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.     

Improved algorithms for path partition and related problems

We study the L_∞ path partition problem: given a path of n weighted vertices and an integer k, remove k−1 edges from the path so that the maximum absolute deviation of the weights of the resulting k sub-paths from their mean is minimized. Previously, the best algorithm solves this problem in O(nklogk) time. We present an O(nk) time algorithm. We also give improved solutions for two related problems: the L_d path partition problem and the web proxies placement problem.     

Optimal stopping for non-linear expectations—Part II

Relying on the stochastic analysis tools developed in Bayraktar and Yao (2011) [1], we solve the optimal stopping problems for non-linear expectations.     

Optimal stopping for non-linear expectations—Part I

We develop a theory for solving continuous time optimal stopping problems for non-linear expectations. Our motivation is to consider problems in which the stopper uses risk measures to evaluate future rewards. Our development is presented in two parts. In the first part, we will develop the stochastic analysis tools that will be essential in solving the optimal stopping problems, which will be presented in Bayraktar and Yao (2011) [1].     

Sample-based polynomial approximation of rational Bézier curves

We present an iteration method for the polynomial approximation of rational B&eacute;zier curves. Starting with an initial B&eacute;zier curve, we adjust its control points gradually by the scheme of weighted progressive iteration approximations. The <em>Lem>_<em>pem>-error calculated by the trapezoidal rule using sampled points is used to guide the iteration approximation. We reduce the <em>Lem>_<em>pem>-error by a predefined factor at every iteration so as to obtain the best approximation with a minimum error. Numerical examples demonstrate the fast convergence of our method and indicate that results obtained using the <em>Lem>₁-error criterion are better than those obtained using the <em>Lem>₂-error and <em>Lem>_<em>∞em>-error criteria.