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We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod-p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields. 

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGL and of the Real bordism spectrum MUR. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height n, the bracket itself will be of chromatic height (n+1). Using these, we deduce a family of exotic multiplications in the π_{**}MGL-module structure of the motivic Morava K-theories, including non-trivial multiplications by 2. These in turn imply the analogous family of exotic multiplications in the π_{\star}MUR-module structure on the Real Morava K-theories. 

Abstract Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors. 

We study modules over the commutative ring spectrum 𝐻𝔽₂∧𝐻𝔽₂, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ_{k} in the category of associative algebras freely kills the higher generators ξ_{k+n}. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative 𝐻𝔽₂∧𝐻𝔽₂-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum. 

For an equivariant commutative ring spectrum R, \pi_0 R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N_\infty ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and \pi_0 R has the structure of an incomplete Tambara functor. However, if R is an N_\infty ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures. 

Abstract Machining-induced residual stresses (MIRS) are a main driver for distortion of thin-walled monolithic aluminum workpieces. Before one can develop compensation techniques to minimize distortion, the effect of machining on the MIRS has to be fully understood. This means that not only an investigation of the effect of different process parameters on the MIRS is important. In addition, the repeatability of the MIRS resulting from the same machining condition has to be considered. In past research, statistical confidence of MIRS of machined samples was not focused on. In this paper, the repeatability of the MIRS for different machining modes, consisting of a variation in feed per tooth and cutting speed, is investigated. Multiple hole-drilling measurements within one sample and on different samples, machined with the same parameter set, were part of the investigations. Besides, the effect of two different clamping strategies on the MIRS was investigated. The results show that an overall repeatability for MIRS is given for stable machining (between 16 and 34% repeatability standard deviation of maximum normal MIRS), whereas instable machining, detected by vibrations in the force signal, has worse repeatability (54%) independent of the used clamping strategy. Further experiments, where a 1-mm-thick wafer was removed at the milled surface, show the connection between MIRS and their distortion. A numerical stress analysis reveals that the measured stress data is consistent with machining-induced distortion across and within different machining modes. It was found that more and/or deeper MIRS cause more distortion.