Let χ be a smooth proper Deligne–Mumford stack over ℂ. One can define twisted orbifold Gromov–Witten invariants of χ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps χg,n,d, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantum K-theory of a complex compact manifold X.

We show that for the second Painlevé equation y″ = 2y3 + ty + α, the Bäcklund transformation group G, which is isomorphic to the extended affine Weyl group of type Â1, operates regularly on the natural projectification χ(c)/ℂ(c, t) of the space of initial conditions, where c = α - 1/2. χ(c)/ℂ(c, t) has a natural model χ[c]/ℂ(t)[c]. The group G does not operate, however, regularly on χ[c]/ℂ(t)[c]. To have a family of projective surfaces over ℂ(t)[c] on which G operates regularly, we have to blow up the model χ[c] along the projective lines corresponding to the Riccati type solutions.

We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.

In this note we will make a few observations on the structure of fields and local rings. The main point is to show that a weaker version of Cohen structure theorem for complete local rings holds for any (not necessarily complete) local ring. The consideration of non-complete case makes the meaning of Cohen’s theorem itself clearer. Moreover, quasi-coefficient fields (or rings) are handy when we consider derivations of a local ring.

Eisenstein führte in seiner Abhandlung [1] einen durchsichtigen, analytischen Beweis des Reziprozitätsgesetzes der Potenzreste im rationalen und Gaussschen Zahlkörper. Zur kurzen Fassung seiner Methode im rationalen Fall, seien m und n positive ganzrationale Zahlen und sei

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In the previous paper [11], introducing the notions of potentials and of capacity associated with a convex function Φ given on a regular functional space we discussed potential theoretic properties of the gradient ∇Φ which were originally introduced and studied by Calvert [5] for a class of nonlinear monotone operators in Sobolev spaces. For example:

• (i) The modulus contraction operates.

• (ii) The principle of lower envelope holds.

• (iii) The domination principle holds.

• (iv) The contraction Tk onto the real interval [0, k] (k > 0) operates.

• (v) The strong principle of lower envelope holds.

• (vi) The complete maximum principle holds.

Let k(≧3) be a positive integer and φ(k) be the Euler function. We denote by % one of the φ(k) characters formed with modulus k, and by % the principal character. Let Lis, %) be the L-series corresponding to % Throughout the paper we use c and c(ε) to denote respectively an absolute positive constant and a positive constant depending on parameter ε( >0) alone, not necessarily the same at their various occurrences. We use the symbol y = O(Z) for positive X when there exists a c satisfying Y ≦ in the full domain of existence of X and Y.

The main purpose of this paper is to derive conclusions about the structure of a nilpotent group of algebra automorphisms and, in the case of a Lie algebra, about the influence of this nilpotence on the structure of the algebra. A motivation for this study is a well known theorem due to Kolchin: A unipotent linear group can be triangularized and is thus nilpotent. The converse is manifestly false, but we have (as an immediate consequence of Theorem 2.7):

A curve c2 of finite length L2 moves on a euclidean plane. Let the number of points of intersection of c2 with the fixed, curve C1 of length Ls1 be n, and the element of kinematic measure of the position of c2 be dK.

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

In this paper we shall give a definition of an analogue for Jordan algebras of the classical notion of a Cartan subalgebra of a Lie algebra. This is based on a notion of associator nilpotency of a Jordan algebra. A Jordan algebra is called associator nilpotent if there exists a positive (odd) integer M such that every associator of order M formed of elements of is 0 (§2).

We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.

For Hunt processes on Rd, strong and weak transience is defined by finiteness and infiniteness, respectively, of the expected last exit times from closed balls. Under some condition, which is satisfied by Lévy processes and Ornstein-Uhlenbeck type processes, this definition is expressed in terms of the transition probabilities. A criterion is given for strong and weak transience of Ornstein-Uhlenbeck type processes on Rd, using their Lévy measures and coefficient matrices of linear drift terms. An example is discussed.

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the Lp–Lq estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

We study the asymptotic behavior at low energy of scattering amplitudes in two dimensional magnetic fields with compact support. The obtained result depends on the total flux of magnetic fields. It should be noted that magnetic potentials do not necessarily fall off rapidly at infinity. The main body of argument is occupied by the resolvent analysis at low energy for magnetic Schrödinger operators with perturbations of lang-range class. We can show that the dimension of resonance spaces at zero energy does not exceed two. As a simple application, we also discuss the scattering by magnetic field with small support and the convergence to the scattering amplitude by δ-like magnetic field.

Let be the Cayley algebra of dimension 8 over the field R of real numbers and let be the set of all 3 × 3 Hermitian matrices

Suppose that Ω is a simply connected domain in the w-plane, w = u + iv, and that w∞ is an accessible boundary point of Ω located at w = ∞.

Bernstein’s theorem states that the following properties are equivalent for a function ψ:]0, ∞ [→ R (which then is called completely monotone):

moreover, the measure σ in iii) is uniquely determined.

In my previous paper “On the theory of semi-local rings,” we saw that if a semi-local ring R with maximal ideals p1…,ph is a subdirect sum of local rings Rm,2) then R is the direct sum of R[p1] (proposition 15, (slr)1)) and that a complete semi-local ring is a direct sum of complete local rings (Remark to proposition 5, (slr)).

Let X be a non-singular projective surface in (k an algebraically closed field of characteristic 0) and C an irreducible curve, which is a set-theoretically complete intersection in X; is it true that C is actually a complete intersection in X?