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We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k -differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics. [ABSTRACT FROM AUTHOR]

In the conventional formulation of general relativity, gravity is represented by the metric curvature of Riemannian geometry. There are also alternative formulations in flat affine geometries, wherein the gravitational dynamics is instead described by torsion and nonmetricity. These so-called general teleparallel geometries may also have applications in material physics, such as the study of crystal defects. In this work, we explore the general teleparallel geometry in the language of differential forms. We discuss the special cases of metric and symmetric teleparallelisms, clarify the relations between formulations with different gauge fixings and without gauge fixing, and develop a method of recasting Riemannian into teleparallel geometries. As illustrations of the method, exact solutions are presented for the generic quadratic theory in 2, 3 and 4 dimensions. [ABSTRACT FROM AUTHOR]

In this paper we consider discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal edges, while some of the singular vertices can be considered as discrete swallowtails. The classification of singularities of discrete nets is quite a difficult task, and our results can be seen as a first step in this direction. We also prove some characterizations of ruled discrete improper affine spheres which are analogous to the smooth case. [ABSTRACT FROM AUTHOR]

For neurological disorders and diseases, functional and anatomical connectomes of the human brain can be used to better inform targeted interventions and treatment strategies. Functional magnetic resonance imaging (fMRI) is a non-invasive neuroimaging technique that captures spatio-temporal brain function through change in blood-oxygen-level-dependent (BOLD) signals over time. FMRI can be used to study the functional connectome through the functional connectivity matrix; that is, Pearson's correlation matrix between time series from the regions of interest of an fMRI image. One approach to analysing functional connectivity is using partial least squares (PLS), a multivariate regression technique designed for high-dimensional predictor data. However, analysing functional connectivity with PLS ignores a key property of the functional connectivity matrix; namely, these matrices are positive definite. To account for this, we introduce a generalisation of PLS to Riemannian manifolds, called R-PLS, and apply it to symmetric positive definite matrices with the affine invariant geometry. We apply R-PLS to two functional imaging datasets: COBRE, which investigates functional differences between schizophrenic patients and healthy controls, and; ABIDE, which compares people with autism spectrum disorder and neurotypical controls. Using the variable importance in the projection statistic on the results of R-PLS, we identify key functional connections in each dataset that are well represented in the literature. Given the generality of R-PLS, this method has the potential to investigate new functional connectomes in the brain, and with future application to structural data can open up further avenues of research in multi-modal imaging analysis. [ABSTRACT FROM AUTHOR]

The affine quermassintegrals associated to a convex body in \mathbb {R}^n are affine-invariant analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, and thus constitute a central pillar of Affine Convex Geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the k-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases k=1 and k=n-1 correspond to the classical Blaschke–Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of k=1,\ldots,n-1, in a single unified framework. In fact, it turns out that ellipsoids are the only local minimizers with respect to the Hausdorff topology. For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty's inequality is interpreted as an integrated form of a generalized Blaschke–Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general L^p-moment quermassintegrals, and interpret the case p=0 as a sharp averaged Loomis–Whitney isoperimetric inequality. [ABSTRACT FROM AUTHOR]

Jan von Plato proposed in 1998 an intuitionist axiomatization of ordered affine geometry consisting of 22 axioms. It is shown that axiom I.7, which is equivalent to a conjunction of four statements, two of which are redundant, can be replaced with a simpler axiom, which is von Plato's Theorem 3.10. [ABSTRACT FROM AUTHOR]

Materials that exhibit synaptic properties are a key target for our effort to develop computing devices that mimic the brain intrinsically. If successful, they could lead to high performance, low energy consumption, and huge data storage. A 2D square array of engineered nanoparticles (ENPs) interconnected by an emergent polymer network is a possible candidate. Its behavior has been observed and characterized using coarse-grained molecular dynamics (CGMD) simulations and analytical lattice network models. Both models are consistent in predicting network links at varying temperatures, free volumes, and E-field ( E ⃗ ) strengths. Hysteretic behavior, synaptic short-term plasticity and long-term plasticity—necessary for brain-like data storage and computing—have been observed in CGMD simulations of the ENP networks in response to E-fields. Non-volatility properties of the ENP networks were also confirmed to be robust to perturbations in the dielectric constant, temperature, and affine geometry. [ABSTRACT FROM AUTHOR]

We propose to use a particular case of Kaniadakis' logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis' logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine setup suggests a rationale for choosing a specific divergence, which we name the Kaniadakis divergence. [ABSTRACT FROM AUTHOR]

