We consider the problems of sufficiency of jets relative to a given closed set. In the non-relative case, criteria for $r$-jets to be $V$-sufficient and $C^0$-sufficient in $C^r$ mappings or $C^{r+1}$ mappings have been obtained. In particular, it is shown that $V$-sufficiency and $C^0$-sufficiency in $C^r$ functions or $C^{r+1}$ functions are equivalent. In this paper we discuss characterisations of $V$-sufficiency and $C^0$-sufficiency in the relative case, corresponding to the above non-relative results. Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative $r$-jets are $V$-sufficient in $C^r$ functions and $C^{r+1}$ functions but not $C^0$-sufficient in $C^r$ functions and $C^{r+1}$ functions, respectively. In addition, we give characterisations of relative finite $V$-determinacy and also relative finite $C^r$ contact determinacy.

In this paper, we give estimates of the solutions to Schrödinger equation on modulation spaces with vector potential of sub-linear growth.

We introduce and study generalized Bishop frames of regular curves, which are generalizations of the Frenet and Bishop frames for regular curves. There are four types B, C, D and F of generalized Bishop frames of regular curves in $\mathbb{E}^4$ up to the change of the order of vectors fixing the first one which is the tangent vector. We prove the following hierarchy among these four types of frames: Let $\gamma$ be a regular curve in $\mathbb{E}^4$. If $\gamma$ admits a frame of type F, which corresponds to the Frenet frame, then $\gamma$ admits a frame of type D. If $\gamma$ admits a frame of type D, then $\gamma$ admits a frame of type C. Since a frame of type B is a Bishop frame, by a result of Bishop, every regular curve in $\mathbb{E}^4$ admits such a frame. It follows that if $\gamma$ admits a frame of type D, then $\gamma$ clearly admits a frame of type B. We also construct examples of regular curves in $\mathbb{E}^4$ which show the hierarchy is strict.

We prove that on a compact contact manifold there are no infinitesimal contact automorphisms, apart from the scalar multiples of the Reeb vector field, provided the manifold carries an admissible Riemannian metric whose Jacobi operator is negative definite on the contact subbundle.

In this work, we prove a regularity criterion to a mathematical model in superfluidity in $\mathbb{R}^n$ for any $n\geq3$.

In the theory of potentials on Riemannian manifolds, Sario et al. [14] introduced the notion of $(p,q)$-biharmonic functions, i.e., $\Delta_q\Delta_pu = 0$, where $- \Delta_q = q - \Delta$ is a linear Schrödinger operator. They showed $(p,q)$-biharmonic classification of Riemannian manifolds and related facts. We show some analogous results by taking infinite networks for Riemannian manifolds. We obtain some new facts relating to the $(p,q)$-version of the growth of $\Delta_qu$ studied in [12]. Our $(p, q)$-biharmonic projection is analogous to that in [14].

We consider the Cauchy problems for systems of nonlinear wave and Klein-Gordon equations, or for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Global existence of solutions under the so-called null condition is already known for these systems. Our aim here is to obtain the global existence for these systems with some kinds of nonlinearities violating the null condition.

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under $\mathscr A$-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the cuspidal and the transverse double point curves and prove that the frontal has finite codimension if and only if both curves are reduced. Finally, we also discuss about the frontal versions of the Marar-Mond formulas and Mond's conjecture.