q-subharmonicity and q-convex domains in ℂ n

In this paper we study q-subharmonic and q-plurisubharmonic functions in ? ⁿ . Next as an application, we give the notion of q-convex domains in ? ⁿ which is an extension of weakly q-convex domains introduced and investigated in [10]. In the end of the paper we show that the q-convexity is the local property and give some examples about q-convex domains.     

Canonical and monophonic convexities in hypergraphs

Known properties of “canonical connections” from database theory and of “closed sets” from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls.     

Conglomerable coherence

We contrast Williams? and Walley?s theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them most: Williams? theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walley?s notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considering also some previous results in the literature, we deduce that Williams? theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walley?s theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.     

New integral representations of nth order convex functions

In this paper we give an integral representation of an n-convex function f in general case without additional assumptions on function f. We prove that any n-convex function can be represented as a sum of two (n+1)-times monotone functions and a polynomial of degree at most n. We obtain a decomposition of n-Wright-convex functions which generalizes and complements results of Maksa and Páles (2009) [13]. We define and study relative n-convexity of n-convex functions. We introduce a measure of n-convexity of f. We give a characterization of relative n-convexity in terms of this measure, as well as in terms of nth order distributional derivatives and Radon-Nikodym derivatives. We define, study and give a characterization of strong n-convexity of an n-convex function f in terms of its derivative f(n+1)(x) (which exists a.e.) without additional assumptions on differentiability of f. We prove that for any two n-convex functions f and g, such that f is n-convex with respect to g, the function g is the support for the function f in the sense introduced by W?sowicz (2007) [29], up to polynomial of degree at most n.     

Convex Partitions of Graphs

A set of vertices S of a graph G is convex if all vertices of every geodesic between two of its vertices are in S. We say that G is k-convex if V(G) can be partitioned into k convex sets. The convex partition number of G is the least k ⩾ 2 for which G is k-convex. In this paper we examine k-convexity of graphs. We show that it is NP-complete to decide if G is k-convex, for any fixed k ⩾ 2. We describe a characterization for k-convex cographs, leading to a polynomial time algorithm to recognize if a cograph is k-convex. Finally, we discuss k-convexity for disconnected graphs.     

Partial Convexity

We study OC-convexity, which is defined by the intersection of conic semispaces of partial convexity. We investigate an optimization problem for OC-convex sets and prove a Krein--Milman type theorem for OC-convexity. The relationship between OC-convex and functionally convex sets is studied. Topological and numerical aspects, as well as separability properties are described. An upper estimate for the Carathéodory number for OC-convexity is found. On the other hand, it happens that the Helly and the Radon number for OC-convexity are infinite. We prove that the OC-convex hull of any finite set of points is the union of finitely many polyhedra.     

Conformity and independence with coherent lower previsions

We define the conformity of marginal and conditional models with a joint model within Walley's theory of coherent lower previsions. Loosely speaking, conformity means that the joint can reproduce the marginal and conditional models we started from. By studying conformity with and without additional assumptions of epistemic irrelevance and independence, we establish connections with a number of prominent models in Walley's theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

A survey of the theory of coherent lower previsions

This paper presents a summary of Peter Walley’s theory of coherent lower previsions. We introduce three representations of coherent assessments: coherent lower and upper previsions, closed and convex sets of linear previsions, and sets of desirable gambles. We show also how the notion of coherence can be used to update our beliefs with new information, and a number of possibilities to model the notion of independence with coherent lower previsions. Next, we comment on the connection with other approaches in the literature: de Finetti’s and Williams’ earlier work, Kuznetsov’s and Weischelberger’s work on interval-valued probabilities, Dempster–Shafer theory of evidence and Shafer and Vovk’s game-theoretic approach. Finally, we present a brief survey of some applications and summarize the main strengths and challenges of the theory.     

On conditionally δ-convex functions

Let X be a real vector space, V a subset of X and δ ≧ 0 a given number. We say that f: V → ? is a conditionally δ-convex function if for each convex combination t ₁ υ ₁ + … + t _n υ _n of elements of V such that t ₁ υ ₁ + … + t _n υ _n ∈ V the following inequality holds true: $$ f(t_1 v_1 + \cdots + t_n v_n ) \leqq t_1 f(v_1 ) + \cdots + t_n f(v_n ) + \delta . $$ We prove that f: V → ? is conditionally δ-convex if and only if there exists a convex function $ \tilde f $ : conv V → [?∞, ∞) such that $$ \tilde f(v) \leqq f(v) \leqq \tilde f(v) + \delta for v \in V. $$ In case X = ? ⁿ some conditions equivalent to conditional δ-convexity are also presented.     

The Goodman–Nguyen relation within imprecise probability theory

The Goodman–Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman–Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.     

The use of Markov operators to constructing generalised probabilities

A new approach to constructing generalised probabilities is proposed. It is based on the models using lower and upper previsions, or equivalently, convex sets of probability measures. Our approach uses sets of Markov operators in the role of rules preserving desirability of gambles. The main motivation being the operators of conditional expectations which are usually assumed to reduce riskiness of gambles. Imprecise probability models are then obtained in the ways to be consistent with those desirability preserving rules. The consistency criteria are based on the existing interpretations of models using imprecise probabilities. The classical models based on lower and upper previsions are shown to be a special class of the generalised models. Further, we generalise some standard extension procedures, including the marginal extension and independent products, which can be defined independently of the existing procedures known for standard models.