Notions Of Monotonicity Research Articles

A framework for generalized monotonicity of fusion functions

The relaxation of the property of monotonicity is a trend in the theory of aggregation and fusion functions and several generalized forms of monotonicity have been introduced, most of which are based on the notion of directional monotonicity. In this paper, we propose a general framework for generalized monotonicity that encompasses the different forms of monotonicity that we can find in the literature. Additionally, we introduce various new forms of monotonicity that are not based on directional monotonicity. Specifically, we introduce dilative monotonicity, which requires that the function increases when the inputs have increased by a common factor, and a general form of monotonicity that is dependent on a function T and a subset of the domain Z. This two new generalized monotonicities are the basis to propose a set of different forms of monotonicity. We study the particularities of each of the new proposals and their links to the previous relaxed forms of monotonicity. We conclude that the introduction of dilative monotonicity complements the conditions of weak monotonicity for fusion functions and that (T,Z)-monotonicity yields a condition that is slightly stronger than weak monotonicity. Finally, we present an application of the introduced notions of monotonicity in sentiment analysis.

Pointwise directional increasingness and geometric interpretation of directionally monotone functions

The relaxation of monotonicity requirements is a trend in the theory of aggregation functions. In the recent literature, we can find several relaxed forms of monotonicity, such as weak, directional, cone, ordered directional and strengthened directional monotonicity. All these forms of monotonicity are global properties in the sense that they are imposed for all the points in the domain of a function. In this work, we introduce a local notion of monotonicity called pointwise directional monotonicity, or directional monotonicity at a point. Based on this concept, we characterize all the previously defined notions of monotonicity and, in the final part of the paper, we present some geometric aspects of the global weaker forms of monotonicity, stressing their relations and singularities.

Curve-based monotonicity: a generalization of directional monotonicity

ABSTRACTIn this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.

Generalized monotonicity in terms of differential inequalities

AbstractThe classical notions of monotonicity and convexity can be characterized via the nonnegativity of the first and the second derivative, respectively. These notions can be extended applying Chebyshev systems. The aim of this note is to characterize generalized monotonicity in terms of differential inequalities, yielding analogous results to the classical derivative tests. Applications in the fields of convexity and differential inequalities are also discussed.

Copulas, stable tail dependence functions, and multivariate monotonicity

Abstract For functions of several variables there exist many notions of monotonicity, three of them being characteristic for resp. distribution, survival and co-survival functions. In each case the “degree” of monotonicity is just the basic one of a whole scale. Copulas are special distribution functions, and stable tail dependence functions are special co-survival functions. It will turn out that for both classes the basic degree of monotonicity is the only one possible, apart from the (trivial) independence functions. As a consequence a “nesting” of such functions depends on particular circumstances. For nested Archimedean copulas the rather restrictive conditions known so far are considerably weakened.

Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity

In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Łukasiewicz implication.

Hirsch polytopes with exponentially long combinatorial segments

In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes. We use this definition to investigate further properties of combinatorial segments. Besides recovering the two stated bounds, we show a refined bound for banner complexes, and study the behavior of the maximal length of combinatorial segments with respect to two usual operations, namely join and one-point suspension. Finally, we show the limitations of combinatorial segments by constructing pure normal simplicial complexes in which all combinatorial segments between two particular facets achieve the length $\Omega(n2^{d})$. This includes vertex-decomposable---therefore Hirsch---polytopes.

The Solvency II Standard Formula, Linear Geometry, and Diversification

We introduce the notions of monotonicity, subadditivity, and homogeneity for functions defined on a convex cone, call functions with these properties diversification functions and obtain the respective properties for the risk aggregation given by such a function. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of nonnegative vectors. Any Lp norm has this property, and any scalar product given by a non-negative positive semidefinite matrix as well. In particular, the Standard Formula is a diversification function, hence a risk measure that preserves homogeneity, subadditivity, and convexity.Another funny application of our viewpoint is a geometric deduction of the Markowitz and Sharpe optimal portfolio.

Complexity and monotonicity results for domination games

In this paper we study Domination Games, a class of games introduced by Fomin, Kratsch, and Müller in [8]. Domination games are a variant of the well-known graph searching games (also called cops and robber games), where a number of cops tries to capture a robber hiding on the vertices of a graph. Variants of these games are often used to provide a game-theoretic characterization of important graph parameters such as pathwidth, treewidth, and hypertreewidth.We are primarily interested in questions concerning the complexity and monotonicity of these games. We show that dominating games are computationally much harder than standard cops and robber games and establish strong non-monotonicity results for various notions of monotonicity that arise naturally in the context of domination games. Answering a question of [8], we show that there are graphs where the shortest winning strategy for a minimal number of cops must necessarily be of exponential length.

Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices

This note contains some observations on primary matrix functions and different notions of monotonicity with relevance toward constitutive relations in nonlinear elasticity. Focusing on primary matrix functions on the set of symmetric matrices, we discuss and compare different criteria for monotonicity. The demonstrated results are particularly applicable to computations involving the true-stress–true-strain monotonicity condition, a constitutive inequality recently introduced in an Arch. Appl. Mech. article by C.S. Jog and K.D. Patil. We also clarify a statement by Jog and Patil from the same article which could be misinterpreted.

Lattices and Lotteries

We consider the consumer problem under uncertainty when the consumer can choose the quantity of a risk-free good and the lottery, or distribution, of a risky good from a set of distributions. These goods are imperfect substitutes in the consumer preferences, with additive preferences a special case. We develop sufficient conditions for the choice of the risky good to be monotone with respect to income, exploring different notions of monotonicity. The sufficient conditions are ordinal, independent of concavity, and do not require differentiability or continuity. Cardinal conditions and conditions from the single good case are not necessary and are not always sufficient. The sufficient conditions are formulated in appropriate value lattices. The framework is flexible and adaptable to handle different uncertainty applications. Examples demonstrate the sufficient conditions and different applications where available lotteries may be finite in number, may have discrete support, or may form a chain or a lattice.

Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity

In search of a meaningful 2-dimensional analog to monotonicity, we introduce two new definitions and give examples of and discuss the relationship between these definitions and others that we found in the literature.

Every maximally monotone operator of Fitzpatrick–Phelps type is actually of dense type

We show that every maximally monotone operator of Fitzpatrick-Phelps type defined on a real Banach space must be of dense type. This provides an affirmative answer to a question posed by Stephen Simons in 2001 and implies that various important notions of monotonicity coincide.

Matrix interval monotonicity

AbstractWe propose new notions of monotonicity for real matrices in order to characterize a particular notion of interval boundedness of various generalized inverses.

MATRIX INTERVAL MONOTONICITY

We propose new notions of monotonicity for real matrices in order to characterize a particular notion of interval boundedness of various generalized inverses. AMS Subject Classification: 15A09, 15A48.

Lattices and Lotteries

This paper considers the consumer problem under uncertainty when the consumer can choose the quantity of a risk-free good and the lottery, or distribution, of a risky good. These goods are imperfect substitutes in the consumer preferences, with additive preferences a special case. We develop sufficient conditions for the choice of the risky good to be monotone with respect to income, exploring different notions of monotonicity. The sufficient conditions are independent of concavity and do not require differentiability or continuity. Cardinal conditions that can be derived from standard assumptions are not necessary and are not always sufficient, and neither are conditions from the single good case. The sufficient conditions are formulated in lattices speci cally constructed for the consumer problem under uncertainty. The framework we develop is fl‡exible and adapt- able to handle different uncertainty applications. Examples demonstrate the sufficient conditions as well as different applications where available lotteries may be fi nite in number, may have discrete support, may form a lattice, or may form a chain.

Causal Directed Acyclic Graphs and the Direction of Unmeasured Confounding Bias

We present results that allow the researcher in certain cases to determine the direction of the bias that arises when control for confounding is inadequate. The results are given within the context of the directed acyclic graph causal framework and are stated in terms of signed edges. Rigorous definitions for signed edges are provided. We describe cases in which intuition concerning signed edges fails and we characterize the directed acyclic graphs that researchers can use to draw conclusions about the sign of the bias of unmeasured confounding. If there is only one unmeasured confounding variable on the graph, then nonincreasing or nondecreasing average causal effects suffice to draw conclusions about the direction of the bias. When there are more than one unmeasured confounding variable, nonincreasing and nondecreasing average causal effects can be used to draw conclusions only if the various unmeasured confounding variables are independent of one another conditional on the measured covariates. When this conditional independence property does not hold, stronger notions of monotonicity are needed to draw conclusions about the direction of the bias.

Degrees of non-monotonicity for restarting automata

In the literature various notions of monotonicity for restarting automata have been studied. Here we introduce two new variants of monotonicity for restarting automata and for two-way restarting automata: left-monotonicity and right-left-monotonicity. It is shown that for the various types of deterministic and nondeterministic (two-way) restarting automata without auxiliary symbols, these notions yield infinite hierarchies, and we compare these hierarchies to each other. Further, as a tool used to simplify some of the proofs, the shrinking restarting automaton is introduced, which is a generalization of the standard (length-reducing) restarting automaton to the weight-reducing case. Some of the consequences of this generalization are also discussed.

Monotonicity and complete monotonicity for continuous-time Markov chains

We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time. To cite this article: P. Dai Pra et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Monotonicity of power and power measures

Monotonicity is commonly considered an essential requirement for power measures; violation of local monotonicity or related postulates supposedly disqualifies an index as a valid yardstick for measuring power. This paper questions if such claims are really warranted. In the light of features of real-world collective decision making such as coalition formation processes, ideological affinities, a priori unions, and strategic interaction, standard notions of monotonicity are too narrowly defined. A power measure should be able to indicate that power is non-monotonic in a given dimension of players' resources if – given a decision environment and plausible assumptions about behaviour – itis non-monotonic.