arXiv research

We solve the multi-criteria benchmarking problem by formalizing it as a social choice problem and identifying conditions for meaningful rankings.

New knot invariants from biquandle arrow weights.

We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollobás-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ri…

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summa…

New invariant for virtual links defined using homology.

Paper detects checkerboard colorability of virtual links using odd writhe and arrow polynomial.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial. The paper is relatively self-contained and it describes virtual knot theory bot…

General equilibrium equations in economics play the same role with many-body Newtonian equations in physics. Accordingly, each solution of the general equilibrium equations can be regarded as a possible microstate of the economic system. Since Arrow's Impossibility Theorem and Rawls' principle of social fairness will p…

We develop a calculus for diagrams of knotted objects. We define Arrow presentations, which encode the crossing informations of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w-tree presentations, which can be seen as `higher order Gauss diagrams'. This Arrow calculus is used to d…

We give a definition of an integer-valued function $\sum_i α_i x ^*_i$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l$ arrows, called relators of Type~($\check{…

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this p…

New knot invariants derived from biquandle quivers.

For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\leq 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_…

In this survey paper we present results about link diagrams in Seifert manifolds using arrow diagrams, starting with link diagrams in $F\times S^1$ and $N\hat{\times}S^1$, where $F$ is an orientable and $N$ an unorientable surface. Reidemeister moves for such arrow diagrams make the study of link invariants possible. T…

We build a concrete and natural model for the strict 2-category of orbifolds. In particular we prove that if one localizes the 2-category of proper etale Lie groupoids at a class of 1-arrows that we call "covers", then the strict 2-category structure drops down to the localization. In our construction the spaces of 1- …

Define quiver representation-valued invariants for classical and virtual knots

New formula for knot invariants simplifies calculations and counts.

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial, the Kuperberg bracket, and the normalised arrow polynomial.

We describe the space of arrow diagram formulas for virtual knot diagrams in the annulus as the kernel of a linear map, inspired from a conjecture due to M. Polyak. As a main application, we slightly improve Grishanov-Vassiliev's theorem for planar chain invariants.

Under risk, Arrow-Debreu equilibria can be implemented as Radner equilibria by continuous trading of few long-lived securities. We show that this result generically fails if there is Knightian uncertainty in the volatility. Implementation is only possible if all discounted net trades of the equilibrium allocation are m…

This paper tabulates prime knot projections up to eight double points.

The paper confirms a conjecture and extends arrow polynomial to twisted links.

We consider an economy where agents' consumption sets are given by the cone $\mathbf{L}^0_+$ of non-negative measurable functions and whose preferences are defined by additive utilities satisfying the Inada conditions. We extend to this setting the results in \citet{Dana:93} on the existence and uniqueness of Arrow-Deb…

The definition of preferences assigned to individuals is a concept that concerns many disciplines, from economics, with the search of an acceptable outcome for an ensemble of individuals, to decision making an analysis of vote systems. We are concerned in the phenomena of good selection and economic fairness. In Arrow'…

The Thistlethwaite theorem is extended to knotoids and linkoids.

We show that univariate and symmetric multivariate Hawkes processes are only weakly causal: the true log-likelihoods of real and reversed event time vectors are almost equal, thus parameter estimation via maximum likelihood only weakly depends on the direction of the arrow of time. In ideal (synthetic) conditions, test…

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for classical knotoids in analogy to corresponding inv…

The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as $\barμ$--invariants. We prove that, analogously as for $\barμ$--invariants, certain residue classes of tree invariants yield link homotopy invariants of c…

Study on multiple linking numbers, extending Gauss diagram formulas.

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus.

We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.

This paper characterizes Milnor invariants using diagrammatic methods.

We show how effective-potential path-integrals methods, stemming on a simple and nice idea originally due to Feynman and successfully employed in Physics for a variety of quantum thermodynamics applications, can be used to develop an accurate and easy-to-compute semi-analytical approximation of transition probabilities…

Breiman discusses two statistical cultures, advocating for more research on 'before' and 'after' the black box.

This paper characterizes Kashiwara-Vergne groups using algebraic structures of knotted tubes.

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. F…

Lie groupoids and their orbit spaces are linked through equivalence classes.

Four-dimensional Einstein Dehn filling is impossible.

Submodularity is studied for convex risk measures, including Expected Shortfall.

I answer an open question left by Gui-Song Li in "On self-intersections of immersed surfaces" (AMS Proceedings, Volume 126, 1998, pp.3721-3726.) The intersection graph $M(i)$ of a generic surface $i:F \to S^3$ is the set of values which are either singularities or intersections. It is a multigraph whose edges are trans…

The purpose of this paper is to discuss how topology and geometry provide, in many instances, the connective tissue that enables logical comprehension. We illustrate this theme with many examples including Venn diagrams, knot diagrams, knot-logical diagrams and an arrow of reference that elucidates self-reference and G…

We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of th…

Machine fairness is impossible to achieve fully due to historical biases.

In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.

Novel cohomology theories for operadic algebras and spaces.

Chemical reactions can be described as the stepwise redistribution of electrons in molecules. As such, reactions are often depicted using `arrow-pushing' diagrams which show this movement as a sequence of arrows. We propose an electron path prediction model (ELECTRO) to learn these sequences directly from raw reaction …

TWM doesn't reduce delta in PDLPs, proving impossibility.

An impossibility result shows limitations in learning symmetries and equivariant functions.