It is shown that any HNN extension is precisely a compression by a projection of a certain amalgamated free product in the framework of operator algebras. As its applications several questions for von Neumann algebras or C^*-algebras arising as HNN extensions are considered.

Let $A$ be a Cohen-Macaulay local ring which contains a field $k$, and let $I \subseteq A$ be an ideal generated by polynomials in a system of parameters of $A$ with coefficients in $k$. In this paper, we shall prove that all the Bass numbers of local cohomology modules are finite for all $j \in{\mathbb Z}$ provided that the residue field is separable over $k$. We also prove that the set of associated prime ideals of those is a finite set under the same hypothesis. Furthermore, we shall discuss finiteness properties of local cohomology modules over regular local rings.

Let $(x_n)$, $(y_n)$ be two martingales adapted to the same filtration $(\mathcal{F}_n)$ satisfying, with probability $1$, \[ |dx_n|\leq|dy_n|,\quad n=0, 1, 2, \ldots . \] For every $K>0,$ we determine the best constant $L=L(K)$ for which the inequality \[ \mathbb{E} |x_n| \leq K\mathbb{E} |y_n|\log|y_n|+L,\quad n=0, 1, 2, \ldots \] holds true. We also prove a similar inequality for harmonic functions.

Let $(\mathbf{R}^n, d_X)$ be a Carnot–Carathéodory metric space generated by a family of smooth vector fields $\{X_i\}_{i=1}^m$ satisfying Hörmander's finite rank condition, and $\mathcal{H}_X= \{(x, \sum_{i=1}^m a_i X_i(x))| x\in\mathbf{R}^n, (a_i)_{i=1}^m\in\mathbf{R}^m\}$ be the horizontal tangent bundle generated by $\{X_i\}_{i=1}^m$. Assume that $H=H(x,p)\in C^1(\mathcal{H}_X)$ is quasiconvex in $p$-variable. We prove that any absolute minimizer $u\in W^{1,\infty}_X(\Omega)$ to $F_\infty(v,\Omega)=\operatorname{ess \sup}_{x\in\Omega} H(x, Xv(x))$ is a viscosity solution of the Aronsson equation \[ \mathcal{A}^{X}[u]:= X(H(x,Xu(x)))\cdot H_p(x, Xu(x)) =0 \quad\hbox{in } \Omega. \]

We describe an algorithmic approach to prove or disprove several recent conjectures for epsilon constants of Galois extensions of $p$-adic fields and number fields. For this approach, we must develop various algorithms for computations in Galois extensions of $p$-adic fields which are of independent interest. Our algorithms for $p$-adic fields are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to $p$-adic numbers.

We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterizations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces.

We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob–Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder–Davis–Gundy inequalities. A connection with some balayage formulae is also established.

A $2$-knot is (the isotopy class of) a $2$-sphere smoothly embedded in $4$-space. The apparent contour of a generic planar projection of a $2$-knot divides the plane into several regions, and to each such region, we associate the number of sheets covering it. The total width of a $2$-knot is defined to be the minimum of the sum of these numbers, where we take the minimum among all generic planar projections of the given $2$-knot. In this paper, we show that a $2$-knot has total width eight if and only if it is an $n$-twist spun $2$-bridge knot for some $n \neq\pm1$.

In this paper, we describe the cohomogeneity one special Lagrangian 3-folds in both the deformed and the resolved conifolds. Our results give an explicit construction of the families of $SO(3)$ and $T^2$-invariant special Lagrangian submanifolds in these conifolds and describe their asymptotic behavior.

For a compact subset $K$ of a Riemann surface, necessary and sufficient conditions are given for a function algebra containing $A(K)$ to be all of $C(K)$. Using these results, several conditions are given on a complex-valued function $f$ so that the algebra generated by $A(K)$ and $f$ is all of $C(K)$. In particular, the results are applied to a harmonic function $f$ to give sufficient conditions for the algebra generated by $A(K)$ and $f$ to be all of $C(K)$. Also, sufficient conditions are given for the algebra $A(K)$ to be a maximal subalgebra of $C(K)$.

Balanced and strongly balanced lattices were introduced in order to generalize the uniform dimension of modular lattices. A description of finite groups with strongly balanced subgroup lattices was given by the authors in (Colloq. Math. 82 (1999), 65–77) and strengthened by Schmidt in (Illinois J. Math. 47 (2003), 515–528). In this paper, a description of finite groups with dually strongly balanced subgroup lattices is given.

A criterion is given for a strong type in a finite rank stable theory $T$ to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for a compact complex analytic space to be “algebraic” (namely Moishezon). The canonical base property for a stable theory states that the type of the canonical base of a stationary type over a realisation is almost internal to the minimal types of the theory. It is conjectured that every finite rank stable theory has the canonical base property. It is shown here, that in a theory with the canonical base property, if $p$ is a stationary type for which there exists a family of types $q_b$, each internal to a nonlocally modular minimal type $r$, and such that any pair of independent realisations of $p$ are “connected” by the $q_b$’s, then $p$ is almost internal to $r$.

We prove the equality statements for the classical symmetrization estimates for harmonic measure. In fact, we prove more general results for $\alpha$-harmonic measure.\break The $\alpha$-harmonic measure is the hitting distribution of symmetric $\alpha$-stable processes upon exiting an open set in $\mathbb{R^n}$ ($0<\alpha<2$, $n\geq2$). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove polarization and symmetrization inequalities for $\alpha$-harmonic measure. We give a complete description of the corresponding equality cases. The proofs involve analytic and probabilistic arguments.

This paper introduces a generalization of Pontryagin duality for locally compact Hausdorff Abelian groups to locally compact Hausdorff Abelian group bundles.

We prove that all ergodic components of an extension of an ergodic system by translations on a nilmanifold $X$ are isomorphic to extensions of this system by translations on subnilmanifolds of $X$.

We show how a standard method of geometric measure theory for providing density estimates may be used in general metric spaces to obtain information on the upper porosity of packing type measures. We also obtain a connection between lower densities and the upper porosity of measures on Euclidean spaces.

We prove a Sturm-type comparison theorem by a geometric study of plane (multi)hedgehogs. This theorem implies that for every $2\pi$-periodic smooth real function $h$, the number of zeros of $h$ in $[0,2\pi[$ is not bigger than the number of zeros of $h+h^{\prime\prime}$ plus $2$. In terms of $N$-hedgehogs, it can be interpreted as a comparison theorem between number of singularities and maximal number of support lines through a point. The rest of the paper is devoted to a series of geometric consequences.

Let $\mathcal{A}$ be the set of automorphisms of the unit disk with $1$ as attractive fixed point. We prove that there exists a single Blaschke product that is universal for every composition operator $C_\phi$, $\phi\in\mathcal{A}$, acting on the unit ball of $H^\infty(\mathbb{D})$.

We study the intertwining relation $XT_\varphi=T_\psi X$ where $T_\varphi $ and $T_\psi$ are the Toeplitz operators induced on the Hardy space $H^2$ by analytic functions $\varphi$ and $\psi$, bounded on the open unit disc~$\mathbb{U}$, and $X$ is a nonzero bounded linear operator on $H^2$. Our work centers on the connection between intertwining and the image containment $\psi(\mathbb{U})\subset\varphi (\mathbb{U})$, as well as on the nature of the intertwining operator $X$. We use our results to study the ``extended eigenvalues'' of analytic Toeplitz operators $T_\varphi$, i.e., the special case $XT_{\lambda\varphi}=T_\varphi X$, where $\lambda$ is a complex number.

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature $\ge\kappa$ is an Alexandrov's space of curvature $\ge\kappa$. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.

We consider a class of transcendental meromorphic functions $f:\mathbb{C}\mapsto\overline{\mathbb{C}}$ with infinitely many poles. Under some regularity assumption on the location of poles and the behavior of the function near the poles, we provide explicite lower bounds for the hyperbolic dimension (Hausdorff dimension of radial points) of the Julia set and upper bounds for the Hausdorff dimension of the set of escaping points in the Julia set. In particular, the Hausdorff dimension of the latter set is less than the Hausdorff dimension of the former set. Consequently, the Hausdorff dimension of the set of escaping points is less than 2, and the area of this set is equal to zero. The functions under consideration may have infinitely many singular values, and we do not even assume them to belong to the class $\mathcal{B}$. We only require the distance between the set of poles and the set of finite singular values to be positive.

A main tool in studying topological properties of sets definable in o-minimal structures is the Cell Decomposition Theorem. The present paper proposes its metric counterpart based on the idea of a Lipschitz cell. In contrast to earlier results, we give an algorithm of a Lipschitz cell decomposition involving only permutations of variables as changes of coordinates.