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We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of p -adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, p -adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a p -adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a p -adic Hilbert space — we consider an algebraic definition of physical states in p -adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a p -adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the p -adic setting, with an affine geometry; therefore, a symmetry transformation of a p -adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a p -adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers. [ABSTRACT FROM AUTHOR]

We consider the problem of defining arc length for a plane curve invariant under a group action. Initially one partitions the curve and sums a distance function applied to consecutive support elements. Arc length then is defined as a limit of approximating distance function sums. Alternatively, arc length is defined using the integral that arises when applying the method to smooth curves. That these definitions agree in the general case for the Euclidean group was an early victory of the Lebesgue integral; the equivalence also is known for the equi-affine group. Here we present a unified treatment for equi-affine, Laguerre, inversive, and Minkowski (pseudo-arc) geometries. These are alike in that arc length corresponds with a geometric average of finite Borel measures. [ABSTRACT FROM AUTHOR]

In this paper we define the class of constant affine mean curvature (CAMC) discrete asymptotic nets, which contains the well-known classes of affine spheres and affine minimal asymptotic nets. This class is defined by considering fields of compatible interpolating quadrics, i.e., quadrics that have common tangent planes at the edges of the net. We show that, for CAMC asymptotic nets, ruled discrete asymptotic nets is equivalent to ruled compatible interpolating quadrics. Moreover, we prove discrete counterparts of some known properties of the Demoulin transform of a smooth CAMC surface. [ABSTRACT FROM AUTHOR]

In this note, we explain how mirror symmetry for basic local models in the Gross–Siebert program can be understood through the nontoric blowup construction described by Gross–Hacking–Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations. [ABSTRACT FROM AUTHOR]

Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of |$3d$| |${\mathcal{N}}=4$| quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the Coulomb branch construction. Beyond finite type these embeddings were not known, and our second result is to construct them for all symmetric Kac-Moody types. We show that these embeddings respect Poisson structures under a mild "goodness" hypothesis. These results give an affirmative answer to a question posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs in studying the geometry of Kac-Moody affine Grassmannian slices, even in finite types. [ABSTRACT FROM AUTHOR]

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface M in a complex Calabi–Yau manifold (that is, a complex n -manifold with a nowhere-vanishing holomorphic n -form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in R n . We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces M bounding compact domains Ω ⊂ Z. That is, we study critical points of the volume functional A (M) where the ordinary volume V (Ω) is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere S 2 n − 1 ⊂ C n does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in "most" directions but non-negative in directions corresponding to deformations of S 2 n − 1 by holomorphic diffeomorphisms. We are led to conjecture a "minimax" characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case n = 3 and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in Z using only a closed "pseudoconvex" real 3 -form on M. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact 3 -forms on M and the moment map for the action of the diffeomorphisms of M. [ABSTRACT FROM AUTHOR]

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension. [ABSTRACT FROM AUTHOR]

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]

A cap in a projective or affine geometry over a finite field is a set of points no three of which are collinear. We give several new constructions for complete caps in affine geometry over the field implying some well-known results. [ABSTRACT FROM AUTHOR]

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules, modules over a ring and affine spaces are revealed and analysed. [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]

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]

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]

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]

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]

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]

We define a new notion of affine subspace concentration conditions for lattice polytopes and prove that these conditions hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle with the extension class c1 (Tx) on Fano toric varieties. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of the plane affine group Aff(2) considered with the natural action on the vector space of two-variable polynomials. The polynomial family {Bm,n(x, y)} is called quasi-monomial with respect to H if the group operators in two different bases {xmyn} and {Bm,n(x, y)} have identical matrices. We obtain a criterion of quasi-monomiality for the case when the group H is generated by rotations and translations in terms of exponential generating function for the polynomial family {Bm,n(x, y)}. [ABSTRACT FROM AUTHOR]

