where we use c |X to denote the projection of c on X. The teaching dimension of C is the smallest number t such that every c ∈ C has a teaching set of size no more than t [GK95]. However, teaching dimension does not always capture the cooperation in teaching and learning, and the notion of recursive teaching dimension has been introduced and studied extensively in the literature [Kuh99, DSZ10, ...

We prove that the bound on the L norms of the Kakeya type maximal functions studied by Cordoba [2], and by Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r log(1/r) is the exact Minkowski dimension of the class of Kakeya sets in R, and prove that the exact Hausdorff dimens...

as an application of hirota bilinear method, perturbation expansion truncated at different levels is used to obtain exact soliton solutions to (2+1)-dimensional nonlinear evolution equation in much simpler way in comparison to other existing methods. we have derived bilinear form of nonlinear evolution equation and using this bilinear form, bilinear backlund transformations and construction of ...

land use planning is the process of allocating different land uses and activities to the special units of land. so we face with two problems of assigning kind of activity and location of it. by considering the role of industries in development, employment and environment, we consider industrial and especially intensive energy industries planning as a multi objective optimization problem. in thi...

This paper establishes an upper bound on the size of a concept class with given recursive teaching dimension (RTD, a teaching complexity parameter.) The upper bound coincides with Sauer’s well-known bound on classes with a fixed VC-dimension. Our result thus supports the recently emerging conjecture that the combinatorics of VC-dimension and those of teaching complexity are intrinsically interl...

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrands projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorf...

In this paper we deal with the dimension of multisequences and related properties. For a given multisequenceW and R ∈ Z+, we define the R−extension ofW . Further we count the number of multisequences W whose R−extensions have maximum dimension and give an algorithm to derive such multisequences. We then go on to use this theory to count the number of Linear Feedback Shift Register(LFSR) configu...