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Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in $\bf R$.…

Using the ideas of abstract algebra, we introduce the basic concepts of abstract probability theory that generalize the Kolmogorov's probability theory, possibility theory and other theories that deal with uncertainty. Based on abstract…

We propose the star-shaped acceptability indexes as generalizations of both the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013) in the same vein as star-shaped risk measures generalize both the classes of coherent…

The problem of advancing coordinatization of mathematics is considered. The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation…

We study a general risk measure called the generalized shortfall risk measure, which was first introduced in Mao and Cai (2018). It is proposed under the rank-dependent expected utility framework, or equivalently induced from the cumulative…

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and…

We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…

We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken…

The main goal of this article is to introduce an economic perspective in the social logic of space. Firstly, we describe the economic model of a linear city to show how depth can generate value by creating local monopolies in less…

State of the art language models return a natural language text continuation from any piece of input text. This ability to generate coherent text extensions implies significant sophistication, including a knowledge of grammar and semantics.…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some…

We present a novel approach to uncertainty quantification in classification tasks based on label-wise decomposition of uncertainty measures. This label-wise perspective allows uncertainty to be quantified at the individual class level,…

A general notion of algebraic conditional plausibility measures is defined. Probability measures, ranking functions, possibility measures, and (under the appropriate definitions) sets of probability measures can all be viewed as defining…

Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $\phi$ on a lattice $L$ which are modular, i.e.,…

$P$-values have been the focus of considerable criticism based on various considerations. Still, the $P$-value represents one of the most commonly used statistical tools. When assessing the suitability of a single hypothesized distribution,…

For the functors acting in the category of compact Hausdorff spaces, we introduce the so-called open multi-commutativity property, which generalizes both bicommutativity and openness, and prove that this property is satisfied by the functor…

We present a general framework for measuring the liquidity risk. The theoretical framework defines a class of risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement…

Traditional macroeconomic models, based on static algebraic systems, fail to capture the dynamics of a bimonetary economy like Argentina's. This paper proposes a framework based on category theory to develop a more flexible and structured…

After a brief review of classical probability theory (measure theory), we present an observation (due to Sorkin) concerning an aspect of probability in quantum mechanics. Following Sorkin, we introduce a generalized measure theory based on…

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the…