Simplicial Complex

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Somayeh Moradi - One of the best experts on this subject based on the ideXlab platform.

  • On the Facet Ideal of an Expanded Simplicial Complex
    Bulletin of the Iranian Mathematical Society, 2018
    Co-Authors: Somayeh Moradi, R. Rahmati-asghar
    Abstract:

    For a Simplicial Complex $$\Delta $$ Δ , the effect of the expansion functor on combinatorial properties of $$\Delta $$ Δ and algebraic properties of its Stanley–Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $$I(\Delta )$$ I ( Δ ) and its Alexander dual which we denote by $$J_{\Delta }$$ J Δ to see how the expansion functor alters the algebraic properties of these ideals. It is shown that for any expansion $$\Delta ^{\alpha }$$ Δ α the ideals $$J_{\Delta }$$ J Δ and $$J_{\Delta ^{\alpha }}$$ J Δ α have the same total Betti numbers and their Cohen–Macaulayness is equivalent, which implies that the regularities of the ideals $$I(\Delta )$$ I ( Δ ) and $$I(\Delta ^{\alpha })$$ I ( Δ α ) are equal. Moreover, the projective dimensions of $$I(\Delta )$$ I ( Δ ) and $$I(\Delta ^{\alpha })$$ I ( Δ α ) are compared. In the sequel for a graph G , some properties that are equivalent in G and its expansions are presented and for a Cohen–Macaulay (respectively, sequentially Cohen–Macaulay and shellable) graph G , we give some conditions for adding or removing a vertex from G , so that the remaining graph is still Cohen–Macaulay (respectively, sequentially Cohen–Macaulay and shellable).

  • homological invariants of the stanley reisner ring of a k decomposable Simplicial Complex
    Asian-european Journal of Mathematics, 2017
    Co-Authors: Somayeh Moradi
    Abstract:

    In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a k-decomposable Simplicial Complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for k-decomposable Simplicial Complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable Simplicial Complex Δ, a formula for the regularity of the Stanley–Reisner ring of Δ is presented. Finally, for a chordal clutter ℋ, an upper bound for reg(I(ℋ)) is given in terms of the regularities of edge ideals of some chordal clutters which are minors of ℋ.

  • on the stanley reisner ideal of an expanded Simplicial Complex
    Manuscripta Mathematica, 2016
    Co-Authors: Rahim Rahmatiasghar, Somayeh Moradi
    Abstract:

    Let Δ be a Simplicial Complex. We study the expansions of Δ mainly to see how the algebraic and combinatorial properties of Δ and its expansions are related to each other. It is shown that Δ is Cohen–Macaulay, sequentially Cohen–Macaulay, Buchsbaum or k-decomposable, if and only if an arbitrary expansion of Δ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley–Reisner ideals of Δ and those of their expansions are compared.

  • Expansion of a Simplicial Complex
    arXiv: Commutative Algebra, 2016
    Co-Authors: Somayeh Moradi, Fahimeh Khosh-ahang
    Abstract:

    For a Simplicial Complex $\Delta$, we introduce a Simplicial Complex attached to $\Delta$, called the expansion of $\Delta$, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a Simplicial Complex and its Stanley-Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen-Macaulayness. Also it is proved that some homological invariants of Stanley-Reisner ring of a Simplicial Complex relate to those invariants in the Stanley-Reisner ring of its expansions.

  • on the stanley reisner ideal of an expanded Simplicial Complex
    arXiv: Commutative Algebra, 2015
    Co-Authors: Rahim Rahmatiasghar, Somayeh Moradi
    Abstract:

    Let $\Delta$ be a Simplicial Complex. We study the expansions of $\Delta$ mainly to see how the algebraic and combinatorial properties of $\Delta$ and its expansions are related to each other. It is shown that $\Delta$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $\Delta$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of $\Delta$ and those of their expansions are compared.

Hugh Thomas - One of the best experts on this subject based on the ideXlab platform.

  • semi stable subcategories for euclidean quivers
    Proceedings of The London Mathematical Society, 2015
    Co-Authors: Colin Ingalls, Charles Paquette, Hugh Thomas
    Abstract:

    In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a Simplicial Complex to any acyclic quiver Q, and this Simplicial Complex allows one to completely determine the canonical decomposition of any dimension vector. This Simplicial Complex has a nice description in the Euclidean case: it is described using an arrangement of convex pieces of hyperplanes, each piece being indexed by a real Schur root or a set of quasi-simple objects.

  • Semi‐stable subcategories for Euclidean quivers
    Proceedings of the London Mathematical Society, 2015
    Co-Authors: Colin Ingalls, Charles Paquette, Hugh Thomas
    Abstract:

    In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a Simplicial Complex to any acyclic quiver Q, and this Simplicial Complex allows one to completely determine the canonical decomposition of any dimension vector. This Simplicial Complex has a nice description in the Euclidean case: it is described using an arrangement of convex pieces of hyperplanes, each piece being indexed by a real Schur root or a set of quasi-simple objects.

