We study the generalized eigenvalue problems (GEPs) derived from modeling the surface acoustic wave in piezoelectric materials with periodic inhomogeneity. The eigenvalues appear in the reciprocal pairs due to periodic boundary conditions in the modeling. By transforming the GEP into a T-palindromic quadratic eigenvalue problem (TPQEP), the reciprocal relationship of the eigenvalues can be maintained. In this paper, we outline four recently developed structure-preserving algorithms, SA, SDA, TSHIRA and GTSHIRA, for solving the TPQEP. Numerical comparisons on the accuracy and the computational costs of these algorithm are presented. The eigenvalues close to unit circle on the complex plane are of interest in the area of filter and sensor designs. Our numerical results show that the Arnoldi-type structure-preserving algorithms TSHIRA and GTSHIRA with re-symplectic and re-bi-isotropic processes

In this paper, we show that there is a Cantor set of initial conditions in the planar 4‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the 4‑body case, and thus settles the last open case of the conjecture.

An irreducible integrable connection (E,∇) on a smooth projective complex variety X is called rigid if it gives rise to an isolated point of the corresponding moduli space M_{dR}(X). According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of X. In this article we study mod-p reductions of irreducible rigid connections and establish results which confirm Simpson’s prediction. In particular, for large p, we prove that p-curvatures of mod-p reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an F-isocrystalline realization for irreducible rigid flat connections. More precisely, we prove that there exist smooth models X_R and (E_R,∇_R) of X and (E,∇), over a finite-type ring R, such that for every Witt ring W(k) of a finite field k and every homomorphism R→W(k), the p-adic completion of the base change (\hat E_{W(k)}, \hat ∇_{W(k)}) on \hat X_{W(k)} represents an F-isocrystal. Subsequently, we show that irreducible rigid flat connections with vanishing p-curvatures are unitary. This allows us to prove new cases of the Grothendieck–Katz p-curvature conjecture. We also prove the existence of a complete companion correspondence for F-isocrystals stemming from irreducible cohomologically rigid connections.

Making use of the extended flux homomorphism defined in [13] on the group Symp§g of symplectomorphisms of a closed ori- ented surface §g of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non- trivial. We also prove that the second homology of the group Ham§g of Hamiltonian symplectomorphisms of §g, equipped with the discrete topology, is very large for all g ≥ 2.

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of h-projective transformations of a closed, connected Riemannian K¨ahler manifold consists of isometries unless the manifold is the complex projective space with the standard Fubini-Study metric (up to a constant).