Dimensional Simplex Research Articles

Upper bounds for Heilbronn's triangle problem in higher dimensions

AbstractWe develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed , any subset of of size contains points that span a simplex of volume at most , points whose convex hull has volume at most , points that span a ‐dimensional simplex of volume at most .

Wasserstein Dictionaries of Persistence Diagrams.

This article presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters Turner et al. (2014), Vidal et al. (2020) of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

Criteria for the Existence of Internal Fixed Points of Lotka-Volterra Quadratic Stochastic Mappings with Homogeneous Tournaments Acting in an (m−1) - Dimensional Simplex

Criteria for the Existence of Internal Fixed Points of Lotka-Volterra Quadratic Stochastic Mappings with Homogeneous Tournaments Acting in an (m−1) - Dimensional Simplex

Refined Ehrhart Series and Bigraded Rings

We study a natural set of refinements of the Ehrhart series of a closed polytope, first considered by Chapoton. We compute the refined series in full generality for a simplex of dimension 𝑑, a cross-polytope of dimension 𝑑, respectively a hypercube of dimension 𝑑 ≤ 3, using commutative algebra. We deduce summation formulae for products of 𝑞-integers with different arguments, generalizing a classical identity due to MacMahon and Carlitz. We also present a characterisation of a certain refined Eulerian polynomial in algebraic terms.

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

The Gordian complexes of knots given by 4-move

The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in [Formula: see text]. Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot [Formula: see text] and for any given natural number [Formula: see text], there exists a family of knots [Formula: see text] such that for any pair [Formula: see text] of distinct elements of the family, the Gordian distance of knots by 4-move is [Formula: see text]. We also show the existence of an arbitrarily high dimensional simplex in the Gordian complexes.

Calibrated simplex-mapping classification.

We propose a novel methodology for general multi-class classification in arbitrary feature spaces, which results in a potentially well-calibrated classifier. Calibrated classifiers are important in many applications because, in addition to the prediction of mere class labels, they also yield a confidence level for each of their predictions. In essence, the training of our classifier proceeds in two steps. In a first step, the training data is represented in a latent space whose geometry is induced by a regular (n - 1)-dimensional simplex, n being the number of classes. We design this representation in such a way that it well reflects the feature space distances of the datapoints to their own- and foreign-class neighbors. In a second step, the latent space representation of the training data is extended to the whole feature space by fitting a regression model to the transformed data. With this latent-space representation, our calibrated classifier is readily defined. We rigorously establish its core theoretical properties and benchmark its prediction and calibration properties by means of various synthetic and real-world data sets from different application domains.

Fusions of Consciousness.

What are conscious experiences? Can they combine to form new experiences? What are conscious subjects? Can they combine to form new subjects? Most attempts to answer these questions assume that spacetime, and some of its particles, are fundamental. However, physicists tell us that spacetime cannot be fundamental. Spacetime, they say, is doomed. We heed the physicists, and drop the assumption that spacetime is fundamental. We assume instead that subjects and experiences are entities beyond spacetime, not within spacetime. We make this precise in a mathematical theory of conscious agents, whose dynamics are described by Markov chains. We show how (1) agents combine into more complex agents, (2) agents fuse into simpler agents, and (3) qualia fuse to create new qualia. The possible dynamics of n agents form an n(n-1)-dimensional polytope with nn vertices-the Markov polytopeMn. The total fusions of n agents and qualia form an (n-1)-dimensional simplex-the fusion simplexFn. To project the Markovian dynamics of conscious agents onto scattering processes in spacetime, we define a new map from Markov chains to decorated permutations. Such permutations-along with helicities, or masses and spins-invariantly encode all physical information used to compute scattering amplitudes. We propose that spacetime and scattering processes are a data structure that codes for interactions of conscious agents: a particle in spacetime is a projection of the Markovian dynamics of a communicating class of conscious agents.

Heterogeneous Graph Convolutional Neural Network via Hodge-Laplacian for Brain Functional Data.

