Mathematical Sciences- Journal Articles
Permanent URI for this collection
Browse
Recent Submissions
Item A restricted 2-plane transform related to Fourier restriction for surfaces of codimension 2(Mathematical Sciences Publishers, 2025) Dendrinos, Spyridon; Mustaţă, Andrei; Vitturi, MarcoWe draw a connection between the affine invariant surface measures constructed by P. Gressman in [30] and the boundedness of a certain geometric averaging operator associated to surfaces of codimension 2 and related to the Fourier Restriction Problem for such surfaces. For a surface given by (ξ, Q1(ξ), Q2(ξ)), with Q1, Q2 quadratic forms on Rd, the particular operator in question is the 2-plane transform restricted to directions normal to the surface, that is T f(x, ξ) := Z Z |s|,|t|≤1 f(x − s∇Q1(ξ) − t∇Q2(ξ), s, t) ds dt, where x, ξ ∈ Rd. We show that when the surface is well-curved in the sense of Gressman (that is, the associated affine invariant surface measure does not vanish) the operator satisfies sharp Lp → Lq inequalities for p, q up to the critical point. We also show that the well-curvedness assumption is necessary to obtain the full range of estimates. The proof relies on two main ingredients: a characterisation of well-curvedness in terms of properties of the polynomial det(s∇2Q1 + t∇2Q2), obtained with Geometric Invariant Theory techniques, and Christ’s Method of Refinements. With the latter, matters are reduced to a sublevel set estimate, which is proven by a linear programming argumentItem Synchronization cluster bursting in adaptive oscillator networks(AIP Publishing, 2024-12-24) Wei, Mengke; Amann, Andreas; Burylko, Oleksandr; Han, Xiujing; Yanchuk, Serhiy; Kurths, Jürgen; Deutsche Forschungsgemeinschaft; China Scholarship Council; Potsdam Institute for Climate Impact Research; National Natural Science Foundation of ChinaAdaptive dynamical networks are ubiquitous in real-world systems. This paper aims to explore the synchronization dynamics in networks of adaptive oscillators based on a paradigmatic system of adaptively coupled phase oscillators. Our numerical observations reveal the emergence of synchronization cluster bursting, characterized by periodic transitions between cluster synchronization and global synchronization. By investigating a reduced model, the mechanisms underlying synchronization cluster bursting are clarified. We show that a minimal model exhibiting this phenomenon can be reduced to a phase oscillator with complex-valued adaptation. Furthermore, the adaptivity of the system leads to the appearance of additional symmetries, and thus, to the coexistence of stable bursting solutions with very different Kuramoto order parameters.Item Quantum stochastic cocycles and completely bounded semigroups on operator spaces(Oxford University Press, 2013-03-06) Lindsay, J. Martin; Wills, Stephen J.; UK-India Education and Research InitiativeAn operator space analysis of quantum stochastic cocycles is undertaken. These are cocycles with respect to an ampliated CCR flow, adapted to the associated filtration of subspaces, or subalgebras. They form a noncommutative analog of stochastic semigroups in the sense of Skorohod. One-to-one correspondences are established between classes of cocycle of interest and corresponding classes of one-parameter semigroups on associated matrix spaces. Each of these “global” semigroups may be viewed as the expectation semigroup of an associated quantum stochastic cocycle on the corresponding matrix space. Proof of the two key characterizations, namely that of completely positive contraction cocycles on a C*-algebra, and contraction cocycles on a Hilbert space, involves all of the analysis undertaken here. As indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint circumvents technical (domain) limitations inherent in the theory of quantum stochastic differential equations.Item Finite groups with at most nine non-central conjugacy classes(Royal Irish Academy, 2024) Heffernan, Robert; MacHale, DesAbstract: We classify all finite groups with at most nine non-central conjugacy classes.Item Canard cascading in networks with adaptive mean-field coupling(American Physical Society, 2024-12-02) Balzer, J.; Berner, R.; Lüdge, K.; Wieczorek, Sebastian; Kurths, J.; Yanchuk, Serhiy; Deutsche ForschungsgemeinschaftCanard cascading (CC) is observed in dynamical networks with global adaptive coupling. It is a slow-fast phenomenon characterized by a recurrent sequence of fast transitions between distinct and slowly evolving quasistationary states. In this Letter, we uncover the dynamical mechanisms behind CC, using an illustrative example of globally and adaptively coupled semiconductor lasers, where CC represents sequential switching on and off the lasers. First, we show that CC is a robust and truly adaptive network effect that is scalable with network size and does not occur without adaptation. Second, we uncover multiple saddle slow manifolds (unstable quasistationary states) linked by heteroclinic orbits (fast transitions) in the phase space of the system. This allows us to identify CC with a novel heteroclinic canard orbit that organizes different unstable quasistationary states into an intricate slow-fast limit cycle. Although individual quasistationary states are unstable (saddles), the CC cycle as a whole is attractive and robust to parameter changes.