Complex Hyperplane Research Articles

Cohomology of ℤ-Local Systems on Complex Hyperplane Arrangement Complements

Abstract We prove a Cohen-Dimca-Orlik-type theorem for rank one ${\mathbb{Z}}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara.

A Novel Approach for Quaternion Algebra Based JSEG Color Texture Segmentation

In this work, a novel colour quantization approach has been applied to the JSEG colour texture segmentation using quaternion algebra. As a rule, the fundamental vectors of the colour space are derived by inverting the three RGB colour directions in the complex hyperplanes. In the proposed system, colour is represented as a quaternion because quaternion algebra provides a very intuitive means of working with homogeneous coordinates. This representation views a colour pixel as a point in the three-dimensional space. A novel quantization approach that makes use of projective geometry and level set methods has been produced as a consequence of the suggested model. The JSEG colour texture segmentation will use this technique. The new colour quantization approach utilises the binary quaternion moment preserving thresholding methodology, and is therefore a splintering clustering method. This method is used to segment the colour clusters found inside the RGB cube and the colour consistency throughout the spectrum and in the space are both considered. The results of the segmentation are compared with JSEG as well as with the most recent standard segmentation techniques. These comparisons show that the suggested quantization technique makes JSEG segmentation more robust.

Some remarks on a theorem of Green

The purpose of this paper is to study holomorphic curves f from C to C3 avoiding four complex hyperplanes and a real subspace of real dimension four in C3. We show that the projection of f into the complex projective space C P^2 does not remain constant as in the complex case studied by Green, which indicates that the complex structure of the avoided hyperplanes is a necessary condition in the Green theorem

Multi-Agent Reinforcement Learning Controller to Maximize Energy Efficiency for Multi-Generator Industrial Wave Energy Converter

Waves in the oceans are one of the most significant renewable energy sources and are an excellent resource to tackle climate challenges through decarbonizing energy generation. Lowering the Levelized Cost of Energy (LCOE) for energy generation from ocean waves is critical for competitiveness with other forms of clean energy like wind and solar. It requires complex controllers to maximize efficiency for state-of-the-art multi-generator industrial Wave Energy Converters (WEC), which optimizes the reactive forces of the generators on multiple legs of WEC. This paper introduces Multi-Agent Reinforcement Learning controller (MARL) architectures that can handle these various objectives for LCOE. MARL can help increase energy capture efficiency to boost revenue, reduce structural stress to limit maintenance cost, and adaptively and proactively protect the wave energy converter from catastrophic weather events preserving investments and lowering effective capital cost. These architectures include 2-agent and 3-agent MARL implementing proximal policy optimization (PPO) with various optimizations to help sustain the training convergence in the complex hyperplane without falling off the cliff. Also, the design for trust assures the operation of WEC within a safe zone of mechanical compliance. As a part of this design, reward shaping for multiple objectives of energy capture and penalty for harmful motions minimizes stress and lowers the cost of maintenance. We achieved double-digit gains in energy capture efficiency across the waves of different principal frequencies over the baseline Spring Damper controller with the proposed MARL controllers.

Gromov–Witten theory and invariants of matroids

We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

Ferrers graphs, D-permutations, and surjective staircases

We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices of Ferrers graphs. Using recent work of Lazar and Wachs we are able to give combinatorial interpretations of the characteristic polynomials of these arrangements in terms of permutation enumeration. For certain infinite families of these hyperplane arrangements, we are able to give generating function formulas for their characteristic polynomials. To do so, we develop a generalization of Dumont’s surjective staircases, and introduce a polynomial which enumerates these generalized surjective staircases according to several statistics. We prove a recurrence for these polynomials and show that in certain special cases this recurrence can be solved explicitly to yield a generating function. We also prove refined versions of several of these results using the theory of complex hyperplane arrangements.

Relational Structure-Aware Knowledge Graph Representation in Complex Space

Relations in knowledge graphs have rich relational structures and various binary relational patterns. Various relation modelling strategies are proposed for embedding knowledge graphs, but they fail to fully capture both features of relations, rich relational structures and various binary relational patterns. To address the problem of insufficient embedding due to the complexity of the relations, we propose a novel knowledge graph representation model in complex space, namely MARS, to exploit complex relations to embed knowledge graphs. MARS takes the mechanisms of complex numbers and message-passing and then embeds triplets into relation-specific complex hyperplanes. Thus, MARS can well preserve various relation patterns, as well as structural information in knowledge graphs. In addition, we find that the scores generated from the score function approximate a Gaussian distribution. The scores in the tail cannot effectively represent triplets. To address this particular issue and improve the precision of embeddings, we use the standard deviation to limit the dispersion of the score distribution, resulting in more accurate embeddings of triplets. Comprehensive experiments on multiple benchmarks demonstrate that our model significantly outperforms existing state-of-the-art models for link prediction and triple classification.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

A stochastic collocation approach for parabolic PDEs with random domain deformations

In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. The stochastic moments of the QoI are computed by employing a collocation method in conjunction with an isotropic Smolyak sparse grid. Theoretical sub-exponential convergence rates as a function to the number of collocation interpolation knots are derived. Numerical experiments are performed and they confirm the theoretical error estimates.

