Related papers: Lattices with non-Shannon Inequalities
The minimal set of Shannon-type inequalities (referred to as elemental inequalities), plays a central role in determining whether a given inequality is Shannon-type. Often, there arises a situation where one needs to check whether a given…
Any unconstrained information inequality in three or fewer random variables can be written as a linear combination of instances of Shannon's inequality I(A;B|C) >= 0 . Such inequalities are sometimes referred to as "Shannon" inequalities.…
A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is…
In 1997, Z.Zhang and R.W.Yeung found the first example of a conditional information inequality in four variables that is not "Shannon-type". This linear inequality for entropies is called conditional (or constraint) since it holds only…
The symmetric difference in Boolean lattices can be defined in two different but equivalent forms. However, it can be introduced also in every bounded lattice with complementation where these two forms need not coincide. We study lattices…
This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of…
A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional…
Let $L$ be a lattice of full rank in $n$-dimensional real space. A vector in $L$ is called $i$-sparse if it has no more than $i$ nonzero coordinates. We define the $i$-th successive sparsity level of $L$, $s_i(L)$, to be the minimal $s$ so…
This paper studies systems of linear difference equations on the lattice $\Z^n$ that are invariant under a finite group of symmetries, and shows that there exist solutions to such systems that are also invariant under this group of…
The suspicion that the existence of a minimal uncertainty in position measurements violates Lorentz invariance seems unfounded. It is shown that the existence of such a nonzero minimal uncertainty in position is not only consistent with…
Conway and Sloane constructed a 4-parameter family of pairs of isospectral lattices of rank four. They conjectured that all pairs in their family are non-isometric, whenever the parameters are pairwise different, and verified this for…
In 1998, Zhang and Yeung found the first unconditional non-Shannon-type information inequality. Recently, Dougherty, Freiling and Zeger gave six new unconditional non-Shannon-type information inequalities. This work generalizes their work…
We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In…
The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean…
We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in…
The suspicion that the existence of a minimal uncertainty in position measurements violates Lorentz invariance seems unfounded. It is shown that the existence of such a nonzero minimal uncertainty in position is not only consistent with…
We describe a nonlinear kagome lattice with nonlinear dynamics described by Klein-Gordon interactions with a scalar unknown at each node, such as might occur in a nonlinear electrical lattice. We show that the dispersion relation has three…
Fundamental solitons pinned to the interface between three semi-infinite one-dimensional nonlinear dynamical chains, coupled at a single site, are investigated. The light propagation in the respective system with the self-attractive on-site…
The profiles of narrow lattice solitons are calculated analytically using perturbation analysis. A stability analysis shows that solitons centered at a lattice (potential) maximum are unstable, as they drift toward the nearest lattice…
We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the…