## Recursos de colección

1. #### On J-orders of elements of $KO(CP^{m})$

Let $KO(CP^{m})$ be the $KO$ -ring of the complex projective space $CP^{m}$ . By means of methods of rational $D$ -series [4], a formula for the $J$ -orders of elements of $KO(CP^{m})$ is given. Explicit formulas are given for computing the $J$ -orders of the canonical generators of $KO(CP^{m})$ and the $J$ -order of any complex line bundle over $CP^{m}$ .

2. #### Invariance of Hochschild cohomology algebras under stable equivalences of Morita type

POGORZAŁY, Zygmunt
There is proved that the Hochschild cohomology algebras of finitedimensional self-injective $K$ -algebras over a field $K$ are invariants of stable equivalences of Morita type.

3. #### Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions

KUBO, Hideo; KUBOTA, Kôji
We give a new a priori estimate for a classical solution of the inhomogeneous wave equation in $R^{n}\times R$ , where $n=2,3$ . As an application of the estimate, we study the asymptotic behavior as $t\rightarrow\pm\infty$ of solutions $u(x,t)$ and $v(x,t)$ to a system of semilinear wave equations: $\partial_{t}^{2}u-\Delta u=|v|^{p}$ , $\partial_{t}^{2}v-\Delta v=|u|^{q}$ in $R^{n}\times R$ , where $(n+1)/(n$ -1 $)$ $0$ , the Cauchy problem for the system has a global solution...

4. #### Half twists of Hodge structures of CM-type

van GEEMEN, Bert
To a Hodge structure $V$ of weight $k$ with CM by a field $K$ we associate Hodge structures $V_{-n/2}$ of weight $k+n$ for $n$ positive and, under certain circumstances, also for $n$ negative. We show that these 'half twists' come up naturally in the KugaSatake varieties of weight two Hodge structures with CM by an imaginary quadratic field.

5. #### A method of energy estimates in $L^{\infty}$ and its application to porous medium equations

ÔTANI, Mitsuharu; SUGIYAMA, Yoshie
The existence of time local $C^{\infty}$ -solutions is shown for Cauchy problem of the porous medium equations. Our arguments rely on the " $L^{\infty}$ -energy method" developed in our previous paper [16] and a new method based on the theory of evolution equations in the $L^{2}$ -framework which enables us to handle with perturbations which can be decomposed into monotone parts and small parts in Sobolev spaces of higher order.

6. #### Correlated sums of $r(n)$

CHAMIZO, Fernando
We prove an asymptotic formula for $\displaystyle \sum_{n\leq N}r(n)r(n+m)$ using the spectral theory of automorphic forms and we specially study the uniformity of the error term in the asymptotic approximation when $m$ varies. The best results are obtained under a natural conjecture about the size of a certain spectral mean of the Maass forms. We also employ large sieve type inequalities for Fourier coefficients of cusp forms to estimate some averages (over $m$ ) of the error term.

7. #### On commutators of foliation preserving homeomorphisms

FUKUI, Kazuhiko; IMANISHI, Hideki
We consider the group of foliation preserving homeomorphisms of afoliated manifold. We compute the first homologies of the groups for codimension one foliations. Especially, we show that the group for the Reeb foliation on the 3-sphere is perfect and the groups for irrational linear foliations on the torus are not perfect.

8. #### Conformally flat 3-manifolds with constant scalar curvature

CHENG, Qing-Ming; ISHIKAWA, Susumu; SHIOHAMA, Katsuhiro
We classify complete conformally flat three dimensional Riemannian manifolds with constant scalar curvature and constant squared norm of Ricci curvature tensor by applying the Generalized Maximum Principle due to H. Omori.

9. #### On the $L_{q}-L_{r}$ estimates of the Stokes semigroup in a two dimensional exterior domain

DAN, Wakako; SHIBATA, Yoshihiro
We proved $L_{q}-L_{r}$ type estimates of the Stokes semigroup in a two dimensional exterior domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

10. #### Boundary behavior of positive solutions of $\Delta u=Pu$ on a Riemann surface

SATŌ, Takeyoshi
The classical Fatou limit theorem was extended to the case of positive harmonic functions on a hyperbolic Riemann surface $R$ by Constantinescu-Cornea. They used extensively the notions of Martin's boundary and fine limit following the filter generated by the base of the subsets of $R$ whose complements are closed and thin at a minimal boundary point of $R$ . We shall consider such a problem for positive solutions of the Schrödinger equation on a hyperbolic Riemann surface.

11. #### On Eisenstein series on quaternion unitary groups of degree 2

HIRAI, Yoshikazu
In this paper we give an explicit Fourier expansion of the Eisenstein series on certain quaternion unitary groups of degree 2 by means of Shimura's method. Moreover using an explicit formula of the Fourier coefficients of holomorphic Eisenstein series and Oda's lifted cusp forms, we give some numerical examples.

