Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

Duplicate form of Carlitz inversions and summation formulae

By applying the duplicate form of Carlitz inversions to three special cases of the q-Saalschütz theorem respectively, we establish three classes of interesting summation formulae.     

The order of hypotraceable oriented graphs

A digraph of order n is hypotraceable if it is nontraceable but all its induced subdigraphs of order n−1 are traceable. Grötschel et al. (1980) [M. Grötschel, C. Thomassen, Y. Wakabayashi, Hypotraceable digraphs, J. Graph Theory 4 (1980) 377–381] constructed an infinite family of hypotraceable oriented graphs, the smallest of which has order 13. We show that there exist hypotraceable oriented graphs of order n for every n≥8 except possibly for n=9,11 and that eight="14" border="0" style="vertical-align:bottom" width="19" alt="View the MathML source" title="View the MathML source" src="http://binary-services.sciencedirect.com/content/image/1-s2.0-S0012365X11000999-si8.gif"> is the only one of order less than 8.Furthermore, we determine all the hypotraceable oriented graphs of order 8 and explain the relevance of these results to the problem of determining, for given k≥2, the maximum order of nontraceable oriented digraphs each of whose induced subdigraphs of order k is traceable.     

Bases in Banach spaces of smooth functions on Cantor-type sets

We suggest a Schauder basis in Banach spaces of smooth functions and traces of smooth functions on Cantor-type sets. In the construction, local Taylor expansions of functions are used.     

The Cuntz semigroup and comparison of open projections

We show that a number of naturally occurring comparison relations on positive elements in a <em>Cem>^?-algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the <em>Cem>^?-algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of—and is weaker than—a comparison notion defined by Peligrad and Zsid&oacute;. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray-von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the <em>Cem>^?-algebra. We use these findings to give a new picture of the Cuntz semigroup.     

The inclusion relation between Sobolev and modulation spaces

The inclusion relations between the <em>Lem><em>pem>-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier <em>eem><em>iem><em>αem>|<em>Dem>| between <em>Lem><em>pem>-Sobolev spaces and modulation spaces are discussed.     

An abstract transmission problem in a thin layer, I: Sharp estimates

We study an elliptic transmission problem in Banach spaces. The problem is considered on the juxtaposition of two intervals, one of which of small length <em>δem>, and models physical phenomena in media constituted by two parts with different physical characteristics. We obtain results of existence, uniqueness, maximal regularity and optimal dependence on the parameter <em>δem> for <em>Lem><em>pem> solutions of the problem. The main tools of our approach are impedance and admittance operators (i.e. Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and <em>Hem>^∞ functional calculus for sectorial operators in Banach spaces.