In this paper we establish the ${{L}^{p}}$ boundedness of a class of singular integrals with rough kernels associated to polynomial mappings.

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio $(-1\,+\,\sqrt{5})/2$ for which when Newton's method for approximating roots is applied with an initial value ${{x}_{0}}\,=\,0$ , all approximates are so-called “best rational approximates”—in this case, of the form ${{F}_{2n}}/{{F}_{2n+1}}$ , where ${{F}_{n}}$ denotes the $n$ -th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z\,+\,{{z}_{n}}}{1\,+\,{{{\bar{z}}}_{n}}z})\,:\,n\,\in \,\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

Using weighted Delsarte surfaces, we give examples of $K3$ surfaces in positive characteristic whose formal Brauer groups have height equal to 5, 8 or 9. These are among the four values of the height left open in the article of Yui [11].

Leray's self-similar solution of the Navier-Stokes equations is defined by

$$u(x,\,t)\,=\,U(y)/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},$$

where $y\,=\,x/\sqrt{2\sigma ({{t}^{*}}\,-\,t)},\,\sigma \,>\,0$ . Consider the equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of ${{\mathbb{R}}^{3}}$ with non-zero boundary condition:

$$-v\,\Delta \,U\,+\,\sigma U\,+\,\sigma y\,\cdot \,\nabla U\,+\,U\,\cdot \,\nabla U\,+\,\nabla P\,=\,0,\,\,\,y\,\in \,\mathcal{D}\,$$

$$\nabla \,\cdot \,U\,=\,0,\,\,\,y\,\in \,\mathcal{D},$$

$$U\,=\,\mathcal{G}(y),\,\,\,y\,\in \,\partial \mathcal{D}.$$

We prove an existence theorem for the Dirichlet problem in Sobolev space ${{W}^{1,2}}(\mathcal{D})$ . This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at $t\,=\,{{t}^{*}}$ with ${{t}^{*}}\,<\,+\infty $ , provided the function $\mathcal{G}(y)$ is permissible.

We use the homological perturbation lemma to give an algebraic proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.

The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$ . We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$ , in particular from the Bloch space into $\text{BMOA}$ .

Let $g:\,{{M}^{2n}}\,\to \,{{M}^{2n}}$ be a smooth map of period $m\,>\,2$ which preserves orientation. Suppose that the cyclic action defined by $g$ is regular and that the normal bundle of the fixed point set $F$ has a $g$ -equivariant complex structure. Let $F\,\pitchfork \,F$ be the transverse self-intersection of $F$ with itself. If the $g$ -signature $\text{Sign(g,}\,\text{M)}$ is a rational integer and $n\,<\,\phi (m)$ , then there exists a choice of orientations such that $\text{Sign}\,\text{(g,}\,\text{M)}\,\text{=}\,\text{Sign}\,\text{F}\,\text{=}\,\text{Sign}(F\,\pitchfork \,F)$ .

In this paper, we give some conditions to judge when a system of Hermitian quadratic forms has a real linear combination which is positive definite or positive semi-definite. We also study some related geometric and topological properties of the moduli space.

This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups $G$ . The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of $G$ is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced.

The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a $G$ -dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a $G$ -dynamical system, is a factor of the dynamical local hull.

Let $N$ be an integer which is larger than one. In this paper we study invariant subspaces of ${{L}^{2}}({{\mathbb{T}}^{N}})$ under the double commuting condition. A main result is an $N$ -dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an $N$ -dimensional version of Lax's theorem.

It is known that any non-archimedean Fréchet space of countable type is isomorphic to a subspace of $c_{0}^{\mathbb{N}}$ . In this paper we prove that there exists a non-archimedean Fréchet space $U$ with a basis $({{u}_{n}})$ such that any basis $({{x}_{n}})$ in a non-archimedean Fréchet space $X$ is equivalent to a subbasis $({{u}_{kn}})$ of $({{u}_{n}})$ . Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of $U$ . In contrast to this, we show that a non-archimedean Fréchet space $X$ with a basis $({{x}_{n}})$ is isomorphic to a complemented subspace of $c_{0}^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: ${{c}_{0}},\,{{c}_{0}}\,\times \,{{\mathbb{K}}^{\mathbb{N}}},\,{{\mathbb{K}}^{\mathbb{N}}},\,c_{0}^{\mathbb{N}}$ . Finally, we prove that there is no nuclear non-archimedean Fréchet space $H$ with a basis $({{h}_{n}})$ such that any basis $({{y}_{n}})$ in a nuclear non-archimedean Fréchet space $Y$ is equivalent to a subbasis $({{h}_{kn}})$ of $({{h}_{n}})$ .

Exponent information is proven about the Lie groups $SU(3),\,SU(4),\,Sp(2)$ , and ${{G}_{2}}$ by showing some power of the $H$ -space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8,\,32,\,64$ , and ${{2}^{8}}$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of 2 for $SU(4)$ and $Sp(2)$ , and off by at most two powers of 2 for ${{G}_{2}}$ .

We prove that a class of perturbations of standard $\text{CR}$ structure on the boundary of threedimensional complex ellipsoid ${{E}_{p,\,q}}$ can be realized as hypersurfaces on ${{\mathbb{C}}^{2}}$ , which generalizes the result of Burns and Epstein on the embeddability of some perturbations of standard $\text{CR}$ structure on ${{S}^{3}}$ .

Voiculescu has previously established the uniqueness of the wave operator for the problem of ${{\mathcal{C}}^{(0)}}$ -perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal $\mathcal{C}$ has the property ${{\lim }_{n\,\to \,\infty }}\,{{n}^{-1/2}}\,\left\| {{P}_{n}} \right\|\mathcal{C}\,=\,0$ , where $\{{{P}_{n}}\}$ is any sequence of orthogonal projections with rank $({{P}_{n}})\,=\,n$ . We prove that the same uniqueness result holds true so long as $\mathcal{C}$ is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

In this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.