Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1]

Structural Analysis Using NX Nastran 9.0

NX Nastran is a powerful Finite Element Analysis (FEA) software package used to solve linear and non-linear models for structural and thermal systems. The software, which consists of both a solver and user interface, breaks down analysis into four files, each of which are important to the end results of the analysis. The software offers capabilities for a variety of types of analysis, and also contains a respectable modeling program. Over the course of ten weeks, I was trained to effectively implement NX Nastran into structural analysis and refinement for parts of two missions at NASA's Kennedy Space Center, the Restore mission and the Orion mission

An asymptotic property of Schachermayer's space under renorming

A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every \ep > 0, there exists \delta > 0 such that for every \ep-separated sequence (x_n) in the unit ball B, and every x in B, there are distinct indices m and n such that ||x + x_m + x_n|| < 3(1 - \delta), respectively, ||x_m + x_n|| < 2(1 - \delta). It is shown that a Banach space constructed by Schachermayer has property 2-beta but cannot be renormed to have property 2-NUC

Networks and Actors: Foreign Office Attitudes Towards European Integration, 1957-73

The history of Britain's relationship with Europe is one which has received significant attention from scholars and laypeople alike, especially in recent times. It has been explored from a wide range of angles and perspectives, all of which offer unique insights into what has often been characterised as an awkward or reluctant relationship. This thesis' contribution employs a specific focus on the attitudes of Foreign Office officials towards European integration in the years 1957-73, and the ways in which these attitudes shaped the foreign policymaking process. The role which Foreign Office officials played in Britain's approach to membership of the EEC was extremely significant, and their attitudes had a profound impact on the policymaking process. In certain cases, these attitudes conflicted with those of their political masters and resulted in serious struggles and confrontations in the corridors of power. This study will examine four case studies in Britain's approach to European integration in the years 1957-73, which cover the most critical junctures in the Foreign Office's approach to European integration across this period. In each case study, the attitudes and actions of the officials most intimately involved in European policy will form the main focus, including an in-depth analysis of how their attitudes had been shaped through their own formative experiences. It will become clear that officials' attitudes towards European integration were exceptionally diverse and were not reflective of a rigid departmental orthodoxy. Foreign Office personnel were increasingly recruited from a wider base of social and educational backgrounds and this in turn created a diplomatic service containing a broad range of views. However, a gulf in attitudes between the elder and younger generations of officials became increasingly evident, with the latter being much more receptive to the principles of European unity after their experiences of the Second World War. The result was a department which increasingly viewed membership of the EEC as the future of Britain's foreign policy strategy

CH, a problem of Rolewicz and bidiscrete systems

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum. We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K2 which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K

Improved Nd Column Chemical Methods and Implications for Reconstruction of Water Mass Circulation Over the Mid-Eocene Climatic Optimum

The mid to late Eocene (49—34 Ma) is characterized as a long-term cooling (7℃) transition from greenhouse to icehouse conditions. However, this long-term cooling period was interrupted by a rapid warming event: the Mid Eocene Climatic Optimum (MECO; 40.6 Ma). During this episode of transient warming, Southern Ocean temperatures gradually rose 4 to 6°C over ~500 ka, and then warmed ~1.5°C more over a brief period at the peak of the MECO (40.0 Ma). After the peak warming, rapid cooling occurred over ~200 ka returning the Southern Ocean waters to pre-MECO temperatures. Such prominent and transient warming may have affected (or been influenced) by a change in deep-water formation and circulation. Seawater Neodymium (Nd) isotope ratios are commonly used as a proxy for ocean circulation, however, due to the multiple column chemistry steps in sample preparation, traces of Nd are frequently lost throughout the methods. Consequently, in some samples, where Nd is scarce, it is difficult to record an isotope signal. Here we dissect our sample preparation procedures in order to best optimize Nd collection for isotope analysis. We test the yield of Nd through the columns and suggest and alternative in order for maximum collection. Additionally, we investigate the potential relationship between the meridional overturning circulation (MOC) and climate change during the MECO using a high-resolution record of water mass Nd isotopes at Ocean Drilling Project Site 464, in the North Pacific. An approximately -1 εNd shift concurrent with warming occurs over an ~500kyr period at about 42Ma could suggest stronger water mass intrusion from the south or an increase in weathering to the surface ventilated source water of the Pacific. Utilizing the methods adjustments compiled here along with generating a dataset over a North-South Pacific transect can provide a more clear answer to the type and magnitude of potential shifts in ocean circulation

Seawater Osmium Isotope Records from Pacific ODP and IODP Sites- Refining the Paleogene Curve and Dating Red Clay Sequences

Reconstructing ancient ocean circulation patterns contributes to a better understanding of Earth’s past climate conditions, as the two are known to be a strongly coupled system. Paleoceanographic reconstructions of the Late Cretaceous and early Cenozoic require enhanced geographic coverage, particularly in the Pacific, in order to better constrain meridional variations in environmental conditions. The challenge with the existing inventory of Pacific deep-sea cores is that they consist almost exclusively of pelagic clay with little existing age control. Recent work indicates that seawater Osmium (Os) isotope analyses provide useful age control for red clay sequences due to its relatively long residence time compared to oceanic mixing. The drawback to using seawater Os isotope stratigraphy to date Paleogene age sediments is that the compilation of existing data has some significant temporal gaps, notably between ~38 and 55 million years ago (Ma). To improve the temporal resolution of the seawater Os isotope curve, we present new data from Ocean Drilling Program (ODP) Site 865, which has excellent biostratigraphic age control over this time interval. Preliminary data indicate a seawater composition consistent with the apparent trend in the few existing data points. We also analyzed the Os isotopic composition at Integrated Ocean Drilling Progam (IODP) Site U1370 to construct an age model for this predominantly pelagic clay section. The 187Os/188Os values generally increase from 0.312 at 64.46 meters below seafloor (mbsf), likely reflecting the Os isotope minimum recorded across the Cretaceous-Paleogene boundary, to 0.531 at 28.26 mbsf, likely correlating to the Eocene/Oligocene interval

NON-LEIBNIZ ALGEBRAS WITH LOGARITHMS DO NOT HAVE THE TRIGONOMETRIC IDENTITY

Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x, y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. [C(x)] 2 + [S(x)] 2 = e whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold