We study the $\mathcal{C}^{\infty}$-smooth functions which are locally definable in an o-minimal expansion of the real exponential field with some additional smoothness conditions. Here, the local definability generalizes the subanalytic setting to more transcendental sets and functions. The focus is set on the locally definable diffeomorphisms between manifolds, for which we prove analogies to classical differential geometric results. Moreover, we investigate the relation between classical diffeomorphy and locally definable diffeomorphy.

Frames for $\mathbb{R}^{n}$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word “frame” has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point which varies smoothly over the manifold. It is well known that the only spheres with a moving basis for their tangent bundle are $S^{1}$, $S^{3}$, and $S^{7}$. On the other hand, after combining the two separate meanings of the word “frame”, we show that the $n$-dimensional sphere, $S^{n}$, has a moving finite unit norm tight frame for its tangent bundle if and only if $n$ is odd. We give a procedure for creating vector fields on $S^{2n-1}$ for all $n\in\mathbb{N}$, and we characterize exactly when sets of such vector fields form a moving finite unit norm tight frame on $S^{2n-1}$. This gives as well a new method for constructing finite unit norm tight frames for Hilbert spaces.

We consider a class of martingales on Cartan–Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan–Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito’s rule.

Given a Hilbert $C^{*}$-module $E$ over a $C^{*}$-algebra $\mathcal{A}$, we give an explicit description for the invariant subspace lattice $\operatorname{lat} \mathcal{L} (E)$ of all adjointable operators on $E$. We then show that the collection $\operatorname{End}_{\mathcal{A} }(E)$ of all bounded $\mathcal{A}$-module operators acting on $E$ forms the reflexive closure for $\mathcal{L} (E)$, i.e., $\operatorname{End}_{\mathcal{A} }(E)=\operatorname{alg} \operatorname{lat} \mathcal{L} (E)$. Finally, we make an observation regarding the representation theory of the left centralizer algebra of a $C^{*}$-algebra and use it to give an intuitive proof of a related result of H. Lin.

We provide an elementary proof of the fact that the space of all isometric minimal immersions $f:M\to\mathbb{S}^{4}$ of a 2-dimensional Riemannian manifold $M$ into $\mathbb{S}^{4}$ with the same normal curvature is, up to congruence, either finite or a circle. Furthermore, we show that if $M$ is compact and the Euler number of the normal bundle of $f$ is nonzero, then there exist at most finitely many noncongruent isometric minimal immersions of $M$ into $\mathbb{S}^{4}$ with the same normal curvature.

We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.

Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions, we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in $\mathbb{P}^{n}$. As a corollary, we get a characterization of the projective hull for connected compact sets in $\mathbb{P}^{n}$ by the existence of analytic discs.

The main result is that the cluster value problem in separable Banach spaces, for the Banach algebras $A_{u}$ and $H^{\infty}$, can be reduced to the cluster value problem in those spaces which are $\ell_{1}$ sums of a sequence of finite dimensional spaces. In particular, we prove that the cluster value problem for $\ell_{1}$ is equivalent to the cluster value problem for $L_{1}(0,1)$.

Following Serre’s initial work, a number of authors have considered twists of quadratic forms on a scheme $Y$ by torsors of a finite group $G$, together with formulas for the Hasse–Witt invariants of the twisted form. In this paper, we take the base scheme $Y$ to be affine and consider non-constant group schemes $G$. Our main result describes these twists by a simple and explicit formula. There is a fundamental new feature in this case—in that the torsor may now be ramified over $Y$. The natural framework for handling the case of a non-constant group scheme over the affine base is provided by the quadratic theory of Hopf-algebras.

In this paper, we reconsider the well-known result of Pego–Weinstein ( Comm. Math. Phys. 2 (1994) 305–349) that soliton solutions to the Korteweg–de Vries equation are asymptotically stable in exponentially weighted spaces. In this work, we recreate this result in the setting of modern well-posedness function spaces. We obtain asymptotic stability in the exponentially weighted space via an iteration argument. Our purpose here is to lay the groundwork to use the $I$-method to obtain asymptotic stability below $H^{1}$, which will be done in a second, forthcoming paper (Asymptotic stability for KdV solitons in weighted $H^{s}$ spaces. Preprint). This will be possible because the exponential approach rate obtained here will defeat the polynomial loss in traditional applications of the $I$-method ( Commun. Pure Appl. Anal. 2 (2003) 277–296, Discrete Contin. Dyn. Syst. 9 (2003) 31–54, Commun. Pure Appl. Anal. 13 (2014) 389–418).

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

For a family of domains in the Sierpinski gasket, we study harmonic functions of finite energy, characterizing them in terms of their boundary values, and study their normal derivatives on the boundary. We characterize those domains for which there is an extension operator for functions of finite energy. We give an explicit construction of the Green’s function for these domains.

For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold $C(M)$ of a Kähler manifold $C(N)$. We show that, if $C(M)$ is biharmonic in $C(N)$, then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if $C(M)$ has a nonzero eigen-section of the Jacobi operator with the eigenvalue $m=\dim M$.

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result see ( J. Geom. Phys. 57 (2007) 1951–1976) to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions $\mathcal{P}_{-,\mathcal{L}_{0}}$, $\mathcal{P}_{+,\mathcal{L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions.

We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_{p}$-norm of the sum of a martingale difference.

We generalize to $FC^{*}$, the class of generalized $FC$-groups introduced in [ Serdica Math. J. 28 (2002) 241–254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an $FC^{*}$-group $G$ coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of $G$ are subgroups and the former coincides with the hypercentre. We also give an example of an $FC^{*}$-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.

We give a new proof for the result of J. Malý and O. Martio, stating that Hölder continuous mappings in $W^{1,n}$ satisfy the Lusin N property. We further generalize this result to a metric setting.