Combinatorial Type Research Articles

Poncelet’s closure theorem and the embedded topology of conic-line arrangements

Abstract In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet’s closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$ -sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components.

DeSide: A unified deep learning approach for cellular deconvolution of tumor microenvironment

Cellular deconvolution via bulk RNA sequencing (RNA-seq) presents a cost-effective and efficient alternative to experimental methods such as flow cytometry and single-cell RNA-seq (scRNA-seq) for analyzing the complex cellular composition of tumor microenvironments. Despite challenges due to heterogeneity within and among tumors, our innovative deep learning-based approach, DeSide, shows exceptional accuracy in estimating the proportions of 16 distinct cell types and subtypes within solid tumors. DeSide integrates biological pathways and assesses noncancerous cell types first, effectively sidestepping the issue of highly variable gene expression profiles (GEPs) associated with cancer cells. By leveraging scRNA-seq data from six cancer types and 185 cancer cell lines across 22 cancer types as references, our method introduces distinctive sampling and filtering techniques to generate a high-quality training set that closely replicates real tumor GEPs, based on The Cancer Genome Atlas (TCGA) bulk RNA-seq data. With this model and high-quality training set, DeSide outperforms existing methods in estimating tumor purity and the proportions of noncancerous cells within solid tumors. Our model precisely predicts cellular compositions across 19 cancer types from TCGA and proves its effectiveness with multiple additional external datasets. Crucially, DeSide enables the identification and analysis of combinatorial cell type pairs, facilitating the stratification of cancer patients into prognostically significant groups. This approach not only provides deeper insights into the dynamics of tumor biology but also highlights potential therapeutic targets by underscoring the importance of specific cell type or subtype interactions.

Boundary rigidity of 3D CAT(0) cube complexes

The boundary rigidity problem is a classical question from Riemannian geometry: if (M,g) is a Riemannian manifold with smooth boundary, is the geometry of M determined up to isometry by the metric dg induced on the boundary ∂M? In this paper, we consider a discrete version of this problem: can we determine the combinatorial type of a finite cube complex from its boundary distances? As in the continuous case, reconstruction is not possible in general, but one expects a positive answer under suitable contractibility and non-positive curvature conditions. Indeed, in two dimensions Haslegrave gave a positive answer to this question when the complex is a finite quadrangulation of the disc with no internal vertices of degree less than 4. We prove a 3-dimensional generalisation of this result: the combinatorial type of a finite CAT(0) cube complex with an embedding in R3 can be reconstructed from its boundary distances. Additionally, we prove a direct strengthening of Haslegrave’s result: the combinatorial type of any finite 2-dimensional CAT(0) cube complex can be reconstructed from its boundary distances.

Combinatorics of Correlated Equilibria

We study the correlated equilibrium polytope PG of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of PG , and show that for ( 2 × n ) -games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic ( 2 × 3 ) -games.

On the rank decoding problem over finite principal ideal rings

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings.In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

Further Rogers-Ramanujan type identities for modified lattice paths

Recently, the authors introduced the modified lattice paths which generalize Agarwal-Bressoud weighted lattice paths. Using these new objects they interpreted combinatorially two basic series identities which led to two new combinatorial Rogers-Ramanujan type identities. In this paper we obtain three more Rogers-Ramanujan type identities for modified lattice paths. This also leads to three new 3-way combinatorial identities.

Rigidity, Tensegrity, and Reconstruction of Polytopes Under Metric Constraints

Abstract We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset {\mathbb {R}}^{d}$ and $Q\subset {\mathbb {R}}^{e}$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack these questions and we verify them in three relevant special cases: $P$ and $Q$ are centrally symmetric, $Q$ is a slight perturbation of $P$, and $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q:V(G_{P})\to {\mathbb {R}}^{e}$ of the edge-graph $G_{P}$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.

The Newton polytope of the Morse discriminant of a univariate polynomial

In this paper we compute the Newton polytope MA of the Morse discriminant in the space of univariate polynomials with a given support set A. Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of MA.

