Embedding multidimensional grids into optimal hypercubes

Let $G$ and $H$ be graphs, with $|V(H)|\geq |V(G)|$, and $f:V(G)\rightarrow V(H)$ a one to one map of their vertices. Let $dilation(f) = max\{ dist_{H}(f(x),f(y)): xy\in E(G) \}$, where $dist_{H}(v,w)$ is the distance between vertices $v$ and $w$ of $H$. Now let $B(G,H)$ = $min_{f}\{ dilation(f) \}$, over all such maps $f$. The parameter $B(G,H)$ is a generalization of the classic and well studied "bandwidth" of $G$, defined as $B(G,P(n))$, where $P(n)$ is the path on $n$ points and $n = |V(G)|$. Let $[a_{1}\times a_{2}\times \cdots \times a_{k} ]$ be the $k$-dimensional grid graph with integer values $1$ through $a_{i}$ in the $i$'th coordinate. In this paper, we study $B(G,H)$ in the case when $G = [a_{1}\times a_{2}\times \cdots \times a_{k} ]$ and $H$ is the hypercube $Q_{n}$ of dimension $n = \lceil log_{2}(|V(G)|) \rceil$, the hypercube of smallest dimension having at least as many points as $G$. Our main result is that $$B( [a_{1}\times a_{2}\times \cdots \times a_{k} ],Q_{n}) \le 3k,$$ provided $a_{i} \geq 2^{22}$ for each $1\le i\le k$. For such $G$, the bound $3k$ improves on the previous best upper bound $4k+O(1)$. Our methods include an application of Knuth's result on two-way rounding and of the existence of spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure

High pressure flame system for pollution studies with results for methane-air diffusion flames

A high pressure flame system was designed and constructed for studying nitrogen oxide formation in fuel air combustion. Its advantages and limitations were demonstrated by tests with a confined laminar methane air diffusion flame over the pressure range from 1 to 50 atm. The methane issued from a 3.06 mm diameter port concentrically into a stream of air contained within a 20.5 mm diameter chimney. As the combustion pressure is increased, the flame changes in shape from wide and convex to slender and concave, and there is a marked increase in the amount of luminous carbon. The height of the flame changes only moderately with pressure

Pollutant emissions from flat-flame burners at high pressures

Maximum flame temperatures and pollutant emission measurements for NOx, CO, and UHC (unburned hydrocarbons) are reported for premixed methane air flat flames at constant total mass flow rate over the pressure range from 1.9 to 30 atm and for equivalence ratios from 0.84 to 1.12. For any given pressure, maxima typically occur in both the temperature and NOx emissions curves slightly to the lean side of stoichiometric conditions. The UHC emissions show minima at roughly the same equivalence ratios. The CO emissions, however, increase continually with increasing equivalence ratio. Flame temperature and NOx emissions decrease with increasing pressure, while the opposite is true for the CO and UHC emissions. The NOx data correlate reasonably well as a function of flame temperature only. Four flameholders, differing only slightly, were used. In general, the temperature and emissions data from these four flameholders are similar, but some differences also exist. These differences appear to be related to minor variations in the condition of the flameholder surfaces

An experimental study of fluidization processes under lunar conditions

Fluidized ash flow in simulated lunar soil

Statistical derivation of design criteria for liquid rocket combustion instability Final report

Statistical correlation between engine design and combustion stability in liquid propellant rocket engine

Two transducer formula for more precise determination of ultrasonic phase velocity from standing wave measurements

A two transducer correction formula valid for both solid and liquid specimens is presented. Using computer simulations of velocity measurements, the accuracy and range of validity of the results are discussed and are compared with previous approximations

Thermally regenerable carbon dioxide absorbent system Final report, 1 May 1964 - 31 Jan. 1966

Carbon dioxide absorption by solid state ion exchange resin

Pre and post processing using the IBM 3277 display station graphics attachment (RPQ7H0284)

A graphical interactive procedure operating under TSO and utilizing two CRT display terminals is shown to be an effective means of accomplishing mesh generation, establishing boundary conditions, and reviewing graphic output for finite element analysis activity