Narrow Results

Coupled map lattices are a popular and computationally simpler model of pattern formation in nonlinear systems. In this work, we investigate three-site interactions with linear multiplicative coupling in one-dimensional coupled logistic maps that cannot be decomposed into pairwise interactions. We observe the transition to synchronization and the transition to long-range order in space. We coarse-grain the phase space in regions and denote them by spin values. We use two quantifiers the flip rate $F(t)$ that quantify departure from expected band-periodicity as an order parameter. We also study a non-Markovian quantity, known as persistence $P(t)$ to study dynamic phase transitions. Following transitions are observed. (a) Transition to two band attractor state: At this transition $F(t)$ as well as $P(t)$ shows a power-law decay in the range of coupling parameters. Here all sites reach one of the bands. The $F(t)$ as well as $P(t)$ decays as power-law with the decay exponent ${\delta }_1=0.46$ and ${\eta }_1=0.28$, respectively. (b) The transition from a fluctuating chaotic state to a homogeneous synchronized fixed point: Here both the quantifiers $F(t)$ and $P(t)$ show power-law decay with decay exponent ${\delta }_2=1$ and ${\eta }_2=0.11$, respectively. We compare the transitions with the case, where pairwise interactions are also present. The spatiotemporal evolution is analyzed as the coupling parameter is varied.

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian ${\mathrm{\Delta }}_p$, which allows us to analyze and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $\lambda ({\mathrm{\Delta }}_p)$, the function $p\mapsto p{(2\lambda ({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ is locally increasing, while the function $p\mapsto 2^{-p}\lambda ({\mathrm{\Delta }}_p)$ is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues ${\lambda }_1({\mathrm{\Delta }}_p),\dots ,{\lambda }_k({\mathrm{\Delta }}_p),\dots ,$ are locally Lipschitz continuous with respect to $p\in [1,+\infty )$. We also establish the monotonicity of $p{(2{\lambda }_k({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ and $2^{-p}{\lambda }_k({\mathrm{\Delta }}_p)$ with respect to $p\in [1,+\infty )$.

These results systematically establish a refined analysis of ${\mathrm{\Delta }}_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min–max form; (3) refine the higher-order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case.

Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min–max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors’ work on discrete Morse theory.

The algorithm of Edelsbrunner for surface reconstruction by "wrapping" a set of points in ℝ³ is described.

It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\ge 2$. In the case of dimension 2, it is shown that BRSC have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also, in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen–Macaulay are determined. Complexity bounds are provided for all the algorithms involved.

In this note, we characterize the Hilbert regularity of the Stanley–Reisner ring $K[\mathrm{\Delta }]$ in terms of the $f$-vector and the $h$-vector of a simplicial complex $\mathrm{\Delta }$. We also compute the Hilbert regularity of a Gorenstein algebra.

It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic lattices, providing a characterization of the lattices of flats of boolean representable simplicial complexes and a decidability condition. We remark that every finite lattice occurs as the lattice of flats of some simplicial complex.

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices.

Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.

In this paper, we define and characterize the f-graphs. Also, we give a construction of f-graphs and importantly we show that the f-graphs obtained from this construction are Cohen–Macaulay.

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.

In this paper, we study almost Cohen–Macaulay bipartite graphs. In particular, we prove that if $G$ is an almost Cohen–Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $deg(v)\le 2$. In particular, if $G$ is an almost Cohen–Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus \{v\}$ is almost Cohen–Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen–Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.

In this paper, we introduce the concept of f-ideals and discuss its algebraic properties. In particular, we give a characterization of all the f-ideals of degree 2.

We classify the ideals of mixed products that are sequentially Cohen-Macaulay.

In this paper we introduce a class of monomial ideals, called k-decomposable ideals. It is shown that the class of k-decomposable ideals is contained in the class of monomial ideals with linear quotients, and when k is large enough, the class of k-decomposable ideals is equal to the class of ideals with linear quotients. In addition, it is shown that a d-dimensional simplicial complex is k-decomposable if and only if the Stanley-Reisner ideal of its Alexander dual is a k-decomposable ideal, where k ≤ d. Moreover, it is shown that every k-decomposable ideal is componentwise k-decomposable.

In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if $\mathcal{C}$ is a clutter such that 〈$\mathcal{C}$〉 is a vertex decomposable simplicial complex or I($\overline{\mathcal{C}}$) is squarefree stable, then $\mathcal{C}$ is chordal.

Let $\mathrm{\Delta }$ be a simplicial complex on $[n]$. The $\mathcal{NF}$-complex of $\mathrm{\Delta }$ is the simplicial complex ${\delta }_{\mathcal{NF}}(\mathrm{\Delta })$ on $[n]$ for which the facet ideal of $\mathrm{\Delta }$ is equal to the Stanley–Reisner ideal of ${\delta }_{\mathcal{NF}}(\mathrm{\Delta })$. Furthermore, for each $k=2,3,\dots$, we introduce the $k$th $\mathcal{NF}$-complex ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })$, which is inductively defined as ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })={\delta }_{\mathcal{NF}}{\delta }_{\mathcal{NF}}^{(k-1)}(\mathrm{\Delta })$ by setting ${\delta }_{\mathcal{NF}}^{(1)}(\mathrm{\Delta })={\delta }_{\mathcal{NF}}(\mathrm{\Delta })$. One can set ${\delta }_{\mathcal{NF}}^{(0)}(\mathrm{\Delta })=\mathrm{\Delta }$. The $\mathcal{NF}$-number of $\mathrm{\Delta }$ is the smallest integer $k>0$ for which ${\delta }_{\mathcal{NF}}^{(k)}(\mathrm{\Delta })\simeq \mathrm{\Delta }$. In the present paper we are especially interested in the $\mathcal{NF}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathcal{NF}$-number of the finite graph $K_n\bigsqcup K_m$ on $[n+m]$, which is the disjoint union of the complete graphs $K_n$ on $[n]$ and $K_m$ on $[m]$ for $n\ge 2$ and $m\ge 2$ with $(n,m)\ne (2,2)$, is equal to $n+m+2$. As a corollary, we find that the $\mathcal{NF}$-number of the complete bipartite graph $K_{n,m}$ on $[n+m]$ is also equal to $n+m+2$.

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polyhedrons and graphs.

Topological behaviour of self-similar spectra for fractal domains is shown and applied to solve electromagnetic problems on fractal geometries, like for example the Sierpinski gasket. Two different mathematical tools are employed: the Topological Calculus,^1,2 which frames a topology-consistent,³ discrete counterpart to domains and operators and the Iterated Function Systems (IFSS)⁴ to produce fractals as limit sets of simple recursion mappings. Topological invariants and Analytical features of a set can be easily extracted from such a discrete model, even for complex geometries like fractal ones.

One of the targets of this work is to show how recursion symmetries of a (pre-) fractal set, mathematically coded by "algebraic" relationships between its parts, are sole responsible for the self-similar distribution of its (laplacian) eigenvalues: no metric information is needed for this property to be observed.

Another primary target is to show how Topological Calculus easily allows for an almost instantaneous discretization of contoinuum equations of any (topological) field theory. Investigating the natural modes of self-similar domains is important to many applications whose core geometry is prefractal or at least highly irregular. Most recently, transport^5,6 and electromagnetic pehnomena were focused: IFS-generated waveguides, resonators^7,8 and antennas⁹ exhibiting multi-band properties. Such complex domains need careful mathematical formulations in order to transfer traditional-geometric properties to them; Topological Calculus is one of such discrete formulations.