Slice Topology Research Articles

A representation formula for slice regular functions over slice-cones in several variables

The aim of this paper is to extend the so called slice analysis to a general case in which the codomain is a real vector space of even dimension, i.e. is of the form {mathbb {R}}^{2n}. This is a new setting which contains and encompasses in a nontrivial way other cases already studied in the literature and which requires new tools. To this end, we define a cone {mathcal {W}}_{mathcal {C}}^d in [{text {End}}({mathbb {R}}^{2n})]^d and we extend the slice topology tau _s to this cone. Slice regular functions can be defined on open sets in left( tau _s,{mathcal {W}}_{mathcal {C}}^dright) and a number of results can be proved in this framework, among which a representation formula. This theory can be applied to some real algebras, called left slice complex structure algebras. These algebras include quaternions, octonions, Clifford algebras and real alternative *-algebras but also left-alternative algebras and sedenions, thus providing brand new settings in slice analysis.

Extension theorem and representation formula in non-axially-symmetric domains for slice regular functions

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of f(q)=\sum_{n\in\mathbb{N}}(q-p)^{*n} a_n , given by a \sigma -ball \Sigma(p,r) , is not open in \mathbb{H} unless p\in\mathbb{R} . This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along these lines, we can establish an extension theorem and a representation formula in a slice domain.

Novel Protocols to Mitigate Network Slice Topology Learning Attacks and Protect Privacy of Users’ Service Access Behavior in Softwarized 5G Networks

In the softwarized fifth generation (5G) networks, authentication protocols are deployed by both the mobile network operators and the third-party service providers to enable secure slice formation at the network side and user validation at the service provider side, respectively. However, the usage of static key-based authentication protocols in 5G networks gives rise to two major problems: (i) network slice topology learning attack during the formation of slice and (ii) exposure of users’ service access behavior to the third-party service providers and device-to-device communication peers. In this article, we propose group anonymous (i) privacy preserving mutual authentication (PPMA) and (ii) privacy preserving one-way authentication (PPOA) protocols to address the aforementioned problems, respectively. We also prove the security of the PPMA and PPOA protocols against the chosen ciphertext attack in the random oracle model. The PPMA and PPOA protocols are implemented using the JPBC library and the computation overhead is measured. For 1024 bits discrete logarithm security, the computation overhead of the PPMA and PPOA protocols is respectively 48.54 and 68.65 percent lower than an existing privacy preserving authentication protocol on an Intel processor. The PPMA and PPOA protocols are generic and can be applied in peer-to-peer authentication scenarios where group anonymity and privacy preservation are considered.

A New Approach to Slice Analysis Via Slice Topology

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

Set valued Aumann–Pettis integrable martingale representation theorem and convergence

It is known in the literature that in the RNP Banach space the set valued uniformly integrable martingale is a regular martingale. In this paper by using a selector approach we provide a weaker condition than uniform integrability of a set valued Aumann–Pettis integrable martingale to be a set valued Aumann–Pettis integrable regular martingale. The Converse is also established. As an application of the aforementioned results, a new convergence result of set valued Pettis integrable martingales in Slice topology is provided.

A Blockchain-Enabled Multi Domain Edge Computing Orchestrator

Today, management and orchestration are considered prime components of the new network management layer. Multi-domain orchestration has helped in simplifying infrastructural operations and enables better scaling and faster deployment of network services. However, resource provisioning considering network optimization and fulfillment of multi-constraint quality of service (QoS) is still a major concern. In this article, an architecture comprising multi-domain edge orchestration (MDEO) entrusted by blockchain designed to solve the problem of multi-constraint QoS is proposed. A dynamic end-to-end (E2E) network slicing algorithm is devised to execute at the MDEO to enable multi-tenant on-demand network infrastructure provisioning isolation and security. The algorithm first calculates an optimum network slice topology and then instantiates the involved virtual network functions. Based on multi-constraint QoS, it fulfills the E2E slice request. Blockchain is deployed to ensure trustworthiness between different telecom operators, introduce transparency and to automate the fulfillment of service-level agreements through smart contracts. In addition, an experimental simulation of the system is performed and the burst and response times of the proposed framework are analyzed using different distributions. The data in both cases demonstrates a lognormal distributed behavior.

Somatotopic organization of rat thalamocortical slices

The thalamocortical slice is widely employed for in vitro studies of cortical circuits. This preparation was developed in order to preserve anatomical and functional connectivity between the ventrobasal thalamus and somatosensory (whisker/barrel) cortex of young mice, and thalamocortical slice experiments have contributed significantly to our understanding of the thalamocortical synapse. Cortical somatotopy within thalamocortical slices, however, has not been characterized, and this greatly limits their use in studies that require identification of cortical areas associated with particular regions of the sensory periphery. To address this shortcoming we used electrophysiological recording and neuroanatomical labeling techniques in rats to mark the position of functionally defined whisker barrels, in vivo. We subsequently processed the brains in a plane appropriate for TC slices and characterized the location of somatotopically identified barrels in relation to other aspects of slice topology. We found that barrels associated with the large mobile whiskers occupy a particular location in TC slices, but that there are certain constraints to studying this portion of the barrelfield in vitro.

Exceptional Sets of Slices for Functions From the Bergman Space in the Ball

AbstractLet BN be the unit ball in and let f be a function holomorphic and L2-integrable in BN. Denote by E(BN, f) the set of all slices of the form , where L is a complex one-dimensional subspace of , for which is not L2-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for f. We give a characterization of exceptional sets which are closed in the natural topology of slices.

Slice convergence of parametrised sums of convex functions in non-reflexive spaces

The objectives of this study of slice convergence are two-fold. The first is to derive results regarding the passage of certain semi–convergences through Young–Fenchel conjugation. These semi–convergences arise from the splitting of the usual slice topology in the primal and dual spaces into (non-Hausdorff) topologies: the upper slice topology ; a topology generating a convergence closely resembling the bounded–weak* upper Kuratowski convergence; along with the respective primal and dual lower Kuratowski topologies. This gives rise to topological convergences not reliant on sequentially–based definitions found in many such studies, and associated topological continuity results for conjugation (in normed spaces), in contrast to the usual sequential continuity exhibited by analogues of Mosco convergence. The second objective is to study the passage of slice convergence through addition. Such sum theorems have been derived in other works and we establish previous theorems from a unified framework as well as obtaining a new result.

Epi/Hypo–Convergence: The Slice Topology and Saddle Points Approximation

We show that the slice convergence of the convex parents of saddle functions implies the epi/hypo–convergence of these saddle functions and hence the convergence of their saddle points. We also obtain conditions for the slice convergence of sums of convex functions. We then apply these results to problems in convex programming, optimal control and Chebyshev approximations.

Stability results for convergence of convex sets and functions in nonreflexive spaces

Let $\Gamma(X)$ be the convex proper lower semicontinuous functions on a normed linear space $X$. We show, subject to Rockafellar's constraints qualifications, that the operations of sum, episum and restriction are continuous with respect to the slice topology that reduces to the topology of Mosco convergence for reflexive $X$. We show also when $X$ is complete that the epigraphical difference is continuous. These results are applied to convergence of convex sets.

Multivalued strong laws of large numbers in the slice topology. Application to integrands

Starting from the multivalued strong law of large numbers in the Wijsman topology recently proved by the present author, we deduce two multivalued strong laws of large numbers in the ‘slice topology’ introduced by Beer. An application to integrands via their epigraphical multifunctions is also provided.