Characterization of stochastic equilibrium controls by the Malliavin calculus

We derive a characterization of equilibrium controls in continuous-time, time-inconsistent control (TIC) problems using the Malliavin calculus. For this, the classical duality analysis of adjoint BSDEs is replaced with the Malliavin integration by parts. This results into a necessary and sufficient maximum principle which is applied to a linear-quadratic TIC problem, recovering previous results obtained by duality analysis in the mean-variance case, and extending them to the linear-quadratic setting. We also show that our results apply beyond the linear-quadratic case by treating the generalized Merton problem.

Life’s Energy and Information: Contrasting Evolution of Volume- versus Surface-Specific Rates of Energy Consumption

As humanity struggles to find a path to resilience amidst global change vagaries, understanding organizing principles of living systems as the pillar for human existence is rapidly growing in importance. However, finding quantitative definitions for order, complexity, information and functionality of living systems remains a challenge. Here, we review and develop insights into this problem from the concept of the biotic regulation of the environment developed by Victor Gorshkov (1935–2019). Life’s extraordinary persistence—despite being a strongly non-equilibrium process—requires a quantum-classical duality: the program of life is written in molecules and thus can be copied without information loss, while life’s interaction with its non-equilibrium environment is performed by macroscopic classical objects (living individuals) that age. Life’s key energetic parameter, the volume-specific rate of energy consumption, is maintained within universal limits by most life forms. Contrary to previous suggestions, it cannot serve as a proxy for “evolutionary progress”. In contrast, ecosystem-level surface-specific energy consumption declines with growing animal body size in stable ecosystems. High consumption by big animals is associated with instability. We suggest that the evolutionary increase in body size may represent a spontaneous loss of information about environmental regulation, a manifestation of life’s algorithm ageing as a whole.

A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics

In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm ∗. Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm ∗ is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship.

Duality of Complex Systems Built from Higher-Order Elements

The duality of nonlinear systems built from higher-order two-terminal Chua’s elements and independent voltage and current sources is analyzed. Two different approaches are now being generalized for circuits with higher-order elements: the classical duality principle, hitherto restricted to circuits built from R-C-L elements, and Chua’s duality of memristive circuits. The so-called storeyed structure of fundamental elements is used as an integrating platform of both approaches. It is shown that the combination of associated flip-type and shift-type transformations of the circuit elements can generate dual networks with interesting features. The regularities of the duality can be used for modeling, hardware emulation, or synthesis of systems built from elements that are not commonly available, such as memristors, via classical dual elements.

Strong Inconsistency in Nonmonotonic Reasoning

Minimal inconsistent subsets of knowledge bases play an important role in classical logics, most notably for repair and inconsistency measurement. It turns out that for nonmonotonic reasoning a stronger notion is needed. In this paper we develop such a notion, called strong inconsistency. We show that—in an arbitrary logic, monotonic or not—minimal strongly inconsistent subsets play the same role as minimal inconsistent subsets in classical reasoning. In particular, we show that the well-known classical duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics if the strong notion of inconsistency is used. We investigate the complexity of various related reasoning problems and present a generic algorithm for computing minimal strongly inconsistent subsets of a knowledge base. We also demonstrate the potential of our new notion for applications, focusing on repair and inconsistency measurement.

An explicit formula for the free exponential modality of linear logic

The exponential modality of linear logic associates to every formula A a commutative comonoid !A which can be duplicated in the course of reasoning. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.