Counting joints in vector spaces over arbitrary fields

We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.Comment: Not intended for publication. References added and other minor edits in this versio

Laser spot welding of laser textured steel to aluminium

Laser welding of dissimilar metals (steel and aluminium) was investigated with the aim to increase the maximum tensile shear load of the Fe-Al joints. The increase was achieved by texturing the surface of steel prior to the laser spot welding process which was performed in a lap-joint configuration with the steel positioned on top of the aluminium and with a texture faced down to the aluminium surface. This configuration enabled an increase of the bonding area of the joints, because the molten aluminium filled in the gaps of the texture, without the need of increasing the process energy which typically leads to the growth of the intermetallic compounds. Different textures (containing hexagonally arranged craters, parallel lines, grid and spiral patterns) were tested with different laser welding parameters. The Fe-Al joints obtained with the textured steel were found to have up to 25% higher maximum tensile-shear load than the joints obtained with the untextured steel

On lines, joints, and incidences in three dimensions

AbstractWe extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for m⩾n, and Θ(m2/3n2/3+m+n) for m⩽n. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9]

Joints tightened

In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $\binom{k}{d-1}$ lines and their $d$-wise intersections to form $\binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.Comment: 10 page

Tight Bound and Structural Theorem for Joints

A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we include some discussion on the constants for the joints of varieties problem.Comment: 39 page

Уточнение механизма повреждаемости металла длительно эксплуатируемых сварных соединений паропроводов

Уточнили особенности повреждаемости метала сварных соединений паропроводов длительно эксплуатируемых в условиях ползучести, состоящих из сталей 15Х1М1Ф и 12Х1МФ. Предложена классификация повреждаемости, состоящая из пяти стадий. Рассмотрены механизмы образования пор ползучести и трещин усталости. Установлено, что наиболее интенсивно поры ползучести образуются на участке неполной перекристаллизации зоны термического влияния сварных соединений из теплоустойчивых сталей.The features of metal damageability of welded joints of steam lines, long operated under creep conditions, made of steels 15H1M1F and 12H1MF were specified. Damageability classification, consisting of five stages was proposed. The formation mechanisms of creep pores and fatigue cracks were considered. It was found that creep pores are formed most intensively in the area of incomplete recrystallization of heat-affected zone of welded joints made of heat-resistant steels. It was revealed that the metal destruction of welded joints has predominantly brittle intergranular nature. In the destruction area, the body of α-phase grains is affected by individual or locally grouped creep pores. It was found that the origination and development of fatigue cracks on the inner surface of steam lines have locally grouped nature. Their development is characterized by the absence of branching. Specification of metal damageability features of long-operated welded joints of steam lines will allow to reduce the damageability intensity, which provides increased lifetime

Lightweight thermally efficient composite feedlines, preliminary design and evaluation

Six liquid hydrogen feedline design concepts were developed for the cryogenic space tug. The feedlines include composite and all-metal vacuum jacketed and non-vacuum jacketed concepts, and incorporate the latest technology developments in the areas of thermally efficient vacuum jacket end closures and standoffs, radiation shields in the vacuum annulus, thermal coatings, and lightweight dissimilar metal flanged joints. The feedline design concepts were evaluated on the basis of thermal performance, weight, cost, reliability, and reusability. It is shown that composite tubing provides improved thermal performance and reduced weight for each design concept considered. Approximately 12 kg (26 lb.) can be saved by the use of composite tubing for the LH2 feedline and the other propulsion lines in the space tug