Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPoset limits can be totally ordered
150
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval.
rate research
Read More
Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by convex sets when there is no restriction on whether the sets are all open or closed. We achieve this by constructing a convex realization for an arbitrary code with $k$ nonempty codewords in $mathbb{R}^{k-1}$. This result justifies the usual restriction of the definition of convex neural codes to include only those that can be realized by receptive fields that are all either open convex or closed convex. We also show that the dimension of our construction cannot in general be lowered.
For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $epsilon$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.
The bloom of complex network study, in particular, with respect to scale-free ones, is considerably triggering the research of scale-free graph itself. Therefore, a great number of interesting results have been reported in the past, including bounds of diameter. In this paper, we focus mainly on a problem of how to analytically estimate the lower bound of diameter of scale-free graph, i.e., how small scale-free graph can be. Unlike some pre-existing methods for determining the lower bound of diameter, we make use of a constructive manner in which one candidate model $mathcal{G^*} (mathcal{V^*}, mathcal{E^*})$ with ultra-small diameter can be generated. In addition, with a rigorous proof, we certainly demonstrate that the diameter of graph $mathcal{G^{*}}(mathcal{V^{*}},mathcal{E^{*}})$ must be the smallest in comparison with that of any scale-free graph. This should be regarded as the tight lower bound.
Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990s.
A binary poset code of codimension M (of cardinality 2^{N-M}, where N is the code length) can correct maximum M errors. All possible poset metrics that allow codes of codimension M to be M-, (M-1)- or (M-2)-perfect are described. Some general conditions on a poset which guarantee the nonexistence of perfect poset codes are derived; as examples, we prove the nonexistence of R-perfect poset codes for some R in the case of the crown poset and in the case of the union of disjoin chains. Index terms: perfect codes, poset codes
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