Arrhythmia detection using multi-lead ECG spectra and Complex Support Vector Machine Classifiers

Electrocardiograms (ECG) are extensively used for the diagnosis of cardiac arrhythmias. This work investigates the use of machine learning classification algorithms for ECG analysis and arrhythmia detection. Four beat types, Normal (N), Premature Ventricular Contraction (PVC), Atrial Premature Contraction (APC) and Right Bundle Branch Block Beat (RBBB) are simultaneously presented to a Complex Support Vector Machine (CSVM) classifier. The ECG signals are obtained from the St Petersburg INCART 12-lead Arrhythmia Database (INCARTDB). The detection of ECG Wave (P, QRS, T) is performed with the Wave Form Database (WFDB) Software Package which is used to read the annotation files and find the R (peak) location. For feature extraction, the Discrete Fourier Transform (DFT) is used. Fifty Fourier coefficients were selected for reconstructing individual ECG beats. This ensures moderate dimensionality reduction and de-noising of the input vector to the classifier. ECG beats classification is performed using CSVM with several training and test datasets. Sequential Minimal Optimization (SMO) is used to train the CSVM and compute the hyperplane parameters associated with both the real and complex hyperplanes. Cross validation is used for finding the best parameter values of the SVM and the two Gaussian RBK kernel functions. The aim of the study is to establish the advantage of CSVM over standard SVM in simultaneously detecting different types of arrhythmias on the basis of multi-lead recordings following signal compression in the Fourier domain. Implementation of the algorithms was performed in MATLAB. The CSVM classification algorithm provided better performance than the standard SVM classifier. The classification accuracy of the proposed scheme is 98.25% using CSVM. Future work will concentrate on the further development of ECG signal pre-processing using adaptive wavelet algorithms as well as classification with Clifford SVMs.

Cubature rules from Hall–Littlewood polynomials

Abstract Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group $SU(n;\mathbb{C})$). By passing to Macdonald’s hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group $Sp (n;\mathbb{H})$). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.

The Tutte polynomial of symmetric hyperplane arrangements

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

Hyperbolicity of the complement of arrangements of non complex lines

The goal of this paper is twofold. We study holomorphic curves f:C ? C3 avoiding four complex hyperplanes and a real subspace of real dimension five in C3 where we study the cases where the projection of f into the complex projective space CP2 is constant. On the other hand, we investigate the kobayashi hyperbolicity of the complement of five perturbed lines in CP2.

Double coverings of arrangement complements and 2-torsion in Milnor fiber homology

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space $$\mathbb {CH}^n$$ . As an application, we study minimal ruled real hypersurfaces in $$\mathbb {CH}^n$$ and also those having constant scalar curvature.

Chern–Schwartz–MacPherson cycles of matroids

We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.

Combinatorial analogs of topological zeta functions

In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function ZL(s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolutions of singularities, and the main result is that ZL(s) does not depend on this choice and depends only on L. Known properties of the topological zeta function provide a source of potential ℂ-realisability test for matroids.

Naturality properties and comparison results for topological and infinitesimal embedded jump loci

We use augmented commutative differential graded algebra (▪) models to study G-representation varieties of fundamental groups π=π1(M) and their embedded cohomology jump loci, around the trivial representation 1. When the space M admits a finite family of maps, uniformly modeled by ▪ morphisms, and certain finiteness and connectivity assumptions are satisfied, the germs at 1 of Hom(π,G) and of the embedded jump loci can be described in terms of their infinitesimal counterparts, naturally with respect to the given families. This approach leads to fairly explicit answers when M is either a compact Kähler manifold, the complement of a central complex hyperplane arrangement, or the total space of a principal bundle with formal base space, provided the Lie algebra of the linear algebraic group G is a non-abelian subalgebra of sl2(C).

Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras H*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FIW-algebras finitely generated inFIW-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

Equidistribution rate for Fekete points on some real manifolds

Let L be a positive line bundle over a compact complex projective manifold X and K be a compact subset of X which is regular in a sense of pluripotential theory. A Fekete configuration of order k is a finite subset of K maximizing a Vandermonde type determinant associated with the power L^k of L. Berman, Boucksom and Witt Nystrom proved that the empirical measure associated with a Fekete configuration converges to the equilibrium measure of K as k tends to infinity. Dinh, Ma and Nguyen obtained an estimate for the rate of convergence. Using techniques from Cauchy-Riemann geometry, we show that the last result holds when K is a real nondegenerate C^5-piecewise submanifold of X such that its tangent space at any regular point is not contained in a complex hyperplane of the tangent space of X at that point. In particular, the estimate holds for Fekete points on some compact sets in R^n or the unit sphere in R^{n+1}.