We study subgroups of SL(3,R)⋉R3$\mathrm{SL}(3,\mathbb {R})\ltimes \mathbb {R}^3$ obtained by adding a translation part to the holonomy of a finite‐volume convex projective surface. Under a natural condition on the translations added to the peripheral parabolic elements, we show that the affine action of the group on R3$\mathbb {R}^3$ has convex domains of discontinuity which are regular, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all such domains arising from a fixed group and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature (CAGC). The proof is based on the analysis of CAGC surfaces developed in a previous work, along with a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. We also show that the moduli space of such groups is a vector bundle over the moduli space of finite‐volume convex projective structures, with rank equal to the dimension of the Teichmüller space. [ABSTRACT FROM AUTHOR]

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in the 1950s and 1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as causal geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard–Cartan theorem of Riemannian geometry to the affine manifold context, as well as a new version of the so-called Lorentzian Hadamard–Cartan theorem using weaker assumptions than global hyperbolicity and timelike 1-connectedness required in the extant version. We also include a general description of pseudoconvexity and disprisonment of broad classes of geodesics in terms of suitable restrictions of the exponential map. The latter description sheds further light on the relation between pseudoconvexity and disprisonment of a given such class on the one hand, and geodesic connectedness by members of that class on the other. [ABSTRACT FROM AUTHOR]

Using the vectors and symmetry of affine geometry induced by the ternary quasigroup satisfying the para-associative laws, we found the conditions under which such quasigroup becomes a ternary group. The obtained results also give a simple characterization of semiabelian n-ary groups. [ABSTRACT FROM AUTHOR]

In this work, we give a detailed description and discussion of the dynamic gravitational equations of the model with Lagrangian of the type ∫ det R μ ν d 4 x as proposed by Eddington time ago but with R μ ν being a non-Riemannian generalization of the Ricci tensor with the end to find the geometrical origin of the Eddington and Weyl conjectures concerning Lagrangian densities (generalized volume) and natural gauge. The Ricci tensor in our case is particularly based on an affine geometry with a generalized compatibility condition previously proposed in [B. McInnes, On the geometrical interpretation of 'non-symmetric' space-time field structures, Class. Quantum Grav. 1 (1984) 105–113; D. J. Cirilo-Lombardo, Non-Riemannian geometry, Born–Infeld models and trace-free gravitational equations, J. High Energy Astrophys. 16 (2017) 1–14]. Specifically, we show that: (i) the geometric action can be taken to a BI-type form considering a totally antisymmetric torsion field, (ii) Weyl's proposal considering a universal gauge linked to a cosmological constant λ appears in the model naturally due to the proposed affine geometry, (iii) the Eddington conjecture that establishes a relationship between metric and curvature or fundamental tensor with constant of proportionality λ (natural gauge) is geometrically verified in the model with generalized affine geometry. [ABSTRACT FROM AUTHOR]

The article introduces various geometric concepts, including affine and projective geometries, as well as spherical and conic geometries, and discusses the relationship between these geometries and fields. It also poses a question about bijections between points in projective spaces that map algebraic curves to algebraic curves and explores different answers depending on the field involved, such as finite fields, real fields, and algebraically closed fields of characteristic 0.

In teleparallel geometries, symmetries are represented by affine frame symmetries that constrain both the (co)frame basis and the spin-connection (which are the primary geometric objects). In this paper, we shall study teleparallel geometries with a single affine symmetry, utilizing the locally Lorentz covariant approach and adopting a complex null gauge. We first introduce an algorithm to study geometries with an affine frame symmetry, which consists of choosing coordinates adapted to the symmetry, constructing a canonical frame, and solving the equations describing the symmetry. All of the constraints on the geometry are determined in the case of a single affine symmetry, but there are additional constraints arising from the field equations for a given theory of teleparallel gravity. In particular, we find that in f(T) teleparallel gravity there will be severe constraints on the geometry arising from the antisymmetric part of the field equations. [ABSTRACT FROM AUTHOR]

In this paper, we give a counterexample to Zariski's cancellation problem for principal ${\mathbb{G}}_{a}$ G a -bundles over certain open subvarieties of non- $\mathbb{A}^{1}$ A 1 -uniruled affine varieties. [ABSTRACT FROM AUTHOR]

Thermodynamics provides a unified perspective of the thermodynamic properties of various substances. To formulate thermodynamics in the language of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile, affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper, affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. For equilibrium systems, it is shown that several thermodynamic quantities can be identified with geometric objects in affine geometry and that several geometric objects can be introduced in thermodynamics. Examples of these include the following: specific heat is identified with the affine fundamental form and a flat connection is introduced in thermodynamic phase space. For nonequilibrium systems, two classes of relaxation processes are shown to be described in the language of an extension of affine geometry. Finally, this affine geometric description of thermodynamics for equilibrium and nonequilibrium systems is compared with a contact geometric description. [ABSTRACT FROM AUTHOR]

