The paper establishes a correspondence between normal distributions and neat foliations on manifolds with boundary.
problem Understanding normal distributions on manifolds with boundary.
method Develops a theory analogous to Stefan and Sussmann's for integrable distributions, focusing on neat foliations.
result A one-to-one correspondence between neatly integrable normal distributions and neat foliations by manifolds with boundary.
Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
NEAT algorithm optimizes stock trading with reduced risk.
problem Maximizing earnings while minimizing risk in stock trading.
method Applied NEAT algorithm to stock trading with multiple technical indicators, using progressive training data and a multi-objective fitness function.
result NEAT model achieved similar returns to Buy & Hold but with lower risk and stability.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
New method for classifying disk embeddings in 4-manifolds.
problem Classifying smooth isotopy classes of neat embeddings of 2-disks in 4-manifolds.
method Using an invariant going back to Dax, constructing a group structure, and relating to mapping class groups.
result The group structure on isotopy classes of neat embeddings is usually not abelian or finitely generated.
In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper a…
Paper applies NEAT for dynamic credit evaluation using streaming data.
problem Dynamic credit evaluation using streaming data.
method Neuroevolution of Augmenting Topologies (NEAT) with enhancements.
result NEAT effectively handles dynamic credit evaluation with streaming data.
We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of 38π2, i.e., they attain all possible volumes of c…
The paper proves a Connes trace theorem for curved noncommutative tori.
problem Recovering scalar curvature in curved noncommutative tori.
method Proving a version of Connes' trace theorem for noncommutative tori of any dimension.
result Establishes a curved version of Connes' integration formula for scalar curvature.
Quantization and reduction for coisotropic A-branes on Hamiltonian manifolds.
problem Quantization and reduction of coisotropic A-branes.
method Definition and construction of coisotropic A-branes, and a fibration over reduced spaces.
result Quantization commutes with reduction for coisotropic A-branes.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
The paper simplifies embedding spaces in manifolds by attaching handles.
problem Embedding spaces in manifolds with knotted spheres.
method Using framed dual spheres and attaching handles to simplify embedding spaces.
result The homotopy type of embedding spaces is significantly simplified.
Method constructs fundamental domains for Picard modular groups.
problem Classify and understand torsion elements in Picard modular groups.
method Systematic construction of coarse fundamental domains.
result Classification of conjugacy classes of torsion elements.
Explicitly describes pluriclosed metrics on compact Lie groups.
problem Characterizing pluriclosed metrics on compact Lie groups.
method Explicit description using root systems and invariant structures.
result Explicit formulas for pluriclosed metrics in terms of root systems.
We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals consid…
In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set Ω⊂Rn, n≥3. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the p-capacitary potentials associated with Ω, for every p suffici…
Different directed acyclic graphs (DAGs) may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Meek (1995) characterizes Markov equivalence classes for DAGs (with no latent variables) by presenting a set of orientation rules that can correctly i…
Let G be an almost simple, simply connected algebraic group defined over a number field k, and let S be a finite set of places of k including all infinite places. Let X be the product over v∈S of the symmetric spaces associated to G(kv), when v is an infinite place, and the Bruhat-Tits buildings ass…
Continuous word representation (aka word embedding) is a basic building block in many neural network-based models used in natural language processing tasks. Although it is widely accepted that words with similar semantics should be close to each other in the embedding space, we find that word embeddings learned in seve…
Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
Develops unisolvent weights for Nédélec second family finite elements in 2D.
problem Finding efficient degrees of freedom for Nédélec second family finite elements.
method Uses techniques of homological algebra to obtain degrees of freedom for differential forms.
result Provides a family of unisolvent and minimal physical degrees of freedom for Nédélec second family finite elements.
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
The paper shows how certain circle families in S1imesD3 relate to sphere families in S2imesD2 and induces nontrivial barbell diffeomorphisms.
problem Understanding the relationship between circle and sphere families in specific 3-manifolds.
method Analyzing the fundamental groups and ambient extensions of circle and sphere families.
result Induces nontrivial barbell diffeomorphisms of S1imesS2imesI. Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
The article constructs stochastic integration in Riemannian manifolds.
problem No specific problem stated; focuses on the construction of stochastic integration.
method Functional-analytic approach to stochastic integration in Riemannian manifolds.
result There are infinitely many stochastic integrals, and they are related by a simple formula.
We investigate the use of the Hurst exponent, dynamically computed over a moving time-window, to evaluate the level of stability/instability of financial firms. Financial firms bailed-out as a consequence of the 2007-2010 credit crisis show a neat increase with time of the generalized Hurst exponent in the period prece…
This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of "universal portfolios." I generalize Cover's benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hin…
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
It is well known that the compactifications of the canonical contact systems living on real jet spaces Jr(1,1), r≥2, are locally universal Goursat distributions, Δr, living on compact manifolds (called Goursat monsters) having open dense jet-like (Jr(1,1)-like) parts. By virtue of the results of …
Method calculates function integrals on complex manifolds.
problem Integrating functions on complex manifolds.
method Digital representation and calculation method.
result Integral calculation on compact manifolds.
Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
Extends integrability to cosymplectic manifolds.
problem Integrability of Hamiltonian systems on cosymplectic manifolds.
method Extended Arnold-Liouville and noncommutative integrability to cosymplectic manifolds, proved a variant of non-commutative integrability for specific fields, constructed action-angle variables.
result Variant of non-commutative integrability for evaluation and Reeb vector fields on cosymplectic manifolds.
The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T∗Q∖Q. If the geodesic flow is toric integrable, the cosphere bundle admit…
The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express th…
The paper proves integral formulas for manifolds with multiple orthogonal distributions.
problem Understanding geometric properties of manifolds with multiple orthogonal distributions.
method Develops integral formulas for Riemannian manifolds with k>2 orthogonal complementary distributions. result Generalizes known formulas for k=2 and applies to manifold splitting and immersions. Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
problem Computing the Connes-Chamseddine cycle for 6D manifolds.
method Using noncommutative integral on 6D manifolds, they compute the cycle.
result The Connes-Chamseddine cycle on 6D manifolds is computed.
Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
Develops integrators for Hamiltonian systems in Jacobi manifolds.
problem Modeling conservative systems with dissipative and thermodynamic phenomena.
method Constructs structure-preserving integrators for Hamiltonian systems in Jacobi manifolds.
result Proposes a numerical integration technique compatible with Jacobi dynamics.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
A symplectic integration of a Poisson manifold (M,Λ) is a symplectic groupoid (Γ,η) which realizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure Λ0 is isomorphic to (M,Λ). This notion was introduced by A. Weinstein in order to quantize Poisson manifolds …
Perfect pairing for tropical cycles on integral affine manifolds.
problem Computing period integrals and versality of Calabi-Yau degenerations.
method Introducing a cap product pairing and using simplicial methods for constructible sheaves.
result The pairing is perfect in degree one for symplectic singularities.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…