Vector spaces as unions of proper subspaces

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.Comment: 8 pages, LaTex; to appear in "Linear Algebra and its Applications

Maximal partial spreads and the modular n-queen problem III

AbstractMaximal partial spreads in PG(3,q)q=pk,p odd prime and q⩾7, are constructed for any integer n in the interval (q2+1)/2+6⩽n⩽(5q2+4q−1)/8 in the case q+1≡0,±2,±4,±6,±10,12(mod24). In all these cases, maximal partial spreads of the size (q2+1)/2+n have also been constructed for some small values of the integer n. These values depend on q and are mainly n=3 and n=4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3,q),q=pk where p is an odd prime and q⩾7, of size n for any integer n in the interval (q2+1)/2+6⩽n⩽q2−q+2

A Comparison of the Effects of Simulation Training and Non-Simulation Training On Self-Efficacy in Providing Women\u27s Health Care

The Veterans Administration (VA) recognizes that proficiency in the core concepts of primary care women\u27s health is required to provide comprehensive primary care for women. A potentially superior form of training that has been recently used for care providers is simulation. The examination of the relationship between simulation training through the Mini-Residency Course and increased self-efficacy among Women\u27s Health Primary Care Providers (WH-PCP) is important, as the Mini-Residency Course is designed specifically to fill knowledge gaps and enhance the participant\u27s knowledge and skill. A single post-test only, two group design was used for this study. The experimental group included those who completed simulation training on how to provide effective, essential healthcare to women veterans. The simulation-based training occurred July, 2012. The study gathered survey data designed to determine the level of self-efficacy of practitioners from a sample who had participated in the Mini-Residency program (Part I, or Parts I and II) and compared the levels of self-efficacy to a sample of practitioners who did not participate in simulations. Limited by a low response rate, the study sample included 23 practitioners. A self-efficacy survey was constructed using Bandura\u27s self-efficacy theory. The self-efficacy score for this analysis used the mean of six discrete skill items. The reliability of this self-efficacy scale was examined using Cronbach\u27s alpha. Results indicated reliability at a = .71. The results failed to demonstrate any statistically significant differences between groups. However, it was noted that a significant result ( p = .10 level) was evident in the differences in mean self-efficacy scores based on standardized patient experience, which suggests the need for future research using a larger sample size

Maximal partial line spreads of non-singular quadrics

For n >= 9 , we construct maximal partial line spreads for non-singular quadrics of for every size between approximately and , for some small constants and . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gacs and SzAnyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles and by Pepe, Roing and Storme

On perfect 1-mathcalEmathcal E-error-correcting codes

We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-$mathcal{E}$-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting $q$-ary Hamming codes as subsets

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = \omega_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on

Some necessary conditions for vector space partitions

Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t,q) that contains m_d spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V(2t,q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.Comment: 19 pages; corrected typos and rewritten introductio