Related papers: Wiring switches to more light bulbs
The Gale-Berlekamp switching game is played on the following device: $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ is an $n \times n$ array of lights is controlled by $2n$ switches, one for each row or column. Given an (arbitrary) initial…
We introduce a lower bound for the independence number of an arbitrary $k$-uniform hypergraph that only depends on the number of vertices and number of edges of the hypergraph.
In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…
We derive and discuss general physical bounds on the electromagnetic scattering and absorption of passive structures. Our theory, based on passivity and power conservation, quantifies the minimum and maximum allowed scattering for an object…
We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…
Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the…
We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.
We improve the test to show the impossibility of a quantum theory based on real numbers by a larger ratio of complex-to-real bound on a Bell-type parameter. In contrast to previous theoretical and experimental proposals the test requires…
Consider $n$ sensors whose positions are represented by $n$ uniform, independent and identically distributed random variables assuming values in the open unit interval $(0,1)$. A natural way to guarantee connectivity in the resulting sensor…
Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove…
Xue and Kumar have established that the number of neighbors required for connectivity of wireless networks with N uniformly distributed nodes must grow as log(N), and they also established that the actual number required lies between…
We consider a flat metric with conical singularities on the sphere. Under the assumption that no partial sum of angle defects is equal to $2\pi$, we draw on the geometry of immersed disks to obtain an explicit upper bound on the number of…
When a light bulb is turned on, light moves away from it at speed $c$, by definition. When light from this bulb illuminates a surface, however, this illumination front is not constrained to move at speed $c$. A simple proof is given that…
A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will define prime and composite graphs, and we will show that if $n \ge 15$, then the grpahs with the maximum number of maximum…
We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.
We give upper bounds for the number of spin 1/2 particles that can be bound to a nucleus of charge Z in the presence of a magnetic field B, including the spin-field coupling. We use Lieb's strategy, which is known to yield N_c<2Z+1 for…
We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large…
Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange which increases the communication size in steps…
Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or…
Reversible circuits have been studied extensively and intensively, and have plenty of applications in various areas, such as digital signal processing, cryptography, and especially quantum computing. In 2003, the lower bound $\Omega(2^n…