We give an explicit parametrization of the set of algebraic points of given degree on Q over the affine equation curve: y²=157(x²-2)(x²+x)(x²+1). This note treat aspecial case of the curves described by Anna ARNTH-JENSEN and Victor FLYNN in [1], where the generators of the Mordell-Weil group explained. [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]

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]

This paper is dedicated to the me­mo­ry of Professor Kowalski who was one of the leading re­sear­chers in the field of dif­fe­ren­tial geometry and especially Rie­mannian and affine geometry. He signif­icantly contributed to rai­sing the level of teaching dif­fe­ren­tial geometry by careful and sys­tematic preparation of lectures for students. Prof. Kowalski is the author or co-author of more than 170 professional articles in internation­ally recognized jour­nals, two monographs, text books for students. Prof. Kowalski collaborated with many mathematicians from other countries, particularly from Belgium, Italy, Ja­pan, Romania, Russia, Morocco, Spain and others. With the death of Professor Oldřich Kowalský mathe­matical community are losing a significant personality and an exceptional colleague, a kind and dedicated teacher, a man of high moral qualities.

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]

We consider the problem of constructing complete caps in affine geometry AG(n, 3) of dimension n over the field F3 of order three. We will take the elements of F3 to be 0, 1 and 2. A cap is a set of points, no three of which are collinear. Using the concept of Pn - set, we give two new methods for constructing complete caps in affine geometry AG(n, 3). These methods lead to some new upper and lower bounds on the possible minimal and maximal cardinality of complete caps in affine geometry AG(n, 3).

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]

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]

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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]

In this work, geometry of tangential and affine configuration chain complex is proposed in generalized form. Initially tangential and affine configuration chain complexes are connected for special case n = 6, lastly this concept is generalized for any case n ∈ N through two types of generalized homomorphisms. [ABSTRACT FROM AUTHOR]

Let K be a field and Kn be the n-dimensional linear space over K, where n > 1. Denote by GL(n, K) the set of all n × n-matrices A over K such that det(A) 6= 0. Put Af(n, K) = n F : Kn → Kn | F(x) = gx + b, g ∈ GL(n, K), b ∈ Kn, ∀x ∈ Kn o, where the elements b, x ∈ Kn are assumed to be written as column-vectors. Let T be a nonempty set with at least two elements. A mapping ϕ : T → Kn will be called a T-parametric figure in Kn . For shortness, it will be called a T-figure. For natural actions of groups G = GL(n, K), Af(n, K) and some their subgroups on Kn, problems of G-equivalence of T- figures in Kn are investigated. Complete systems of global G-invariants of a T-figure in the space Kn for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of G-invariants are investigated. [ABSTRACT FROM AUTHOR]

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised. [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]

Dropout is used in deep learning to prevent overlearning. It is a method of learning by invalidating nodes randomly for each layer in the multi-layer neural network. Let V 1 , V 2 , ... , V n be mutually disjoint node sets (layers). A multi-layer neural network can be regarded as a union of the complete bipartite graphs K | V i | , | V i + 1 | on two consecutive node sets V i and V i + 1 for i = 1 , 2 , ... , n - 1 . The dropout method deletes a random sample of activations (nodes) to zero during the training process. A random sample of nodes also causes irregular frequencies of dropout edges. A dropout design is a combinatorial design on dropout nodes from each layer which balances frequencies of selected edges. The block set of a dropout design is B = { { C 1 | C 2 | ... | C n } | C i ⊆ V i , C i ≠ ∅ , 1 ≤ i ≤ n } having a balancing condition in consecutive t sub-blocks C i , C i + 1 , ... , C i + t - 1 , see [3]. If | V i | and | C i | are constants for 1 ≤ i ≤ n , then the dropout design is called uniform. If a uniform dropout design satisfies the circulant property, then the design can be extended to a design with as many layers as you need. In this paper, we describe a construction for uniform dropout designs of circulant type by using affine geometries. [ABSTRACT FROM AUTHOR]

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]