We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let M be an infra-nilmanifold and ƒ: M → M be a continuous map. Suppose MK is a regular covering of M which is a compact nilmanifold with π1(MK = K. Assume that f*(K) ⊂ K. Then ƒ has a lifting . We prove a question raised by McCord, which is for any with an essential fixed point class, fix =1. As a consequence, we obtain the following averaging formula for Nielsen numbers

In the theory of functions meromorphic in |z| < + ∞, Iversen [4] proved the following: If w = f(z) is meromorphic in |z| < + ∞ and has an essential singularity at z = ∞, then any inverse function-element of this function with the centre w0 can be continued analytically to any point w ≠ wo, except possibly this point w, in any disc having the centre at the point w and containing the point w0.

We study Toeplitz operators on the harmonic Bergman spaces on bounded smooth domains. Two classes of symbols are considered; one is the class of positive symbols and the other is the class of uniformly continuous symbols. For positive symbols, boundedness, compactness, and membership in the Schatten classes are characterized. For uniformly continuous symbols, the essential spectra are described.

Let f be a Cr-diffeomorphism, r ≦ 1, on a compact differentiable manifold M with dim M ≧ 2. In [9] F. Takens introduced the concept of extended f-orbits and conjectured the following.

In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with S1-action.

Au 19e siècle le terme de géométrie algébrique désignait exclusivement la géométrie birationnelle. Aujourd’hui on n’accepterait pas cette définition, parce que l’objet de la géométrie algébrique est celui des invariants biréguliers plutôt que birationnels. Quelle que soit la définition adoptée, il est intéressant de connaître la structure de groupes des automorphismes birationnels. Mais celle-ci est très peu connue (Mumford [10]). La question suivante a été posée depuis longtemps; déterminer à conjugaison près tous les sous-groupes algébriques connexes et maximaux du groupe de Cremona Crn. Le groupe Crn est, par définition, le groupe des automorphismes du corps des fonctions rationnelles à n indéterminées. Si n = 1. Cr1 coïncide avec le groupe des automorphismes de P1 Enriques a résolu ce problème pour n = 2 (Enriques [5]). On croit généralement (par exemple Godeaux [8]) que Enriques et Fano l’ont fait pour n = 3 (Enriques et Fano [6]). Après avoir collaboré avec Enriques, G. Fano a travaillé seul sur ce problème et il a laissé des résultats très intéressants dont la plupart des démonstrations ne semble pas rigoureuse. Demazure [4] a étudie les; sous-groupes algébriques de rang maximum n de Crn. Il a démontré qu’il y a une correspondence bijective entre les sous-groupes algébriques de rang maximum n de Crn et les systèmes d’Enriques. Puisque les derniers sont de nature combinatoire, c’est un dictionaire géométrique-combinatoire.

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

In [5] we defined the notion of separable extensions of a ring as a generalization of that of separable algebras. In this paper we try to introduce the notion of ‘central’ separable algebras to separable extensions. Such extensions are viewed in §2. But we do not know that it is suitable or not to call them central separable extensions. As the Morita theorem for projective generators is fundamental to examine the central separable algebras we generalize it in §1. Let R be a ring and let A and B be left R-modules. We assume that R-module B is isomorphic to a direct summand of a finite direct sum of copies isomorphic to A.

In this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum (sln, ∧Vn) link invariant, where ∧Vn is the set of fundamental representations of Uq(sln). In the case of an oriented link diagram composed of [k, 1]-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general [i,j]-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

In this paper we consider the periodic problems for certain nonlinear parabolic differential equations in domains with periodically moving boundaries. The typical problem, which is going to be discussed in the present paper, is to solve the following:

Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t є R} with stationary increments which has a self-similar property, that is, there exists a constant H (for the Brownian motion H = 1/2, in general 0 < H < 1 for Gaussian processes) called the exponent of self-similarity of the process, such that, for any c > 0, two processes are subject to the same law (see [10]).

The class of normal functionals

is, as is well known, adapted to the domain of Lévy’s Laplacian and plays important roles in the works by P. Lévy and T. Hida (cf. [1], [2] and [8]), where Bx denotes one-dimensional parameter white noise and denotes the renormalization of

Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕n≥0I(n)tn is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).

Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.

We study the behavior of Hilbert-Kunz multiplicity for powers of an ideal, especially the case of stable ideals and ideals in local rings of dimension 2. We can characterize regular local rings by certain equality between Hilbert-Kunz multiplicity and usual multiplicity.

We show that rings with “minimal” Hilbert-Kunz multiplicity relative to usual multiplicity are “Veronese subrings” in dimension 2.

In this paper, we shall give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. Let D be a division quaternion algebra over Q.

It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of Berndtsson-Andersson type.

Dans l’espace de Dirichlet il existe des mesures des condensateurs et inversement un certain espace de fonctions est léespace de Dirichlet séil y existe des mesures des condensateurs. Dans ce mémoire nous considérons le problème d’existence des mesures des condensateurs par rapport ä un noyau positif et continu au sens large, symétrique ou non.

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

Consider a Riemann surface X equipped with a projective structure, that is, a covering of X with coordinate neighborhoods U and corresponding (holomorphic) local coordinates {t} such that in the intersection U ∩ U′ of any two such coordinate neighborhoods U and U′ change of local coordinates is mediated by a fractional linear transformation