Five-Minute Mathematics

by Ehrhard Behrends

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

Genres: Mathematics, Science, Nonfiction

380 pages, Paperback

First published January 1, 2008

Reviews

[Michael Kotsarinis · Rating 3 out of 5]

Μια συλλογή μικρών άρθρων από εφημερίδα με λίγο περισσότερη ανάπτυξη σε ορισμένα από αυτά. Παρουσιάζονται διάφορες πτυχές των μαθηματικών, μερικές πιο ενδιαφέρουσες για το ευρύ κοινό, μερικές λιγότερο. Δεν είναι άσχημη συλλογή αλλά προσωπικά η μεταπήδηση από το ένα θέμα στο άλλο και η ανομοιόμορφη ανάπτυξη δεν με ικανοποίησαν. Επίσης σε ορισμένα άρθρα δεν ήταν τελείως ξεκάθαρο που στόχευαν. Για όσους θέλουν μια γρήγορη ματιά σε μια πλειάδα μαθηματικών θεμάτων είναι καλό αλλά υπάρχουν πολύ καλύτερα βιβλία για εισαγωγή στον κόσμο των μαθηματικών.

A collection of short newspaper articles on mathematics. It is OK but there are better books out there to get you acquainted with the world of numbers.

[Stefania Pezzato · Rating 5 out of 5]

Interessante raccolta di articoli sugli argomenti matematici più disparati; utile per far capire l'importanza di questa materia, e comprensibile anche per chi vi è negato (come me!). Lo consiglio a chiunque.

[Evi S · Rating 4 out of 5]

4,5/5 ⭐

Ήταν πολύ καλό, περιέχει 100 μικρές "ιστορίες" μαθηματικών και σε κάνει να έρθεις πιο κοντά με αυτά κάτι που σίγουρα χρειάζεται. Επίσης μπορείς να διαβάσεις οποίο κεφάλαιο θες αφού το κάθε ένα είναι 2 σελίδες περίπου.

[Manolis · Rating 3 out of 5]

Κάποια θέματα τα αναλύει με όμορφο και κατανοητό τρόπο, άλλα όχι και τόσο ικανοποιητικά, γενικά δεν ξεχωρίζει από πολλά άλλα αντίστοιχα, σίγουρα κάποιος που δεν έχει εντρυφύσει στο θέμα θα το βρει ενδιαφέρον και με αρκετά topics..

[Maurizio Codogno · Rating 3 out of 5]

Invidia. Tanta invidia. Tra il 2003 e il 2005 il quotidiano tedesco Die Welt ha ospitato una rubrica settimanale dal nome "cinque minuti di matematica" (Fünf Minuten Mathematik); cento brevissimi articoli raccolti in questo libro. L'invidia è naturalmente al pensiero che almeno all'estero qualcuno abbia pensato che si puo parlare di matematica su un giornale, mentre in Italia la cosa è impossibile, e purtroppo non credo proprio che un libro come questo verrà nemmeno tradotto. Qui si tratta più che altro di divulgazione matematica, prendendo spesso a spunto le cose di tutti i giorni come le probabilità di vincere al lotto tedesco (probabilità molto maggiori di quelle del Superenalotto, per la cronaca). Dimostrazioni chiaramente non ce ne sono, e le formule sono spesso tralasciate, probabilmente perché si spaventano anche i lettori tedeschi. Se devo fare un appunto, i capitoletti mi sembrano spesso terminare di colpo e in tono minore; ma ad ogni modo siamo sempre a un prodotto che noi ce lo sogniamo.