A note on the 1-prevalence of continuous images with full Hausdorff dimension

We consider the Banach space consisting of real-valued continuous functions on an arbitrary compact metric space. It is known that for a prevalent (in the sense of Hunt, Sauer and Yorke) set of functions the Hausdorff dimension of the image is as large as possible, namely 1. We extend this result by showing that `prevalent' can be replaced by `1-prevalent', i.e. it is possible to \emph{witness} this prevalence using a measure supported on a one dimensional subspace. Such one dimensional measures are called \emph{probes} and their existence indicates that the structure and nature of the prevalence is simpler than if a more complicated `infinite dimensional' witnessing measure has to be used.Comment: 8 page

The Hausdorff dimension of graphs of prevalent continuous functions

We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function

On the use of multiple layer thicknesses within laser powder bed fusion and the effect on mechanical properties

Laser Powder Bed fusion is capable of rapid production of parts, from conception, compared with traditional manufacturing methods. This said, the time taken to fabricate a single part can still be significant – typically many hours. Processing thicker layers, and hence fewer total layers, in the Laser Powder Bed Fusion process, is an effective way to reduce build times. However, mechanical performance can suffer as a result of this strategy. This study proposes and demonstrates a method to enable the interlacing of multiple layer thicknesses within one part, allowing for finer layers within regions where they are specifically required, whilst maintaining overall component integrity for specific load cases. Thicker layers are used within regions with lower property requirements in order to optimise an overall part for improved production rate. The design of interfaces between two disparate layer thickness regions could also be tailored for control of material properties and such will be investigated in an independent study. Ti6Al4V LPBF samples are fabricated, characterised by way of tensile testing, porosity analysis and microstructural analysis. The study demonstrate parts can be additively built using multiple layer thickness regions with consistent ultimate tensile strength (1110–1135 MPa) and varying penalties to ductility, depending on layer thickness and interface design (elongation to failure reductions up to 40% in the most extreme case)

Failure modes in dual layer thickness Laser Powder Bed Fusion components using a novel post-mortem reconstruction technique

To exploit the design freedoms of Powder Bed Fusion, parameters can be varied within sub-volumes of components to achieve the optimal part for both service conditions and manufacturing productivity. This involves prioritising mechanical strength in areas of structural significance and high volumetric build rates in areas of low structural significance. In theory, a component with similar mechanical behaviour to that seen in standard Laser Powder Bed Fusion parts can be built in significantly less time and at a reduced cost. In practice however, the boundary between such regions is yet to be understood and discretising components into sub-volumes can induce interfacial defects. In this study, an in-depth analysis of interfaces between disparate layer thickness volumes in single components has been explored, to gain information vital to solving interface quality issues so that LPBF design freedoms can be fully exploited. A novel 3D reconstruction technique has been demonstrated to characterise transient plastic behaviour of interfacial pores post-fracture. This technique enables post-mortem evaluation of additively manufactured parts and tracking of pore deformation during subsequent mechanical testing. X-ray Computed Tomography (XCT) identified interfacial pores up to 170 µm Feret diameter, with a voxel resolution of 6 µm. Micro tensile testing with in-situ microscopy exhibited a real-time mechanical response, observing evidence that these interfacial defects lead to fracture at interface locations. The 3D reconstruction technique found that pores constricted 10.0 – 14.1% in the x direction and 10.3 – 14.6% in the y direction after fracture – normal to the loading direction. These findings contribute towards improving Additively Manufactured biomedical implants and airframe components with reduced time and cost

Influence of the initial chemical conditions on the rational design of silica particles

The influence of the water content in the initial composition on the size of silica particles produced using the Stöber process is well known. We have shown that there are three morphological regimes defined by compositional boundaries. At low water levels (below stoichiometric ratio of water:tetraethoxysilane), very high surface area and aggregated structures are formed; at high water content (>40 wt%) similar structures are also seen. Between these two boundary conditions, discrete particles are formed whose size are dictated by the water content. Within the compositional regime that enables the classical Stöber silica, the structural evolution shows a more rapid attainment of final particle size than the rate of formation of silica supporting the monomer addition hypothesis. The clearer understanding of the role of the initial composition on the output of this synthesis method will be of considerable use for the establishment of reliable reproducible silica production for future industrial adoption

Signal-to-noise measures for magnetic resonance imagers

The signal-to-noise ratio (SNR) in magnetic resonance imaging represents one of the system operating variables that must be determined both for evaluating the performance of different imaging protocols on a particular machine, and for monitoring machine performance as part of a routine quality control (QC) program. Utilizing a phantom and set of automated analysis programs currently under development, this study evaluated several ways of measuring image signal and noise and demonstrated the importance of utilizing measured voxel volumes as opposed to nominal volumes in the calculation of SNR. The NEMA proposed standard for SNR is compared with several other SNR measures and is recommended as the measure to be used in routine SNR reporting. The importance of utilizing other SNR measures in addition to the NEMA proposed standard for routine QC is discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31085/1/0000762.pd