Teaching Children to be Mathematicians Versus Teaching About Mathematics

The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child's work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is ‘Turtle Geometry’, which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a ‘cybernetic turtle’ controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical performance.

     

Band-Limited Localized Parseval Frames and Besov Spaces on Compact Homogeneous Manifolds

In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (= band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-known φ-transform on Euclidean spaces.     

Beitrag zur Festigkeitsuntersuchung eingetauchter Körper

Ohne Zusammenfassung     

Thermologic investigations of three species of <Emphasis Type="Italic">Amorphophallus</Emphasis>

Thermologic investigations were carried out on three species of Amorphophallus: A. konjac, A. paeoniifolius and A. titanum, all the three strongly thermogenic. Moreover, their breeding system is described as protogynous, the heat production occurs in the appendix and male florets, no warming is seen in the female florets and pollen is shed after the end of heat dissipation. All the three have large, impressive inflorescences developed from big corms and have considerable sizes. During their inflorescence, they have a strong scent like rotting meat with carrion smell. Amorphophallus konjac (K. Koch) has a large, exposed appendix that produces a disgusting scent during the day of the female phase of blooming. The appendix produces about 3 W for several hours, and the temperature elevation is about 2.9 K. The low temperature elevation is attributed to a high surface area and a high evaporative heat loss from the appendix. During the male phase of blooming, a second episode of thermogenesis occurs during the same time of day, apparently from the male florets, reaching a maximum of 1.6 W. Amorphophallus paeoniifolius (Dennst.) Nicolson has a spadix that varies considerably from that of A. konjac and A. titanum with an amorphous upper end of the appendix like a shrunken red pepper instead of cone-like appendices for the two others. It shows thermogenic temperature increases of up to +9.1 K in the male florets and +2.6 K for a short time in the appendix. Amorphophallus titanum (Becc.) Becc. ex Arcang is the largest inflorescence of the world, growing up to 300 cm high and 250 cm across. A much smaller plant was observed during its thermogenic period by means of infrared (IR) thermography, IR thermometry, and thermometric data logger. The temperature maximum showed 36.6 °C at ambient 24.0 °C, which means a temperature difference of about +12.6 K. In the morning of the next day, all temperatures are back to ambient at about 24 °C. Estimates of the heat production (about 74 W) were made from the geometric data and special assumptions with respect to the heat transfer.     

Adaptive nonparametric regression on spin fiber bundles

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning, for instance, the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008, 2010) and Geller et al. (2008, 2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.     

Packing seagulls

A seagull in a graph is an induced three-vertex path. When does a graph G have k pairwise vertex-disjoint seagulls? This is NP-complete in general, but for graphs with no stable set of size three we give a complete solution. This case is of special interest because of a connection with Hadwiger’s conjecture which was the motivation for this research; and we deduce a unification and strengthening of two theorems of Blasiak [2] concerned with Hadwiger’s conjecture. Our main result is that a graph G (different from the five-wheel) with no three-vertex stable set contains k disjoint seagulls if and only if

1. |V (G)|≥3k
2. G is k-connected
3. for every clique C of G, if D denotes the set of vertices in V (G)\C that have both a neighbour and a non-neighbour in C then |D|+|V (G)\C|≥2k, and
4. the complement graph of G has a matching with k edges.

We also address the analogous fractional and half-integral packing questions, and give a polynomial time algorithm to test whether there are k disjoint seagulls.     

Generating Multiset Permutations in Constant Time

An algorithm is presented that generates multiset permutations taking constant time between each permutation.     

Laser-induced retardation of radiation pulses under arbitrary collisional relaxation of low-frequency coherence

The effect of thermal atomic motion in gaseous media on the features of electromagnetically induced transparency is studied. Relatively general and compact analytical expressions are derived for the absorption coefficient and refractive index of a probe wave in the presence of high-power driving radiation that is resonant with respect to the adjacent atomic transition allowing for the relaxation of low-frequency coherence. It is demonstrated that the shape of the spectrum is similar to laser-induced structures in a continuum. This representation of resonances allows the electromagnetically induced transparency to be studied on relatively general assumptions and a significant retardation of radiation pulses and their delays to be demonstrated.     

Graph Minors. XX. Wagner''s conjecture

In a previous paper (J. Combin. Theory 48 (1990) 255) we showed that for any infinite set of (finite) graphs drawn in a fixed surface, one of the graphs is isomorphic to a minor of another. In this paper we extend that result in two ways:

• we generalize from graphs to hypergraphs drawn in a fixed surface, in which each edge has two or three ends, and

• the edges of our hypergraphs are labelled from a well-quasi-order, and the minor relation is required to respect this order.

This result is another step in the proof of Wagner's conjecture, that for any infinite set of graphs, one is isomorphic to a minor of another.