Combinatorial dimension in fractional Cartesian products

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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Totally Split-Decomposable Metrics of Combinatorial Dimension Two

Annals of Combinatorics ◽

10.1007/pl00001294 ◽

2001 ◽

Vol 5 (1) ◽

pp. 99-112 ◽

Cited By ~ 6

Author(s):

A. Dress ◽

K.T. Huber ◽

K.T. Huber

Keyword(s):

Combinatorial Dimension

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Cartesian products as profinite completions

International Mathematics Research Notices ◽

10.1155/imrn/2006/72947 ◽

2006 ◽

Cited By ~ 8

Author(s):

M. Kassabov ◽

N. Nikolov

Keyword(s):

Cartesian Products ◽

Profinite Completions

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The distinguishing number of Cartesian products of complete graphs

European Journal of Combinatorics ◽

10.1016/j.ejc.2007.11.018 ◽

2008 ◽

Vol 29 (4) ◽

pp. 922-929 ◽

Cited By ~ 19

Author(s):

Wilfried Imrich ◽

Janja Jerebic ◽

Sandi Klavžar

Keyword(s):

Complete Graphs ◽

Cartesian Products ◽

Distinguishing Number

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On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

Analysis and Applications ◽

10.1142/s021953051650024x ◽

2016 ◽

Vol 15 (05) ◽

pp. 731-770 ◽

Cited By ~ 6

Author(s):

D. P. Hewett ◽

A. Moiola

Keyword(s):

Maximal Regularity ◽

Cantor Sets ◽

Characteristic Functions ◽

Bessel Potential ◽

Empty Interior ◽

Cartesian Products ◽

Sobolev Regularity ◽

Full Classification ◽

Differential And Integral Equations

This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.

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THE FERMION DOUBLING PROBLEM AND NONCOMMUTATIVE GEOMETRY

Modern Physics Letters A ◽

10.1142/s0217732300001389 ◽

2000 ◽

Vol 15 (19) ◽

pp. 1279-1286 ◽

Cited By ~ 74

Author(s):

A. P. BALACHANDRAN ◽

T. R. GOVINDARAJAN ◽

B. YDRI

Keyword(s):

Noncommutative Geometry ◽

Field Theories ◽

Fuzzy Sphere ◽

Cartesian Products

We propose a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its Cartesian products. Its relation to the Ginsparg–Wilson approach is also clarified.

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