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We establish the subadditivity of strongly operator convex functions on (0 , ∞) and (− ∞ , 0). By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on (− ∞ , 0). We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on (0 , ∞). Moreover, we study strongly operator convex functions on (a , ∞) with − ∞ < a and on the left half-line (− ∞ , b) with b < ∞. We demonstrate that any nonconstant strongly operator convex function on (a , ∞) is strictly operator decreasing, and any nonconstant strongly operator convex function on (− ∞ , b) is strictly operator monotone. Consequently, for a strongly operator convex function g on (a , ∞) or (− ∞ , b) , we provide lower bounds for | g (A) − g (B) | whenever A − B > 0. [ABSTRACT FROM AUTHOR]

An asymptotic hypothesis test for value-at-risk subadditivity is introduced and studied. The test is derived based on an equivalent formulation of the value-at-risk subadditivity inequality in terms of the distribution of the underlying risks' sum. Its size is considered mathematically, and its power and p-value are studied empirically for different dependence structures, strength of dependence, marginal distributions, sample sizes, number of risks and value-at-risk confidence levels. [ABSTRACT FROM AUTHOR]

A contemporary business challenge for bundling theory is the distribution of digital content such as media, entertainment, software, and other information goods. Consumers can use a number of devices to interact with software, online services, music, video, news, and other forms of digital content. For instance, Netflix videos and Kindle ebooks, initially accessed only on television sets and computers, respectively, are now also consumed on smartphones and tablets. Print products are now distributed and consumed both in print and digitally. Firms offer product line designs that include prices for single device access as also bundle discounts for multi-device access, so that consumers can choose a device or even multiple devices. This paper provides guidelines regarding such multi-device product line design and pricing. A starting point is to note that there are many ways to practice bundling, from offering prices for single-device consumption as well as a bundle discount (mixed bundling), to forcing multi-device prices (pure bundling), and designs in between (partial bundling). Picking the best design, and associated prices, is a complex problem due to how single-device demand functions relate to multi-device demand. Recognizing that some dual-device purchases can occur even when there is no bundle discount, the paper develops intuition around the net gain or loss incurred from giving a bundle discount to entice single-device access buyers to dual sales. One surprising finding is that inducing dual sales through bundle discounts can be profitable even when intent to consume multiple times is quite weak, because in this case dual sales would not occur organically. Less surprisingly, such inducement is also profitable when multi-consumption intent is strong, and least attractive when the intent is moderate. When one of the devices is an emerging one or has weak own demand in the short term, then it can be useful to offer a partial bundle, tying sales for the stronger device into a bundle comprising access to both devices. When device valuations are such that consumers generally agree on the rank-ordering of the devices (e.g., if a location-based app offers greater value on a smartphone than on a tablet), then it is best to employ some type of bundling, unless the intent for multi-device consumption is proportionally lower among low-value consumers than for high-value consumers. [ABSTRACT FROM AUTHOR]

We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances. [ABSTRACT FROM AUTHOR]

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities.

An asymptotic hypothesis test for value-at-risk subadditivity is introduced and studied. The test is derived based on an equivalent formulation of the value-at-risk subadditivity inequality in terms of the distribution of the underlying risks’ sum. Its size is considered mathematically, and its power and p-value are studied empirically for different dependence structures, strength of dependence, marginal distributions, sample sizes, number of risks and value-at-risk confidence levels.

In Shulman (J Funct Anal 263:1468–1484, Theorem 2.2) the author estimated the cardinality of the global image of a subadditive map from a group to an arbitrary power set and applied the result to functional equations of Levi-Civita type (the relations of this result to covering a group by subgroups will be shortly discussed in the present paper). Here we will establish the metric stability of this theorem. Namely it will be shown that if a set-valued map F from a group G to a metric space (X , d) satisfies the condition d F (g h) , F (g) ∪ F (h) < δ , for any g , h ∈ G and some δ > 0 , and the cardinality of each F(g) does not exceed n ∈ N , then the set F (G) : = ∪ g ∈ G F (g) can be covered by n (n + 3) / 2 balls of radius (3 · 6 n - 1 - 1) δ . [ABSTRACT FROM AUTHOR]

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall. [ABSTRACT FROM AUTHOR]

