Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
New proof of Yamabe invariant for RP^3 using harmonic functions.
problem Yamabe invariant of RP3 method Using harmonic functions
result New proof of Yamabe invariant for RP3 Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
New integer-valued functions for Legendrian knots.
problem Understanding Legendrian knots better.
method Using Legendrian fronts to derive integer-valued linear functions.
result Introduced new invariants similar to Arnold's basic invariant.
Analogous zeta function for twisted Alexander invariants defined.
problem Defining a zeta function for twisted Alexander invariants.
method Modeling random walks on knot diagrams and interpreting Alexander polynomials and Jones polynomials as zeta functions.
result Analogous zeta function expression for twisted Alexander invariants.
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
New spectral invariants distinguish Joyce orbifolds from other manifolds.
problem Distinguishing Joyce orbifolds from other G2-structures. method Introducing and computing two new spectral invariants for Joyce orbifolds.
result These invariants are more effective than existing invariants for distinguishing Joyce orbifolds.
We develop Fourier methods to expand translation-invariant kernels.
problem Constructing orthonormal expansions for translation-invariant kernels.
method Fourier analytic technique to derive explicit expansions.
result Explicit expansions for various kernels (Matérn, Cauchy, Gaussian).
The paper studies invariant functions and their relation to Landsberg surfaces.
problem Investigating the geometry of invariant functions and their applications to Landsberg surfaces.
method Investigating the geometry of S-invariant functions and their associated vertical subdistribution, and relating the holonomy distribution to these subdistributions. result For Landsberg surfaces, if the flag curvature is S-invariant, it is constant, and the surface is Riemannian. In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial, the affine index polynomial and the zero polynomial.
Study Hamiltonian diffeomorphisms on symplectic manifolds and properties of invariant convex functions.
problem Properties of invariant convex functions under Hamiltonian diffeomorphisms.
method Analysis of the adjoint action and properties of invariant convex functions.
result Continuous convex functions invariant under Hamiltonian diffeomorphisms are also invariant under strict rearrangements.
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
New L-functions for 3-manifolds connect to Witten invariants and relate to generalized Bernoulli polynomials.
problem Understanding L-functions for 3-manifolds and their invariants. method Using Mellin transforms and asymptotic techniques, proving entire functions and their values.
result Linear relations between L-function values at negative integers, generalizing known zeta functions. In this paper we characterize a function defined on the set of edges of a triangulated surface such that there is a spherical angle structure having the function as the edge invariant (or Delaunay invariant). We also characterize a function such that there is a hyperbolic angle structure having the function as the edge…
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random var…
Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.
problem Eta invariant of (2,3,5) nilmanifolds.
method Study of eta function and invariant of a self-adjoint differential operator of Heisenberg order two.
result Formula expressing the eta function of (2,3,5) nilmanifolds in terms of elementary functions.
New findings on how certain functionals behave in random variable spaces.
problem Understanding when law-invariant convex functionals simplify to the mean.
method Analyzing a broad class of random variable spaces and mild semicontinuity assumptions.
result The expectation functional is the only law-invariant convex functional that collapses to the mean under certain conditions.
New invariant distinguishes singular knots and links.
problem Classifying singular knots and links.
method Using oriented singquandles and weight functions at crossings.
result Distinguishes singular granny knot from singular square knot.
The paper explores non-metrizability of projective deformations of Finsler sprays.
problem Investigating non-metrizability of projective deformations of Finsler sprays.
method Analyzing projective deformation of Finsler sprays by holonomy invariant functions and proving non-metrizability for most cases.
result For most values of λ and holonomy invariant nontrivial functions P, the projective deformation is not Finsler metrizable.
The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.
problem Hamiltonian flows on surface group representations induced by invariant multi-functions.
method Introducing subsurface deformation and proving Poisson commutativity of induced invariant multi-functions.
result Hamiltonian flows on character varieties are of subsurface deformation type and Poisson commute if supporting subsurfaces are disjoint.
