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1. Rational homotopy type of projectivization of the tangent bundle of certain spaces

   https://doi.org/10.1108/AJMS-02-2024-0029  

   Gastinzi, Jean Baptiste; Ndlovu, Meshach (August 2024, Arab Journal of Mathematical Sciences)

   PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces. Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of $P(E)$and establish relationships with the base space. FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model ofP(E). We also provide the homogeneous space ofP(E)whenn = 2. Finally, we prove the formality ofP(E)over a homogeneous space of equal rank. Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles. Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling. Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits. Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles. 

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2. THE BOUNDARY OF THE p -RANK STRATUM OF THE MODULI SPACE OF CYCLIC COVERS OF THE PROJECTIVE LINE

   https://doi.org/10.1017/nmj.2022.12  

   OZMAN, EKIN; PRIES, RACHEL; WEIR, COLIN (December 2022, Nagoya Mathematical Journal)

   Abstract. We study the p-rank stratification of the moduli space of cyclic degree ! covers of the projective line in characteristic p for distinct primes p and !. The main result is about the intersection of the p-rank 0 stratum with the boundary of the moduli space of curves. When ! = 3 and p ≡ 2 mod 3 is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type (r,s), and p-rank f satisfying the clear necessary conditions. 

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3. Behavioral fingerprinting of Internet‐of‐Things devices

   https://doi.org/10.1002/widm.1337  

   Bezawada, Bruhadeshwar; Ray, Indrakshi; Ray, Indrajit (September 2019, WIREs Data Mining and Knowledge Discovery)

   Abstract Rapid advances in the Internet‐of‐Things (IoT) domain have led to the development of several useful and interesting devices that have enhanced the quality of home living and industrial automation. The vulnerabilities in the IoT devices have rendered them susceptible to compromise and forgery. The problem of device authentication, that is, the question of whether a device's identity is what it claims to be, is still an open problem. Device fingerprinting seems to be a promising authentication mechanism. Device fingerprinting profiles a device based on information available about the device and generate a robust, verifiable and unique identity for the device. Existing approaches for device fingerprinting may not be feasible or cost‐effective for the IoT domain due to the resource constraints and heterogeneity of the IoT devices. Due to resource and cost constraints, behavioral fingerprinting provides promising directions for fingerprinting IoT devices. Behavioral fingerprinting allows security researchers to understand the behavioral profile of a device and to establish some guidelines regarding the device operations. In this article, we discuss existing approaches for behavioral fingerprinting of devices in general and evaluate their applicability for IoT devices. Furthermore, we discuss potential approaches for fingerprinting IoT devices and give an overview of some of the preliminary attempts to fingerprint IoT devices. We conclude by highlighting the future research directions for fingerprinting in the IoT domain. This article is categorized under:Application Areas > Science and TechnologyApplication Areas > InternetTechnologies > Machine LearningApplication Areas > Industry Specific Applications 

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4. An update of the development of motor behavior

   https://doi.org/10.1002/wcs.1682  

   Franchak, John_M; Adolph, Karen_E (June 2024, WIREs Cognitive Science)

   Abstract This primer describes research on the development of motor behavior. We focus on infancy when basic action systems are acquired—posture, locomotion, manual actions, and facial actions—and we adopt a developmental systems perspective to understand the causes and consequences of developmental change. Experience facilitates improvements in motor behavior and infants accumulate immense amounts of varied everyday experience with all the basic action systems. At every point in development, perception guides behavior by providing feedback about the results of just prior movements and information about what to do next. Across development, new motor behaviors provide new inputs for perception. Thus, motor development opens up new opportunities for acquiring knowledge and acting on the world, instigating cascades of developmental changes in perceptual, cognitive, and social domains. This article is categorized under:Cognitive Biology > Cognitive DevelopmentPsychology > Motor Skill and PerformanceNeuroscience > Development 

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5. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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