1,303 research outputs found
The purity of set-systems related to Grassmann necklaces
Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky
introduced in 1998 the notion of weakly separated sets in . Moreover, they raised several conjectures on the purity for this
symmetric relation, in particular, on the Boolean cube . In
0909.1423[math.CO] we proved these purity conjectures for the Boolean cube
, the discrete Grassmanian , and some other
set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for
weakly separated collections inside a positroid which contain a Grassmann
necklace defining the positroid. We denote such set-systems as
. In this paper we give an alternative (and
shorter) proof of the purity of and present a
stronger result. More precisely, we introduce a set-system
complementary to , in
a sense, and establish its purity. Moreover, we prove (Theorem~3) that these
two set-systems are weakly separated from each other. As a consequence of
Theorem~3, we obtain the purity of set-systems related to pairs of weakly
separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise
a conjecture on the purity of both the interior and exterior of a generalized
necklace.Comment: 13 pages, 3 figure
Non-classical measurement theory: a framework forbehavioral sciences
Instances of non-commutativity are pervasive in human behavior. In this paper, we suggest that psychological properties such as attitudes, values, preferences and beliefs may be suitably described in terms of the mathematical formalism of quantum mechanics. We expose the foundations of non-classical measurement theory building on a simple notion of orthospace and ortholattice (logic). Two axioms are formulated and the characteristic state-property duality is derived. A last axiom concerned with the impact of measurements on the state takes us with a leap toward the Hilbert space model of Quantum Mechanics. An application to behavioral sciences is proposed. First, we suggest an interpretation of the axioms and basic properties for human behavior. Then we explore an application to decision theory in an example of preference reversal. We conclude by formulating basic ingredients of a theory of actualized preferences based in non-classical measurement theory.non-classsical measurement ; orthospace ; state ; properties ; non-commutativity
Non-classical expected utility theory with application to type indeterminacy
In this paper we extend Savage's theory of decision-making under uncertainty from a classical environment into a non-classical one. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expected utility. We also propose an application in simple game context in the spirit of Harsanyi.non-classical ; uncertainty ; decision-making
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