arXiv research

Research examines octonionic slice regular functions and their automorphisms and invariants.

New proof of Yamabe invariant for RP^3 using harmonic functions.

Paper describes invariants of slice regular functions' automorphism group.

Study beta function for convex billiard maps, linking spectral invariants.

New integer-valued functions for Legendrian knots.

Analogous zeta function for twisted Alexander invariants defined.

Introduced coloring-allowed invariants of planar knotoids with the coloring number.

Study on algebraic curves' invariants and vanishing criteria.

New spectral invariants distinguish Joyce orbifolds from other manifolds.

We develop Fourier methods to expand translation-invariant kernels.

The paper studies invariant functions and their relation to Landsberg surfaces.

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.

Group-invariant neural networks improve approximation accuracy for symmetric functions.

New $L$-functions for 3-manifolds connect to Witten invariants and relate to generalized Bernoulli polynomials.

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.

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holono…

Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.

New findings on how certain functionals behave in random variable spaces.

New invariant distinguishes singular knots and links.

The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.

Investigates invariant hulls of functionals on manifolds.

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.

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.

New approach to quantum knot invariants using perturbed Gaussian generating functions.

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

Researchers develop a method to infer reference measures from observed functionals.

Study invariant minimizers in convex functions under amenable groups.

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.

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.

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 $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(Ω_{x})$-invariant fun…

New principles for collapsing law-invariant functionals to means, extending beyond convexity.

Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.

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…

The number of functionally independent scalar invariants of arbitrary order of a generic pseudo--Riemannian metric on an $n$--dimensional manifold is determined.