AN APPLICATION OF RECURSION THEORY TO ANALYSIS

Mauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$-null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $-ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

Depiction of semi-algebraic and semi-analytic set by means of particle swarm optimization

We present an algorithm of particle swarm optimization (PSO) which can depict the boundary of semi-algebraic or semi-analytic set. The algorithm imitates physical phenomena: the pair-annihilation of particle and antiparticle (or electron and hole). The pairs of particle and antiparticles are attracted by the boundary of the boundary of semi-algebraic or semi-analytic set, led by a special penalty function. The accumulation of these pairs around the optima will act as the "group best set" in the PSO algorithm (which also attracts the other pair) and it finally depicts the boundary.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

A Note on Regression and Reassurance

In this brief work, Winnicott expresses his belief that, in teaching psychoanalysis, analysts must continue to speak against the use of reassurance in their clinical work. He believes that reassurance is much better discussed in terms of countertransference, and that the analytic set-up is in itself reassuring.

Analytic cycles in flip passages and in instanton moduli spaces over non-Kählerian surfaces

Let [Formula: see text] ([Formula: see text]) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with [Formula: see text] and let [Formula: see text] be a pure [Formula: see text]-dimensional analytic set. We prove a general formula for the homological boundary [Formula: see text] of the Borel–Moore fundamental class of [Formula: see text] in the boundary of the blown up moduli space [Formula: see text]. The proof is based on the holomorphic model theorem of [A. Teleman, Instanton moduli spaces on non-Kählerian surfaces, Holomorphic models around the reduction loci, J. Geom. Phys. 91 (2015) 66–87] which identifies a neighborhood of a boundary component of [Formula: see text] with a neighborhood of the boundary of a “blown up flip passage”. We then focus on a particular instanton moduli space which intervenes in our program for proving the existence of curves on class VII surfaces. Using our result, combined with general properties of the Donaldson cohomology classes, we prove incidence relations between the Zariski closures (in the considered moduli space) of certain families of extensions. These incidence relations are crucial for understanding the geometry of the moduli space, and cannot be obtained using classical complex geometric deformation theory.

Non-standard real-analytic realizations of some rotations of the circle

We extend some aspects of the smooth approximation by conjugation method to the real-analytic set-up, and create examples of zero entropy, uniquely ergodic, real-analytic diffeomorphisms of the two-dimensional torus that are metrically isomorphic to some (Liouvillian) irrational rotations of the circle.

Analytic in planar domains functions with preassigned asymptotic set

In 1954 M. Heins proved that, for every analytic set

Tameness of Complex Dimension in a Real Analytic Set

Given a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.