Abstract P3049: Exploring The Therapeutic Potential Of Aav-mediated Restoration Of Col6a2 In A Mouse Model Of Ullrich Muscular Dystrophy

Ullrich muscular dystrophy is a genetic disease characterized by joint contractures and muscle weakness which leads to loss of ambulation and respiratory insufficiency early in life. This disease is caused by mutations in any of the three major genes of collagen VI - Col6a1, Col6a2, or Col6a3. Together, these three genes are required to generate a complex quaternary protein assembly that is secreted to help form the basement membrane. Currently there is no cure or greater treatment for Ullrich muscular dystrophy although 2 of the genes (Col6a1, Col6a2) are within the size limit of adeno-associated virus (AAV) and could be addressed by gene therapy. However, it is unclear what cell type in muscle generates the bulk of Collagen VI during development, or if it could be replaced at a later time by gene therapy from expression in myofibers or from fibroblasts. Given that mutations in the Col6a2 gene are the most common cause of Ullrich muscular dystrophy, here we generated Col6a2 gene deleted (Col6a2 -/- ) mice which showed the anticipated histopathologic evidence of dystrophic disease as young adults. Adult mice were then subject to retro-orbital injection of MyoAAV-Col6a2, which resulted in robust expression of Col6a2 within the myofibers of all skeletal muscles analyzed (and heart), although this was ineffective in restoring the collagen VI complex within the basement membrane. Further characterization into the effects of Col6a2 re-expression from the myofibers versus the fibroblasts in these gene targeted mice is underway with the goal of defining a combinatorial (with Col6a1 and Col6a3) or cell type restricted infection protocol to successfully achieve full restoration of Collagen VI in muscle.

Many regular triangulations and many polytopes

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A Tropical Count of Real Bitangents to Plane Quartic Curves

A smooth tropical plane quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Plücker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.

Product decompositions of moment-angle manifolds and B-rigidity

AbstractA simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

Double modals in contemporary British and Irish speech

This article reports on the use of double modals, a non-standard syntactic feature, in the contemporary speech of the UK and Ireland. Most data on the geographic extent of the feature and its combinatorial types come from surveys or acceptability ratings or from older attestations focused on northern England, Scotland or Northern Ireland, with relatively few attestations in naturalistic data and from England and Wales. Manual verification of double modals in a large corpus of geolocated Automatic Speech Recognition transcripts from YouTube videos of local government channels from the UK and Ireland shows that the feature exhibits a larger inventory of combinatorial types than has previously been found and is attested in speech from throughout the UK and Ireland. The development may be related to ongoing changes in the semantic space occupied by modal auxiliaries in English.

Optimal deployment of fast-charging stations for electric vehicles considering the sizing of the electrical distribution network and traffic condition

The conventional vehicle fleet worldwide has contributed to the degradation of air quality due to CO2 emissions. Consequently, it has migrated from internal combustion to electric vehicles (EVs). However, it is essential to ensure the deployment of electric vehicle charging station infrastructures (EVCSI) to guarantee their interoperability for the development of electric mobility. Moreover, the sustainability of EVCSI depends not only on the capacity to meet demand but also on the adequate number of terminals in the different public charging stations (CS) to reduce waiting times for battery recharging. Then to achieve an optimal sizing of charging stations, it is crucial to foresee the maximum number of vehicles that could use the different CS during a time interval. The sizing of CS must respond to real mobility constraints and technical conditions, such as the capacity of vehicular flow, the capacity of the roads according to their geometry, the trajectories marked by the users, and the possible exit of operation of some CS. Therefore, this paper addresses the problem considering four fundamental axes, which are: stochastic analysis of heterogeneous vehicular flow, a solution to the transportation problem with the capacitated multicommodity flow problem and Hungarian algorithm, analysis of the optimal number of terminals considering loading times, and finally the proposed EVCSI will be validated using the CymDist software for electrical engineering. Consequently, the computational complexity of the model is of a combinatorial type and is defined as NP-hard given the multiple variables and constraints within the transportation problem.

