We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of [8] and [10], it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then $T^*G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.

In this paper we introduce a general notion of a symmetric cone, valid for the finite and infinite dimensional case, and prove that one can deduce the seminegative curvature of the Thompson part metric in this general setting, along with standard inequalities familiar from operator theory. As a special case, we prove that every symmetric cone from a JB-algebra satisfies a certain convexity property for the Thompson part metric: the distance function between points evolving in time on two geodesics is a convex function. This provides an affirmative answer to a question of Neeb [22].

In this paper, we prove the following. Let $(R, \mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with multiplicity $e$ and embedding dimension $v=e+d-k$, where $k \geq 3$ and $e-k>1$. If $\lambda(\mathfrak{m}^3/J\mathfrak{m}^2)=1$ and $\mathfrak{m}^3\subseteq J\mathfrak{m}$, where $J$ is a minimal reduction of $\mathfrak{m}$, then $3 \leq s \leq \tau +k-1$, where $s$ is the degree of the $h$-polynomial of $R$ and $\tau$ is the Cohen-Macaulay type of $R$.

Existence and $\mathit{SO}(3) \times \mathit{SO}(3)$-congruence of Lagrangian immersion from oriented 2-dimensional Riemannian manifold to the Riemannian product of 2-spheres are studied. In particular, we will show that two minimal Lagrangian immersions are $\mathit{SO}(3) \times \mathit{SO}(3)$-congruent if and only if the corresponding angle functions are coincide.

Generalized eigenfunctions of the odd-dimensional ($n \geq 3$) relativistic Schrödinger operator $\sqrt{-\Delta}+V(x)$ with $|V(x)| \leq C\langle x\rangle^{-\sigma}$, $\sigma>1$, are considered. We compute the integral kernels of the boundary values $R^{\pm}(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, and prove that the generalized eigenfunctions $\varphi^{\pm}(x,k):=\varphi_0(x,k)-R^{\mp}(|k|)V\varphi_0(x,k)$ ($\varphi_0(x,k):=e^{ix \cdot k}$) are bounded for $(x,k)\in\mathbb{R}^n\times\{k\mid a\leq |k|\leq b\}$, where $[a,b]\subset(0,\infty)\setminus\sigma_p(H)$. This fact, together with the completeness of the wave operators, enables us to obtain the eigenfunction expansion for the absolutely continuous spectrum.

On considère les fonctions propres généralisées de l'opérateur relativiste de Schrödinger $\sqrt{-\Delta}+V(x)$ où $|V(x)| \leq C\langle x\rangle^{-\sigma}$ en dimension impaire ($n \geq 3$). On calcule les noyaux intégraux associés aux valeurs limites $R^{\pm}(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, et on prouve que les fonctions propres généralisées $\varphi^{\pm}(x,k):=\varphi_0(x,k)-R^{\mp}(|k|)V\varphi_0(x,k)$ ($\varphi_0(x,k):=e^{ix\cdot k}$) sont bornées pour $(x,k)\in\mathbb{R}^n\times\{k\mid a\leq |k|\leq b\}$, où $[a,b]\subset(0,\infty)\setminus\sigma_p(H)$. Ce résultat, associé à la complétude des opérateurs d'onde, nous permet d'obtenir le développement en fonction propres pour le spectre absolument continu.

In this paper, we give an example of a finitely generated 3-dimensional $\mathbb{C}$-algebra which has infinitely generated Derksen invariant as well as Makar-Limaonv invariant.

By a Frobenius system on a finite group $G$, we mean the data, for each maximal solvable subgroup $M$ of $G$, of a normal subgroup $\mathcal{F}(M)$ of $M$, satisfying some of the properties of a Frobenius kernel, and subject to certain additional conditions. We prove that a finite group with a Frobenius system is either solvable (in which case we get a complete description), or isomorphic to $\mathit{SL}_{2}(K)$ (for $K$ a finite field of characteristic 2) or to a Suzuki group. The respective possibilities for the mapping $\mathcal{F}$ are then determined. This extends a previous result of ours (Nagoya Math. J. 165 (2002), 117--121) by removing the condition that each $M/\mathcal{F}(M)$ be abelian. Curiously enough, the Feit-Thompson Theorem is used in the proof.

We discuss some estimates of the tail distributions of the supremum and the quadratic variation of a local martingale. The assumption made so far in the literature on exponential moments involving a quasi left continuous local martingale is improved.

Our purpose in this paper is to prove existence of wave operator $\mathcal{W}_{+}$ or inverse wave operator $\widetilde{\mathcal{W}}_{+}$ for nonlinear Schrödinger equations with quadratic nonlinearities in three space dimensions. Our results show that the mapping $\mathcal{W}_{+} \widetilde{\mathcal{W}}_{+}$ is well defined and are improvement of results on the range of inverse wave operator obtained in [6].

We study large time behavior of the solutions to the initial value problem for the generalized Burgers equation. It is known that the solution tends to a self-similar solution to the Burgers equation at the rate $t^{-1} \log t$ in $L^{\infty}$ as $t \to \infty$. The aim of this paper is to show that the rate is optimal under suitable assumptions and to obtain the second asymptotic profile of large time behavior of the solutions.

Let $A$ be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over $A$ whose orbit closures are regular varieties.

In this article we prove that every entire curve in the complement of a generic hypersurface of degree $d\geq 586$ in $\mathbb{P}_{\mathbb{C}}^{3}$ is algebraically degenerated, i.e. there exists a proper subvariety which contains the entire curve.

The main aim of this paper is to compute the character table of $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$, where $\sigma$ is the graph automorphism of $\mathrm{G}_{2}(3^{2n+1})$ such that the fixed-point subgroup $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ is the Ree group of type $\mathrm{G}_{2}$. As a consequence we explicitly construct a perfect isometry between the principal $p$-blocks of $\mathrm{G}_{2}(3^{2n+1})^{\sigma}$ and $\mathrm{G}_{2}(3^{2n+1})\rtimes\langle \sigma \rangle$ for prime numbers dividing $q^2-q+1$.