It is an old idea to consider whether a function on $\mathbb{R}^N$ that is smooth in each variable separately is in fact jointly smooth. It turns out that some uniformity of estimates in each variable is necessary for such a result. More recently, there have been studies of functions that are smooth along integral curves of certain vector fields. Depending on the commutator properties of the vector fields, different types of results may be obtained. Another recent idea is that if one has smoothness along all curves then the uniformity hypothesis may be dropped. In the present paper we explore all these approaches to the problem in a variety of new norms. We present new, simpler proofs of some classical results. We also explore new theorems in the real analytic category.

In this paper, we survey progress on the general theory for path integrals as envisioned by Feynman. We introduce a new class of spaces ${\bf{KS}}^p(\R^n)$ for $1 \le p \le \infty$ and $n \in \N$, and their Sobolev counterparts, ${\mathbf{KS}}^{m, p}(\R^n)$, for $1 \le p \le \infty,\; m \in \N$, which allow us to construct the path integral in the manner originally intended by Feynman. Each space contains all of the standard Lebesgue spaces, ${\bf{L}}^{ p}(\R^n)$ (respectively Sobolev spaces, ${\bf{W}}^{m, p}(\R^n)$), as compact dense embeddings. More importantly, these spaces all provide finite norms for nonabsolutely integrable functions. We show that both the convolution and Fourier transform extend as bounded linear operators. This allows us to construct the path integral of quantum mechanics in exactly the manner intended by Feynman. Finally, we then show how a minor change of view makes it possible to construct Lebesgue measure on (a version of) $\mathbb{R}^{\infty}$ which is no more difficult than the same construction on $\mathbb{R}^{n}$. This approach allows us to construct versions of both Lebesgue and Gaussian measure on every separable Banach space, which has a basis.

If $A$ is a ${\bf \Sigma}^{0}_{\xi}$ set and $A_{n}$ $(n \omega)$ are Borel sets then we call $\{A_{n} \colon n \omega\}$ a presentation of $A$ if $A = \bigcup_{n \omega}A_{n}$ and $A_{n}$ $(n \omega)$ have lower Borel class than $A$ has. We show that for $2 \leq \xi \omega_{1}$ it is not possible to assign a presentation to ${\bf \Sigma}^{0}_{\xi}$ sets in a monotone way; i.e., it is not possible to define functions $f_{n} \colon {\bf \Sigma}^{0}_{\xi} \rightarrow {\bf \Pi}^{0}_{\xi}$ $(n \omega)$ such that for every $A \in {\bf \Sigma}^{0}_{\xi}$ we have $A = \bigcup_{n \omega}f_{n}(A)$ and $A, A' \in {\bf \Sigma}^{0}_{\xi}$, $A \subseteq A'$ implies $f_{n}(A) \subseteq f_{n}(A')$ $(n \omega)$. This answers a question of M\'arton Elekes in the negative. We also show the nonexistence of monotone presentation for Borel functions.

We study a relation between the porosity of sets in Euclidean spaces and the Darboux property of (relative) Fréchet derivatives.

We relate the concepts of entropy and pressure to that of KMS states for $C^*$-algebras. Several different definitions of entropy are known in our days: the one we present here is quite natural, extending the usual one for Dynamical Systems in Thermodynamic Formalism Theory, being basically obtained from transfer operators (also called Ruelle operators) and having the advantage of being very easily introduced. We also present a concept of pressure as a min-max principle. Later on, we consider the concept of a KMS state as an equilibrium state for a potential, in the context of $C^*$-algebras, and we show that there is a relation between equilibrium measures and KMS states for certain algebras arising from a continuous transformation.

The purpose of this work is to ascertain when arithmetic operations with periodic functions whose domains may not coincide with the whole real line preserve periodicity.

We investigate continuous selfmappings of a real compact interval, which are, in a sense, close to iterable ones, that is embeddable into continuous iteration semigroup. This is a continuation of a research initiated by W.~Jarczyk. We present some necessary and sufficient conditions of weak almost iterability and $M-$weak almost iterability. Classes of functions investigated here are generalizations of a class of almost iterable functions introduced by W. Jarczyk. This refers to the problem posed by E.~Jen.

