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.
New formulas for measuring geometric properties of definable sets.
Defines tangent spaces on causal sets using partial derivatives and metrics.
In this thesis, we consider semi-algebraic sets over a real closed field defined by quadratic polynomials. Semi-algebraic sets of are defined as the smallest family of sets in that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the bool…
In this paper we prove that every definable set has a definable triangulation which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial stratification of the simplicial complex. We also distinguish a class T of regularity conditions and give a universal construction of a definable triangulation with …
We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Paw…
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…
Study concordance of decompositions from defining sequences in 3-sphere.
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by d…
Defines smoothness of definable sets in o-minimal structures.
We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …
We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney -regularity of a stratification…
We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
The paper extends a link criterion for Lipschitz normal embeddings to definable sets in o-minimal structures.
In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.
This study provides a new mathematical structure for Koopman eigenfunctions.
To each unit complex number with positive imaginary part there is defined a Tristram-Levine knot signature function. The set of all such signature functions is linearly independent as a set of functions defined on the set of all knots. The set of averaged signature functions forms a linearly independent set of homomoro…
We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…
A new multi-armed bandit framework with credal sets for uncertain outcomes.
We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain st…
The algebraic -groups $L_*(\A,X)$ are defined for an additive category $\A$ with chain duality and a -set , and identified with the generalized homology groups $H_*(X;\LL_{\bullet}(\A))$ of with coefficients in the algebraic -spectrum $\LL_{\bullet}(\A)$. Previously such groups had only been defined for…
New groups defined from knot diagrams, invariant under Reidemeister moves.
Defines and extends Lie algebroid prolongations in convenient settings.
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
New invariants defined for knots and links using quandle representations.
Two of the authors have defined the class as the class of all subsets of a smooth manifold that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If is Riemanian and is a group of isometries acting transitively on the sphere bundle , we def…
Generalizes quandle constructions and defines a multiplication that results in an abelian group.
Paper proves generalizations of Bernstein's theorem in higher dimensions.
Defines invariants for reflection groups and connects them to Frobenius structures.
Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X-Z, and a nowhere zero section on X-Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by …
In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators(we will call it the "algebraic" version in this article). In 2012, Przytycki defined another homology theory for pre-Yang-Baxter operators which has a nice graphic visualization(we will call it the "graphic" version i…
The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle we define the generalized component $\gencomp (E)$ as the set of Clifford bundles which have finite distance to . If , are the associated generalized Dirac ope…
New invariants for virtual knots and links defined via quiver representations.
Paper studies planar extensions in o-minimal structures.
We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized s…
Invariant for 3-manifolds with torus boundary defined.
We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hofer-like norm [5] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positiv…
In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding p…
We address the problem of synthetic gene design using Bayesian optimization. The main issue when designing a gene is that the design space is defined in terms of long strings of characters of different lengths, which renders the optimization intractable. We propose a three-step approach to deal with this issue. First, …
Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
In this paper, we have tried to apply the concepts of fuzzy sets to Lie groups and its relative concepts. First, we define a fuzzy submanifold after reviewing fuzzy manifold definition. In main section, we defined the Lie group and some its relative concepts such as fuzzy transformation group,…
A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactificat…
We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vani…
New method extends conformal prediction to multivariate settings using optimal transport.
We study the complex geometry of generalized Kepler manifolds, defined in Jordan theoretic terms, introduce Hilbert spaces of holomorphic functions defined by radial measures, and find the complete asymptotic expansion of the corresponding reproducing kernels for Kähler potentials, both in the flat and bounded setting.
Self-affine tiles homeomorphic to a ball proven for a specific digit set.
Neural networks adapt to any input dimensionality.