12. #### On the Seifert form at infinity associated with polynomial maps

NÉMETHI, András
If apolynomial map $f$ : $C^{n}\rightarrow C$ has anice behaviour at infinity (e.g. it is a "good polynomial"), then the Milnor fibration at infinity exists; in particular, one can define the Seifert form at infinity $\Gamma(f)$ associated with $f$ . In this paper we prove a Sebastiani-Thom type formula. Namely, if $f$ : $C^{n}\rightarrow C$ and $g:c^{m}\rightarrow C$ are "good" polynomials, and we define $h=f$ $\oplus$ $g$ : $C^{n+m}\rightarrow C$ by $h(x,y)=f(x)+g(y)$ , then $\Gamma(h)=(-\mathrm{I})^{mn}\Gamma(f)$ $\otimes\Gamma(g)$ . This is the global analogue of the local result, proved independently by K. Sakamoto and P. Deligne for isolated hypersurface singularities.

13. #### Galois covering singularities I

TSUCHIHASHI, Hiroyasu
We give a necessary condition for Galois covering singularities to be logterminal or $\log$ -canonical singularities, which is also sufficient under acertain restriction on the branch loci of the covering maps. We also give a method constructing explicitly resolutions of 2-dimensional Abel covering singularities.

14. #### Invariants for representations of Weyl groups and two-sided cells

GYOJA, Akihiko; NISHIYAMA, Kyo; SHIMURA, Hiroyuki
The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played an important role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc arguments. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set $\mathscr{M}(\mathscr{G})$ of each two-sided cell.

15. #### Ineffability of $\mathscr{P}_\kappa \lambda$ for $\lambda$ with small cofinality

USUBA, Toshimichi
We study ineffability, the Shelah property, and indescribability of $\mathscr{P}_\kappa \lambda$ when ${\rm cf}(\lambda)\kappa$ . We prove that if $\lambda$ is a strong limit cardinal with ${\rm cf}(\lambda)\kappa$ then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over $\mathscr{P}_\kappa \lambda$ are the same, and that it can be precipitous. Furthermore we show that $\Pi^1_1$ -indescribability of $\mathscr{P}_\kappa \lambda$ is much stronger than ineffability if $2^\lambda=\lambda^{\kappa}$ .

16. #### Asymptotically holomorphic embeddings of presymplectic manifolds

MORIYAMA, Takayuki
We apply Donaldson-Auroux's asymptotically holomorphic methods to construct asymptotically holomorphic embeddings of presymplectic closed manifolds of constant rank with integral form into Grassmannians ${\rm Gr}(r,N)$ . In particular, we obtain asymptotically holomorphic embeddings into the projective spaces $\bm{C}{\rm P}^{N-1}$ such that the pull-back of the Fubini-Study form is cohomologous to ${k\omega}/{2\pi}$ for large integers $k$ . Moreover, we can construct asymptotically holomorphic immersions along the symplectic distribution of presymplectic manifolds into the projective spaces.

17. #### Continued fractions and certain real quadratic fields of minimal type

KAWAMOTO, Fuminori; TOMITA, Koshi
The main purpose of this article is to introduce the notion of real quadratic fields of minimal type in terms of continued fractions with period $\ell$ . We show that fundamental units of real quadratic fields that are not of minimal type are relatively small. So, we see by a theorem of Siegel that such fields have relatively large class numbers. Also, we show that there exist exactly $51$ real quadratic fields of class number $1$ that are not of minimal type, with one more possible exception. All such fields are listed in the table of Section 8.2. Therefore we...

18. #### The Log-effect for p-evolution type models

CICOGNANI, Massimo; HIROSAWA, Fumihiko; REISSIG, Michael
The goal of the paper is to study the Log-effect for special $p$ -evolution type models. The loss of regularity is related in an optimal way due to some unboundedness conditions for the derivatives of coefficients up to the second order with respect to $t$ . Some counter-examples show that these conditions are sharp. We present the state of art of methods to construct such counter-examples.

19. #### On nonseparable Erdös spaces

DIJKSTRA, Jan J.; VAN MILL, Jan; VALKENBURG, Kirsten I. S.
In 2005, Dijkstra studied subspaces $\mathscr{E}$ of the Banach spaces $\ell^p$ that are constructed as `products' of countably many zero-dimensional subsets of $\bm{R}$ , as a generalization of Erdös space and complete Erdös space. He presented a criterion for deciding whether a space of the type $\mathscr{E}$ has the same peculiar features as Erdös space, which is one-dimensional yet totally disconnected and has a one-dimensional square. In this paper, we extend the construction to a nonseparable setting and consider spaces $\mathscr{E}_\mu$ corresponding to products of $\mu$ zero-dimensional subsets of $\bm{R}$ in nonseparable Banach spaces. We are able to generalize both...

20. #### Homotopy groups of the spaces of self-maps of Lie groups

MARUYAMA, Ken-ichi; ŌSHIMA, Hideaki
We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, $\SU(3)$ , $\Sp(2)$ , and $G_2$ . We use the cell structures of these Lie groups and the standard methods of homotopy theory.

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