On upper bounds on the smallest size of a saturating set in a projective plane

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi _{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $\Pi _{q}$ with $2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset \Pi_{q}$ is $(1,\mu)$-saturating if for every point $Q$ of $\Pi _{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $\mu$, counted with multiplicity. The multiplicity of a secant $\ell$ is computed as ${\binom{{\#(\ell \,\cap S)}}{{2}}}.$ The following upper bound on the smallest size $s_{\mu }(2,q)$ of a $(1,\mu)$-saturating set in $\Pi_{q}$ is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,\mu)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde

Welfare finance reform: District states could face funding challenges

Welfare finance reform could have a substantial fiscal impact on the budgets of some states in the Tenth district, although the ability to cope with reform will vary from state to state. This article discusses some of the proposed changes in welfare funding and the effect they may have on the states in the Tenth District. Each state's ability to offset potential reductions in welfare funding will depend on their own budget resources and whether additional funds are provided to states with limited resources and higher administrative costs.Federal Reserve District, 10th ; Welfare

An emerging computer sector in the Tenth District

Computer industries have been an important facet of the nation's current economic expansion. During the 1990s, the computer sector has added almost three-quarters of a million jobs across the country, creating high-paying new positions and spawning new opportunities for the American entrepreneur.Computers ; Federal Reserve District, 10th

What's hampering job growth in the District's services sector?

Employment growth in the Tenth District has fallen behind the national rate in 1999 for the first time in ten years. Although all economic sectors have been experiencing slower job growth, the services sector, due to its size, has played perhaps the most important role in the slowdown of overall employment growth in the district. While services employment elsewhere in the nation continues to grow rapidly, the district has witnessed very little job expansion in services so far in 1999 (Chart 1). In fact, the district services sector has added jobs during the first seven months of this year at less than a third of the rate enjoyed by the nation as a whole.Employment (Economic theory) ; Service industries ; Federal Reserve District, 10th

Good times continue for Tenth District banks

Commercial banks in Tenth District states continued to perform well in 1997. Profitability remained high, loan quality improved, and loan growth and deposit growth were both strong. Banks in district states also outperformed banks nationwide by all four performance measures.Federal Reserve District, 10th

The Tenth District's brain drain: who left and what did it cost?

Most of the Tenth Federal Reserve District states experienced a brain drain, or an outmigration of highly educated people, during the last half of the 1980s. Fortunately, the recent tide of migration appears to have turned for some district states. Yet, it is still important for policymakers to understand the full impact of a brain drain on a state's economy. Highly educated people are prone to move, based on their region's economic performance relative to other parts of the country. Thus, current favorable migration trends in the district could easily be reversed.Federal Reserve District, 10th ; Intellectual property ; Labor mobility

How important are small businesses in the Tenth District?

Small businesses were widely recognized in the 1980s as the driving force behind economic growth. One benchmark study estimated that small firms accounted for 80 percent of all new jobs in the nation in the late 1970s. Analysts predicted that small businesses would continue to provide most new jobs as the economy became more service-oriented and therefore dominated by firms that tended to be smaller in scale. ; The desire to promote small businesses as an engine of job growth was partly responsible for efforts in the mid-1980s to overhaul the tax system and deregulate business. Since then, election campaigns and congressional debates have continued to place small businesses and job growth together in the public eye. ; Today, improved access to additional data allows a comprehensive look at the importance of small businesses in the Tenth District economy. Small firms continue to be the primary source of total jobs in the district, accounting for 60 percent of all jobs in 1994. But not all sectors of the district economy favor small firms. For instance, small firms in 1994 accounted for less than a third of all manufacturing jobs in the district. Therefore, in their efforts to promote job growth policymakers should not overlook the contribution that large firms can make in some key sectors.Small business ; Federal Reserve District, 10th

Exporting trends in Tenth District manufactured goods

The Tenth District has consistently outperformed the nation in the 1990s in the growth of manufactured exports. Foreign exports account for a large share of the district's sales of manufactured goods and have also helped the district retain many of its factory jobs.Exports ; Federal Reserve District, 10th