The potentially large spatial footprint of earthquake disasters and the increased concentration of population and values in dense urban areas call for an explicit consideration of seismic risk at a regional, building portfolio level. The relation between the building‐level seismic risk and the portfolio‐level seismic risk is helpful if one wants to meet a specific regional seismic risk tolerance level by specifying seismic risk targets for individual buildings. We examine four types of common risk measures and point to the importance of subadditivity, a risk measure mathematical property, for deriving a conservative upper‐bound relation between the building‐specific and the building portfolio seismic risks. Subadditive risk measures, such as the Expected Shortfall, allow estimates of conservative upper bounds of the portfolio‐level seismic risk. In a case study, we show that nonsubadditive risk measures commonly used in earthquake engineering can lead to counter‐intuitive and nonconservative perceptions of the regional seismic risk, especially when one extrapolates from individual buildings to the entire building portfolio. We also illustrate the advantages of subadditive risk measures for regional seismic risk assessment. [ABSTRACT FROM AUTHOR]

Under the presupposition that human time perception is distorted in intertemporal choice, this study constructs a time scale in the framework of axiomatic measurement. First, the conditions (homogeneity of degree one or two) to identify the form of a time scale are proposed so that one can determine whether the hyperbolic or exponential is a more suitable function for modeling people's discounting. Homogeneity of degree one implies that subjective time delay is measured by a power scale and its discount function becomes a power-exponential type, whereas homogeneity of degree two implies that subjective time delay is measured by a log scale and its discount function becomes a hyperbolic type. Second, a method is shown for constructing a model that can reflect subadditive and superadditive discounting. A non-commutative and non-associative concatenation operation is provided to generate a time string consisting of subdivided durations that incorporates the effect of repeated delay with a subdivided duration. The discounting model is yielded by substituting these strings in an exponential model equipped with a generalized time scale, and the determination of subadditivity or superadditivity depends on whether the discount function value of a product by the non-commutative and non-associative operation is, respectively, smaller or greater than that of a product by an operation of the usual extensive structure. [ABSTRACT FROM AUTHOR]

The concepts of generalized finite subadditivity and generalized finite superadditivity with respect to a pair of monotone measures are proposed. As special cases, they cover the concepts of subadditivity and superadditivity of monotone measures, respectively. By means of these new structural characteristics of monotone measures, we show the generalized sub(super-)additivity of the pan-integrals, and describe the generalized order relations among the Choquet, pan- and concave integral in general context concerning an ordered pair of monotone measures. The generalized Hölder and Minkowski inequalities of the pan-integrals are also presented. The previous results related to the Choquet, pan- and concave integral for sub(super-)additive monotone measures are recovered. [ABSTRACT FROM AUTHOR]

A popular rule of thumb, usually called "heuristic technique" in Behavioral Economics, for determining the likelihood insensitivity regions of probability weighting functions (pwf's) is based on searching for points at which the pwf's are twice their values at half the points. Although this technique works remarkably well for many commonly used pwf's, it sometimes fails to provide the correct answer. In order to cover the class of pwf's for which the heuristic technique does not work, in this paper we propose, discuss, and illustrate an extension of the technique into what we call the "slicing method," which is capable of finding the subadditivity and insensitivity regions of any continuous pwf. [ABSTRACT FROM AUTHOR]

The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We argue that this bilinear cover inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bipartite bilinear sets with box constraints. [ABSTRACT FROM AUTHOR]

The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by adapting a proof technique from Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI. [ABSTRACT FROM AUTHOR]

We prove an analogue of Fekete's subadditivity lemma for functions of several real variables which are subadditive in each variable taken singularly. This extends both the classical case for subadditive functions of one real variable, and a similar result for functions of integer variables. While doing so, we prove that the functions with the property mentioned above are bounded in every closed and bounded subset of their domain. The arguments expand on those in Chapter 6 of E. Hille's 1948 textbook. [ABSTRACT FROM AUTHOR]

The aim of this article is to provide characterizations for subadditivitylike growth conditions for the so-called associated weight functions in terms of the defining weight sequence. Such growth requirements arise frequently in the literature and are standard when dealing with ultradifferentiable function classes defined by Braun-Meise-Taylor weight functions since they imply or even characterize important and desired consequences for the underlying function spaces, e.g. closedness under composition. [ABSTRACT FROM AUTHOR]

We consider the following question: If a simplicial complex Γ has d-homology, then does the corresponding d-cycle always induce cycles of smaller dimension that are not boundaries? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree. [ABSTRACT FROM AUTHOR]