This study proposes a novel heterogeneous graph convolutional neural network (HGCNN) to handle complex brain fMRI data at regional and across-region levels. We introduce a generic formulation of spectral filters on heterogeneous graphs by introducing the k-th Hodge-Laplacian (HL) operator. In particular, we propose Laguerre polynomial approximations of HL spectral filters and prove that their spatial localization on graphs is related to the polynomial order. Furthermore, based on the bijection property of boundary operators on simplex graphs, we introduce a generic topological graph pooling (TGPool) method that can be used at any dimensional simplices. This study designs HL-node, HL-edge, and HL-HGCNN neural networks to learn signal representation at a graph node, edge levels, and both, respectively. Our experiments employ fMRI from the Adolescent Brain Cognitive Development (ABCD; n=7693) to predict general intelligence. Our results demonstrate the advantage of the HL-edge network over the HL-node network when functional brain connectivity is considered as features. The HL-HGCNN outperforms the state-of-the-art graph neural networks (GNNs) approaches, such as GAT, BrainGNN, dGCN, BrainNetCNN, and Hypergraph NN. The functional connectivity features learned from the HL-HGCNN are meaningful in interpreting neural circuits related to general intelligence.

Criteria for internal fixed points existence of discrete dynamic Lotka–Volterra systems with homogeneous tournaments

Purpose of the work is to study the dynamics of the asymptotic behavior of trajectories of discrete Lotka–Volterra dynamical systems with homogeneous tournaments operating in an arbitrary (𝑚 − 1)-dimensional simplex. It is known that a dynamic system is an object or a process for which the concept of a state is uniquely defined as a set of certain quantities at a given time, and a law describing the evolution of initial state over time is given. Mainly in questions of population genetics, biology, ecology, epidemiology and economics, systems of nonlinear differential equations describing the evolution of the process under study often arise. Since the Lotka–Volterra equations often arise in life phenomena, the main purpose of the work is to study the trajectories of discrete dynamical Lotka–Volterra systems using elements of graph theory. Methods. In the paper cards of fixed points are constructed for quadratic Lotka–Volterra mappings, that allow describing the dynamics of the systems under consideration. Results. Using cards of fixed points of a discrete dynamical system, criteria for the existence of fixed points with odd nonzero coordinates are given in a particular case, and these results on the location of fixed points of Lotka–Volterra systems are generalized accordingly in the case of an arbitrary simplex. The main results are theorems 5–9, which allow us to describe the dynamics of these systems arising in a number of genetic, epidemiological and ecological models. Conclusion. The results obtained in the paper give a detailed description of the dynamics of the trajectories of Lotka–Volterra maps with homogeneous tournaments. The map of fixed points highlights a specific area in the simplex that is most important and interesting for studying the dynamics of these maps. The results obtained are applicable in environmental problems, for example, to describe and study the cycle of biogens.

PI-type fully symmetric quadrature rules on the 3-, …, 6-simplexes

We consider fully symmetric quadrature rules with positive weights, and with nodes lying inside the 3 , … , 6 dimensional simplex (so-called PI-type). PI-type fully symmetric quadrature rules up to 20-th order on the tetrahedron, 16-th order on 4-simplex, 10-th order on 5- and 6-simplexes are presented. The number of nodes of the presented quadrature rules for the corresponding orders does not exceed the known ones, and most of them are new. In the calculation we applied the modified Levenberg-Marquardt methods for solving nonlinear equations with convex constraints. The corresponding programs are implemented in MAPLE-FORTRAN environment, and the weights and nodes are first calculated using a FORTRAN program with an accuracy of 10 − 25 and refined up to accuracy of 10 − 50 using a MAPLE program.

Configurable Open-source Data Structure for Distributed Conforming Unstructured Homogeneous Meshes with GPU Support

A general multi-purpose data structure for an efficient representation of conforming unstructured homogeneous meshes for scientific computations on CPU and GPU-based systems is presented. The data structure is provided as open-source software as part of the TNL library (https://tnl-project.org/). The abstract representation supports almost any cell shape and common 2D quadrilateral, 3D hexahedron and arbitrarily dimensional simplex shapes are currently built into the library. The implementation is highly configurable via templates of the C++ language, which allows avoiding the storage of unnecessary dynamic data. The internal memory layout is based on state-of-the-art sparse matrix storage formats, which are optimized for different hardware architectures in order to provide high-performance computations. The proposed data structure is also suitable for meshes decomposed into several subdomains and distributed computing using the Message Passing Interface (MPI). The efficiency of the implemented data structure on CPU and GPU hardware architectures is demonstrated on several benchmark problems and a comparison with another library. Its applicability to advanced numerical methods is demonstrated with an example problem of two-phase flow in porous media using a numerical scheme based on the mixed-hybrid finite element method (MHFEM). We show GPU speed-ups that rise above 20 in 2D and 50 in 3D when compared to sequential CPU computations, and above 2 in 2D and 9 in 3D when compared to 12-threaded CPU computations.

