Related papers: Transseries for beginners
Higher-order network analysis uses the ideas of hypergraphs, simplicial complexes, multilinear and tensor algebra, and more, to study complex systems. These are by now well established mathematical abstractions. What's new is that the ideas…
Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the $20^{\text{th}}$ century, they have since then spread to…
Multisets are an intuitive extension of the traditional concept of sets that allow repetition of elements, with the number of times each element appears being understood as the respective multiplicity. Recent generalizations of multisets to…
We give a presentation of Conway's surreal numbers focusing on the connections with transseries and Hardy fields and trying to simplify when possible the existing treatments.
This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a…
Time-series data exists in every corner of real-world systems and services, ranging from satellites in the sky to wearable devices on human bodies. Learning representations by extracting and inferring valuable information from these time…
In this paper we show an alternative way of defining Fourier Series and Transform by using the concept of convolution with exponential signals. This approach has the advantage of simplifying proofs of transforms properties and, in our view,…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space…
We investigate compositional iteration of fractional order for transseries. For any large positive transseries $T$ of exponentiality 0, there is a family $T^{[s]}$ indexed by real numbers $s$ corresponding to teration of order $s$. It is…
Prices of commodities or assets produce what is called time-series. Different kinds of financial time-series have been recorded and studied for decades. Nowadays, all transactions on a financial market are recorded, leading to a huge amount…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them.…
Multisets are sets that allow repetition of elements. As such, multisets pave the way to a number of interesting possibilities of theoretical and applied nature. In the present work, after revising the main aspects of traditional sets, we…
We show that the composition of omega-series by surreal numbers, or more generally by elements of any confluent field of transseries, is monotonic in its second argument. In particular, omega-series and LE-series interpreted as functions…
We use series expansions to study dynamics of equilibrium and non-equilibrium systems on networks. This analytical method enables us to include detailed non-universal effects of the network structure. We show that even low order…
This work presents an introduction to feature-based time-series analysis. The time series as a data type is first described, along with an overview of the interdisciplinary time-series analysis literature. I then summarize the range of…
In many real-world application, e.g., speech recognition or sleep stage classification, data are captured over the course of time, constituting a Time-Series. Time-Series often contain temporal dependencies that cause two otherwise…