Investigates invariant hulls of functionals on manifolds.
problem Understanding meaningful functionals on manifolds through reparameterizations.
method Uses inner-variations to transform arbitrary functionals into invariant realizations.
result Explicit computations for volume functional in N-dimensional manifolds, especially in N=2. Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine t…
New hyperbolic knots with convex Upsilon invariants constructed.
problem Constructing knots with convex Upsilon invariants.
method Combinatorial method for (1,1)-knots and connected sum operation. result Infinitely many mutually non-concordant hyperbolic knots with convex Upsilon invariants.
In a natural way, the local diffeomorphisms of a manifold onto itself act on the reference frame bundles of any order and on the bundles associated with them. Due to the transitivity, the invariants by diffeomorphisms of an associated bundle correspond to the real functions on the orbit space of the action of the jet g…
In theoretical analysis of deep learning, discovering which features of deep learning lead to good performance is an important task. In this paper, using the framework for analyzing the generalization error developed in Suzuki (2018), we derive a fast learning rate for deep neural networks with more general activation …
We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…
The study bounds Dehn functions of coabelian subgroups using a second BNSR invariant.
problem Bounding the Dehn function of coabelian subgroups in hyperbolic groups.
method Using an area-radius pair for a finitely presented group and a second BNSR invariant.
result Finitely presented coabelian subgroups of hyperbolic groups have polynomially bounded Dehn functions.
New approach to quantum knot invariants using perturbed Gaussian generating functions.
problem Developing universal quantum knot invariants.
method Introducing generating functions of the form PeG where G is quadratic and P is a perturbation, and developing a calculus for such functions. result The rank one invariant ZD dominates sl2-colored Jones polynomials and relates to knot genus and Whitehead doubling. Quantum invariants from Uhsl(2∣1) are q-holonomic.
problem Understanding quantum invariants from a specific quantum group.
method Demonstrated q-holonomic property through quantum group representations.
result Existence of an underlying field theory for these quantum invariants.
Researchers develop a method to infer reference measures from observed functionals.
problem Tackles the challenge of identifying or recovering a reference measure from observed functionals.
method Uses the property of law-invariant functionals defining lower or upper supporting sets in dual spaces of signed measures.
result Illustrates the methodology with examples and develops a modification for Value-at-Risk.
Study invariant minimizers in convex functions under amenable groups.
problem Finding invariant minimizers in convex functions invariant under amenable groups.
method Analyze smallest closed invariant convex subsets and apply to invariant optimality problem.
result Clarifies relations between equivariant neural networks and statistical theorems.
We consider a simple and overarching representation for permutation-invariant functions of sequences (or multiset functions). Our approach, which we call Janossy pooling, expresses a permutation-invariant function as the average of a permutation-sensitive function applied to all reorderings of the input sequence. This …
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graph…
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the…
Given an action of a Lie group on a smooth manifold, we discuss the induced action on the Hochschild cohomology of smooth functions, and notions of invariance on this space. Depending on whether one considers invariance of cochains or invariance of cohomology classes, two different spaces of invariants arise. We perfor…
We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neuma…
We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We…
The paper analyzes counterfactual invariance and its relation to conditional independence.
problem Understanding the relationship between counterfactual invariance and conditional independence.
method Theoretical analysis of existing definitions, graphical implications, and mathematical proofs.
result Counterfactual invariance implies conditional independence, but not the other way around.
The knot invariant Upsilon, defined by Ozsvath, Stipsicz, and Szabo, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2].…
Let M be a smooth manifold of dimension 2n, and let OM be the dense open subbundle in ∧2T∗M of 2-covectors of maximal rank. The algebra of DiffM-invariant smooth functions of first order on OM is proved to be isomorphic to the algebra of smooth Sp(Ωx)-invariant fun…
New principles for collapsing law-invariant functionals to means, extending beyond convexity.
problem Conditions for law-invariant functionals to reduce to means.
method Establishing collapse to the mean principles for non-convex functionals.
result General principles apply beyond convexity, including quasiconvex and Choquet integrals.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
We study Heegaard Floer homology and various related invariants (such as the h-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the h-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Fl…