From the simplest point of view, transseries concern manipulations on formal series, or a new kind of expansion for real-valued functions. But transseries constitute much more than that---they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries---also known as the transline---is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: dynamical systems, model theory, computer algebra, surreal numbers. This paper is an exposition for the non-specialist mathematician. All a mathematician needs to know in order to apply transseries.

The present paper is a survey on $C^{k,1}$ functions. Both theoretical and numerical results related to this class of nonsmooth functions are presented. We also give few new results and pose some open problems for further investigation.

A function $f: {\mathbb R} \rightarrow {\mathbb R}$ is vertically [horizontally] rigid for $C \subseteq (0,\infty)$ if $graph(cf)$ [$graph(f(c\;\cdot))$] is isometric with $graph(f)$ for every $c \in C$. $f$ is vertically [horizontally] rigid if this applies to $C= (0,\infty)$. Balka and Elekes have shown that a continuous function $f$ vertically rigid for an uncountable set $C$ must be of the form $f(x)=px+q$ or $f(x)=pe^{qx}+r$, in this way confirming Jancović's conjecture saying that a continuous $f$ is vertically rigid if and only if it is of one of these forms. We prove that their theorem actually applies to every $C \subseteq (0,\infty)$ generating a dense subgroup of $((0,\infty),\cdot)$, but not to any smaller $C$. A continuous $f$ is shown to be horizontally rigid if and only if it is of the form $f(x)=px+q$. In fact, $f$ is already of that kind if it is horizontally rigid for some $C$ with $card(C \cap ((0,\infty) \setminus \{1\})) \ge 2$

Consider a fixed differentiable curve $K$ (which is the boundary of a convex domain) and a family indexed by $\lambda\in [0,1]$ of differentiable curves $L = L_\lambda$ such that they are the boundary of convex (connected) domains $A_\lambda$. Suppose that for $\lambda_1<\lambda_2$ we have $A_{\lambda_1}\subset A_{\lambda_2}$. Then, the number of $n$-Poncelet pairs is given by $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd$(m,n)=1$. The curve $K$ does not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion.

Itô's Formula is the stochastic analogue of the change of variable formula for deterministic integrals. It is a useful tool in dealing with stochastic integration. In this paper, using Henstock's approach, we derive two versions of Itô's Formula. Henstock's or generalized Riemann approach has been successful in giving an alternative definition of stochastic integral, which is more explicit, intuitive and less measure theoretic. Henstock's approach provides a simpler and more direct proof of Itô's Formula, although we do not claim that it is a generalization of the classical results.

An almost isometry-invariant set $A\subset \mathbb{R}$ satisfies $|gA\ \Delta \ A| < \mathbf{c}$ for any isometry $g$ acting on $\mathbb{R}$, where ${\bf c}$ is the cardinality of the continuum. A shading is any set $S\subseteq \mathbb{R}$ in which $\frac{\mu(S\cap I)}{\mu(I)}$ has the same constant value for every finite interval $I$, for any Banach measure $\mu$. (A Banach measure is a finitely additive, isometry-invariant extension of the Lebesgue measure to $2^{\mathbb{R}}$.) In this paper we prove several theorems that show how these two types of sets are related. We also prove several sum and difference set results for almost isometry-invariant sets. Finally, we completely solve a problem involving subsets of Archimedean sets first posed by R. Mabry and partially solved by K. Neu.

We generalize a classical Steinhaus theorem replacing addition by any two variable function which is differentiable. We deal with category and measure case. Measure case was done before in papers [3], [9]. We give a proof which can be easily generalized to category case. We also obtain a variety of examples of nonmeasurable sets which have interesting algebraic properties.

In 2000 J.Borsík, J. Doboš, and M. Repický characterized sums of quasi-continuous functions with closed graphs. More precisely, they showed that such a sum must be Baire one star, and proved that each Baire one star function defined on a separable metric space which is Baire in the strong sense is the sum of three quasi-continuous functions with closed graphs. They showed also that not every Baire one star function defined on $\mathbb{R}$ is the sum of two quasi-continuous functions with closed graphs, and asked for characterization of such sums. The goal of this article is to present the required characterization.

In the whole article, the function is understood as the real valued function of real variables. It is well known that the construction of semicontinuous functions is possible either as a limit (in the sense of pointwise convergence) of the monotonous sequence of continuous functions or with assistance of the certain system of associated sets. In the first case a good understanding of the constructed function is obtained. However in the second case the Darboux property of semicontinuous function is assured when certain requirements on the system of associated sets are satisfied. Therefore the combination of both approaches seems to be the optimal method when a semicontinuous function with Darboux property is constructed. Such a combined method is used when the main theorem of this paper is proven. Later, the validity of the main theorem helps us to address the problem published by J. G. Ceder and T. L. Pearson in [2].

Romanovski introduced a procedure, Romanovski's lemma, to construct the Denjoy integral without the use of transfinite induction. Here we give two versions of Romanovski's lemma which hold in general topological spaces. We provide several applications in various areas of mathematics.

The main aim of this paper is to prove that the Nörlund logarithmic means $t_{n}^{\kappa }f$ of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that $t_{n}^{\kappa }f$ converges in measure on $I$ for every function $f\in L(I)$ and $t_{n,m}^{\kappa }f$ converges in measure on $I^{2}$ for every function $f\in L\ln ^{+}L(I^{2}).$ Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.

For a natural number $n$, selecting a $2n$-interval symmetric wavelet set by making use of a result of Arcozzi, Behera and Madan [J. Geom. Anal. 13 (2003), 557-579], we construct a family of symmetric wavelet sets accumulating at the origin. Such a family of wavelet sets is also obtained from a family of symmetric six-interval wavelet sets provided by them in the same paper. Three-interval wavelet sets are employed in having a family of wavelet sets accumulating at the origin which are non-symmetric. Further, we obtain a larger class of $H^{2}$-wavelet sets accumulating at the origin, which include the one given by Behera in [Bull. Polish Acad. Sci. Math. 52 (2004), 169-178]. Finally, non-MSF non-MRA wavelets are obtained through the selected family of $2n$-interval symmetric wavelet sets.

We prove the ideal versions of Lunina's Theorem on convergence and divergence sets of real continuous functions defined on a metric space for $F_\sigma$-ideals and ideals with Baire property.

What are necessary and sufficient conditions in order that a function may be an indefinite integral in the Riemann sense? The problem has been explicitly posed in a short note [3] published by Erik Talvila in 2008 in THIS EXCHANGE. Since neither Erik nor I have been able to find a solution in the literature I propose the following solution which is the sole subject of the paper.

Here we give a positive answer to the so-called Product Measure Problem under the relatively simple hypothesis that the measure in one of the factors is inner regular and its support with the induced Hausdorff topology is locally metrizable. No special hypothesis on the other topological measure space is required. The proof is inspired by the rather imprecise conjecture that the open sets of a topological space must satisfy some restrictions in order to support any strictly positive $\sigma $-finite measure.

In this note we show that a measure can be constructed on an arbitrary set by iterated arbitrary mass distribution over arbitrary subsets of the set.

We present a convergence theorem for the Henstock-Kurzweil integral of functions taking values in a locally convex topological vector space, which is sequentially complete with respect to its weak topology.

We apply Hardy-Littlewood's Tauberian theorem to obtain an estimate on the harmonic expansion of a complex measure on the unit sphere, using a monotonicity property for positive harmonic functions.

This note corrects a very misleading typographical error in the paper "Change of variable theorems for the KH integral", Real Analysis Exchange 35(1), pp. 167--194.