We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

Let $x\,=\,\left( {{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}} \right)\,\in \,{{\mathbb{R}}^{n}}$ and ${{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x\,=\,\left( {{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{1}}}}{{x}_{1}},\,.\,.\,.\,,\,{{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{n}}}}{{x}_{n}} \right)$, where $\text{ }\lambda \,>\text{0}$ and $1\,\le \,{{\alpha }_{1}}\,\le \,\cdot \,\cdot \,\cdot \,\le \,{{\alpha }_{n}}$. Denote $\left| \alpha \right|\,=\,{{\alpha }_{1}}+\,\cdot \,\cdot \,\cdot \,+{{\alpha }_{n}}$. We characterize those functions $A\left( x \right)$ for which the parabolic Calderón commutator

1

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$

is bounded on ${{L}^{2}}\left( {{\mathbb{R}}^{n}} \right)$, where $K\left( {{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x \right)\,=\,{{\text{ }\!\!\lambda\!\!\text{ }}^{-\,\left| \alpha \right|\,-\,1}}K\left( x \right)$, $K$ is smooth away fromthe origin and satisfies a certain cancellation property.

We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.

If $V\,\to \,X$ is a vector bundle of fiber dimension $k$ and $Y\,\to \,X$ is a finite sheeted covering map of degree $d$, the implications for the Euler class $e(V)$ in ${{H}^{k}}(X)$ of $V$ implied by the existence of an embedding $Y\,\to \,V$ lifting the covering map are explored. In particular it is proved that $d{{d}^{\prime }}\text{e(V)}\text{=}\text{0}$ where ${{d}^{\prime }}$ is a certain divisor of $d\,-\,1$, and often ${{d}^{\prime }}=1$.

We give a “wall-crossing” formula for computing the topology of the moduli space of a closed $n$-gon linkage on ${{\mathbb{S}}^{2}}$. We do this by determining the Morse theory of the function ${{\rho }_{n}}$ on the moduli space of $n$-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first $\left( n\,-\,1 \right)$ side-lengths are fixed. We obtain a Morse function on the $\left( n\,-\,2 \right)$-torus with level sets moduli spaces of $n$-gon linkages. The critical points of ${{\rho }_{n}}$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ${{\rho }_{n}}$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Let ${{j}_{v,1}}$ be the smallest (first) positive zero of the Bessel function ${{J}_{v}}(z),\,v\,>\,-\,1$, which becomes zero when $v$ approaches −1. Then $j_{v,1}^{2}$ can be continued analytically to $-2\,<\,v\,<\,-1$, where it takes on negative values. We show that $j_{v,1}^{2}$ is a convex function of $v$ in the interval $-2\,<\,v\,\le \,0$, as an addition to an old result [Á. Elbert and A. Laforgia, SIAM J. Math. Anal. 15(1984), 206–212], stating this convexity for $v\,>\,0$. Also the monotonicity properties of the functions $\frac{j_{v,1}^{2}}{4(v+1)},\,\frac{j_{v,1}^{2}}{4(v+1)\sqrt{v+2}}$ are determined. Our approach is based on the series expansion of Bessel function ${{J}_{v}}(z)$ and it turned out to be effective, especially when $-2\,<\,v\,<\,-1$.

Let ${{\mathcal{S}}_{r}}$ be the collection of all axially symmetric functions $f$ in the Sobolev space ${{H}^{1}}\left( {{\mathbb{S}}^{2}} \right)$ such that $\int_{{{\mathbb{S}}^{2}}}{{{x}_{i}}{{e}^{2f\left( x \right)}}\,dw\left( \text{x} \right)}$ vanishes for $i\,=\,1,\,2,\,3$. We prove that

$$\underset{f\in {{\mathcal{S}}_{r}}}{\mathop \inf }\,\frac{1}{2}\int_{{{\mathbb{S}}^{2}}}{{{\left| \nabla f \right|}^{2}}\,dw\,+\,2\,\int_{{{\mathbb{S}}^{2}}}{f\,dw\,-\,\log \,\int_{{{\mathbb{S}}^{2}}}{{{e}^{2f}}\,dw\,>\,-\infty ,}}}$$

and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange’s method we obtain the main term and smaller order terms which contain periodic fluctuations.

In this paper we prove the convergence of arithmetic means of sequences generated by a periodic function $\varphi (X)$, moreover if $\varphi (X)$ satisfies a suitable symmetry condition, we prove that their limit is $\varphi (0)$. Applications of previous results are given to study other means of sequences and the behaviour of a class of recursive series.

