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Juris Steprāns

Publications and source records attributed to Juris Steprāns.

27 records · Page 2Linked to original sources
arXiv1995

Orthogonal Families of Real Sequences

For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real sequences x such that |x(0)|^p + |x(1)|^p + .. converges. For Hilbert space, l_2, any family of pairwise orthogonal sequences must be countable. Thm 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of l_p for every p>2. It was already known that there exists a family of continuum many pairwise orthogonal elements of real sequences. Thm 2. There exists a perfect maximal orthogonal family of elements of real sequences. Abian raised the question of what are the possible cardinalities of maximal orthogonal families. Thm 3. In the Cohen real model there is a maximal orthogonal set cardinality omega_1, but there is no maximal orthogonal set of cardinality k with ω_1< k < c. Thm 4. For any countable standard model M of ZFC and cardinal k in M such that M satisfies k^ω=k, there exists a ccc generic extension M[G] such that the continuum of M[G] is k and in M[G] for every infinite cardinal i less than or equal to k there is a maximal orthogonal family of cardinality i. Thm 5. (MA_k(σ-centered)) Suppose cardinality of X is less than or equal to k, X contains only finitely many elements of l_2, and for every distinct pair x,y in X the inner product (x,y) converges. Then there exists a z such that z is orthogonal to every element of X. Thm 6.(a) There exists X which is a maximal orthogonal family in l_2 such that for all n with 1\leq n \leqωthere exists Y of cardinality n with (X union Y) a maximal orthogonal family in $\rr^\om$. Furthermore, every maximal orthogonal family containing $X$ is countable. (b) There exists a perfect maximal orthogonal family P such that (P intersect l_2) is a maximal orthogonal family in $l_2$.

math.LO↗
arXiv1995

Decomposing with smooth sets

A subset of Euclidean space will be said to be $n$-smooth if it has an $n$-dimensional tangent plane at each of its points. Let ${\frak d}_n$ denote the least number $n$-smooth sets into which $n+1$-dimensional Euclidean space can be decomposed. For each $n$ it is shown to be consistent that ${\frak d}_n > {\frak d}_{n+1} $. Moreover, the inequalities ${\frak d}_{n+1}^+ \geq ${\frak d}_n$ are established where ${\frak d}_1$ is defined to be the continuum. The cardinal invariant ${\frak d}_2$ is shown to be the same as the least $κ$ such that each continuous function from the reals to the reals can be decomposed into $κ$ differentiable functions.

math.LO↗
arXiv1994

Cardinal Invariants Associated with Hausdorff Capacities

This is a revision (and partial retraction) of my previous abstarct. Let $λ(X)$ denote Lebesgue measure. If $X\subseteq [0,1]$ and $r \in (0,1)$ then the $r$-Hausdorff capacity of $X$ is denoted by $H^r(X)$ and is defined to be the infimum of all $\sum_{i=0}^\infty λ(I_i)^r$ where $\{I_i\}_{i\inω}$ is a cover of $X$ by intervals. The $r$ Hausdorff capacity has the same null sets as the $r$-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given $r < 1$, it is possible to enlarge a model of set theory, $V$, by a generic extension $V[G]$ so that the reals of $V$ have Lebesgue measure zero but still have positive $r$-Hausdorff capacity.

math.LO↗
arXiv1994

Decomposing Baire class 1 functions into continuous functions

Let dec be the least cardinal kappa such that every function of first Baire class can be decomposed into kappa continuous functions. Cichon, Morayne, Pawlikowski and Solecki proved that cov(Meager) <= dec <= d and asked whether these inequalities could, consistently, be strict. By cov(Meager) is meant the least number of closed nowhere dense sets required to cover the real line and by d is denoted the least cardinal of a dominating family in omega^omega. Steprans showed that it is consistent that cov(Meager) not= dec. In this paper we show that the second inequality can also be made strict. The model where dec is different from d is the one obtained by adding omega_2 Miller - sometimes known as super-perfect or rational-perfect - reals to a model of the Continuum Hypothesis. It is somewhat surprising that the model used to establish the consistency of the other inequality, cov(Meager) not= dec, is a slight modification of the iteration of super-perfect forcing.

math.LO↗
arXiv1993

Addendum to ``Maximal Chains in $\fomom$ and Ultrapowers of the Integers''

Upon presenting the proof of Theorem 3.3 in "Maximal chains in $$ and ultrapowers of the integers" I discovered that it is not entirely correct and certainly some details should be added. I have therefore written an addendum to the paper and made it available by ftp. Unfortunately the published version will be somewhat wanting.

math.LO↗
arXiv1993

Somewhere trivial automorphisms

It is shown to be consistent that there is a non-trivial autohomeomorphism of beta N while all such autohomeomorphisms are trivial on some open set. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.

math.LO↗
arXiv1993

Dominating functions and graphs

A graph is called dominating if its vertices can be labelled with integers in such a way that for every function f: omega-> omega the graph contains a ray whose sequence of labels eventually exceeds f. We obtain a characterization of these graphs by producing a small family of dominating graphs with the property that every dominating graph must contain some member of the family.

math.LO↗
arXiv1993

Sums of Darboux and continuous functions

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 function $F$ there is a nowhere constant continuous function $f$ such that $F+f$ is still Darboux. This answers questions raised by B.~Kirchiem and T.~Natkaniec who have shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere constant, continuous function fails to be Darboux.

math.LO↗