A major hypothesis about conditionals is the Equation in which the probability of a conditional equals the corresponding conditional probability: p(if A then C) = p(C|A). Probabilistic theories often treat it as axiomatic, whereas it follows from the meanings of conditionals in the theory of mental models. In this theory, intuitive models (system 1) do not represent what is false, and so produce errors in estimates of p(if A then C), yielding instead p(A & C). Deliberative models (system 2) are normative, and yield the proportion of cases of A in which C holds, i.e., the Equation. Intuitive estimates of the probability of a conditional about unique events: If covid-19 disappears in the USA, then Biden will run for a second term, together with those of each of its clauses, are liable to yield joint probability distributions that sum to over 100%. The error, which is inconsistent with the probability calculus, is massive when participants estimate the joint probabilities of conditionals with each of the different possibilities to which they refer. This result and others under review corroborate the model theory. [ABSTRACT FROM AUTHOR]

The work introduces a family of new risk measures, “VaR to the power of t”. The aim of the work is to study the properties of this family of measures and to derive formulas to calculate them. The study used methods for assessing financial risks by risk measures VaR and ES. As a result, the author proposed a new tool to measure catastrophic financial risks — “VaR to the power of t”. The study proved that for the measuring, it is sufficient to calculate the common risk measure VaR with the confidence probability changed in a certain way. The author concludes that this family of measures should find application in solving the problem of penetrating risk events with low probabilities, but with catastrophic financial losses. The study results may be of use to the regulator to assess the capital adequacy of financial institutions. If t > 1, these measures prove to be more conservative risk measures of catastrophic losses than the known risk measures VaR, ES and GlueVaR.

In this paper, we present an analytical description of emergence from the density matrix framework as a state of knowledge of the system, and its generalized probability formulation. This description is based on the idea of fragile systems, wherein the observer modifies the system by the measurement (i.e., the observer effect) in order to detect possible emergent behavior. We propose the use of a descriptor, based on quantum mutual information, to calculate if subsystems of systems have inner correlations. This may contribute to a definition of emergent systems in terms of emergent information.

In this paper, the subadditivity and superadditivity of convex functions and concave functions on positive operators are proved by applying the function orders preserving or reversing operator inequalities. Moreover, a version of Jensen’s operator inequality without operator convexity is applied to obtain some results on subadditivity or superadditivity of a class of non-negative functions which are not necessarily convex or concave. [ABSTRACT FROM AUTHOR]

Let (X, d) be a metric space and Iso(X, d) the associated isometry group. We study the subadditivity of the minimal displacement function f : → Iso(X, d) R for different metric spaces. When (X, d) is ultrametric, we prove that the minimal displacement function is subadditive. We show, by a simple algebraic argument, that subadditivity does not hold for the direct isometry group of the hyperbolic plane. The same argument can be used for other metric spaces. [ABSTRACT FROM AUTHOR]

The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ϵ within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the definition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples. [ABSTRACT FROM AUTHOR]

We provide several improvements and corrections of results dealing with the smallest universal integral I S based on a semicopula S. We deal with a transformation of the integral into another semicopula-based universal integral for continuous and commutative semicopulas, monotone convergence theorems with new proofs and Fréchet-type mappings' properties corrected. During this way we discover several fine properties of the integral functional I S. We also discuss further convergences of measurable functions and the corresponding integrals in the light of the newest results from the literature answering several open questions stated in our previous articles of the series. [ABSTRACT FROM AUTHOR]

Recall the classical hypothesis testing setting with two sets of probability distributions $P$ and $Q$. One receives either $n$ i.i.d. samples from a distribution $p \in P$ or from a distribution $q \in Q$ and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on $p$ or $q$ , but only on the sets $P$ and $Q$. We consider an adaptive generalization of this model where the choice of $p \in P$ and $q \in Q$ can change in each sample in some way that depends arbitrarily on the previous samples. In other words, in the $k^{th}$ round, an adversary, having observed all the previous samples in rounds $1,\ldots,k-1$ , chooses $p_{k} \in P$ and $q_{k} \in Q$ , with the goal of confusing the hypothesis test. We prove that even in this case, the optimal exponential error rate can be achieved by a simple maximum-likelihood test that depends only on $P$ and $Q$. We then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements. For example, it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication (LOCC). The basic idea is that in our setup, the deleterious effects of entanglement can be simulated by an adaptive classical adversary. We prove a quantum Stein’s Lemma in this setting: In many circumstances, the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states. In particular, our arguments yield an alternate proof of Li and Winter’s recent strengthening of strong subadditivity for von Neumann entropy. [ABSTRACT FROM AUTHOR]

Capital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures. [ABSTRACT FROM AUTHOR]

