8,599 research outputs found
On Saturated -Sperner Systems
Given a set , a collection is said to
be -Sperner if it does not contain a chain of length under set
inclusion and it is saturated if it is maximal with respect to this property.
Gerbner et al. conjectured that, if is sufficiently large with respect to
, then the minimum size of a saturated -Sperner system
is . We disprove this conjecture
by showing that there exists such that for every and there exists a saturated -Sperner system
with cardinality at most
.
A collection is said to be an
oversaturated -Sperner system if, for every
, contains more
chains of length than . Gerbner et al. proved that, if
, then the smallest such collection contains between and
elements. We show that if ,
then the lower bound is best possible, up to a polynomial factor.Comment: 17 page
Saturation in the Hypercube and Bootstrap Percolation
Let denote the hypercube of dimension . Given , a spanning
subgraph of is said to be -saturated if it does not
contain as a subgraph but adding any edge of
creates a copy of in . Answering a question of Johnson and Pinto, we
show that for every fixed the minimum number of edges in a
-saturated graph is .
We also study weak saturation, which is a form of bootstrap percolation. A
spanning subgraph of is said to be weakly -saturated if the
edges of can be added to one at a time so that each
added edge creates a new copy of . Answering another question of Johnson
and Pinto, we determine the minimum number of edges in a weakly
-saturated graph for all . More generally, we
determine the minimum number of edges in a subgraph of the -dimensional grid
which is weakly saturated with respect to `axis aligned' copies of a
smaller grid . We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and
Computin
Geometric erogdicity of a bead-spring pair with stochastic Stokes forcing
We consider a simple model for the
uctuating hydrodynamics of a
exible polymer
in dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a
stochastic Stokes
uid velocity field. This is a generalization of previous models which
have used linear spring forces as well as white-in-time
uid velocity fields.
We follow previous work combining control theoretic arguments, Lyapunov functions, and hypo-elliptic diffusion theory to prove exponential convergence via a Harris
chain argument. To this, we add the possibility of excluding certain "bad" sets in phase
space in which the assumptions are violated but from which the systems leaves with a
controllable probability. This allows for the treatment of singular drifts, such as those
derived from the Lennard-Jones potential, which is an novel feature of this work
Cracking the Code on Stem: A People Strategy for Nevada\u27s Economy
Nevada has in place a plausible economic diversification strategy—and it’s beginning to work. Now, the state and its regions need to craft a people strategy. Specifically, the state needs to boost the number of Nevadans who possess at least some postsecondary training in the fields of science, technology, engineering, or math—the so-called “STEM” disciplines (to which some leaders add arts and design to make it “STEAM”).
The moment is urgent—and only heightened by the projected worker needs of Tesla Motors’ planned “gigafactory” for lithium-ion batteries in Storey County.
Even before the recent Tesla commitment, a number of the more high-tech industry sectors targeted by the state’s new economic diversification strategy had begun to deliver significant growth. Most notable in fast-growing sectors like Business IT Ecosystems (as defined by the Governor’s Office for Economic Development) and large sectors like Health and Medical Services, this growth has begun to increase the demand in Nevada for workers with at least a modicum of postsecondary training in one or more STE M discipline.
However, there is a problem. Even though many available opportunities require no more than the right community college certificate, insufficient numbers of Nevadans have pursued even a little STEM training. As a result, too few Nevadans are ready to participate in the state’s emerging STEM economy. The upshot: Without concerted action to prepare more Nevadans for jobs in STEM-intensive fields, skills shortages could limit growth in the state’s most promising target industries and Nevadans could miss out on employment that offers superior paths to opportunity and advancement.
Which is the challenge this report addresses: Aimed at focusing the state at a critical moment, this analysis speaks to Nevada’s STEM challenge by providing a new assessment of Nevada’s STEM economy and labor market as well as a review of actions that leaders throughout the state—whether in the public, private, civic, or philanthropic sectors—can take to develop a workforce capable of supporting continued growth through economic diversification
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