For $f : [0,1] \to \mathbb{R}^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^{\omega}$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n \omega} f(|y_{n} - x_{n}|) \infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\mathbf{E}_{f}$. Our results indicate that for every $1 \leq p q \infty$, the order $\leq_{B}$ of Borel reducibility on the set of equivalence relations $\mathbf{E}:\mathbf{E}_{Id^\mathcal{p}} \leq_B \mathbf{E} \leq_B \mathbf{E}_{Id^\mathcal{q}}$ is more complicated than expected, e.g.\ consistently every linear order of cardinality continuum embeds into it

A \textquoteleft pressure' functional $\Phi^s (T_1,\ldots ,T_N)$, defined as the limit of sums of singular value functions of products of linear mappings $(T_1,\ldots ,T_N)$, is central in analysing fractal dimensions of self-affine sets. We investigate the continuity of $\Phi^s$ with respect to the linear mappings $(T_1,\ldots ,T_N)$ which underlie the self-affine sets.

We prove a representation theorem for bounded linear functionals with domain the set of all real-valued, quasi-continuous functions defined on a closed interval; thus, giving a characterization of a class of bounded linear functionals.

Let $C^{(n)}[0,1]$ stand for the Banach space of functions $f:[0,1]\rightarrow \mathbb{R}$ with continuous $n$ -th derivative. We prove that if $B_{1},B_{2}$ are open balls in $C^{(n)}[0,1]$ then the set $B_{1}\cdot B_{2}=\{f\cdot g:f\in B_{1},g\in B_{2}\}$ has non-empty interior in $C^{(n)}[0,1].$ This extends the result of [1] dealing with the space of continuous functions on $[0,1]$.

The notion of smoothness (according to Riemann) is introduced for functions of two variables and some of their properties are established. As an application we prove the uniform smoothness of an everywhere continuous sum of a double trigonometric series in the complex form which is obtained by twice term-by-term integration, over every variable rectangle $[0,x] \times [0,y] \subset [0,2\pi]$ of a double trigonometric series in the complex form absolutely converging at some point. An analogous consideration is given to a double trigonometric series in the real form, the absolute values of whose coefficients form a converging series.

We give results about almost continuous multi-valued functions and a characterization of compact almost continuous $M$-retracts of the Hilbert cube $Q$, where almost continuity is in the sense of Stallings instead of Husain. For instance, each connectivity or almost continuous point to closed-set valued multi-function $f:I \to I$, where $I=[0\,,\,1]$, has a fixed point; i.e., a point $x\in I$ such that $x\in f(x)$. When $Y$ is a compact subset of $Q$, a sufficient condition is given for a continuous multifunction $r:Y\to Y$, with $x\in r(x)$ $\forall x\in Y$, to have an almost continuous multi-valued extension $r:Q \to Y$.

Weighted Hardy-type inequalities between suitable amalgams $\ell^q(L^{p},u)$ and $\ell^{\bar q}(L^{\bar p},v)$ are characterized. The Hardy-type operator involved in the inequalities involves functions which are not necessarily non-negative.

Let $\{\varphi_n(x)$, $n=1,2,\dots\}$ be an arbitrary complete orthonormal system (ONS) on the interval $I:=[0,1)$ that consists of a.e. bounded functions. Then there exists a rearrangement $\{ \varphi_{\sigma_1(n)}$, $n=1,2, \dots\}$ of the system $\{\varphi_n(x)$, $n=1,2,\dots\}$ that has the following property: for arbitrary nonnegative, continuous and nondecreasing on $[0,\infty)$ function $\phi(u)$ such that $u\phi (u)$ is a convex function on $[0,\infty)$ and $\phi (u) = o(\ln u)$, $u \to \infty$, there exists a function $f \in L(I^2)$ such that $\int_{I^2} | f(x,y) |$ $\phi( | f(x,y) | )\;dx\; dy \infty$ and the sequence of the square partial sums of the Fourier series of $f$ with respect to the double system $\{ \varphi_{\sigma_1 (m)}(x)\varphi_{\sigma_1 (n)}(y)$, $m,n \in\N \}$ on $I^2$ is essentially unbounded in measure on $I^2$.

We prove that the set of all functions in $C[0,1]$, with countably many points at which the derivative does not exist, is ${\pmb \Pi}^1_1$--complete, in particular non--Borel. We obtain the classical Mazurkiewicz's theorem and the recent result of Sofronidis as corollaries from our result.

In this paper we introduce a notion of the path Darboux property. We examine the basic properties and investigate relationships between the path Darboux property and path continuity.

The main results presented in this paper concern generalized continuous multivalued mappings. An attempt has been made to formulate sufficient conditions under which convergence of nets of multivalued mappings preserves generalized continuity.

In this paper we apply some elements of real analysis theory to the distinguishing of the certain subclass of the class of functions possessing Sharkovsky property. The main aim of this is connected with the answer to the following question: what kind of assumption should we impose on Sharkovsky function $f$ in order to have that there exists a ring of functions possessing Sharkovsky property containing $f$?