scholarly journals Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings

2021 ◽  
Vol 104 (3) ◽  
Author(s):  
Malayaja Chutani ◽  
Bosiljka Tadić ◽  
Neelima Gupte
Keyword(s):  
Simplicial Complexes ◽  
High Dimensional ◽  
Kuramoto Oscillators

scholarly journals Large random simplicial complexes, I

2016 ◽  
Vol 08 (03) ◽  
pp. 399-429 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Probability Measure ◽  
Simplicial Complex ◽  
Parameter Space ◽  
Simplicial Complexes ◽  
High Dimensional ◽  
Topological Properties ◽  
Convex Domains ◽  
Dimensional Parameter ◽  
Random Simplicial Complexes ◽  
Simply Connected

In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

scholarly journals Mixing in High-Dimensional Expanders

2017 ◽  
Vol 26 (5) ◽  
pp. 746-761 ◽  
Author(s):  
ORI PARZANCHEVSKI
Keyword(s):  
Arbitrary Dimension ◽  
Simplicial Complexes ◽  
Similar Type ◽  
High Dimensional ◽  
Hodge Laplacian ◽  
Laplace Spectrum ◽  
Open Questions

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to asmixing, orpseudo-randomness). Recently, an analogue of this lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

A GENERAL SMALL CANCELLATION THEORY

2000 ◽  
Vol 10 (01) ◽  
pp. 1-172 ◽  
Author(s):  
JONATHAN P. MCCAMMOND
Keyword(s):  
Simplicial Complexes ◽  
High Dimensional ◽  
Small Cancellation ◽  
Small Cancellation Theory

In this article, a generalized version of small cancellation theory is developed which is applicable to specific types of high-dimensional simplicial complexes. The usual results on small cancellation groups are then shown to hold in this new setting with only slight modifications. For example, arbitrary dimensional versions of the Poincaré construction and the Cayley complex are described.

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