We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D\,=\,{{2}^{h}}c$ where $c\,>\,1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be ${{2}^{h}}$. At the end of the paper, we also address the case where $D\,=\,c$ is odd and the central norm of $\sqrt{D}$ is equal to 2.

We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in ${{\mathbb{R}}^{n}}$, where $n\,=\,k\left( k\,+\,3 \right)/2$

We show that the injective Kobayashi–Royden differential metric, as defined by Hahn, is upper semicontinous.

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${{\mathcal{A}}_{\theta }}$, and we show that the ${{C}^{*}}$ algebra $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ generated by these Toeplitz operators is an extension of ${{\mathcal{A}}_{\theta }}$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $K{{K}^{1}}\left( {{\mathcal{A}}_{\theta }},\mathbb{C} \right)$. We also prove an index theorem for $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

If $f$ is a real-valued function on $\left[ -\pi ,\,\pi \right]$ that is Henstock-Kurzweil integrable, let ${{u}_{r}}(\theta )$ be its Poisson integral. It is shown that ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,o\left( 1/\left( 1-r \right) \right)$ as $r\,\to \,1$ and this estimate is sharp for $1\,\le \,p\,\le \,\infty $. If $\mu $ is a finite Borel measure and ${{u}_{r}}(\theta )$ is its Poisson integral then for each $1\,\le \,p\,\le \,\infty $ the estimate ${{\left\| {{u}_{r}} \right\|}_{p}}\,=\,O\left( {{\left( 1-r \right)}^{1/p-1}} \right)$ as $r\,\to \,1$ is sharp. The Alexiewicz norm estimates $\left\| {{u}_{r}} \right\|\,\le \,\left\| f \right\|$$\left( 0\,\le \,r\,<\,1 \right)$ and $\left\| {{u}_{r}}-f \right\|\,\to 0\left( r\to 1 \right)$ hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when $u$ is in the harmonic Hardy space associated with the Alexiewicz norm and when $f$ is of bounded variation.

The orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group ${{S}_{n}}$, in each homogenous degree $n$. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of ${{S}_{n-1}}$. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in ${{S}_{n}}$.

It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $A\,\mapsto \,PAQ\,+\,R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective.

Let $R$ be a commutative Noetherian integral domain with field of fractions $Q$. Generalizing a forty-year-old theorem of E. Matlis, we prove that the $R$-module $Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal and one-dimensional. Moreover, if $X$ is an injective module over a commutative Noetherian ring such that $X$ has Krull dimension, then the Krull dimension of $X$ is at most 1.

We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field $\mathbb{I}$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincaré-type equations and the connection between the solutions of these functional equations and the generalized Heine series.

It is shown, using the Borwein–Preiss variational principle that for every continuous convex function $f$ on a weakly compactly generated space $X$, every ${{x}_{0}}\in X$ and every weakly compact convex symmetric set $K$ such that $\overline{\text{span}}K=X$, there is a point of Gâteaux differentiability of $f$ in ${{x}_{0}}+K$. This extends a Klee's result for separable spaces.

Let $f:X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that dim $Y\le m$. It is shown that for every positive integer $p$ with $p\le m+k+1$ there exists a dense ${{G}_{\delta }}-\text{subset}\,\mathcal{H}\left( k,m,p \right)$ of $C\left( X,{{\mathbb{I}}^{k+p}} \right)$ with the source limitation topology such that each fiber of $f\Delta g$, $g\in \mathcal{H}\left( k,m,p \right)$, contains at most $\max \left\{ k+m-p+2,1 \right\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f:X\to Y$ be a $k$-dimensional map between compact metric spaces with dim $Y\le m$. Then: (1) there exists a map $h:X\to {{\mathbb{I}}^{m+2k}}$ such that $f\Delta h:\,X\to Y\times {{\mathbb{I}}^{m+2k}}$ is 2-to-one provided $k\ge 1$; (2) there exists a map $h:X\to {{\mathbb{I}}^{m+k+1}}$ such that $f\Delta h:X\to Y\times {{\mathbb{I}}^{m+k+1}}$ is $\left( k+1 \right)$-to-one.

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

The Abhyankar–Sathaye Embedded Hyperplane Problem asks whether any hypersurface of ${{\mathbb{C}}^{n}}$ isomorphic to ${{\mathbb{C}}^{n-1}}$ is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of ${{\mathbb{C}}^{n}}$. Generalizing the approach adopted by Kaliman, Vénéreau, and Zaidenberg, which consists in using almost nothing but the acyclicity of ${{\mathbb{C}}^{n-1}}$, we solve this problem for hypersurfaces given by polynomials of $\mathbb{C}\left[ x,y,{{z}_{1}},...,{{z}_{k}} \right]$ as in the title.

This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma $-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma $-Lipschitz over their effective domain.

This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q\left( z \right)$ and the topology of the complement of the hypersurface defined by $Q\left( z \right)=0$.

We define a uniform structure on the set of discrete sets of a locally compact topological space on which a locally compact topological group acts continuously. Then we investigate the completeness of these uniform spaces and study these spaces by means of topological dynamical systems.

It is proved here that a ring $R$ is right pseudo-Frobenius if and only if $R$ is a right Kasch ring such that the second right singular ideal is injective.