Left and right compatibility of strict orders with fuzzy tolerance and fuzzy equivalence relations

The notion of extensionality of a fuzzy relation w.r.t. a fuzzy equivalence was first introduced by Hohle and Blanchard. Belohlavek introduced a similar definition of compatibility of a fuzzy relation w.r.t. a fuzzy equality. In [14] we generalized this notion to left compatibility, right compatibility and compatibility of arbitrary fuzzy relations and we characterized them in terms of left and right traces introduced by Fodor. In this note, we will again investigate these notions, but this time we focus on the compatibility of strict orders with fuzzy tolerance and fuzzy equivalence relations

On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been introduced. In this paper we show that, under some additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fass`o and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux–Nijenhuis coordinates and symplectic connections

Electromagnetic compatibility overview

An assessment of the electromagnetic compatibility impact of the Satellite Power System is discussed. The discussion is divided into two parts: determination of the emission expected from SPS including their spatial and spectral distributions, and evaluation of the impact of such emissions on electromagnetic systems including considerations of means for mitigating effects

Market leadership through technology – Backward compatibility in the U.S. Handheld Video Game Industry

The introduction of a new product generation forces incumbents in network industries to rebuild their installed base to maintain an advantage over potential entrants. We study if backward compatibility moderates this process of rebuilding an installed base. Using a structural model of the U.S. market for handheld game consoles, we show that backward compatibility lets incumbents transfer network effects from the old generation to the new to some extent but that it also reduces supply of new software. We examine the tradeoff between technological progress and backward compatibility and find that backward compatibility matters less if there is a large technological leap between two generations. We subsequently use our results to assess the role of backward compatibility as a strategy to sustain market leadership

Backward Compatibility to Sustain Market Dominance – Evidence from the US Handheld Video Game Industry

The introduction of a new product generation forces incumbents in network industries to rebuild their installed base to maintain an advantage over potential entrants. We study if backward compatibility can help moderate this process of rebuilding an installed base. Using a structural model of the US market for handheld game consoles, we show that backward compatibility lets incumbents transfer network effects from the old generation to the new to some extent but that it also reduces supply of new software. We also find that backward compatibility matters most shortly after the introduction of a new generation. Finally, we examine the tradeoff between technological progress and backward compatibility and find that backward compatibility matters less if there is a large technological leap between two generations. We subsequently use our results to assess the role of backward compatibility as a strategy to sustain a dominant market position

Proper subspaces and compatibility

Let $\mathcal{E}$ be a Banach space contained in a Hilbert space $\mathcal{L}$. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambicki\v{\i}, we say that a bounded operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with respect to the inner product of $\mathcal{L}$. By a proper subspace $\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $\mathcal{L}$, then $\mathcal{S}$ belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition to ensure that a proper subspace is compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$ which is also a proper subspace. We give a characterization of the compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several examples are provided that illustrate different situations between proper and compatible subspaces.Comment: 18 page

Compatibility of quantum states

We introduce a measure of the compatibility between quantum states--the likelihood that two density matrices describe the same object. Our measure is motivated by two elementary requirements, which lead to a natural definition. We list some properties of this measure, and discuss its relation to the problem of combining two observers' states of knowledge.Comment: 4 pages, no figure