Satoshi Murai - One of the best experts on this subject based on the ideXlab platform.

  • Uniformly Cohen-Macaulay Simplicial Complexes and almost Gorenstein* Simplicial Complexes
    Journal of Algebra, 2016
    Co-Authors: Naoyuki Matsuoka, Satoshi Murai
    Abstract:

    Abstract In this paper, we study Simplicial Complexes whose Stanley–Reisner rings are almost Gorenstein and have a-invariant zero. We call such a Simplicial Complex an almost Gorenstein* Simplicial Complex. To study the almost Gorenstein* property, we introduce a new class of Simplicial Complexes which we call uniformly Cohen–Macaulay Simplicial Complexes. A d-dimensional Simplicial Complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the Simplicial Complex Δ ∖ { F } is Cohen–Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these Simplicial Complexes, and show that almost Gorenstein* Simplicial Complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* Simplicial Complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for Simplicial Complexes of dimension ≤2.

  • Uniformly Cohen-Macaulay Simplicial Complexes and almost Gorenstein* Simplicial Complexes
    arXiv: Commutative Algebra, 2014
    Co-Authors: Naoyuki Matsuoka, Satoshi Murai
    Abstract:

    In this paper, we study Simplicial Complexes whose Stanley-Reisner rings are almost Gorenstein and have $a$-invariant zero. We call such a Simplicial Complex an almost Gorenstein* Simplicial Complex. To study the almost Gorenstein* property, we introduce a new class of Simplicial Complexes which we call uniformly Cohen-Macaulay Simplicial Complexes. A $d$-dimensional Simplicial Complex $\Delta$ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet $F$ of $\Delta$, the Simplicial Complex $\Delta \setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate fundamental algebraic, combinatorial and topological properties of these Simplicial Complexes, and show that almost Gorenstein* Simplicial Complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* Simplicial Complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for Simplicial Complexes of dimension $\leq 2$.

  • Uniformly Cohen-Macaulay Simplicial Complexes
    arXiv: Commutative Algebra, 2014
    Co-Authors: Naoyuki Matsuoka, Satoshi Murai
    Abstract:

    In this paper, we introduce a new class of Simplicial Complexes which we call uniformly Cohen-Macaulay Simplicial Complexes. A $d$-dimensional Simplicial Complex $\Delta$ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet $F$ of $\Delta$, the Simplicial Complex $\Delta \setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate fundamental algebraic, combinatorial and topological properties of these Simplicial Complexes. We prove that the uniformly Cohen-Macaulay property is a topological property and that the $h$-vector of a uniformly Cohen-Macaulay Simplicial Complex satisfies an analogue of Stanley's inequality for Ehrhart $\delta$-vectors. Also, by applying this property, we study the almost Gorenstein property for Stanley-Reisner rings.

Frédo Durand - One of the best experts on this subject based on the ideXlab platform.

  • Texture design using a Simplicial Complex of morphable textures
    ACM Transactions on Graphics, 2005
    Co-Authors: Wojciech Matusik, Matthias Zwicker, Frédo Durand
    Abstract:

    We present a system for designing novel textures in the space of textures induced by an input database. We capture the structure of the induced space by a Simplicial Complex where vertices of the simplices represent input textures. A user can generate new textures by interpolating within individual simplices. We propose a morphable interpolation for textures, which also defines a metric used to build the Simplicial Complex. To guarantee sharpness in interpolated textures, we enforce histograms of high-frequency content using a novel method for histogram interpolation. We allow users to continuously navigate in the Simplicial Complex and design new textures using a simple and efficient user interface. We demonstrate the usefulness of our system by integrating it with a 3D texture painting application, where the user interactively designs desired textures.

Ankit Sharma - One of the best experts on this subject based on the ideXlab platform.

  • weighted Simplicial Complex a novel approach for predicting small group evolution
    Pacific-Asia Conference on Knowledge Discovery and Data Mining, 2017
    Co-Authors: Ankit Sharma, Terrence J Moore, Ananthram Swami, Jaideep Srivastava
    Abstract:

    The study of small collaborations or teams is an important endeavor both in industry and academia. The social phenomena responsible for formation or evolution of such small groups is quite different from those for dyadic relations like friendship or large size guilds (or communities). In small groups when social actors collaborate for various tasks over time, the actors common across collaborations act as bridges which connect groups into a network of groups. Evolution of groups is affected by this network structure. Building appropriate models for this network is an important problem in the study of group evolution. This work focuses on the problem of group recurrence prediction. In order to overcome the shortcomings of two traditional group network modeling approaches: hypergraph and Simplicial Complex, we propose a hybrid approach: Weighted Simplicial Complex (WSC). We develop a Hasse diagram based framework to study WSCs and build several predictive models for group recurrence based on this approach. Our results demonstrate the effectiveness of our approach.