Well-posedness for the diffusive 3D Burgers equations with initial data in H1/2H^{1/2}

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 201

Operator-Valued Frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on $\mathbb{R}$, restricted to $E = (-1/2, 1/2)$, forms a Hilbert-space frame for $L^2(E)$. For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group $H_n$, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of $L^2(E)$ for suitable subsets $E$ of $H_n$ in terms of representations of $H_n$

A high resolution microscopy study of biological components for the incorporation in opto-electronic hybrid devices.

Optical microscopy and scanning probe microscopy techniques have been utilised to acquire high resolution topography and fluorescence images of several biological samples. Applying these techniques to patterned samples and single molecules allow the optical properties of a sample to be investigated near to and below the diffraction limit, allowing emission properties to be correlated with those of topography. Optically active biological samples outside of their cellular environment are prone to photo-degredation and in measuring them a challenge is to ensure that optical measurements can be made before the onset of damage to the fluorophore. In this study two forms of fluorescence microscope have been utilised with scanning probe techniques of AFM and SNOM. These techniques have been used alongside microcontact printed arrays of fluorescent proteins and photosynthetic light harvesting complexes to address the accuracy of the printing technique and it's applicablity to biological components for future bionanotechnological applications. Furthermore, the periodicity associated with the arrays has been applied to the techniques to address the relative resolutions of the microscopes as well as the samples being a drive behind implimenting biologically friendly components/techniques to the microscopes (such as liquid cells). Larger structures from photosynthetic bacteria have also been addressed in this study in the form of chlorosomes which are model structures for light harvesting in low light conditions. Studies on the spectral properties of populations of 3 species have been conducted in this work with fluorescence microscopy and it has been shown that populations show small local variations in fluorescence. Furthermore it has been shown that the developed scanning fluorescence technique can be applied to photo senstitive samples successfully with only a small number of cases where spectral properties were affected by the measurement technique. Using high resolution microscopy techniques this research shows the surface patterning techniques in conjunction with biological samples to have mixed success depending on the sample. It also shows spectral measurements on newly discovered chlorosomes with little photo degredation. It further shows the role that the microscopy techniques have in analysing biological systems in different configurations on substrates

Rosa Olvera v. Wynn Las Vegas; and Sedgwick CMS, 139 Nev. Adv. Op. 41 (Sept. 28, 2023)

NRS 616C.390 allows for workers’ compensation claims to be reopened if there is a change of circumstance related to the injury that the claimant can demonstrate by a preponderance of the evidence. This Opinion clarifies that, if there were multiple body parts implicated in the injury, then a claim needs to be reopened only for the body parts that the claimant has demonstrated a change in circumstances

Design and Demonstration of a Miniature Lidar System for Rover Applications

Public awareness of harmful human environmental effects such as global warming has increased greatly in recent years and researchers have increased their efforts in gaining more knowledge about the Earth's atmosphere. Natural and man-made processes pose threats to the environment and human life, so knowledge of all atmospheric processes is necessary. Ozone and aerosols are important factors in many atmospheric processes and active remote sensing techniques provide a way to analyze their quantity and distribution. A compact ground-based lidar system for a robotic platform meant for atmospheric aerosol measurements was designed, tested, and evaluated. The system will eventually be deployed for ozone and aerosol measurements in Mars and lunar missions to improve our knowledge and understanding of atmospheres on Mars and the Moon. Atmospheric testing was performed to test the operability of the receiver system to acquire the lidar return signal from clouds and aerosols

The Selection of Talent for Stenography and Typing

H. C. Link states that The application of psychological tests in those fields where their value has been verified, is the only method short of the laborious and costly method of trial and error, which makes it possible to discover the exact ability, both innate and acquired, of an individual. Unless these facts are known, it becomes impossible to assign the individual to the work for which he is best fitted or to give him the training which he deserves. Once the potential and actual ability of an individual has been discovered the vocational selection or training of that individual can be decided with a measurable degree of intelligence

Hydrodynamic impacts of tectonics in prehistoric Ohiwa Harbour, North Island, New Zealand

Ohiwa Harbour is an estuarine lagoon located in the eastern Bay of Plenty, North Island, New Zealand. Ohiwa Harbour is bounded by two sand spits, Ohope and Ohiwa Spits. This study assessed the likelihood of a resistant barrier underlying Ohiwa Spit, which would control the inlet and spit locations. The observed depths of layers unable to be penetrated by a vibrocorer on Ohiwa Spit supported this idea, along with the discovery of a rock outcrop on the eastern harbour entrance using sidescan SONAR. Vibrocoring on Ohiwa Spit added to the knowledge of the prehistoric evolution of Ohiwa Harbour, and an attempt to infer various subsidence events on Ohiwa Spit were made. Fining upwards coarse sand sequences with dominant shell material were found in the cores. These sequences could be related to a change in harbour hydrodynamics, or recent subsidence events in Ohiwa Harbour, such as the 0.6m subsidence of the Waimana Fault 636 to 575cal yrs BP. A sharp change in the core profile was observed at 1.4m in core C and 1.7m in core C2, marked by increased grain size and an abundance of shell material, mostly Austrovenus stutchburyi. This could be related to a change in wave energy in the harbour or an erosional contact associated with subsidence of Ohiwa Spit. Comparison of radiocarbon dated shells in this study with Murdoch (2005) on Ohope Spit suggests that more subsidence has occurred at Ohiwa Spit than Ohope Spit; this may be associated with a fault through the harbour entrance. Subsidence associated with earthquakes, and erosion associated with at least four tsunami events in the last 6000 years has increased the depth and extent of Ohiwa Harbour, increasing its volume. This study used numerical modeling to determine the hydrodynamic impacts of past catastrophic events (earthquake related subsidence and volcanic eruptions) on Ohiwa Harbour. Ten scenarios along with present conditions were modelled. • In scenarios where sea level was modified only (scenarios 1-5 and 8), tidal range and surface elevation increased within the harbour, but not at the harbour entrance. The entrance hydrodynamics were also influenced by its width; as Ohope Spit accreted, the surface elevation and flow velocities increased within the tidal inlet • Flow speeds increased at the harbour entrance by up to 50% following subsidence in scenarios 9 and 10 • A drop in sea level during subsidence (scenario 6) resulted in a smaller tidal prism and hence reduced flow velocities at the inlet, despite the subsidence • Scenario 7 involved subsidence and the breaching of Ohope Spit, which would divert flow away from the entrance and result in reduced flow velocities at the entranc