23 research outputs found

### Un nuovo integrale per il problema delle primitive

We introduce a new type of integral, which solves the problem of finding antiderivatives but which does not contain the improper integral

### Integrals and Banach spaces for finite order distributions

summary:Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: HÃ¶lder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem

### Remarks on the first return integral

Some pathological properties of the first-return integrals are explored. In particular it is proved that there exist Riemann improper integrable functions which are first-return recoverable almost everywhere, but not first-return integrable, with respect to each trajectory. It is also proved that the usual convergence theorems fail to be true for the first-return integrals

### On the first return integrals

Some pathological properties of the first-return integrals are explored. In particular it is proved that there exist Riemann improper integrable functions which are first-return recoverable almost everywhere, but not first-return integrable, with respect to each trajectory. It is also proved that the usual convergence theorems fail to be true for the first-return integrals

### Giovanni Battista Guccia: pioneer of international cooperation in mathematics

This book examines the life and work of mathematician Giovanni Battista Guccia, founder of the Circolo Matematico di Palermo and its renowned journal, the Rendiconti del Circolo matematico di Palermo. The authors describe how Guccia, an Italian geometer, was able to establish a mathematical society in Sicily in the late nineteenth century, which by 1914 would grow to become the largest and most international in the world, with one of the most influential journals of the time. The book highlights the challenges faced by Guccia in creating an international society in isolated Palermo, and places Gucciaâ€™s activities in the wider European context through comparisons with the formation of the London Mathematical Society and the creation of Mittag-Lefflerâ€™s Acta Mathematica in Stockholm. Based on extensive searches in European archives, this scholarly work follows both historical and scientific treads, and will appeal to those interested in the history of mathematics and science in general

### Multipliers for generalized Riemann integrals in the real line

summary:We use an elementary method to prove that each $BV$ function is a multiplier for the $C$-integral