Related papers: A model with no magic sets

It is consistent that for every function f:R x R-> R there is an uncountable set A subseteq R and two continuous functions f_0,f_1:D(A)-> R such that f(alpha, beta) in {f_0(alpha, beta),f_1(alpha, beta)} for every (alpha, beta) in A^2,…

The goal of this paper is twofold. In addition to the results stated in the next paragraph, we present some classical results on absoluteness relevant to functional analysis that are well known to logicians but not nearly as well advertised…

We prove that there exists a constant $\varepsilon > 0$ with the following property: if $K \subset \mathbb{R}^{2}$ is a compact set which contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\mathrm{dim}_\mathrm{H} K…

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m…

The object of this paper is studying some properties of meromorphic functions which satisfy in the condition \[Re(zf(z)) > \alpha|z^2f'(z)+zf(z)| .\] Parallel results for some related classes are also obtained.

We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null…

Let $\Gamma^\infty$ be the set of all universally Baire sets of reals. Inspired by recent work of the second author and Nam Trang, we introduce a new technique for establishing generic absoluteness results for models containing…

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…

We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain,…

We show in ZFC that there is no set of reals of size continuum which can be translated away from every set in the Marczewski ideal. We also show that in the Cohen model, every set with this property is countable.

For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.

A function F:R^2->R is sup-measurable if F_f:R->R given by F_f(x)=F(x,f(x)), x in R, is measurable for each measurable function f:R->R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable…

It is shown that that for every Darboux function $F$ there is a non-constant continuous function $f$ such that $F+f$ is still Darboux. It is shown to be consistent --- the model used is iterated Sacks forcing --- that for every Darboux…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal…

Answering a question asked by K.C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a set $M \subset \operatorname{graph} f$ nowhere dense in $\operatorname{graph} f$ such that there does not exist…

We prove that for a continuum $K\subset \mathbb R^n$ the sum $K^{+n}$ of $n$ copies of $K$ has non-empty interior in $\mathbb R^n$ if and only if $K$ is not flat in the sense that the affine hull of $K$ coincides with $\mathbb R^n$.…

We prove the consistency of ZF+DC+"there are no mad families"+"there exists a non-meager filter on $\omega$" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the…

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system…

We show that there exists an entire function which has neither fixed points nor invariant Baker domains. The question whether such a function exists was raised by Buff.