4,070 research outputs found
Chains of modular elements and shellability
Let L be a lattice admitting a left-modular chain of length r, not
necessarily maximal. We show that if either L is graded or the chain is
modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable).
This proves a conjecture of Hersh. Under certain circumstances, we can find
shellings of higher skeleta. For instance, if the left-modular chain consists
of every other element of some maximum length chain, then L itself is
shellable. We apply these results to give a new characterization of finite
solvable groups in terms of the topology of subgroup lattices.
Our main tool relaxes the conditions for an EL-labeling, allowing multiple
ascending chains as long as they are lexicographically before non-ascending
chains. We extend results from the theory of EL-shellable posets to such
labelings. The shellability of certain skeleta is one such result. Another is
that a poset with such a labeling is homotopy equivalent (by discrete Morse
theory) to a cell complex with cells in correspondence to weakly descending
chains.Comment: 20 pages, 1 figure; v2 has minor fixes; v3 corrects the technical
lemma in Section 4, and improves the exposition throughou
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
A new meta-module for efficient reconfiguration of hinged-units modular robots
We present a robust and compact meta-module for edge-hinged modular robot units such as M-TRAN,
SuperBot, SMORES, UBot, PolyBot and CKBot, as well as for central-point-hinged ones such as Molecubes and
Roombots. Thanks to the rotational degrees of freedom of these units, the novel meta-module is able to expand
and contract, as to double/halve its length in each dimension. Moreover, for a large class of edge-hinged robots the
proposed meta-module also performs the scrunch/relax and transfer operations required by any tunneling-based
reconfiguration strategy, such as those designed for Crystalline and Telecube robots. These results make it possible to
apply efficient geometric reconfiguration algorithms to this type of robots. We prove the size of this new meta-module to
be optimal. Its robustness and performance substantially improve over previous results.Peer ReviewedPostprint (author's final draft
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Switcher-random-walks: a cognitive-inspired mechanism for network exploration
Semantic memory is the subsystem of human memory that stores knowledge of
concepts or meanings, as opposed to life specific experiences. The organization
of concepts within semantic memory can be understood as a semantic network,
where the concepts (nodes) are associated (linked) to others depending on
perceptions, similarities, etc. Lexical access is the complementary part of
this system and allows the retrieval of such organized knowledge. While
conceptual information is stored under certain underlying organization (and
thus gives rise to a specific topology), it is crucial to have an accurate
access to any of the information units, e.g. the concepts, for efficiently
retrieving semantic information for real-time needings. An example of an
information retrieval process occurs in verbal fluency tasks, and it is known
to involve two different mechanisms: -clustering-, or generating words within a
subcategory, and, when a subcategory is exhausted, -switching- to a new
subcategory. We extended this approach to random-walking on a network
(clustering) in combination to jumping (switching) to any node with certain
probability and derived its analytical expression based on Markov chains.
Results show that this dual mechanism contributes to optimize the exploration
of different network models in terms of the mean first passage time.
Additionally, this cognitive inspired dual mechanism opens a new framework to
better understand and evaluate exploration, propagation and transport phenomena
in other complex systems where switching-like phenomena are feasible.Comment: 9 pages, 3 figures. Accepted in "International Journal of
Bifurcations and Chaos": Special issue on "Modelling and Computation on
Complex Networks
An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u
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