finally, we prove that tame squarefree ideals are of fiber type. [ABSTRACT FROM AUTHOR]

In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains. [ABSTRACT FROM AUTHOR]

We showthat all complements of cuspidal hyperplane sections of smooth projective cubic surfaces have isomorphic A²-cylinders. As a consequence, we derive that the A²-cancellation problem fails in every dimension greater than or equal to 2. [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]

We consider the quantum affine vertex algebra Vc(glN) associated with the rational R-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras Ac(glN) of the completed double Yangian DY~c(glN) at the level c∈C, associated with the reflection equation, and we employ their structure to construct examples of quasi Vc(glN)-modules. Finally, we use the quasi module map, together with the explicit description of the center of Vc(glN), to obtain formulae for families of central elements in the completed algebra A~c(glN). [ABSTRACT FROM AUTHOR]

Define on the set G:=R+×R the operation (t,a)∗(s,b)=(ts,tb+a). (G,∗) is a non-commutative group with the neutral element (1, 0). We consider a non-commutative translation equation F(η,F(ξ,x))=F(η∗ξ,x), η,ξ∈G, x∈I, F(1,0)=id, where I is an open interval and F:G×I→I is a continuous mapping. This equation can be written in the form: F((t,a),F((s,b),x))=F((ts,tb+a),x), t,s∈R+, x∈I. For t=1 the family {F(t,a)} defines an additive iteration group, however for a=0 it defines a multiplicative iteration group. We show that if F(t, 0) for some t≠1 has exactly one fixed point xt, (F(t,0)-id)(xt-id)≥0 and for an a>0F(1,a)>id, then there exists a unique homeomorphism φ:I→R such that F((s,b),x)=φ-1(sφ(x)+b) for s∈R+ and b∈R. [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]

Abstract We study 321-avoiding affine permutations, and prove a formula for their enumeration with respect to the inversion number by using a combinatorial approach. This is done in two different ways, both related to Viennot's theory of heaps. First, we encode these permutations using certain heaps of monomers and dimers. This method specializes to the case of affine involutions. For the second proof, we introduce periodic parallelogram polyominoes, which are new combinatorial objects of independent interest. We enumerate them by extending the approach of Bousquet-Mélou and Viennot used for classical parallelogram polyominoes. We finally establish a connection between these new objects and 321-avoiding affine permutations. [ABSTRACT FROM AUTHOR]

Abstract We obtain, in any dimension N and for a large range of values of θ , a Bernstein theorem for the fourth-order partial differential equation of affine maximal type u i j D i j w = 0 , w = [ det ⁡ D 2 u ] − θ assuming the completeness of Calabi's metric. This contains the results of Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] for affine maximal equations and of Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] for Abreu's equation. In particular, we extend the result of Zhou from 2 ≤ N ≤ 4 to 2 ≤ N ≤ 5. [ABSTRACT FROM AUTHOR]

Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view. [ABSTRACT FROM AUTHOR]

The classification of affine line bundles on a compact complex space is a difficult problem. We study the affine analogue of the Picard functor and the representability problem for this functor. Let X be a compact complex space with H0(OX) = ℂ. Fix c ∈ NS(X), x0 ∈ X. We introduce the affine Picard functor Picaff...c : Anop → Set which assigns to a complex space T the set of families of linearly X0-framed affine line bundles on X parameterized by T. Our main result states that the functor Picaff...c is representable if and only if the map h0 : Pic(X) → ℕ is constant. If this is the case, the space which represents this functor is a linear space over Picc(X) whose underlying set ... H¹..., where ... is a Poincaré line bundle normalized at x0. The main idea idea of the proof is to compare the representability of Picaff...c to the representability of a functor considered by Bingener related to the deformation theory of p-cohomology classes. Our arguments show in particular that, for p = 1, the converse of Bingener's representability criterion holds. [ABSTRACT FROM AUTHOR]