We compute Stokes matrices for generalised Airy equations and prove that they are regular unipotent (up to multiplication with the formal monodromy). This class of differential equations was defined by Katz and includes the classical Airy equation. In addition, it includes differential equations which are not rigid. Our approach is based on the topological computation of Stokes matrices of the enhanced Fourier--Sato transform of a perverse sheaf due to D'Agnolo, Hien, Morando and Sabbah.

Let $p$ be a prime number, $W$ an absolutely unramified $p$-adically complete discrete valuation ring with perfect residue field, and $X$ a curve over the field of fractions of $W$ of genus greater than one. In the present paper, we study the ramification of torsion points on the curve $X$. A consequence of the main result of the present paper is nonexistence of ramified torsion point on $X$ in the case where $p$ is greater than three, the Jacobian variety $J$ of $X$ has good reduction over $W$, and the special fiber of the good model of $J$ is superspecial. This consequence generalizes a theorem proved by Coleman.

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ for the infinite family of graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others.

The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form $f(n)=p f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the number of odd elements in the set of $s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler measure of the associated Laurent polynomial.

Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix-valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian from the solution of a first-order system attached to a given Hamiltonian, consisting of ordinary differential equations parametrized by a set of complex numbers, under certain conditions for the solutions. This inverse problem is a generalization of the inverse problem for two-dimensional canonical systems.

We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.

We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah.The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $t\to\infty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.

The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field.

As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field.

In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.