Related papers: Two encounters with the p-adic Stark conjecture
Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…
We measure the rotation periods of 19 stars in the {\it Kepler} transiting planetary systems, $P_{\rm rot, astero}$ from asteroseismology and $P_{\rm rot, phot}$ from photometric variation of their lightcurve. Two stars exhibit two clear…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences…
In this review I will describe a number of recent advances in extragalactic astronomy. First of all I will describe our current best estimates of the star formation history of the Universe. Then I will describe measurements of local…
The years 1998 to 2008 were very exciting years for cosmology. It was a pleasure to accept this invitation to describe my contributions to the development of our knowledge and understanding of the universe over the course of the past…
Karl Popper had proposed an experiment to test the standard interpretation of quantum mechanics. The proposal survived for many year in the midst of no clear consensus on what results it would yield. The experiment was realized by Kim and…
Peculiar measurements can be obtained on systems that undergo both pre- and post-selection. We prove a conjecture from [1] on logical Pre- and Post-Selection (PPS) paradoxes for a restricted case. We prove that all of these paradoxes admit…
The paper contains a proof of the Fontaine-Jannsen conjecture based on a crystalline version of the p-adic Poincar'e lemma (different proofs were found earlier by Faltings, Niziol and Tsuji).
The theory of Weil-Stark elements is used to develop an axiomatic approach to the formulation of refined versions of Stark's Conjecture. This gives concrete new results concerning leading terms of Artin $L$-series and arithmetic properties…
Let $K/k$ be an abelian extension of number fields with a distinguished place of $k$ that splits totally in $K$. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in $K$, called the Stark unit,…
Viewing Eratosthenes sieve as a discrete dynamic system, we show that every admissible instance of every admissible constellation of gaps arises and persists in Eratosthenes sieve. For an admissible constellation of length J, we show that…
We prove a 1973 conjecture due to Erd\H{o}s on the existence of Steiner triple systems with arbitrarily high girth.
We survey progress on the Kakeya conjecture in Euclidean space, with an emphasis on developments that have occurred since the previous surveys by Wolff and Katz-Tao.
This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential…
From the time the first stars formed over 13 billion years ago to the present, star formation has had an unexpectedly dynamic history. At first, the star formation rate density increased dramatically, reaching a peak 10 billion years ago…
In 1981 Icko Iben Jr published a paper entitled 'The carbon star mystery: why do the low mass ones become such, and where have all the high mass ones gone?', where he discussed the discrepancy between the theoretical expectation and its…
In this paper, we prove Sp\"ath's Character Triple Conjecture for $p$-solvable groups. This is a conjecture proposed by Sp\"ath during the reduction process of Dade's Projective Conjecture to quasisimple groups. In addition, as suggested by…
A new method for determining the stellar rotation period is proposed here, based on the detection of starspots during transits of an extra-solar planet orbiting its host star. As the planet eclipses the star, it may pass in front of a…
We study two kinds of conjectural bounds for the prime gap after the k-th prime $p_k$: (A) $p_{k+1} < (p_k)^{1+1/k}$ and (B) $p_{k+1}-p_k < \log^2 p_k - \log p_k - b$ for $k>9$. The upper bound (A) is equivalent to Firoozbakht's conjecture.…
We consider P\'olya urns with infinitely many colours that are of a random walk type, in two related version. We show that the colour distribution a.s., after rescaling, converges to a normal distribution, assuming only second moments on…