The quotient of normal variables has power-law decay, applied to asset price tails.
problem Understanding the distribution of relative price changes in finance.
method Analyzing the quotient of normal distributions and deriving power-law decay densities.
result Relative price changes follow a power-law distribution with density f(x)≃f0x−2. The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
The paper studies hyperbolic quotients of projection complexes and their actions.
problem Understanding the structure and properties of quotients of projection complexes.
method Analyzing the quotient of projection complexes by normal subgroups and studying the resulting actions.
result The quotient complex is δ-hyperbolic under certain conditions, and the quotient group is acylindrically hyperbolic.
This paper classifies ball quotients of the complex projective plane.
problem Understanding the structure of the complex projective plane as a ball quotient.
method Analyzing the branch locus as a line arrangement and smooth normal-crossing curves.
result The orbifold structure of (P2,D) is isomorphic to either the Deligne-Mostow example or a certain degree 9 cover. Stratifies singular hyperkahler quotients into smooth manifolds.
problem Singular hyperkahler quotients by non-free actions.
method Topological stratification, global Poisson structures, local normal form.
result Quotients are locally isomorphic to linear complex-symplectic reductions.
It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…
The paper shows commensurators of certain subgroups are discrete.
problem Understanding commensurators of specific subgroups in Lie groups.
method Analyzing commensurators of thin normal subgroups with abelian quotients.
result The commensurator of certain subgroups is discrete.
Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.
problem Understanding compact quotients of reductive homogeneous spaces and their implications.
method Analyzing normal bundles and sphere bundles associated with these spaces, proving homotopy triviality conditions.
result Many reductive homogeneous spaces do not admit compact quotients, resolving conjectures.
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
Article studies symmetry in smooth vector bundles using advanced operations.
problem Symmetry phenomena in smooth vector bundles after two iterations of the normal functor.
method Developed theory of pullback and quotient for double vector bundles and morphisms, focusing on naturality of the normal functor.
result Expected symmetry is obtained through universal behavior and compatibility of operations.
In this paper we show that every invariant Finsler metric on Lie group G, induces an invariant Finsler metric on quotient group G/H in the natural way, where H is a closed normal Lie subgroup of G.
In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free ab…
The study verifies a conjecture about homogeneous quotients of manifolds with positive curvature.
problem Verifying the Homogeneity Conjecture for manifolds with positive curvature.
method Examining globally homogeneous Riemannian quotients of homogeneous manifolds, focusing on those admitting a positive curvature metric.
result Further evidence supports the Homogeneity Conjecture for certain manifolds with positive curvature.
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
The paper studies Coxeter quotients of surface braid groups.
problem Analyzing Coxeter quotients of surface braid groups.
method Analyzing Coxeter-type quotients of surface braid groups by normal closure and commutator relations.
result Characterized Coxeter quotients of surface braid groups.
Adaptive feature normalization improves model robustness to extraneous variables.
problem Degrading model performance due to extraneous variables in deep learning.
method Adaptive feature normalization using instance normalization instead of batch normalization.
result Adaptive normalization leads to significant performance gains across different datasets and architectures.
The study examines the topology of map germs and their images.
problem Understanding the topology of map germs and their images.
method Using the topology of the link to analyze the normal and non-normal images.
result Normal images of map germs are quotient singularities.
In the first part of this paper we prove that the mapping class subgroups generated by the D-th powers of Dehn twists (with D≥2) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal s…
Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a sel…
In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type A as a quotient algebra of the Yokonuma-Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…
Random quotients of mapping class groups have rigid properties.
problem Rigidity of random quotients of mapping class groups.
method Generalization of Ivanov's theorem and use of hierarchically hyperbolic groups.
result Automorphisms and commensurators of random quotients coincide with the groups themselves.
Study compares Kähler quotients of torus actions under varying moment maps.
problem Comparing Kähler quotients of torus actions under varying moment maps.
method Analyzes the transformation of Kähler quotients as moment maps change, proving bimeromorphic transformations and desingularizations.
result Each nondegenerate singular Kähler quotient has a partial and rational desingularization.
We prove the "End Curve Theorem," which states that a normal surface singularity (X,o) with rational homology sphere link Σ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An "end-curve function" is an analytic function $(X,o)\to (\C,0)$ whose ze…
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group G in a way that the quotient space M/G has nonempty boundary. Let π:M→M/G denote the quotient map and B be any boundary stratum of M/G. Via a specific soul co…
The paper extends quasimorphisms on subgroups to larger groups.
problem Extending quasimorphisms from subgroups to larger groups.
method Provides a general sufficient condition for extendability of quasimorphisms on subgroups.
result New results for quasimorphisms on normal subgroups, including bi-Lipschitz equivalence of stable commutator length and group-theoretic Dehn filling.