The theory of evidence (TE) has been largely used for many applications. This theory is a generalization of probability distribution and offers a mathematical representation for two types of uncertainty-based information: 1) discord and 2) nonspecificity. Several measures have already been developed to quantify these two types of uncertainty. They have been called total uncertainty (TU) measures since they quantify both types of uncertainty. The generalized Hartley measure and the maximum entropy have been the only measures so far that satisfy a list of properties very desirable for practical applications. Recently, two new measures of nonspecificity and TU based on belief intervals have been proposed. These two measures do not satisfy the properties of additivity, superadditivity, and subadditivity in the TE. The present critique is about these shortcomings and provides a more complete analysis of those uncertainty measures with respect to a list of desired properties. A potential consequence of an ill-characterized measure may yield selecting an inappropriate rule for decision-making in the processing chain from data to information to decisions. [ABSTRACT FROM AUTHOR]

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We consider instances of long‐range percolation on Zd and Rd, where points at distance r get connected by an edge with probability proportional to r−s, for s ∈ (d,2d), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distance D(x,y) between x and y in the limit as |x − y|→∞. For the model on Zd we show that, in probability as |x|→∞, the distance D(0,x) is squeezed between two positive multiples of (logr)Δ, where Δ:=1/log2(1/γ) for γ: = s/(2d). For the model on Rd we show that D(0,xr) is, in probability as r→∞ for any nonzero x∈Rd, asymptotic to φ(r)(logr)Δ for φ a positive, continuous (deterministic) function obeying φ(rγ) = φ(r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly‐exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain. [ABSTRACT FROM AUTHOR]

We show that CMt simplicial complexes, a notion generalizing Buchsbaum-ness, of small codimension must have large depth, proving more precise results in the codimension 2 case. In the paper, we show that the CMt property is a topological invariant of a simplicial complex. [ABSTRACT FROM AUTHOR]

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function—results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature. [ABSTRACT FROM AUTHOR]

Transitive fuzzy relations play a central role in fuzzy research activity. A special type of this class of fuzzy relations is known as indistinguishability operators. Many works have focused their efforts on the study of the properties of such operators. One celebrated problem is to find out the class of functions that allows to aggregate a collection of indistinguishability operators into a new single one. Characterizations of this class of functions have been obtained and the relationship with those functions that aggregate a collection of extended pseudo-metrics into a single one has been revealed in the literature. Moreover, fuzzy preorders are a class of fuzzy transitive relations that extend the notion of indistinguishability operators to the asymmetric context. In this paper we show that there is an equivalence between functions that aggregate fuzzy preorders and those functions that merge extended quasi-pseudo-metrics. The new provided characterizations reveal that, in essence, such functions must be monotone and subadditive. Special attention is paid to the case of fuzzy partial orders (a special case of fuzzy preorder) showing that there is a correspondence between those functions aggregating fuzzy partial orders and those that aggregate extended quasi-metrics. Since all the aforesaid fuzzy relations are transitive we also provide new information about those functions that preserve the class of transitive fuzzy relations in terms of those that preserve the so-called ordinary triangular triplets and those that preserve the triangle inequality. The potential applicability of the exposed theory to multi-criteria decision making problems has been discussed. [ABSTRACT FROM AUTHOR]

Traditionally, the interval and delay effects have been identified and considered as the same anomaly in the context of intertemporal choice, when individuals or groups of individuals make their decisions about reward preferences. This has supposed that most studies on this topic have been focused on the delay effect and, consequently, that the discount functions provided by the existing literature have considered only this effect. This is the case of hyperbolic discounting, which has been used to describe the delay, but not the interval effect. Therefore, the main objective of this paper is to carry out a detailed analysis of both anomalies, which will allow us to mathematically relate them, thus finding their analogies and differences. To do this, we will first analyze the concept of delay effect and later the different definitions of the interval effect. The main conclusion of this paper is twofold. On the one hand, if the benchmark for valuation is fixed, the delay effect coincides with the so-called decreasing interval effect. On the other hand, if the assessment reference point is the beginning of each interval, both anomalies are different. These findings make necessary to redefine the concept of interval effect. Finally, we will analyze the relationship between the interval effect, the delay effect and the subadditivity

The framework of this paper is intertemporal choice and, more specifically, the so-called delay effect. Traditionally, this anomaly, also known as decreasing impatience, has been revealed when individuals reverse their preferences over monetary or non-monetary rewards. In this manuscript, we will analyze the delay effect by using preference relations and discount functions. The treatment of the delay effect with discount functions exhibits several scenarios for this paradox. Thus, the objective of this paper is to deduce the different expressions of the delay effect and their mathematical characterizations by using discount functions in stationary and dynamic settings. In this context, subadditivity will be derived as a particular case of decreasing impatience. Finally, we will introduce a new discount function, the so-called asymmetric exponential discount function, able to describe decreasing impatience.