scholarly journals New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space

2014 ◽  
Vol 53 (1) ◽  
pp. 182-198 ◽  
Author(s):  
Robert Thijs Kozma ◽  
Jenő Szirmai
2019 ◽  
Vol 206 (1) ◽  
pp. 1-25
Author(s):  
Robert Thijs Kozma ◽  
Jenő Szirmai
Keyword(s):  

2019 ◽  
Vol 7 ◽  
Author(s):  
MATTHEW JENSSEN ◽  
FELIX JOOS ◽  
WILL PERKINS

We prove a lower bound on the entropy of sphere packings of $\mathbb{R}^{d}$ of density $\unicode[STIX]{x1D6E9}(d\cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\unicode[STIX]{x1D6FA}(d\cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_{d}(1))\log (2/\sqrt{3})d\cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh.


1989 ◽  
Vol 98 (3) ◽  
pp. 499-509 ◽  
Author(s):  
J. A. Rush
Keyword(s):  

MRS Bulletin ◽  
1997 ◽  
Vol 22 (10) ◽  
pp. 19-27 ◽  
Author(s):  
Wei William Lee ◽  
Paul S. Ho

Continuing improvement of microprocessor performance historically involves a decrease in the device size. This allows greater device speed, an increase in device packing density, and an increase in the number of functions that can reside on a single chip. However higher packing density requires a much larger increase in the number of interconnects. This has led to an increase in the number of wiring levels and a reduction in the wiring pitch (sum of the metal line width and the spacing between the metal lines) to increase the wiring density. The problem with this approach is that—as device dimensions shrink to less than 0.25 μm (transistor gate length)—propagation delay, crosstalk noise, and power dissipation due to resistance-capacitance (RC) coupling become significant due to increased wiring capacitance, especially interline capacitance between the metal lines on the same metal level. The smaller line dimensions increase the resistivity (R) of the metal lines, and the narrower interline spacing increases the capacitance (C) between the lines. Thus although the speed of the device will increase as the feature size decreases, the interconnect delay becomes the major fraction of the total delay and limits improvement in device performance.To address these problems, new materials for use as metal lines and interlayer dielectrics (ILD) as well as alternative architectures have been proposed to replace the current Al(Cu) and SiO2 interconnect technology.


2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.


10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.


10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.


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