Using the notion of modulus of continuity at a point of a mapping between metric spaces, we introduce the notion of extensively bounded mappings generalizing that of Lipschitz mappings. We also introduce a metric on it which becomes a norm if the codomain is a normed linear space. We study its basic properties. We also discuss a linearization of an extensively bounded mapping into a bounded linear mapping. As an application, we introduce the notion of extensively bounded operator ideals. We also discuss extensively bounded finite rank and extensively bounded compact mappings and their corresponding operator ideals.

The Financial system has witnessed rapid technological changes. The rise of Bitcoin and other crypto assets based on Distributed Ledger Technology mark a fundamental change in the way people transact and transmit value over a decentralized network, spread across geographies. This has created regulatory and tax policy blind spots, as governments and tax administrations take time to understand and provide policy responses to this innovative, revolutionary, and fast-paced technology. Due to the breakneck speed of innovation in blockchain technology and advent of Decentralized Finance, Decentralized Autonomous Organizations and the Metaverse, it is unlikely that the policy interventions and guidance by regulatory authorities or tax administrations would be ahead or in sync with the pace of innovation. This paper tries to explain the principles on which crypto assets function, their underlying technology and relates them to the tax issues and taxable events which arise within this ecosystem. It also provides instances of tax and regulatory policy responses already in effect in various jurisdictions, including the recent changes in reporting standards by the FATF and the OECD. This paper tries to explain the rationale behind existing laws and policies and the challenges in their implementation. It also attempts to present a ballpark estimate of tax potential of this asset class and suggests creation of global public digital infrastructure that can address issues related to pseudonymity and extra-territoriality. The paper analyses both direct and indirect taxation issues related to crypto assets and discusses more recent aspects like proof-of-stake and maximal extractable value in greater detail.

Problem statement: Standardisation of AI fairness rules and benchmarks is challenging because AI fairness and other ethical requirements depend on multiple factors such as context, use case, type of the AI system, and so on. In this paper, we elaborate that the AI system is prone to biases at every stage of its lifecycle, from inception to its usage, and that all stages require due attention for mitigating AI bias. We need a standardised approach to handle AI fairness at every stage. Gap analysis: While AI fairness is a hot research topic, a holistic strategy for AI fairness is generally missing. Most researchers focus only on a few facets of AI model-building. Peer review shows excessive focus on biases in the datasets, fairness metrics, and algorithmic bias. In the process, other aspects affecting AI fairness get ignored. The solution proposed: We propose a comprehensive approach in the form of a novel seven-layer model, inspired by the Open System Interconnection (OSI) model, to standardise AI fairness handling. Despite the differences in the various aspects, most AI systems have similar model-building stages. The proposed model splits the AI system lifecycle into seven abstraction layers, each corresponding to a well-defined AI model-building or usage stage. We also provide checklists for each layer and deliberate on potential sources of bias in each layer and their mitigation methodologies. This work will facilitate layer-wise standardisation of AI fairness rules and benchmarking parameters.

Assuming certain conditions on the spectral measures of centered stationary Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the probability of the event that their zero count in an interval (resp., nodal length in a square domain) is larger than $n$, where $n$ is much larger than the expected value of the zero count in that interval (resp., nodal length in that square domain), is exponentially small in $n$.

For a submodule $N$ of an $R$-module $M$, a unique product of prime ideals in $R$ is assigned, which is called the generalized prime ideal factorization of $N$ in $M$, and denoted as ${\mathcal{P}}_M(N)$. But for a product of prime ideals ${{\mathfrak{p}_1} \cdots {\mathfrak{p}_{n}}}$ in $R$ and an $R$-module $M$, there may not exist a submodule $N$ in $M$ with ${\mathcal{P}}_{M}(N) = {{\mathfrak{p}_1} \cdots {\mathfrak{p}_{n}}}$. In this article, for an arbitrary product of prime ideals ${{\mathfrak{p}_1} \cdots {\mathfrak{p}_{n}}}$ and a module $M$, we find conditions for the existence of submodules in $M$ having ${{\mathfrak{p}_1} \cdots {\mathfrak{p}_{n}}}$ as their generalized prime ideal factorization.

This paper describes privacy-preserving approaches for the statistical analysis. It describes motivations for privacy-preserving approaches for the statistical analysis of sensitive data, presents examples of use cases where such methods may apply and describes relevant technical capabilities to assure privacy preservation while still allowing analysis of sensitive data. Our focus is on methods that enable protecting privacy of data while it is being processed, not only while it is at rest on a system or in transit between systems. The information in this document is intended for use by statisticians and data scientists, data curators and architects, IT specialists, and security and information assurance specialists, so we explicitly avoid cryptographic technical details of the technologies we describe.

Suppose $D$ is a finite dimensional C*-algebra carrying a continuous action $\overline{\Pi}$ of the circle group $\mathbb{T}$. We study the quantum symmetry group of $D$, taking $\overline{\Pi}$ into account. We show that they are braided compact quantum groups $\mathbb{G}$ over $\mathbb{T}$. Here, the R-matrix, $\mathbb{Z}\times\mathbb{Z}\ni (m,n)\to \zeta^{-m\cdot n}\in\mathbb{T}$ for a fixed $\zeta\in \mathbb{T}$, governs the braided structure. In particular, if $\overline{\Pi}$ is trivial, $\zeta=1$ or $D$ is commutative, then $\mathbb{G}$ coincides with Wang's quantum group of automorphisms of $D$. Moreover, we show that the bosonisation of $\mathbb{G}$ corresponds to the quantum symmetry group of the crossed product C*-algebra $D\rtimes\mathbb{Z}$, where the $\mathbb{Z}$-action is generated by $\overline{\Pi}_{\zeta{^{-1}}}$.