Completing and Extending Shellings of Vertex Decomposable Complexes

We say that a pure $d$-dimensional simplicial complex $\Delta$ on $n$ vertices is shelling completable if $\Delta$ can be realized as the initial sequence of some shelling of $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable. In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if $\Delta$ is a vertex decomposable complex then there exists an ordering of its ground set $V$ such that adding the revlex smallest missing $(d+1)$-subset of $V$ results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as $k$-vertex decomposable complexes. We also show that if $\Delta$ is a $d$-dimensional complex on at most $d+3$ vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.

Exploration of spinodal decomposition in multi-principal element alloys (MPEAs) using CALPHAD modeling

Researchers attributed the orderly arranged nanoscale phases observed in many multi-principal element alloys (MPEAs) to spinodal/spinodal-mediated phase transformation pathways. However, spinodal decomposition is not well understood in multicomponent systems. Although the theoretical background is available, CALPHAD databases were not used to explore the miscibility gap in MPEAs. In this work, we develop a CALPHAD framework utilizing the Hessian of free energy to study the stability of solid solutions in MPEAs. In particular, we utilize the geometry of higher dimensional Gibbs simplex in conjunction with the Hessian to calculate concentration modulations in early stages of spinodal decomposition. We apply this framework to a diverse set of multi-phase MPEAs that have been studied in the literature including TiZrNbTa (BCC/BCC), Fe15Co15Ni20Mn20Cu30 (FCC/FCC), Al0.5NbTa0.8Ti1.5V0.2Zr (BCC/B2), AlCo0.4Cr0.6FeNi (BCC/B2) and Al0.5Cr0.9FeNi2.5V0.2 (FCC/L12). We show that the MPEA systems are unstable only to certain concentration modulations which could be further explored to design microstructurally engineered alloys.

Linkedness of Cartesian products of complete graphs

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

Triangle inequalities in inner-product spaces

Tereshin's and Panaitopol's are known inequalities involving the median, circumradius and sides of the triangle. In this short note we generalize the inequalities to inner-product spaces. As an application we derive inequality for the median and the radius of the circumscribed sphere of an $n$-dimensional simplex.

Numerical integration on higher dimensional simplicial and curved finite elements

We present a formula which evaluates lower degree monomials over higher dimensional simplices by means of integration of higher degree monomials over an interval, triangle or tetrahedron. Further, we show how to apply some higher order quadrature formulae on curved elements using a one-to-one mapping from the reference simplicial element to a curved element.Finally, we demonstrate that the non-zero Jacobian does not imply that this mapping is one-to-one.

Algorithmic Symplectic Packing

In this article we explore a symplectic packing problem where the targets and domains are 2n-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to Z n , and we require the embeddings to induce isomorphisms between first homology groups. In this case, Miller Maley, Mastrangeli, and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they determined the simplex packing widths in dimension four for up to k = 12 simplices, along with lower bounds for higher values of k. We present a modified algorithmic approach that allows us to determine the k-simplex packing widths for up to k = 13 simplices in dimension four and up to k = 8 simplices in dimension six. Moreover, our approach determines all simplex-multisets that allow for optimal packings.

The arbitrariness of the semi-angle-bisectors of a triangle

Figure 1 shows a triangle ABC with the midpoints A′, B′ and C′ of its sides. The line segments AA′, BB′ and CC′ are called the medians, and the point G of their intersection the centroid. The line segments AG, BG and CG will be called, for lack of a better name, the semi-medians. It is interesting that the medians of any triangle can serve as the side lengths of some triangle. This property of the medians is referred to as the median triangle theorem in [1, §473, page 282], and is discussed, together with generalisations to tetrahedra and higher dimensional simplices, in [2].

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.