Twists of Hooley's Δ\Delta-function over number fields

We prove tight estimates for averages of the twisted Hooley $\Delta$-function over arbitrary number fields

Generalised divisor sums of binary forms over number fields

Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin's conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization of this problem, in particular replacing $\chi$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin's conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number

On the distribution of the density of maximal order elements in general linear groups

In this paper we consider the density of maximal order elements in $\mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $\mathbb{F}_q$ of size $q=p^r$, we compute the expected value of the said density and establish that it follows a distribution law.Comment: 20 pages, substantial corrections. Accepted for publication at The Ramanujan Journa

Finite saturation for unirational varieties

We import ideas from geometry to settle Sarnak's saturation problem for a large class of algebraic varieties

Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over Q that contains a conic defined over Q

Counting rational points on smooth cubic surfaces

We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.Comment: 11 pages, minor revisio

A Global Administrative Law Bibliography

Additive Manufacturing (AM) comprises a family of different technologies that build up parts by adding materials layer by layer at a time based on a digital 3D solid model. After thirty years of development, AM has become a mainstream manufacturing process with more materials and new technologies involved in this process. Undoubtedly, the most dramatic and challenging development of group of technologies has been the printing of metals. Nowadays, the use of AM for the production of parts for final products continues to grow. Organizations around the world are successfully applying the technology to the production of finished goods. AM allows design optimization and produces customized parts on-demand with almost similar material properties with the conventional manufactured parts. It does not require the use of coolants, fixtures, cutting tools and other assisting resources. The advantages of AM over conventional manufacturing can change the world of industry and lead to a new industrial revolution. In this research after reviewing mostly the different technologies and materials used in metallic AM, the application of them in a component of a passenger car engine is described. A criticality analysis is carried out in order to decide which AM development of the parts that compose the final product is more significant for the efficiency of the overall product. Based on that development a sustainability analysis is performed consisting of the analysis of the environmental impacts, the production cost analysis and the societal impact. But what has been derived from the analysis is that despite the lower environmental impact compared with the casting as a conventional method of forming of metals, AM is costly for the production of a small number of industrial products and its societal impact needs further investigation. In fact, the cost depends on the production volume, the batch size as well as the high price of the material powders and the building rates of the machines. In the future, with more advanced machines and cheaper material input the cost of metallic AM is going to drop dramatically. In spite of all the progress, the application of metallic AM is still not widespread. Since the materials as well as its technologies are still evolving, a better and more promising future is foreseen for metallic AM. Keywords: Additive Manufacturing, Direct Digital Manufacturing, Mass Production, Automotive, Internal Combustion Engine, Turbocharger, Development, Sustainability Analysis, Lifecycle AssessmentAdditive Manufacturing (AM) består av en familj av olika teknologier som bygger upp komponenter genom att lägga till material lager efter lager ett lager i taget baserat på en digital 3D solid modell. Efter trettio år av utveckling, har AM blivit en mainstream tillverkningsprocess med fler material och nya teknologier involverade i denna process. Utan tvekan har den mest dramatiska och utmanande utvecklingen inom denna grupp av teknologier varit tryckningen av metaller. Nuförtiden fortsätter användningen av AM för tillverkning av delar till slutprodukter att växa. Företag runtom i världen använder tekniken framgångsrikt för produktionen av färdiga varor. AM tillåter designoptimering och tillverkar kundanpassade delar on-demand med nästan samma materialegenskaper som konventionellt tillverkade delar. Det behövs inte användning av kylmedel, fixturer, skärverktyg och andra källor. Fördelarna med AM jämfört med konventionell tillverkning kan förändra den industriella världen och leda till en ny industriell revolution. Efter att mestadels ha gått igenom de olika tekniker och material som används i metallisk AM, beskrivs i denna forskning applikationen av dem i en komponent av motorn till en personbil. En kritikalitet analys görs för att bestämma vilken AM utveckling av de delar som utgör den slutgiltiga produkten som är viktigast för effektiviteten av den totala produktionen. Baserat på denna utveckling utförs en hållbarhetsanalys som består av en analys av miljöpåverkan, produktionskostnad och de samhälleliga effekterna. Men det som har härletts från analysen är att trots den lägre miljöpåverkan i jämförelse med gjutning som en konventionell metod för formning av metaller, är AM kostsamt för produktionen av ett fåtal industriella produkter och dess samhälleliga effekter behöver studeras ytterligare. I själva verket beror kostnaden på produktionsvolymen, satsstorleken samt det höga priset på materialpulvren och byggnadstakten av maskinerna. I framtiden, med mer avancerade maskiner och billigare material kommer kostnaden för metallisk AM att sjunka dramatiskt. Trots alla utveckling, är applikationen av metallisk AM fortfarande inte utbredd. Eftersom materialen samt dess teknik fortfarande är under utveckling förutses en bättre och mer lovande framtid för metallisk AM. Nyckelord: Additive Manufacturing, Direct Digital Manufacturing, massproduktion, bilindustrin, förbränningsmotor, Turbocharger, utveckling, hållbarhetsanalys, livscykelanalys