An integral formula is derived for the spherical functions on the symmetric space ${G}/{K\,=\,{\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;}\;$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition $G\,=\,KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space ${\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;$ using the Plancherel formula.

In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section 2 the equivalence between Shimogaki’s theorem and some martingale inequalities will be established, and in Section 3 the equivalence between Boyd’s theorem and martingale inequalities with change of probability measure will be established.

We derive a necessary and sufficient condition for $\text{HNN}$-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of $\text{HNN}$-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of $\text{HNN}$-extensions of nilpotent groups with cyclic associated subgroups.

Given a $\text{TQFT}$ in dimension $d\,+\,1$, and an infinite cyclic covering of a closed ($d\,+\,1$)-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a $\text{TQFT}$ and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an ${{S}^{1}}$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a $\text{TQFT}$ associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the $\text{TQFT}$ associated to $G$ in its unmodified form.

A line singularity is a function germ $f:\,\left( {{\text{C}}^{n+1}},\,0 \right)\,\to \,\text{C}$ with a smooth 1-dimensional critical set $\sum \,=\,\left\{ \left( x,\,y \right)\,\in \,\text{C}\,\times \,{{\text{C}}^{n}}\,|\,y\,=\,0 \right\}$. An isolated line singularity is defined by the condition that for every $x\,\ne \,0$, the germ of $f$ at $\left( x,\,0 \right)$ is equivalent to $y_{1}^{2}\,+\,\cdot \,\cdot \,\cdot \,+\,y_{n}^{2}$. Simple isolated line singularities were classified by Dirk Siersma and are analogous of the famous $A\,-\,D\,-E$ singularities. We give two new characterizations of simple isolated line singularities.

We study the splitting of Fermat Jacobians of prime degree $\ell $ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell $. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell $-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

We define embedding of an $n$-dimensional normed space into ${{L}_{-p}},\,0\,<\,p\,<\,n$ by extending analytically with respect to $p$ the corresponding property of the classical ${{L}_{p}}$-spaces. The well-known connection between embeddings into ${{L}_{p}}$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in ${{L}_{-p}}$ if and only if $\parallel x{{\parallel }^{-p}}$ is a positive definite distribution. We show that the technique of embedding in ${{L}_{-p}}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $P\in [n-3,n),\mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{X}_{i}} \right|}^{-p}})\ge \mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{Y}_{i}} \right|}^{-p}})$, where ${{X}_{1}},...,{{X}_{n}}\,\text{and}\,{{Y}_{1}},...,{{Y}_{n}}$ are jointly $q$-stable symmetric random variables, $0\,<\,q\,\le \,2$, so that, for some $k\,\in \,\mathbb{N},\,1\,\le \,k\,<\,n$, the vectors $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{k}} \right)$ and $\left( {{X}_{k+1}},\,.\,.\,.\,,{{X}_{n}} \right)$ have the same distributions as $({{Y}_{1}},...,{{Y}_{k}})\,\,\text{and}\,\,({{Y}_{k+1}},...,{{Y}_{n}})$, respectively, but ${{Y}_{i}}\,\text{and}\,{{Y}_{j}}$ are independent for every choice of $1\,\le \,i\,\le \,k,\,k\,+\,1\,\le \,j\,\le \,n$.

In this paper we study the boundedness of the commutators $[b,\,{{S}_{n}}]$ where $b$ is a $\text{BMO}$ function and ${{S}_{n}}$ denotes the $n$-th partial sum of the Fourier-Bessel series on $(0,\,\infty )$. Perturbing the measure by $\text{exp(}2\text{b)}$ we obtain that certain operators related to ${{S}_{n}}$ depend analytically on the functional parameter $b$.

Let ${{A}_{i}},\,{{B}_{i}}$ and ${{X}_{i}}\,(i\,=\,1,\,2,\ldots ,\,n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right)$ which satisfy the relation $f\,(t)g(t)\,=\,t$ for all $t$ in $\left[ 0,\infty \right)$, then

$${{\left\| \left| \,{{\left| \sum\limits_{i=1}^{n}{A_{i}^{*}{{X}_{i}}{{B}_{i}}} \right|}^{r}} \right| \right\|}^{2}}\,\le \,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{A_{i}^{*}f{{(\left| X_{i}^{*} \right|)}^{2}}{{A}_{i}}} \right)}^{r}} \right| \right\|\,\,\left\| \left| {{\left( \sum\limits_{i=1}^{n}{B_{i}^{*}g{{(\left| {{X}_{i}} \right|)}^{2}}\,{{B}_{i}}} \right)}^{r}} \right| \right\|$$

for every $r\,>\,0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.