相关论文: The Work of R.E. Borcherds
The objective of this paper is to report on recent progress on Strichartz estimates for the Schr\"odinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in…
We commend the authors for an exciting paper which provides a strong contribution to the emerging field of probabilistic numerics (PN). Below, we discuss aspects of prior modelling which need to be considered thoroughly in future work.
This is a survey article to be part of the Encyclopedia of Mathematical Physics, to be published by Elsevier in the beginning of 2006.
Bornmann and Leydesdorff (in press) proposed methods based on Web-of-Science data to identify field-specific excellence in cities where highly-cited papers were published more frequently than can be expected. Top performers in output are…
The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the…
In 1841, Ferdinand Minding published ``Ueber die Bestimmung des Grades einer durch Elimination hervorgehenden Gleichung'' in Crelle's Journal. His main theorem represents the first (implicit) appearance of the BKK bound (and even some of…
We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.
We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. L\"uneburg, of when $\mathbb Z[\theta]$ is the ring of integers of $\mathbb Q(\theta)$, and apply our results to some concrete $2$-towers of…
The ring of ad\`eles of a global field and its group of units, the group of id\`eles, are fundamental objects in modern number theory. We discuss a formalization of their definitions in the Lean 3 theorem prover. As a prerequisite, we…
We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices.…
The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their…
This is a revised and slightly expanded version. We point out that in the previous summary, "without cohomology" should really read "almost without cohomology" because of the proof of Lemma 2, that the idea to consider effective motives…
We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.
A few new indices to characterize the scientific output of scientists are defined in the paper. These indices are compared with -index and its alternative indices using some proven assertions. The gd-indices are introduced as extensions of…
The purpose of this note is to present an alternative proof of a result by H. Smith and C. Sogge showing that in odd dimension of space, local (in time) Strichartz estimates and exponential decay of the local energy for solutions to wave…
We show that the greater the scientific wealth of a nation, the more likely that it will tend to concentrate this excellence in a few premier institutions. That is, great wealth implies great inequality of distribution. The scientific…
This note announces recent exciting progress on the frontier between algebraic topology and probability theory. It is intended for a journal which publishes such announcements (without an abstract, typically in Russian). A description of a…
Formulas for calculating the Riesz function, introduced by Marcel Riesz in connection with the Riemann hypothesis, are derived; and the behavior of the Riesz function is discussed.
We present a new elementary proof of a theorem due to Harald Bohr, which states that an unbounded, analytic, and almost periodic function in a half-plane can be written as the sum of two analytic functions: the first is unbounded and…
Mathematics has become inescapable in modern, digitized societies: there is hardly any area of life left that isn't affected by it, and we as mathematicians play a central role in this. Our actions affect what others, in particular our…