Nearly Geodesic Riemannian Cubics in SO(3)

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are known for Riemannian cubics, and the asymptotics of Riemannian cubics in SO(3) are reasonably well understood. The mathematical properties and medium-term behaviour of Riemannian cubics in SO(3) are known to be be extremely rich, but there are numerical methods for calculating Riemannian cubic splines in practice. What is missing is an understanding of the short-term behaviour of Riemannian cubics, and it is this that is important for applications. The present paper fills this gap by deriving approximations to nearly geodesic Riemannian cubics in terms of elementary functions. The high quality of these approximations depends on mathematical results that are specific to Riemannian cubics

Topics in Multiplicative Number Theory

This thesis is comprised of four articles in multiplicative number theory, a subfield of analytic number theory that studies questions related to prime numbers and multiplicative functions. A central principle in multiplicative number theory is that multiplicative structures, such as the primes or the values of a multiplicative function, should not correlate with additive structures of various types. The results in this thesis can be interpreted as instances of this principle. In the first article, we consider the problem of finding almost primes in almost all short intervals, which is a natural approximation to the problem of finding primes in short intervals. We show that almost all intervals of nearly optimal length contain a product of exactly three primes. For products of exactly two primes, we improve a result of Harman. The proofs are based on careful analysis of Dirichlet polynomials related to almost primes. The second article is about the Goldbach problem for a sparse subset of the primes. Vinogradov famously showed that any large odd number is the sum of three primes, so it is natural to study the same problem with the summands coming from a subset of the primes. Improving a result of Matomäki, we show that a special set of primes, consisting of primes representable as one plus the sum of two squares, satisfies the ternary Goldbach problem. We also establish a number of other additive results for this same set of primes. The proofs use sieve methods and transference principles for additive equations in primes. We also study the Möbius function and its autocorrelations. A famous conjecture of Chowla asserts that products of shifts of the Möbius function should have mean zero. In the third article, together with T. Tao we settle a logarithmic version of this conjecture in all the cases involving an odd number of shifts. This complements Tao's earlier result that the two-point Chowla conjecture holds with logarithmic weights. Lastly, in the fourth article, we study binary correlations of multiplicative functions with logarithmic weights. We prove an asymptotic formula for these correlations for a wide class of multiplicative functions, extending an earlier result of Tao. We then derive a number of applications regarding the largest prime factors of consecutive integers, including a logarithmic version of a conjecture of Erdős and Turán. Moreover, we prove a new estimate for character sums over reducible quadratic polynomials.Tämä väitöskirja koostuu neljästä artikkelista multiplikatiivisessa lukuteoriassa, joka on alkulukuja ja multiplikatiivisia funktioita tutkiva analyyttisen lukuteorian haara. Keskeinen periaate multiplikatiivisessa lukuteoriassa on, että multiplikatiivisten objektien (kuten alkulukujen tai multiplikatiivisten funktioiden arvojen) ei pitäisi korreloida additiivisten objektien kanssa. Tämän väitöskirjan tulokset voidaankin tulkita kyseisen periaatteen ilmentyminä. Ensimmäisessä artikkelissa tarkastelemme melkein alkulukujen löytämistä melkein kaikilta lyhyiltä väleiltä; tämä on luonnollinen approksimaatio alkulukujen löytämiselle lyhyiltä väleiltä. Osoitamme, että melkein kaikki välit, joiden pituus on lähes optimaalisen lyhyt, sisältävät tasan kolmen alkuluvun tulon. Tasan kahden alkuluvun tulojen tapauksessa parannamme Harmanin tulosta. Todistukset perustuvat melkein alkulukuihin liitettyjen Dirichlet'n polynomien tarkkaan analysointiin. Toinen artikkeli koskee Goldbach-ongelmaa eräälle harvalle osajoukolle alkulukuja. Vinogradov osoitti kuuluisassa työssään, että jokainen riittävän suuri pariton luku on kolmen alkuluvun summa, joten on luonnollista tarkastella vastaavaa ongelmaa alkulukujen osajoukoille. Parantaen Matomäen tulosta osoitamme, että vastaus ternääriseen Goldbach-oneglmaan on positiivinen niiden alkulukujen joukolle, jotka voidaan esittää ykkösen ja kahden neliöluvun summana. Osoitamme myös useita muita additiivisia tuloksia samalle alkulukujen osajoukolle. Todistukset käyttävät seulamenetelmiä sekä ns. transferenssiperiaatteita additiivisille yhtälöille alkulukujen joukossa. Tutkimme myös Möbiuksen funktiota ja sen autokorrelaatioita. Chowlan kuuluisa konjektuuri väittää, että Möbiuksen funktioiden translaatioiden tuloilla pitäisi olla keskiarvo nolla. Kolmannessa artikkelissa yhdessä T. Taon kanssa ratkaisemme logaritmisen version tästä konjektuurista kaikissa tapauksissa, joissa translaatioiden määrä on pariton. Tämä täydentää Taon aikaisempaa tulosta, jonka mukaan kahden pisteen Chowlan konjektuuri pätee logaritmisilla painoilla. Lopuksi neljännessä artikkelissa tutkimme multiplikatiivisten funktioiden binäärisiä korrelaatioita logaritmisilla painoilla. Todistamme asymptoottisen kaavan näille korrelaatioille, joka pätee laajalle luokalle multiplikatiivisia funktioita ja parantaa Taon aikaisempaa tulosta. Johdamme sitten useita sovelluksia koskien peräkkäisten lukujen suurimpia alkutekijöitä -- mukaan lukien logaritmisen version eräästä Erdősin ja Turánin konjektuurista. Lisäksi todistamme uuden arvion karakterisummille yli jaollisen toisen asteen polynomin arvojen

The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces