A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The paper explores different realizations of complex Lie groups using various number fields.
In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra . It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculate…
Introduces Lorentzian Cayley form solving geometric puzzle.
In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…
Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras and , i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…
Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…
The study embeds infinite-dimensional geometric structures in Cayley graphs.
Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.
Hurwitz transformations are defined as specific automorphisms of a Cayley-Dickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to qu…
Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to…
Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the -equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,…
The paper examines alternative definitions of exceptional Lie groups using quaternion numbers.
The paper studies the graph geometry of finite groups, creating a dataset and analyzing its properties.
The paper explores alternative definitions of complex Lie groups using real numbers.
We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result…
Fundamental weight systems identified as quantum states.
In the quatenions ($\H$, $\H'$, $\H^{C}$) and octonions ($\gC$, $\gC^\prime$, $\gC^C$), we show some results on the conjugacy of two pure imaginary non-zero elements with same norm.
Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivi…
We show that, if is a random subgroup of a finitely generated free group , only inner automorphisms of may leave invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally of groups which are hyperbolic relative to slender subgroups. These results fol…
We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an …
This paper explores the geometric and topological properties of Poncelet porism for triangles.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
Investigates second-order conditions for Cayley forms in eight dimensions.
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…
Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.
We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is unif…
We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spect…
We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane . By exhibiting this symmetric space as a submanifold of the Grassmannian of -dimensional subspaces of the fundamental representation of , techniques and constructions …
In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex nu…
Abstract: Proves relative versions of group splitting results.
New complex connects graph separability to group properties.
Calibration in 16D disproves Federer's product question.
Deformations of singular Cayley submanifolds studied.
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids…
In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a C…
Desingularizes conically singular Cayley submanifolds.
Decomposes skein algebras for surfaces.
We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph …
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by "looking" at the…
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
The study examines discrete curvature notions on Cayley graphs of certain groups.
We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…
New methods for delta-moves on algebraically split links identified.
Study Cayley fibrations on Bryant-Salamon manifolds.
Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayle…
Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge, we present two main contributions: (1) A new efficient retraction ma…