Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.     

Differential Criterion of Complete Regular Local Rings

Abstract

In the paper by Furuya and Niitsuma [Furuya, M., Niitsuma, H. (2002a). On m -adic higher differentials and regularities of Noetherian complete local rings. J. Math. Kyoto Univ. 42(1):33–40], we gave a regularity criterion of complete Noetherian local rings in terms of the concept of m -adic higher differentials with some assumptions of separability on the residue fields of the local rings. The aim of this paper is to try to weaken “the separability assumptions” in a premise of the theorems in Furuya and Niitsuma (2002a).     

Nonlocal higher order evolution equations

In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u _t (x, t) = (?1) ^n?1 (J * Id ? 1) ⁿ (u(x, t)), x ∈ ? ^N , which is the nonlocal analogous to the higher order local evolution equation v _t = (?1) ^n?1(Δ) ⁿ v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity.     

Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

A magnet spring model

The paper discusses an elementary spring model representing the motion of a magnet suspended from the ceiling at one end of a vertical spring which is held directly above a second magnet fixed on the floor. There are two cases depending upon the north–south pole orientation of the two magnets. The attraction or repelling force induced by the magnets follow an inverse quartic law and thus we are led to a nonlinear model suitable for discussion in a beginning differential equations course. Spring models are common fare in such courses, but usually only linear models with simple sinusoidal forcing are considered. The resultant model is autonomous and thus an energy approach permits a full phase portrait of the resultant motions in the phase plane. These phase portraits show interesting behaviour of the system, reinforcing one's natural physical intuition. The computer algebra system Mathematica is employed here, although almost any other system would suffice. Such a system permits almost effortless calculations and can generate the graphics needed to thoroughly investigate the model.     

Photonic time‐stretch digitizer and its extension to real‐time spectroscopy and imaging

Real‐time wideband digitizers are the key building block in many systems including oscilloscopes, signal intelligence, electronic warfare, and medical diagnostics systems. Continually extending the bandwidth of digitizers has hence become a central challenge in electronics. Fortunately, it has been shown that photonic pre‐processing of wideband signals can boost the performance of electronic digitizers. In this article, the underlying principle of the time‐stretch analog‐to‐digital converter (TSADC) that addresses the demands on resolution, bandwidth, and spectral efficiency is reviewed. In the TSADC, amplified dispersive Fourier transform is used to slow down the analog signal in time and hence to compress its bandwidth. Simultaneous signal amplification during the time‐stretch process compensates for parasitic losses leading to high signal‐to‐noise ratio. This powerful concept transforms the analog signal's time scale such that it matches the slower time scale of the digitizer. A summary of time‐stretch technology's extension to high‐throughput single‐shot spectroscopy, a technique that led to the discovery of optical rouge waves, is also presented. Moreover, its application in high‐throughput imaging, which has recently led to identification of rogue cancer cells in blood with record sensitivity, is discussed.     

Stabile Isotope in den Interaktionen von Parasiten und Wirten bei Höheren Pflanzen

Abstract

An overview is given about the δ¹³C- and δ5D-values in the organic material of hemi- and full- parasitic higher plants and of their hosts.

Hemiparasites have to direct the content of the host xylem into their tissues by lowering their water potential. They achieve this by active water secretion or by lowering the stomatal resistance. In the latter case, the intercellular CO₂-concentration in the chlorenchyme of the parasite is increased. This causes a reduction of the ¹³C-content (δ¹³C-value of the parasite more negative than of the host). The dry matter of the mistletoes is always richer in deuterium than that of the host. The reason for this fact is unknown.

In the case hemiparasites and their hosts show differences in the ¹³CO₂-discrimination during the photosynthetic CO₂-fixation, an eventual transfer of organic material from host to parasite can be checked. By this way a holoparasitic mistletoe, Tristerix aphyllus, could be demonstrated.

In contrast to mistletoes on C3-hosts, such on CAM-hosts have a lower deuterium content in the dry mass than the hosts.

Holoparasites get all their organic material from the hosts and mirror, therefore, in their δ¹³C-values those of the hosts. Their deuterium content in the organic material is always higher than the one in the host. The reason is unknown.     

Cumulants in the 3-dimensional Ising,O（2） and O（4） spin models

Based on the universal properties of a critical point in different systems and that the QCD phase transitions fall into the same universality classes as the 3-dimensional Ising, O（2） or O（4） spin models, the critical behavior of cumulants and higher cumulant ratios of the order parameter from the three kinds of spin models is studied. We found that all higher cumulant ratios change dramatically the sign near the critical temperature. The qualitative critical behavior of the same order cumulant ratio is consistent in these three models.