This study considers a new decomposition of an extended divergence on a foliation by deformed probability simplexes from the information geometry perspective. In particular, we treat the case where each deformed probability simplex corresponds to a set of q-escort distributions. For the foliation, different q-parameters and the corresponding α -parameters of dualistic structures are defined on each of the various leaves. We propose the divergence decomposition theorem that guides the proximity of q-escort distributions with different q-parameters and compare the new theorem to the previous theorem of the standard divergence on a Hessian manifold with a fixed α -parameter. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a fundamental substructure of maximal caps in the affine geometry AG(4, 3) that we call demicaps. Demicaps provide a direct link to particular partitions of AG(4, 3) into 4 maximal caps plus a single point. The full collection of 36 maximal caps that are in exactly one partition with a given cap C can be expressed as unions of two disjoint demicaps taken from a set of 12 demicaps; these 12 can also be found using demicaps in C. The action of the affine group on these 36 maximal caps includes actions related to the outer automorphisms of S 6 . [ABSTRACT FROM AUTHOR]

We prove that if μ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on RPM¹ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case. [ABSTRACT FROM AUTHOR]

We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build homogeneous coherent configurations (also known as association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs give rise to association schemes. The Bose-Mesner algebra of the coherent configuration is then isomorphic to the adjacency algebra of the graph. The latter is more easily computed, and comes with a "small" set of generators, so we are able to produce examples of Bose-Mesner algebras with particularly simple presentations. When a group G acts "strongly transitively", in a certain sense, on a graph, we show that a distance function as above exists canonically; moreover, when the graph is a building, we show that strong transitivity is equivalent to the usual condition of transitivity on pairs of incident chambers and apartments. As an example, we show that the action of the Mathieu group M 24 on its usual geometry is not strongly transitive. We study affine planes in detail. These are not buildings, yet the machinery developed allows us to state and prove some results which are directly analogous to classical facts in the theory of projective planes (which are buildings). In particular, we prove a variant of the Ostrom-Wagner Theorem on Desarguesian planes which holds in both the projective and the affine case. [ABSTRACT FROM AUTHOR]

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ (2 , 2) -invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end. [ABSTRACT FROM AUTHOR]

In this paper we introduce general transfer operators between high-order and loworder refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling. [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]

The aim of this paper is to generalize the convexity preserving properties of sets by direct and inverse images initially stated for set-valued ratios of affine functions to a broader class of fractional type set-valued functions which strictly includes the first one. [ABSTRACT FROM AUTHOR]

In this paper we discuss in some depth the main theorems pertaining to Carnot's theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed. [ABSTRACT FROM AUTHOR]

We argue that the ideas of Bertrand Russell, a famous English philosopher and logician, have a bearing on the philosophical foundations of one of the sub–fields of AI, called qualitative spatial reasoning. The research conducted within that field focuses on non–numerical reasoning about regions of space designed to mimic human spatial behaviour and thus avoid the artificiality of the numerical approach. After briefly describing the main characteristics of this field, we analyse Russell's works on geometry. We show that despite major differences in how the subject matter is treated, these publications do have a common core that is related to the non–numerical, qualitative parts of geometry. Therefore, we argue that Russell should be viewed as a forefather of qualitative spatial reasoning on par with Whitehead or Leśniewski. Moreover, we believe that the efforts within qualitative spatial reasoning should be geared more towards the types of geometry he describes. [ABSTRACT FROM AUTHOR]

We display a symmetric monoidal equivalence between the stable ∞‐category of filtered spectra, and quasi‐coherent sheaves on A1/Gm, the quotient in the setting of spectral algebraic geometry of the flat affine line by the canonical action of the flat multiplicative group scheme. Via a Tannaka duality argument, we identify the underlying spectrum and associated graded functors with pull‐backs of quasi‐coherent sheaves along certain morphisms of stacks. [ABSTRACT FROM AUTHOR]

The connection between the intrinsic geometry of the submanifold with its extrinsic geometry has been extensively developed in recent decades. When a manifold is endowed with a geometric structure, we have more opportunities to explore its geometric properties. 10.3390/math9141669 6 Deshmukh S., Ishan A., Belova O., Al-Shaikh S. Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds. [Extracted from the article]

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]

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]

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 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]

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]

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]

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]

A talented mathematician and patron of the arts and sciences, he included it in his I Kitab al-Istikmal i (book of perfection), a compendium of Greek and Islamic mathematics. To celebrate the event, the Hong Kong postal authorities issued a circular $10 stamp with a geometric design involving triangles and circles, together with a souvenir sheet that features Ceva's theorem, a result from plane and affine geometry. In July 2016, the 57th International Mathematical Olympiad (IMO) was held in Hong Kong, attended by over 600 young mathematicians from 109 countries who were anxious to test their problem-solving skills. [Extracted from the article]

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]

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]

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 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]