The Grothendieck ring of a finite group has been studied by Swan ([5], [6]). At the end of [6] he determined completely the structure of the Grothendieck ring G(Z) of a cyclic p-group over the ring of rational integers Z.

Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima proved

THEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such that

F(g(T1,…,Tm), T1,…,T)= 0.

Given a semilocal 1-dimensional Cohen-Macauly ring A, J. Lipman in [10] gives an algorithm to obtain the integral closure Ā of A, in terms of prime ideals of A. More precisely, he shows that there exists a sequence of rings A = A0 ⊂ A1 ⊂… ⊂ Ai ⊂…, where, for each i, i ≥ 0, Ai+1 is the ring obtained from Ai by “blowing-up” the Jacobson radical ℛ i of Ai+ i.e. Ai+l = ∪n(ℛin:ℛin). It turns out that ∪ {Ai;i≥0} = Ā (cf. [10, proof of Theorem 4.6]) and, if Ā is a finitely generated A-module, the sequence {Ai; i ≥ 0} is stationary for some m and Am = Ā, so that

This paper is concerned with the problem of the determination of unknown scalars involved in the algorithm of computing the generalized Green functions of reductive groups G over a finite field. In the previous paper, we have treated the case where G = SLn. In this paper, we determine the scalars in the case where G is a classical group Sp2n or SON for arbitrary characteristic.

Let l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

We compute the global log-canonical thresholds (lct) of del Pezzo surfaces of degrees ≥ 2 with du Val singularities.

We consider a problem whether Kodaira’s -Lemma holds on weakly 1-complete Kähler manifolds or not. This problem was proposed by S. Nakano. We prove that the Lemma holds for some class of complex quasi-tori ℂn/Γ, and it does not hold for the other class of them. Every complex quasi-tori is weakly 1-complete and complete Kähler. Then we get a negative answer for the above problem.

Let R be an orientable, compact Riemannian space with the metric ds2=gij(x)dxidxj and consider the backward diffusion equation

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

In [5, 6, 7] I introduced the concept of Einstein-Hermitian vector bundle. Let E be a holomorphic vector bundle of rank r over a complex manifold M. An Hermitian structure h in E can be expressed, in terms of a local holomorphic frame field s1, …, sr of E, by a positive-definite Hermitian matrix function (hij) defined by

Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.

For a certain class of hypergeometric functions $_{3}F_{2}$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.

Many of the properties of analytic functions can be proved in a purely topological manner, so that such properties are then valid for the larger class of functions which are topologically equivalent to analytic functions. The importance of such functions has been recognized fairly recently, particularly in the theory of partial differential equations, where certain solutions have been shown to possess the topological properties of analytic functions, i.e., interiority and continuity, but not necessarily the analytic properties of complex differentiability and integrability.

We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.

MM. Ozeki et Hano ont récemment démontré [3] que tout sous-groupe connexe de Lie du groupe linéaire GL(n, R), n≧2, peut être réalisé comme groupe d’holonomie d’une certaine connexion linéaire dans un espace affine de dimension n.

Let (X, O) be a germ of a normal surface singularity, π: → X be the minimal resolution of singularities and let A = (ai,j) be the n × n symmetrical intersection matrix of the exceptional set of In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme , and defines a map from the set of irreducible components of to the set of exceptional components of the minimal resolution of singularities of (X,O). He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition

• • For any couple (Ei,Ej) of distinct exceptional components, we define Numerical Nash condition (NN(i,j)). We have that (NN(i,j)) implies In this paper we prove that (NN(i,j)) is always true for at least the half of couples (i,j).

• • The condition (NN(i,j)) is true for all couples (i,j) with i ≠ j, characterizes a certain class of negative definite matrices, that we call Nash matrices. If A is a Nash matrix then the Nash map N is bijective. In particular our results depend only on A and not on the topological type of the exceptional set.

• • We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.

• • We give infinitely many other classes of singularities where Nash Conjecture is true.

The proofs are based on my old work [8] and in Plenat [10].

In this paper we treat the problem of determining the rate of growth of entire functions which are solutions of first order algebraic differential equations whose coefficients are arbitrary entire functions (i.e. equations of the form Ω(z, y, dy/dz) = 0, where Ω(z, y, dy/dz) = is a polynomial in y and dy/dz, whose coefficients fkJ(z) are entire functions).

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

Let A well known theorem of Hardy and Ramanujan states: v(n) = (1 = 0(1))log log n holds for all n if we neglect a sequence of density 0 [5]. Define for 2 ≤j≤v(n)