By Gromov's mapping theorem for bounded cohomology, the projection of a group to the quotient by an amenable normal subgroup is isometric on group homology with respect to the ℓ1-semi-norm. Gromov's description of the diffusion of cycles also implicitly produces efficient cycles in this situation. We present an e…
The study characterizes subgroups of braid groups and their finite quotients.
problem Understanding the relationship between braid groups and their congruence subgroups.
method Characterization and computation of finite quotients of braid groups.
result Explicit generators and free subgroups are found for certain braid groups.
In this paper we calculate fundamental groups (and some of their quotients) of complements of four toric varieties branch curves. For these calculations, we study properties and degenerations of these toric varieties and the braid monodromies of the branch curves in CP2. The fundamental groups related to th…
Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of …
For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution fro…
The paper studies cyclic covers of rational surfaces and their Hodge structures.
problem Understanding the Hodge structures of cyclic covers of rational surfaces.
method Generalization of Esnault-Viehweg method to analyze monodromy actions.
result The monodromy action splits into direct sums for specific cyclic covers.
We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundl…
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. The paper studies orbifold splice quotients and log covers of surface pairs.
problem Understanding orbifold splice quotients and log covers of surface pairs.
method Analyzes orbifold homology and constructs pairs with universal abelian log covers.
result Computes orbifold homology from resolutions and constructs orbifold splice quotients.
Flat semigroups can represent normal weighted homogeneous surface singularities.
problem Representability of flat semigroups in normal weighted homogeneous surface singularities.
method Study of numerical semigroups associated with surface singularities and prove representability conditions.
result A numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.
Study of quotient groups of mapping class groups by power subgroups.
problem Characterizing quotient groups of mapping class groups by power subgroups.
method Analyzing quotient groups Modgn/Modgn[p] for different values of p and n. result Found infinite normal subgroups and Kähler subgroups in specific quotient groups.
We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compac…
Lie algebras of quotient groups defined under specific conditions.
problem Conditions for Lie differentiation of quotient groups.
method Diffeological group theory, tangent structure, Lie functor instantiation.
result Lie algebra structure on quotient groups derived from Lie algebras of parent groups.
The study examines the structure of certain subgroups of quasi-isometry groups of Euclidean spaces, proving their nontriviality and properties.
problem Investigating the algebraic and dynamical structure of certain subgroups of quasi-isometry groups of Euclidean spaces.
method Analyzing the normal subgroups \(H\) and \(H_\alpha\) of \(QI(\mathbb{R}^n)\) and proving their properties.
result The centers of \(QI(\mathbb{R}^n)/H\) and \(QI(\mathbb{R}^n)/H_\alpha\) are trivial, while these quotients admit nontrivial torsion elements.
For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R−r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the …
Improves data normality with robust transformations.
problem Skewed data distribution.
method Modified Box-Cox and Yeo-Johnson transformations with robust parameter estimation.
result Transformed data approximates normality in the center with outliers.
This paper improves the Diversification Quotient (DQ) for better risk management.
problem Improving portfolio diversification measurement.
method Empirical estimation of DQ using VaR and ES, with asymptotic properties verified.
result Empirical DQ estimators are more robust and have better asymptotic properties.
The paper studies coherent sheaves on subvarieties of Hopf manifolds.
problem Understanding coherent sheaves on subvarieties of Hopf manifolds.
method Proves a version of GAGA theorem, shows natural algebraic structure, and uses quotient and embedding properties.
result Any reflexive coherent sheaf on M is filtrable. The connected covering spaces of a connected and locally path-connected topological space X can be classified by the conjugacy classes of those subgroups of π1(X,x) which contain an open normal subgroup of π1(X,x), when endowed with the natural quotient topology of the compact-open topology on based loops. Ther…
The study examines subgroups of braid groups related to symmetric groups.
problem Characterizing subgroups of braid groups that are extensions of symmetric groups.
method Analyzing normal subgroups of braid groups and their quotient structures.
result There are exactly 8 commensurability classes of such subgroups for n≥4. We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with C-positive normal curvature, if there is a closed basic 1-form φ such that ΔBφ=qCφ, then the foliation is transversally isometric to the quotient of a q-sphere.