Light as an artistic tool of designing the architectural and urban environment

The article examines light as a tool of architectural and lighting design and a mean of artistic expression, which creates new social messages and influences environmental objects and the architectural environment. Within the scope of the tool, techniques of artistic lighting are researched and formed, traditional and innovative lighting tools of emotional and figurative formation of the urban and architectural environment from the standpoint of artistic design are determined. Traditional means and devices are lenses, optics, developer, LED and craft lamps. Innovative means and equipment are projectors, lasers, video mapping, theatrical elements, musical accompaniment, interaction through websites and mobile devices, management of the cloud environment using system controls, copywriting based on social messages and senses. As a result of the study, light methods of artistic design were considered, specifically transformation, digital art, emotional disclosure, city landmark, keeper of history, optical metaphor, fusion with nature, and types of artistic lighting were formulated. The analysis of the examples also allowed us to determine specific types of artistic lighting and light artistic effects: functional, atmospheric, optical, media or digital, combinatorial type with shadow and caustic effects. For each type of lighting, unique techniques and means of environment organization are formulated for each of the three levels: urban, architectural and object-spatial. The usage of methods, techniques and tools is possible both combined and separately. Their harmonious implementation and usage form an emotional interactive lighting architectural and urban planning environment with its own atmosphere and rules of perception. Successful operation of certain methods and means within each of the formulated types of environment lighting could express emotionality and social messages. Their organic interaction is possible applying an artistic approach based on the combination of philosophy and social messages with aesthetics and modern technologies and that encourages social responsibility and self-organization.

Optimal Planning of Electric Vehicle Charging Stations Considering Traffic Load for Smart Cities

The massive introduction of electric vehicles as a mobility alternative requires deploying an infrastructure of charging stations for electric cars (ICSEC). This new mobility concept will mitigate the environmental harm caused by the emission of CO2 generated by conventional internal combustion mobility methods. The sustainability of the ICSEC depends not only on the capacity to meet the demand for charging batteries for electric vehicles (EV) but also on an adequate number of public/private charging stations (CS) distributed in a geolocalized area. It is noted that the distribution of CS must respond to a set of real mobility constraints, such as vehicular flow capacity, road capacity, and trajectories. The planning, intelligent location of public charging stations (PCS), and contingency analysis will enable us to study the increase in demand for electrical distribution substations (EDS). Therefore, the present model explains the need to plan the massive introduction of EVs by observing the user’s conditions at the trajectory level through finite resource allocation processes, segmentation, and minimum spanning trees, by observing heterogeneous vehicular flow criteria through microscopic analysis, to understand the space–time relationship of vehicular flow in georeferenced scenarios. Consequently, the computational complexity of the model is of the combinatorial type, and it is defined as NP-Hard given the multiple variables and constraints within the transportation problem.

Combinatorics and Real Lifts of Bitangents to Tropical Quartic Curves

Smooth algebraic plane quartics over algebraically closed fields of characteristic different than two have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real.

A classification of combinatorial types of discriminantal arrangements

Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the $\textit{discriminantal arrangements}$. Athanasiadis proved a conjecture by Bayer and Brandt providing a full description of the combinatorics of discriminantal arrangements in the case of $\textit{very generic}$ arrangements. Libgober and Settepanella described a sufficient geometric condition for given arrangements to be $\textit{non very generic}$ in terms of the notion of dependency for a certain arrangement. Settepanella and the author generalized the notion of dependency introducing $r$-sets and $K_\mathbb{T}$-vector sets, and provided a sufficient condition for non very genericity but still not convenient to verify by hand. In this paper we give a classification of the $r$-sets, and a more explicit and tractable condition for non very genericity.

Reconstructing simplicial polytopes from their graphs and affine 2-stresses

A conjecture of Kalai from 1994 posits that for an arbitrary 2 ≤ k ≤ ⌊d/2⌋, the combinatorial type of a simplicial d-polytope P is uniquely determined by the (k − 1)-skeleton of P (given as an abstract simplicial complex) together with the space of affine k-stresses on P. We establish the first non-trivial case of this conjecture, namely, the case of k = 2. We also prove that for a general k, Kalai’s conjecture holds for the class of k-neighborly polytopes.

Nonautonomous dynamics: classification, invariants, and implementation

The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently - jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classi cation of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector elds on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14,17], as shown in a recent paper by the authors [5].