We prove that there is no smooth irreducible reduced rational curve of degree $e$, $2 \le e \le 11$, on general hypersurfaces of degree $7$ in $\mathbb{P}^5$.

A double torus knot $K$ is a knot embedded in a Heegaard surface $H$ of genus 2, and $K$ is non-separating if $H \setminus K$ is connected. In this paper, we determine the genus of a non-separating double torus knot that is a band-connected sum of two torus knots. We build a bridge between an algebraic condition and a geometric requirement (Theorem 5.5), and prove that such a knot is fibred if (and only if) its Alexander polynomial is monic, i.e. the leading coefficient is $\pm 1$. We actually construct fibre surfaces, using T. Kobayashi's geometric characterization of a fibred knot in our family. Separating double torus knots are also discussed in the last section.

In this note we shall give a description of the $K$-ring of a quasi-toric manifolds in terms of generators and relations. We apply our results to describe the $K$-ring of Bott-Samelson varieties.

Every locally trivial action of the additive group of complex numbers on a factorial affine variety has finitely generated ring of invariants. A criterion is given for such an action on complex four space to be conjugate to a translation. Restrictions on the nature of the singularities of the variety defined by the ring of invariants of triangular actions are noted.

We study the asymptotic behavior of solutions to the viscous Burgers equation by presenting a new asymptotic approximate solution. This approximate solution, called a diffusion wave approximate solution to the viscous Burgers equation of $k$-th order, is expanded in terms of the initial moments up to $k$-th order. Moreover, the spatial and time shifts are introduced into the leading order term to capture precisely the effect of the initial data on the long-time behavior of the actual solution. We also show the optimal convergence order in $L^p$-norm, $1\leq p\leq \infty$, of the diffusion wave approximate solution of $k$-th order. These results allow us to obtain the convergence of any higher order in $L^p$-norm by taking such a diffusion wave approximate solution with order $k$ large enough.

We consider the Cauchy problem for the operator $D_t^2-D_xa(t,x)D_x$ in the Gevrey classes. We show that if the coefficient $a(t,x)$ is given by a finite sum of non negative functions then the Cauchy problem is well posed in the wider Gevrey class for the larger powers. We also give an example showing that the order of the Gevrey class obtained here is optimal.

In this article we give an explicit construction of versal Galois coverings for any given finite subgroup of $\mathit{GL}(n,\mathbb{Z})$. By this construction we give a positive answer to Question 1.4 in [5].

The first step in the fundamental Clifford Theoretic Approach to General Block Theory of Finite Groups reduces to: $H$ is a subgroup of the finite group $G$ and $b$ is a block of $H$ such that $b({}^{g} b)=0$ for all $g\in G-H$. We extend basic results of several authors in this situation and place these results into current categorical and character theoretic equivalences frameworks.

Non-local elliptic problem, $-\Delta v = \lambda \bigl(e^{v}\big/\bigl(\int_{\Omega}e^{v} dx \bigr)^{p}\bigr)$ with Dirichlet boundary condition is considered on $n$-dimensional bounded domain $\Omega$ with $n \geq 3$ for $p>0$. If $\Omega$ is the unit ball, $3 \leq n \leq 9$ and $2/n \leq p \leq 1$, we have infinitely many bendings in $\lambda$ of the solution set in $\lambda-v$ plane. Finally if $\Omega$ is an annulus domain and $p \geq 1$, we show that a solution exists for all $\lambda>0$.

Every stationary subgroup of the quasiconformal mapping class group of a Riemann surface acts on the Teichmüller space discontinuously if the surface satisfies a certain geometric condition. In this paper, we construct such a Riemann surface that the quasiconformal mapping class group is non-stationary but it still acts on the Teichmüller space discontinuously.

Shinji Miura gave certain multivariable polynomials that express an Affine curve for a given algebraic function field $F$ and its degree one place $\mathcal{O}$, if $F$ contains such an $\mathcal{O}$. Suppose the equations contain $t$ ($\geq 2$) variables, and that the pole orders at $\mathcal{O}$ are $a_1,\ldots,a_t \geq 1$, where $\operatorname{GCD}\{a_1,\ldots,a_t\}=1$. If \[ \frac{a_i}{d_i} \in \frac{a_1}{d_{i-1}}{\mathbb N}+ \cdots +\frac{a_{i-1}}{d_{i-1}}\mathbb{N}, \quad d_i=\operatorname{GCD}\{a_1,\ldots,a_{i}\} \] for each $i=2,\ldots,t$, by arranging $a_1,\ldots,a_{t}$, then we say that the orders $a_1,\ldots,a_{t}$ are telescopic. On the other hand, the number $t'$ ($\geq t-1$) of the equations in the Miura canonical form is determined by $a_1,\ldots,a_{t}$. If $t'=t-1$, then we say that $a_1,\ldots,a_{t}$ are complete intersection. It is known that the telescopic condition implies the complete intersection condition. However, the converse was open thus far. This paper solves the conjecture in the affirmative by giving its proof.

Representation formulas of the solutions to the Cauchy problems for first order systems of the forms $\partial u/\partial t- \sum_{j=1}^{d} A_j(t) \partial u/ \partial x_j -A_0(t) u=f$ are established. The coefficients $A_j$'s are assumed to be matrix-valued functions of the forms $A_j(t) = \alpha_j(t) I + \beta_j(t) M_j$, where $\alpha_j(t), \beta_j(t)$, $j=1,\ldots,d$, are real-valued continuous functions, the eigenvalues of the matrices $M_j$, $j=1,\ldots,d$, are real, and the commutators $[M_j, M_{\ell}] = 0$ for all $j,\ell =0,1,\ldots,d$. No restrictions on the multiplicities of the characteristic roots are imposed.

In this paper we obtain existence and uniqueness of solutions of forward stochastic differential equations driven by compensated Poisson random measures. To this end, an Itô-Ventzell formula for jump processes is proved and the flow properties of solutions of stochastic differential equations driven by compensated Poisson random measures are studied.

We introduce the notion of an $n$-trivialization and a compatible curving and construct the splitting of bundle gerbe modules to define the twisted Chern classes and the twisted Chern character for bundle gerbe modules in terms of algebraic topology. Moreover, we prove that the latter coincides with the twisted Chern character due to Bouwknegt-Carey-Mathai-Murray-Stevenson [1] if the bundle gerbe is given an $n$-trivialization and a compatible curving.