8,400 research outputs found
Some criteria for maximal abstract monotonicity
In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel's duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators
Monotone Linear Relations: Maximality and Fitzpatrick Functions
We analyze and characterize maximal monotonicity of linear relations
(set-valued operators with linear graphs). An important tool in our study are
Fitzpatrick functions. The results obtained partially extend work on linear and
at most single-valued operators by Phelps and Simons and by Bauschke, Borwein
and Wang. Furthermore, a description of skew linear relations in terms of the
Fitzpatrick family is obtained. We also answer one of Simons problems by
showing that if a maximal monotone operator has a convex graph, then this graph
must actually be affine
Well-posedness via Monotonicity. An Overview
The idea of monotonicity (or positive-definiteness in the linear case) is
shown to be the central theme of the solution theories associated with problems
of mathematical physics. A "grand unified" setting is surveyed covering a
comprehensive class of such problems. We elaborate the applicability of our
scheme with a number examples. A brief discussion of stability and
homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte
Quantifying quantum coherence and non-classical correlation based on Hellinger distance
Quantum coherence and non-classical correlation are key features of quantum
world. Quantifying coherence and non-classical correlation are two key tasks in
quantum information theory. First, we present a bona fide measure of quantum
coherence by utilizing the Hellinger distance. This coherence measure is proven
to fulfill all the criteria of a well defined coherence measure, including the
strong monotonicity in the resource theories of quantum coherence. In terms of
this coherence measure, the distribution of quantum coherence in multipartite
systems is studied and a corresponding polygamy relation is proposed. Its
operational meanings and the relations between the generation of quantum
correlations and the coherence are also investigated. Moreover, we present
Hellinger distance-based measure of non-classical correlation, which not only
inherits the nice properties of the Hellinger distance including contractivity,
and but also shows a powerful analytic computability for a large class of
quantum states. We show that there is an explicit trade-off relation satisfied
by the quantum coherence and this non-classical correlation
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