Samy Zafrany

We show that every analytic filter is generated by a prefilter, every filter is generated by a prefilter, and if is a prefilter then the filter generated by it is also . The last result is unique for the Borel classes, as there is a -complete prefilter P such that the filter generated by it is -complete. Also, no complete coanalytic filter is generated by an analytic prefilter. The proofs use König's infinity lemma, a normal form theorem for monotone analytic sets, and Wadge reductions.

For every μ < ω1, let Iμ be the ideal of all sets S⊆ ωμ whose order type is <ωμ. If μ = 1, then I1 is simply the ideal of all finite subsets of ω, which is known to be Σ02-complete. We show that for every μ < ω1, Iμ is Σ02μ-complete. As corollaries to this theorem, we prove that the set WOωμ of well orderings R⊆ω × ω of order type <ωμ is Σ02μ-complete, the set LPμ of linear orderings R⊆ ω × ωthat have a μ-limit point is Σ02μ+1-complete. Similarly, we determine the exact complexity of the set LTμ of trees T⊆ <ωω of Luzin height <μ, the set WRμ of well-founded partial orderings of height <μ, the set LRμ of partial orderings of Luzin height <μ, the set WFμ of well-founded trees T⊆ <ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.

Fuzzy logics based on triangular norms and their corresponding conorms are investigated. An affirmative answer to the question whether in such logics a specific level of satisfiability of a set of formulas can be characterized by the same level of satisfiability of its finite subsets is given. Tautologies, contradictions and contingencies with respect to such fuzzy logics are studied, in particular for the important cases of min-max and Lukasiewicz logics. Finally, fundamental t-norm-based fuzzy logics are shown to provide a gradual transition between minmax and Lukasiewicz logics. Key words: Fuzzy Logics, Min-max logic, Lukasiewicz Logic, Triangular Norms, Satisfiability. AMS-Classification: 03B52, 03B50, 03B05 0

Basic undergraduate course  on Fourier series, Fourier transforms, and Laplace transforms. It is an expanded and polished version of the authors' notes for a one-semester course intended for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. The material is self contained with numerous exercises and various examples of applications.

\begin{abstract}
Let $X$ be a perfect separable complete metric space.
Let $G=(V,E)$ be a simple graph such that $V\subseteq X$.
Let $\vec{E}=\setn{(x,y)}{\{x,y\}\in E}$.
We call $G$ {\em analytic} if $\vec{E}$ is an analytic subset of $X^2$.
Similarly, $G$ is {\em $\sigma$-analytic}
if $\vec{E}$ belongs to the $\sigma$-algebra
generated by all the analytic subsets of $X^2$.
We call $G$ a {\em weighted graph} if an ordinal is assigned to every edge.
Let $G$ be an analytic connected weighted graph. We prove that
under some restrictions $G$ has a $\sigma$-analytic minimal
spanning tree $T$ (i.e, if we replace one edge in $T$ by a lighter edge,
then $T$ stops being a spanning tree).
We also give examples that show that this result is optimal.
\end{abstract}
Key words:
weighted graphs, minimal spanning tree, analytic set,
uniformization, projective hierarchy.
AMS-Classification: 03B52, 03B50, 03B05

\begin{abstract}
We prove a topological version of a combinatorial theorem due to
R.~A.~Brualdi and J.~S.~Pym. Their theorem is a generalization
of the well known Cantor-Bernstein Theorem as well as of
other reformulations of it
(Banach's mapping theorem and \Konig's matching theorem).
Let $G=(V,E)$ be a simple graph such that $V$
is a perfect separable complete metric space.
Let $A,B\subseteq V$ be disjoint,
and let $A{*}B$ be the set of all paths $p=(x_1,x_2,\ldots,x_n)$ in $G$,
such that $x_1\in A$ and $x_n\in B$.
A {\em linking} $f:A\leadsto B$ is a subset $f\subseteq A{*}B$
such that for each $a\in A$ there exists exactly one path $p\in f$
whose initial vertex is $a$.
We call $f$ {\em injective} if every distinct
$p,q\in f$ are vertex-disjoint,
and we call $f$ {\em bijective} if, in addition, for every $b\in B$
there is a path in $f$ that ends with $b$.
The topological version states: if $f:A\leadsto B$ and $g:B\leadsto A$
are Borel injective linkings, then there exists a Borel
bijective linking $h:A\leadsto B$.
We also consider possible extensions
of this result in various directions.
\end{abstract}
Key words: topological graph, Borel set, linking, perfect matching
AMS-Classification: 04A15, 05C10, 03E15

This volume is a an enhanced digital edition of the original book. It provides the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications