A model with no magic sets

We will prove that there exists a model of ZFC+``c= omega_2'' in which every M subseteq R of cardinality less than continuum c is meager, and such that for every X subseteq R of cardinality c there exists a continuous function f:R-> R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R-> R

Klasyczne i nowoczesne źródła finansowania majątku w przedsiębiorstwie

Sprawozdawczość finansowa podlega w ostatnich latach nieustającej krytyce. Uważa się, że nie jest ona w stanie sprostać rosnącym potrzebom i wymaganiom inwestorów, którzy nie zadowalają się już suchą informacją z raportów finansowych. Decyzje o lokowaniu kapitału wymagają obecnie danych o charakterze finansowym i niefinansowym, dobrowolnych ujawnień informacji niewymaganych prawem, o znacząco rozszerzonym zakresie, pozwalających na przewidywanie sytuacji finansowej i wyników jednostki w przyszłości. W erze społeczeństwa informacyjnego raport finansowy jednostki gospodarczej stopniowo ewoluuje w kierunku raportowania biznesowego. Inwestorzy potrzebują danych o znacznie większej przejrzystości, zrozumiałych i często wspomaganych informacjami opisowymi. Prezentowana publikacja stanowi głos w dyskusji nad ewolucyjnym charakterem, kształtem, kierunkami rozwoju oraz perspektywami współczesnej rachunkowości.Publikacja finansowana ze środków Rektora Uniwersytetu Łódzkiego

On additive almost continuous functions under CPAprismgame

We prove that the Covering Property Axiom CPAprismgame, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f from R to R whose graph is of measure zero. We also show that, under CPAprismgame, there exists a Hamel basis H for which the set E+(H), of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin\u27s axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that CPAprismgame implies its seemingly stronger version, in which \omega1-many games are played simultaneously

Set Theoretic Real Analysis

This article is a survey of the recent results that concern real functions (from Rn into R) and whose solutions or statements involve the use of set theory. The choice of the topics follows the author\u27s personal interest in the subject, and there are probably some important results in this area that did not make to this survey. Most of the results presented here are left without the proofs

Lipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski C1 Interpolation

We present a simple argument that for every continuous function f : R → R its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the C 1 free interpolation theorem, that for every continuous function f : R → R there exists a continuously differentiable function g : R → R which agrees with f on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with the Lebesgue measure theory

On the Composition of Derivatives

We show that there exists a derivative f : [0, 1] → [0, 1] such that the graph of f ◦ f is dense in [0, 1]2 , so not a Gδ-set. In particular, f ◦ f is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative

Continuous images of big sets and additivity of s0 under CPAprism

We prove that the Covering Property Axiom CPAprism, which holds in the iterated perfect set model, implies the following facts. There exists a family G of uniformly continuous functions from R to [0,1] such that G has cardinality \omega1 \u3c \continuum and for every subset S of R of cardinality \continuum there exists a g in G with g[S]=[0,1]. The additivity of the Marczewski\u27s ideal s0 is equal to \omega1 \u3c \continuum