Further Revelations on L.S.D.

In a recent paper the author showed that the distribution ofleading significant digits, l.s.d., resulting from successivemultiplications is logarithmic. In this paper these resultsare extended by establishing, still without any assumptionsof invariance to scaling, that this distribution remains invariantunder all further arithmetic operations—both multiplicativeand additive. The fact that 30% of decimal numbers have l.s.d.1 persists.     

A. I. Mal'tsev's problem on operations on groups

In response to A. I. Mal'tsev's problem (1948), associative operations are constructed on the class of all groups; these operations are distinct from the direct and free products and are hereditary for subgroups of the factors.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 225–249, 1989.     

Speech on L. V. Kantorovich

An extended version of the talk at the memorial session dedicated to L. V. Kantorovich on January 11, 2004 at the Euler International Mathematical Institute (St. Petersburg). __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 24–29.     

Distribution theory on P.C.F. fractals

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.     

论高斯

高斯(C.F.Gauss,1777—1855)生于德国Brunswick,1795—98求学于Gttinsen大学。1807—55任天文观测台台长及Gttingen大学教授。 大家都知道高斯的名言:“数学是科学的女王,算术是数学的女王”这里“算术”是在古希腊人的意义下理解的,指的初等数论,而区别于近代的解析数论。     

Energy measures on p.c.f. self-similar sets

On connected post critically finite (p.c.f.) self-similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy, and as an application we also compute the L²$L^2$ dimensions of these measures on some p.c.f. self-similar sets.     

Some remarks on number theory. II.

Like the previous paper of the same title [5] this note contains disconnected remarks on number theory.     

The Influence of P. A. Florensky on N. N. Luzin

The study of the early history of the Moscow school of the theory of functions has been greatly enhanced by the discovery of correspondence between one of its founders, N. N. Luzin, and the priest, theologian, philosopher, and scientist P. A. Florensky. The correspondence reveals that Luzin experienced a profound spiritual crisis in 1905 when his materialist worldview collapsed. This crisis continued for three years and was finally resolved when Luzin had a decisive encounter with the religious philosophy of Florensky. After this, Luzin's interest in mathematics gradually revived until he was able, by 1909, to commit himself to a career in mathematics.Copyright 1998 Academic Press.Les études concernant l'histoire des débuts de l'école moscovite de la théorie des functions ont été grandement éclairées par la découverte d'une correspondance entre l'un de ses fondateurs, N. N. Luzin, et P. A. Florenski, un prêtre, théologien, philosophe, et savant. Cette correspondance révèle que Luzin a traversé une crise spirituelle profonde à partir de 1905, lorsque sa perspective matérialiste du monde s'est effondrée. Cette crise a duré trois ans, et a trouvé sa résolution dans la rencontre décisive de Luzin avec la philosophie religieuse de Florenski. Par la suite, l'intérêt de Luzin pour les mathématiques s'est progressivement ravivé, jusqu'à ce qu'il soit en mesure de décider, en 1909, de consacrer sa vie aux mathématiques.Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.MSC 1991 subject classification: 01A30, 01A60, 01A80, 03-03     

A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge in the upper half-plane in the case of L² coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szegö-Kolmogorov-Krein condition is satisfied. Thus, we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.     

Geodesic flows on Riemannian g.o. spaces

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.     

D. G. near rings on certain groups

All the distributive near rings in which the index ofR inR is a prime have been determined. Certain d. g. near rings in whichR is the only normal subgroup ofR are also considered. It is also shown that if (R,+) is finite and not perfect then there is a distributive near ring in whichR ²0.