arXiv research

PCL framework optimizes climate risk management across three clusters.

Euclidean systems and real PK arrangements linked via geometry.

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…

New infinite $K(π,1)$ arrangements found in higher dimensions.

Study shows certain toric arrangements have minimal topological complements.

Study a specific line arrangement and compute its fundamental group via braid monodromy.

Study restricts line arrangements with odd points using topological arguments.

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is…

Paper introduces EEMs for pricing contingent claim returns.

Sensorimotor contingency theory offers a promising account of the nature of perception, a topic rarely addressed in the robotics community. We propose a developmental framework to address the problem of the autonomous acquisition of sensorimotor contingencies by a naive robot. While exploring the world, the robot inter…

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some couples of arran…

The paper revisits and applies FTAP to life insurance and annuities pricing.

New method determines arrangement combinatorics from Milnor fiber boundary.

Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.

The icosidodecahedral arrangement is introduced by M. Yoshinaga (arXiv:1902.06256) as the first known example that is a hyperplane arrangement whose Milnor fiber has torsions in first integral homology. In this note, we prove that the icosidodecahedral arrangement is $K(π,1)$, hence so is its Milnor fiber.

Survey on hyperplane arrangements and their topology.

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…

Study conic-line arrangements of degree 7, finding their topology and components.

Study stability of contingent claim solutions under probabilistic perturbations.

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, t…

Bordifications of hyperplane arrangements yield complexes with homotopy type of wedges of spheres.

When applied to high-dimensional datasets, feature selection algorithms might still leave dozens of irrelevant variables in the dataset. Therefore, even after feature selection has been applied, classifiers must be prepared to the presence of irrelevant variables. This paper investigates a new training method called Co…

Two unique conic-line arrangements with degree 9 are found.

Paper proposes a method to estimate project cost contingency reserves considering various types of uncertainty.

Link projections with the same circle arrangement can be transformed by specific moves.

Generalizes cohomology ring result for combinatorial line arrangements.

Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the arrangements themselves, and define when a collection of arrangements is "finitely genera…

Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.

An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart, Ortner 2004) so-called *intersection schemes* were introduced. Building up on res…

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arr…

In this note we present examples of $K(π,1)$-arrangements which admit a restriction which fails to be $K(π,1)$. This shows that asphericity is not hereditary among hyperplane arrangements.

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding cert…

Closed-form pricing method for multi-asset options.

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

We note an intimate connection between the Lefschetz Theorem for c-arrangements, and a theorem of Hironaka relating the complement of an arrangement to its boundary manifold. This results in a generalization of Hironaka's result.

We price a contingent claim liability using the utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of positive proportional transaction cost in two c…

Motivated by applications to perverse sheaves, we study combinatorics of two cell decompositions of the symmetric product of the complex line, refining the complex stratification by multiplicities. Contingency matrices, appearing in classical statistics, parametrize the cells of one such decomposition, which has the pr…

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnik…

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size…

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…

Study on constraints for topological and smooth realizations of line arrangements and configurations.

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the fundamental group of their complements. It is derived from the peripheral struct…

A new framework for graph representation learning.

We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have disconnected realization spaces.

The paper defines and implements risk-indifference pricing for American-style contingent claims.

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backwar…