arXiv research

Enhances psyquandle invariants for singular and pseudoknots.

Enhanced invariant for linkoids using quivers.

In-degree quiver polynomials for surface-links computed.

It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are …

Community detection is a central problem of network data analysis. Given a network, the goal of community detection is to partition the network nodes into a small number of clusters, which could often help reveal interesting structures. The present paper studies community detection in Degree-Corrected Block Models (DCB…

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie-Yamaguti algebra. This identity is a consequence of the defining identities for Lie-Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit…

In reinforcement learning, the state of the real world is often represented by feature vectors. However, not all of the features may be pertinent for solving the current task. We propose Feature Selection Explore and Exploit (FS-EE), an algorithm that automatically selects the necessary features while learning a Factor…

String structures in degree four are associated with cancellation of anomalies of string theory in ten dimensions. Fivebrane structures in degree eight have recently been shown to be associated with cancellation of anomalies associated to the NS5-brane in string theory as well as the M5-brane in M-theory. We introduce …

The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $π$, given some mixing conditions on $π$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary rep…

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the inf…

We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a q…

The main result of this paper is that, off of a `fundamental class' in degree 1, the linearized Legendrian contact homology obeys a version of Poincare duality between homology groups in degrees k and -k. Not only does the result itself simplify calculations, but its proof also establishes a framework for analyzing coh…

Study shows non-trivial homology classes in certain symmetric manifolds.

In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (p…

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number…

The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.

In this paper we study the problem of exact recovery of the pure-strategy Nash equilibria (PSNE) set of a graphical game from noisy observations of joint actions of the players alone. We consider sparse linear influence games --- a parametric class of graphical games with linear payoffs, and represented by directed gra…

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then…

Study quadratic one-forms on logarithmic Higgs bundles on pointed curves.

This paper introduces the notion of ``relative gerbes'' for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are further classified by the relative integral cohomology in degree three.

The $L^p$-cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.

BNSR invariants are contained in the complement of tropical varieties.

We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.

New algorithm determines dimensions of hit spaces in polynomial algebra.

This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.

Let ${\cal F}$ be a complex affine Reeb foliation of dimension $1$ on the Hopf manifold ${\Bbb S}^{n+1}\times {\Bbb S}^1$. We prove that its foliated Dolbeault cohomology in degree $1$ is isomorphic to ${\Bbb C}$ by giving an explicit generator.

Decomposable arrangements have simpler topological and combinatorial properties.

Dealing with uncertainty in Bayesian Network structures using maximum a posteriori (MAP) estimation or Bayesian Model Averaging (BMA) is often intractable due to the superexponential number of possible directed, acyclic graphs. When the prior is decomposable, two classes of graphs where efficient learning can take plac…

This study uses Tsallis entropy to analyze diversification and integration in Italian stock market companies.

We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental group. In degree two we clarify the relation between hyperbolic and atoroidal cla…

We prove that Hitchin's generalized Kaehler structure on the moduli space of instantons over a compact, even generalized Kaehler four-manifold may be obtained by generalized Kaehler reduction, in analogy with the usual Kaehler case. The underlying reduction of Courant algebroids is a realization of Donaldson's $μ$-map …

Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generaliz…

The paper explores the geometry of algebraic numbers and their roots.

In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| < \min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359…

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…

Study on $L^p$ cohomology and Hodge decomposition for ALE manifolds.

We argue that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di)operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of that (di)operads i…

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection be…

Symplectic structures simplified for compact manifolds.

The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.

A test for sparsity in Bayesian networks helps choose algorithms.

This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three. Furthermore, by using the concept of relative gerbes, the pre-quantization of Lie…

An equivariant bundle gerbe à la Meinrenken over a $G$-manifold $M$ is known to be a special type of $S^1$-gerbe over the differentiable stack $[M/G]$. We prove that the natural morphism relating the Cartan and simplicial models of equivariant cohomology in degree 3 maps the Dixmier-Douady class of an equivariant bundl…

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no close conjugate points at infinity. Our argument builds on an existence result d…

Develops differential KO-character to determine real vector bundles in multiples of 8.

The study identifies two minimal orbits of foliations on complex projective plane and explores their properties.

We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degr…

We show that the isomorphism induced by the inclusion of pairs $(X,\emptyset)\subset (X,Y)$ between the relative bounded cohomology of $(X,Y)$ and the bounded cohomology of $X$ is isometric in degree at least 2 if the fundamental group of each connected component of $Y$ is amenable. As an application we provide a self-…