Applied Mathematical Methods in Theoretical Physics, 2nd edn., by Michio Masujima

When Oliver Cromwell's decision to place the Post Office under the control of Parliament was ratified by Charles II in 1660, the main business of the Post Office was communication—transmission of the written word by letter. Since these days the Post Office has become responsible for many other services, either on its own account, or as an agent for other Government Departments; communication, however, is still its main business, although many new forms of it have been added during the last hundred years.

The first new form was that of the transmission of the written word in coded form by the electric telegraph; this was followed by transmission of the spoken word by telephone. Transmission of pictorial information came next: first in the slow-speed form of picture telegraphy, then in the high-speed form of television. The latest form is data transmission, the data being mostly numerical, which is associated with the growth of electronic computers over the last decade or so; the administrative and technical arrangements for data transmission by the Post Office are still in the very early stages, but this service will undoubtedly grow.     

Multiple single-point imaging (mSPI) as a tool for capturing and characterizing MR signals and repetitive signal disturbances with high temporal resolution: The MRI scanner as a high-speed camera

In this paper we aim to lay down and demonstrate the use of multiple single-point imaging (mSPI) as a tool for capturing and characterizing steady-state MR signals and repetitive disturbances thereof with high temporal resolution.     

Quantification of superparamagnetic iron oxide using inversion recovery balanced steady-state free precession

Cellular and molecular MRI trafficking studies using superparamagnetic iron oxide (SPIO) have greatly improved non-invasive investigations of disease progression and drug efficacy, but thus far, these studies have largely been restricted to qualitative assessment of hypo- or hyperintense areas near SPIO. In this work, SPIO quantification using inversion recovery balanced steady-state free precession (IR-bSSFP) was demonstrated at 3 T by extracting R₂ values from a monoexponential model (P. Schmitt et al., 2004). A low flip angle was shown to reduce the apparent recovery rate of the IR-bSSFP time course, thus extending the dynamic range of quantification. However, low flip angle acquisitions preclude the use of traditional methods for combining RF phase-cycled images to reduce banding artifacts arising from off-resonance due to B₀ inhomogeneity. To achieve R₂ quantification of SPIO, we present a new algorithm applicable to low flip angle IR-bSSFP acquisitions that is specifically designed to identify on-resonance acquisitions. We demonstrate in this work, using both theoretical and empirical methods, that the smallest estimated R₂ from multiple RF phase-cycled acquisitions correspond well to the on-resonance time course. Using this novel minimum R₂ algorithm, homogeneous R₂ maps and linear R₂ calibration curves were created up to 100 μg(Fe)/mL with 20° flip angles, despite substantial B₀ inhomogeneity. In addition, we have shown this technique to be feasible for pre-clinical research: the minimum R₂ algorithm was resistant to off-resonance in a single slice mouse R₂ map, whereas maximum intensity projection resulted in banding artifacts and overestimated R₂ values. With the application of recent advances in accelerated acquisitions, IR-bSSFP has the potential to quantify SPIO in vivo, thus providing important information for oncology, immunology, and regenerative medicine MRI studies.     

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.     

METAPHORS AND OTHER LINGUISTIC POINTERS TO CHILDREN'S MENTAL REPRESENTATIONS

The mental representations that 6- and 7-year-old pupils form as a result of their interactions with their teacher's verbal, written, pictorial and concrete material representations has to be inferred from the language they use. In this study many pupils seem to have mental representations which capture surface characteristics of a particular teachers’ representation and use metaphoric language associated with that representation when describing their calculations. Pupils’ use of ‘you’ is characteristic of those who adopt a representation-specific procedure, whilst ‘if’ and ‘like’ are linguistic pointers to their use of generic examples to describe a procedure. Individual pupils show a preference for the same style of mental representation when describing images and procedures in mathematical and non-mathematical contexts.     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.