Semiconvex geometry

Flood, Joe

doi:10.1017/s1446788700017973

Cited by 14 publications

(11 citation statements)

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“…We first recall (see e.g. [28,8,20,6,11]) that such convex structures can equivalently be described uniformly as algebras of a monad, namely of the distribution monad D, see Theorem 4. Such an algebra map gives an interpretation of each formal convex combination r 1 x 1 +· · ·+r n x n , where r 1 + · · · + r n = 1, as a single element of X.…”

Section: Introductionmentioning

confidence: 99%

Convexity, Duality and Effects

Jacobs

2010

IFIP Advances in Information and Communication Technology

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This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call 'convex functors'; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are 'dual' adjunctions for convex sets, one time with the Boolean truth values {0, 1} as dualising object, and one time with the probablity values [0,1]. The third adjunction is between effect algebras and convex functors.

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Section: Introductionmentioning

confidence: 99%

Convexity, Duality and Effects

Jacobs

2010

IFIP Advances in Information and Communication Technology

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“…The possibility of embedding a barycentric algebra in a cone makes calculations much easier as already remarked by J. Flood [6]. The proof of Lemma 2.8 illustrates this advantage when compared with Neumann's original proof in the unordered case.…”

Section: Standardmentioning

confidence: 80%

“…The embedding of a barycentric algebra as a convex subset in the abstract cone C A is due to J. Flood [6]. The surprising lemma 2.8 is due to W. Neumann [45] in the unordered case.…”

Section: Standardmentioning

confidence: 99%

Mixed powerdomains for probability and nondeterminism

Keimel¹,

Plotkin²

2017

Logical Methods in Computer Science ; Volume 13

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We consider mixed powerdomains combining ordinary nondeterminism and probabilistic nondeterminism. We characterise them as free algebras for suitable (in)equational theories; we establish functional representation theorems; and we show equivalencies between state transformers and appropriately healthy predicate transformers. The extended nonnegative reals serve as 'truth-values'. As usual with powerdomains, everything comes in three flavours: lower, upper, and order-convex. The powerdomains are suitable convex sets of subprobability valuations, corresponding to resolving nondeterministic choice before probabilistic choice. Algebraically this corresponds to the probabilistic choice operator distributing over the nondeterministic choice operator. (An alternative approach to combining the two forms of nondeterminism would be to resolve probabilistic choice first, arriving at a domain-theoretic version of random sets. However, as we also show, the algebraic approach then runs into difficulties.)Rather than working directly with valuations, we take a domain-theoretic functionalanalytic approach, employing domain-theoretic abstract convex sets called Kegelspitzen; these are equivalent to the abstract probabilistic algebras of Graham and Jones, but are more convenient to work with. So we define power Kegelspitzen, and consider free algebras, functional representations, and predicate transformers. To do so we make use of previous work on domain-theoretic cones (d-cones), with the bridge between the two of them being provided by a free d-cone construction on Kegelspitzen.

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“…and an one-to-one correspondence ρ : S → ρ(S) = U ⊂ V such that ρ(P λ (x, y)) = λ ρ(x) + (1 − λ )ρ(y) for all x, y ∈ S, λ ∈ [0; 1]. For more details, the readers can refer to [8,20].…”

Section: Embedding Theoremmentioning

confidence: 99%

Approach for a metric space with a convex combination operation and applications

Thuan

2016

Journal of Mathematical Analysis and Applications

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Abstract. In this paper, we embed metric space endowed with a convex combination operation, named convex combination space, into a Banach space and the embedding preserves the structures of metric and convex combination. For random element taking values in this kind of space, applications of embedding are also established. On the one hand, some nice properties of expectation such as representation of expected value through continuous affine mappings, the linearity of expectation will be given. On the other hand, the notion of conditional expectation will be also introduced and discussed. Thanks to embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, convergences of martingales and ergodic theorem.Mathematics Subject Classifications (2000): 60B05, 60F15, 51K05.

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Semiconvex geometry

Cited by 14 publications

References 1 publication

Convexity, Duality and Effects

Convexity, Duality and Effects

Mixed powerdomains for probability and nondeterminism

Approach for a metric space with a convex combination operation and applications

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