Department of Mathematical Sciences Articles
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Item Unbounded Derivations in Algebras Associated with Monothetic Groups(Cambridge, 2021-12) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of ScienceGiven an infinite, compact, monothetic group G we study decompositions and structure of unbounded derivations in a crossed product C∗ -algebra C(G)⋊Z obtained from a translation on G by a generator of a dense cyclic subgroup. We also study derivations in a Toeplitz extension of the crossed product and the question whether unbounded derivations can be lifted from one algebra to the other.Item Kloosterman sums over finite Frobenius rings(IMPAN, 2021) Nica, Bogdan; Mathematical Sciences, School of ScienceWe study Kloosterman sums in a generalized ring-theoretic context, that of finite commutative Frobenius rings. We prove a number of identities for twisted Kloosterman sums, loosely clustered around moment computations. Our main result is the computation of the fourth moment of twisted Kloosterman sums, for non-primitive twists.Item On sprays with vanishing χ-curvature(World Scientific, 2021) Shen, Zhongmin; Mathematical Sciences, School of ScienceEvery Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the χ-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvature always have vanishing χ-curvature. Then we establish the Beltrami Theorem for sprays with χ=0.Item Nonlinear spectrums of Finsler manifolds(Springer, 2022-01) Kristály, Alexandru; Shen, Zhongmin; Yuan, Lixia; Zhao, Wei; Mathematical Sciences, School of ScienceIn this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce faithful dimension pairs by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik–Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry–Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik–Schnirelmann dimension pair.Item Chambers in the symplectic cone and stability of symplectomorphism group for ruled surface(arXiv, 2022-02-14) Buse, Olguta; Li, Jun; Mathematical Sciences, School of ScienceWe continue our previous work to prove that for any non-minimal ruled surface $(M,\omega)$, the stability under symplectic deformations of $\pi_0, \pi_1$ of $Symp(M,\omega)$ is guided by embedded $J$-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio $\mu$ between the section and fiber goes to infinity, there is a topological colimit of $Symp(M,\omega_{\mu}).$ Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in $Symp(M,\omega) \cap \rm Diff_0(M),$ where $\rm Diff_0(M)$ is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.Item A Note on the Use of the Nonlinear Dirac Equation for Soliton Particle Models(2022-05) Luke, Jon C.When nonlinear Dirac equations are to be used in classical soliton models, it is essential that certain paradoxical properties of spinors be avoided. This result can be achieved if Dirac’s equation is formulated in a slightly different way, where the dependent variable is a matrix rather than a column vector. It is helpful to understand the change in the context of a Clifford algebra.Item Homological Stability for Spaces of Commuting Elements in Lie Groups(Oxford, 2021-03) Ramras, Daniel A.; Stafa, Mentor; Mathematical Sciences, School of ScienceIn this paper, we study homological stability for spaces Hom(Zn,G) of pairwise commuting n-tuples in a Lie group G. We prove that for each n⩾1, these spaces satisfy rational homological stability as G ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, Comm(G) and BcomG, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases. Our proofs use the theory of representation stability—in particular, the theory of FIW-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.Item Riemann–Hilbert approach to the elastodynamic equation: half plane(Springer, 2021-05) Its, Alexander; Its, Elizabeth; Mathematical Sciences, School of ScienceWe show, how the Riemann–Hilbert approach to the elastodynamic equations, which have been suggested in our preceding papers, works in the half plane case. We pay a special attention to the emergence of the Rayleigh waves within the scheme.Item Special α-limit sets(American Mathematical Society, 2020) Kolyada, Sergiǐ; Misiurewicz, Michał; Snoha, L’ubomírWe investigate the notion of the special α-limit set of a point. For a continuous selfmap of a compact metric space, it is defined as the union of the sets of accumulation points over all backward branches of the map. The main question is whether a special α-limit set has to be closed. We show that it is not the case in general. It is unknown even whether a special α-limit set has to be Borel or at least analytic (it is in general an uncountable union of closed sets). However, we answer this question affirmatively for interval maps for which the set of all periodic points is closed. We also give examples showing how those sets may look like and we provide some conjectures and a problem.Item A Value Region Problem for Continued Fractions and Discrete Dirac Equations(Project Euclid, 2020-06) Klimek, Slawomir; Mcbride, Matt; Rathnayake, Sumedha; Sakai, Kaoru; Mathematical Sciences, School of ScienceMotivated by applications in noncommutative geometry we prove several value range estimates for even convergents and tails, and odd reverse sequences of Stieltjes type continued fractions with bounded ratio of consecutive elements, and show how those estimates control growth of solutions of a system of discrete Dirac equations.