We study the shock structure and the asymptotic behaviour of some flux across the origin in one-dimensional Burgers turbulence, the entropy solution to the inviscid Burgers equation, with random initial velocity for the uniformly distributed particles on the positive half line. We assume, in contrast to other works on Burgers turbulence, initially a vacuum state on the negative half line. We also obtain some asymptotic estimates for the concave majorant of Brownian motion.

Fix a line bundle $\xi$ on a connected smooth complex projective curve $X$ of genus at least three. Let $\mathcal{N}$ denote the moduli space of all stable vector bundles over $X$ of rank $n$ and determinant $\xi$. We assume that $n\geq 3$ and coprime to $\operatorname{degree}(\xi)$; If $\operatorname{genus}(X)\leq 4$, then we also assume that $n \geq 4$. We prove that $H^i(\mathcal{N}, \End(T\mathcal{N}\mkern2mu)) = H^i(X, \mathcal{O}_X)$ for $i= 0,1$.

We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties.

We give a global existence theorem to systems of quasilinear wave equations in three space dimensions, especially for the multiple-speed cases. It covers a wide class of quadratic nonlinearities which may depend on unknowns as well as their first and second derivatives. Our proof is achieved through total use of pointwise and $L^2$-estimates concerning unknowns and their first and second derivatives.

In the context of Connes' spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. Connes' construction of spectral triples for group algebras is a covariant functor from the category of discrete groups with length functions to that of spectral triples. Several interesting lines for future study of the categorical properties of spectral triples and their variants are suggested.

Let $T$ be a measure preserving transformation on $X \subset \mathbb{R}^d$ with a Borel measure $\mu$ and $R_E$ be the first return time to a subset $E$. If $(X,\mu)$ has positive pointwise dimension for almost every $x$, then for almost every $x$ \[ \limsup_{r \to 0^+} \frac{\log R_{B(x,r)}(x)}{-\log \mu(B(x,r))} \le 1, \] where $B(x,r)$ the the ball centered at $x$ with radius $r$. But the above property does not hold for the neighborhood of the `skewed' ball. Let $B(x,r;s) = (x - r^s, x + r)$ be an interval for $s >0$. For arbitrary $\alpha \ge 1$ and $\beta \ge 1$, there are uncountably many irrational numbers whose rotation satisfy that \[ \limsup_{r \to 0^+} \frac{\log R_{B(x,r;s)}(x)}{-\log \mu (B(x,r;s))} = \alpha \quad \text{and}\quad \liminf_{r \to 0^+} \frac{\log R_{B(x,r;s)}(x)}{-\log \mu (B(x,r;s))} = \frac{1}{\beta} \] for some $s$.

For a ribbon knot we define the notion of its ribbon number. In this paper we estimate the ribbon number for a ribbon knot by using the Jones polynomial. As a corollary we determine the ribbon number of the Kinoshita-Terasaka knot.

We exhibit several finite groups that are not active sums of cyclic subgroups. We show that this is the case for groups with $H_{1}G$ of odd order and $H_{2}G$ of even order. As particular examples of this we have the alternating groups $A_n$ for $n\geq 4$, some special and some projective linear groups. Our next set of examples consists of $p$-groups where the normalizer and the centralizer of every element coincide. We also have an example of a 2-group where the above conditions are not satisfied; thus we had to devise an ad hoc argument. We observe that the examples of $p$-groups given also provide groups that are not molecular.

We consider Dehn surgery creating Klein bottles. It is shown that if a non-cabled knot, which is not the figure eight knot, admits two Dehn surgeries creating Klein bottles, then the geometric intersection number of these slopes is bounded by four.

We show that the closure of a surface braid represented by a 4-chart with at most one crossing is a ribbon surface.

In this paper we construct a complete set of representatives of the irreducible representations of the party algebra. which is the centralizer of the unitary reflection group $G(r, 1, k)$ in the endomorphism ring of the tensor space $V^{\otimes n}$ under the condition that $k\geq n$ and $r>n$.