Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty \right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$, and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$, for some integer $m\,\ge \,2$. Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$, be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$, let

$$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$

An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$.

A bounded linear operator $T$ on a Banach space $X$ is an abstract backward shift if the nullspace of $T$ is one dimensional, and the union of the null spaces of ${{T}^{k}}$ for all $k\,\ge \,1$ is dense in $X$. In this paper it is shown that the commutant of an abstract backward shift is an integral domain. This result is used to derive properties of operators in the commutant.

Formulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian. Connections with the concept of subdifferential regularity are also established.

We calculate the additive and multiplicative structure of the ring $K\left( S\left( {{T}^{*}}\mathbb{R}{{P}^{2n+1}} \right) \right)$ using the eta invariant.

Let $G$ be a real Lie group and $X$ a real analytic manifold. Suppose that $G$ acts analytically on $X$ with finitely many orbits. Then the orbits are subanalytic in $X$. As a consequence we show that the micro-support of a $G$-equivariant sheaf on $X$ is contained in the conormal variety of the $G$-action.

The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains that pinch at an exponential rate, weaker, ${{L}^{2}}$ bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity.

Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U\,=\,G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

We prove that the ${{L}^{2}}$ restriction theorem, and ${{L}^{p}}\,\to \,{{L}^{{{p}'}}}\,,\,\frac{1}{p}\,+\,\frac{1}{{{p}'}}\,=\,1$, boundedness of the surface averages imply certain geometric restrictions on the underlying hypersurface. We deduce that these bounds imply that a certain number of principal curvatures do not vanish.

The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type I and Type II spectral flow.

We shall provide a characterization of all isoparametric hypersurfaces ${M}'s$ in a real space form $\tilde{M}\left( c \right)$ by observing the extrinsic shape of geodesics of $M$ in the ambient manifold $\tilde{M}\left( c \right)$.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

In this note we give a proof of Lomonosov’s extension of Burnside’s theoremto infinite dimensional Banach spaces.

Let $G$ be a hermitian quaternionic group. We determine complementary series for representations of $G$ induced from super-cuspidal representations of a Levi factor of the Siegel maximal parabolic subgroup of $G$.

A counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of $A\left[ {Z}/{pZ}\;\,\oplus \,{Z}/{pZ}\; \right]$ is computed when $A$ is an unramified principal Artin local ring.

We observe that any set of uniqueness for the Dirichlet space $\mathcal{D}$ is a set of uniqueness for the class $S$ of normalized univalent holomorphic functions.

The Four Color Theorem says that the faces (or vertices) of a plane graph may be colored with four colors. Vizing’s Theorem says that the edges of a graph with maximum degree $\Delta$ may be colored with $\Delta \,+\,1$ colors. In 1972, Kronk and Mitchem conjectured that the vertices, edges, and faces of a plane graph may be simultaneously colored with $\Delta \,+\,4$ colors. In this article, we give a simple proof that the conjecture is true if $\Delta \,\ge \,6$.

An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\text{PSL}\left( 2,\,\mathbb{Z} \right)$ of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

Soient $A$ un anneau noethérien, $B$ un anneau régulier essentiellement de type fini sur $A$. Si la cohomologie d’André-Quillen ${{H}^{q}}\left( A,\,B,\,B \right)\,=\,0$ pour tout $q\,\ge \,2$ alors $A$ est un anneau régulier.