In his paper [2], [3], D. A. Hejhal investigated the variational theory of linear polynomic functions. In this paper we are concerned with the variational theory of higher-order differential equations. To be more precise, consider a compact Riemann surface having genus g > 1. As is well known, we can choose a projective coordinate covering U = (Ua, za). Fix this coordinate covering of X. We shall be concerned with linear ordinary differential operators of order n defined in each projective coordinate open set Ua

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers for some i > 1 and k ≥ 0, then for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if for some i > 1 and k ≥ 0, then for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that the graded Betti numbers for all i ≥ 1 if and only if I(k) and I(k+1) have a linear resolution. Here I(d) is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

Let Xℝ ⊂ ℝN a real analytic set such that its complexification Xℂ ⊂ ℂN is normal with an isolated singularity at 0. Let fℝ : Xℝ → ℝ a real analytic function such that its complexification fℂ : Xℂ → ℂ has an isolated singularity at 0 in Xℂ. Assuming an orientation given on to a connected component A of we associate a compact cycle Γ(A) in the Milnor fiber of fℂ which determines completely the poles of the meromorphic extension of or equivalently the asymptotics when T → ±∞ of the oscillating integrals . A topological construction of Γ(A) is given. This completes the results of [BM] paragraph 6.

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

Lusztig’s algorithm of computing generalized Green functions of reductive groups involves an ambiguity on certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$$(1/4<H\leqslant 1/2)$. Under Hörmander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.

L. Carleson ([6]) proved the following theorem:

Let u be a finite continuous function in the unit open ball B with center zero in the complex plane.

We denote by D the unit disc {z; |z| < 1} and by the totality of one to one conformal mappings z′ = s(z) of D onto itself. A meromorphic function f(z) in D is normal if and only if the family is a normal family in D in the sense of Montel. We denote by the totality of the normal meromorphic functions in D.

In the study of differential equations from the standpoint of the automorphism pseudogroups, the differential invariants of the pseudogroups play an important role.

A general study of pseudogroups and their differential invariants originated with Sophus Lie. He applied his study to the classification of ordinary and partial differential equations. So as to study differential equations from his point of view, it is very important to write the given differential equation by the differential invariants of the automorphism pseudogroup. That is to say, the geometric structure of a differential equation is contained in the expression of the equation by its differential invariants.

Let M be a compact, oriented Riemannian manifold of dimension dr and let Γ be the fundamental group of M. For a finite dimensional representation ρ of Γ on a vector space F, Ray and Singer [10] have defined the analytic torsion T(M, ρ) as follows: We denote by E the vector bundle over M with typical fibre F defined by the representation ρ.

In this paper we give certain linear relations between the Fourier coefficients of Siegel modular forms that are obtained from Ikeda lifts.

The group of extensions Ext(A, Ga) and Ext(A, Gm) of an abelian variety A by the additive or multiplicative group Ga, Gm have been investigated in detail ([9], [10], [GACC]). On the other hand, F. Oort [8] and M. Miyanishi [6] recently studied the groups Ext(Ga, A) and Ext(Gm, A). The purpose of the present paper is to clarify the relationship between these groups. Our results show that the latter groups can be derived from the former.

Let β > 0 and , define and where a2n denotes Mhaskar-Rahmanov-Staff number for Wβ. Let be the leading coefficients of the nth orthonormal polynomial corresponding to Wβ,cn and write . It is shown that if c > 1 and β is a positive even integer then has an asymptotic expansion. Also when 0 < c < 1, asymptotic expansions of recursion coefficients of the truncated Hermite weights are given.

Let L = K be the composite of two imaginary quadratic fields and K. Suppose that the discriminants of and K are relatively prime. For any primitive ray class character χ of L, we shall compute L(1, χ) for the Hecke L-function in L. We write for the conductor of χ and C for the ray class modulo . Let c ε C be any integral ideal prime to . We write as g-module where g, n and ϑL are, respectively, the ring of integers in k, an ideal in k and the differente of L. Let where T(χ) is the Gaussian sum and, as in (3.2),

The boundary behavior of solutions of the heat equation (temperature functions) is investigated. It is proved that a temperature function is identically equal to zero if it vanishes of finite order at some lateral boundary point where it attains a local minimum.

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.