We find the asymptotic behavior for large time of solutions to the dispersive equations of Schrödinger type $u_t-\frac{i}{\rho }|{\partial }_x{|}^{\rho }u=0,(t,x)\in R\times R,$where $\rho \ge 2$. We obtain some estimates of solutions of linear problem and apply them to nonlinear problems with power nonlinearities of order $p\ge 3$. The nonexistence of wave operator and existence of the modified wave operator for the critical nonlinearity $i\lambda |u{|}^{2}u$ are studied.

In this paper we construct explicitly a special Lagrangian fibration in the Taub-NUT space. The Taub-NUT space is a complex 2-fold with a Ricci-flat metric and it is well known to physicists. For this space, we find $S^1$-invariant special Lagrangian submanifolds by using moment map techniques and show that a family of special Lagrangian submanifolds give a fibration of the Taub-NUT space. We also study a topology of special Lagrangian fibers using explicit description of special Lagrangians.

The aim of this work is to show the existence of the wave operator and its inverse among Kirchhoff equations and free wave equations.

We prove that if $\nu$ and $\mu$ are arbitrary (signed) Borel measures (on the unit circle) such that $M_{+}\nu (x)=M_{+}\mu (x)$ for each $x$, where $M_{+}$ is the one-sided maximal operator (without modulus in the definition), then $\nu =\mu$. The proof is constructive and it shows how $\nu$ can be recovered from $M_{+}\nu$ in the unique way.

We introduce the skein index, which is an integer-valued index of a link invariant. It is used to compare link invariants and to find a skein relation. We give the complete list for link invariants of skein index less than or equal to two, and discuss the skein indices of operator invariants, and in particular quantum invariants and quandle coloring numbers. By determining the skein index of the JKSS invariant, we construct a new skein relation and give a formula for the invariant of a closed 2-braid.

We prove that a locally compact paracompact space is suborderable if and only if it has a continuous weak selection. This fits naturally into the pattern of the van Mill and Wattel's characterization [15] of compact orderable spaces, and provides a further partial positive answer to a question of theirs. Several applications about the orderability and suborderablity of locally compact spaces are demonstrated. In particular, we show that a locally compact paracompact space has a continuous selection for its Vietoris hyperspace of nonempty closed subsets if and only if it is a topologically well-orderable subspace of some orderable space.

We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, $\text{SU}(3)$, $\text{Sp}(2)$, and $G_2$. We use the cell structures of these Lie groups and the standard methods of homotopy theory.

In 2005, Dijkstra studied subspaces $E$ of the Banach spaces ${\ell }^p$ that are constructed as `products' of countably many zero-dimensional subsets of $R$, as a generalization of Erdös space and complete Erdös space. He presented a criterion for deciding whether a space of the type $E$ 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 $E_{\mu }$ corresponding to products of $\mu$ zero-dimensional subsets of $R$ in nonseparable Banach spaces. We are able to generalize both Dijkstra's criterion and his classification of closed variants of $E$. We can further generalize the latter to complete spaces and we find that a one-dimensional complete space $E_{\mu }$ is homeomorphic to a product of complete Erdös space with a countable product of discrete spaces. Among the applications, we find coincidence of the small and large inductive dimension for $E_{\mu }$.

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.

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 study real quadratic fields with period $\ell$ of minimal type in order to find real quadratic fields of class number $1$, and first examine the case where $\ell \le 4$. In particular we obtain a result on Yokoi invariants $m_d$ and class numbers $h_d$ of real quadratic fields $Q(\sqrt{d})$ with period $4$ of minimal type.

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 $\text{Gr}(r,N)$. In particular, we obtain asymptotically holomorphic embeddings into the projective spaces $C{\text{P}}^{N-1}$ 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.

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