The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

Some Invariance Principles for Random Vectors in the Generalized Domain of Attraction of the Multivariate Normal Law

For independent identically distributed random vectors belonging to the generalized Domain of Attraction of the multivariate normal law, we define two partial sum processes analogous to that of Donsker's Theorem. We prove that each converges in distribution to a Brownian Motion in the space of continuous functions. One process uses nonrandom operator normalization, and the other is a studentization of the first, using normalization by the empirical covariance operator.     

Structural and magnetic characterization of Mn3IrGe and Mn3Ir(Si1−xGex): experiments and theory

The structural and magnetic properties of a new ternary Ir-Mn-Ge phase, Mn₃IrGe, as well as the solid solution Mn₃Ir(Si_1−xGe_x), 0?x?1, have been investigated by means of X-ray and neutron powder diffraction, magnetization measurements and first principles calculations. The crystal structure is cubic, of the AlAu₄-type (an ordered form of the β-Mn structure), Z=4, space group P2₁3, and the unit-cell dimension varies linearly with the silicon content. For all compositions, antiferromagnetic ordering is found below a critical temperature of about 225 K. The magnetic structure is noncollinear, as a result of frustrated magnetic interactions on a triangular network of Mn atoms, on which the moments rotate 120° around the triangle axes. The magnitude of the magnetic moment at 10 K is 3.39(4) μ_B for Mn₃IrGe. The theoretical calculations reproduce with very good accuracy the magnitudes as well as the directions of the experimentally observed magnetic moments.     

First‐principles study of metal/nitride polar interfaces: Ti/TiN

We have examined the optimal interface structure, ideal work of adhesion and bonding character of polar Ti(110)/TiN(111) interfaces by first‐principles density‐functional plane‐wave pseudopotential calculations. Both Ti‐ and N‐terminated interfaces, including six different interface structures, were calculated. The interface structure for each termination, continuing the TiN crystal structure across the interface, has the largest work of adhesion. Although both terminations yield substantial adhesion energies in the range 3–7 J m^?2, the N‐terminated interface is ～4 J m^?2 stronger than the Ti‐terminated interface. Analysis of the interfacial electronic structure shows that the Ti‐terminated interface is a mixed strong, metallic and weak covalent character, whereas the N‐terminated interface is a polar covalent bond similar to the Ti/TiC interface. Further study of the separation of the optimal interface shows that the cleavages will never fracture at the interface due to the strong bonding, which is consistent with the experimental results. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficiency of composite vs. conically shaped chromatographic columns

The theory of composite chromatographic columns has been re-examined and general equations relating their efficiency to those of individual columns were derived. The derivations were based on elementary principles without reference to the principle of variance additivity. Derived also, were general equations for the efficiency of columns with continuously varying diameters, HETP and other properties from which formulae for special cases of practical interest were deduced. An example comparing the efficiencies of two columns, one conical and the other composite was given. The calculations show clearly that the conical column is superior in efficiency for the same total length and total retention volume.     

Analysis of the pungent principles of ginger and grains of paradise by high-performance liquid chromatography using electrochemical detection

Summary The pungent principles of ginger and grains of paradise are examined using a reversed-phase HPLC column. An electrochemical detector is used to selectively detect the phenolic gingerols and shogaols. The retention indices of the compounds are compared using methanowater and acetonitrile-water as eluents.Presented at the 14th International Symposium on Chromatography London, September, 1982     

Chemical reaction dynamics of PeCB and TCDD decomposition: A tight‐binding quantum chemical molecular dynamics study with first‐principles parameterization

The decomposition reaction dynamics of 2,3,4,4′,5‐penta‐chlorinated biphenyl (2,3,4,4′,5‐PeCB), 3,3′,4,4′,5‐penta‐chlorinated biphenyl (3,3′,4,4′,5‐PeCB), and 2,3,7,8‐tetra‐chlorinated dibenzo‐p‐dioxin (2,3,7,8‐TCDD) was clarified for the first time at atomic and electronic levels, using our novel tight‐binding quantum chemical molecular dynamics method with first‐principles parameterization. The calculation speed of our new method is over 5000 times faster than that of the conventional first‐principles molecular dynamics method. We confirmed that the structure, energy, and electronic states of the above molecules calculated by our new method are quantitatively consistent with those by first‐principles calculations. After the confirmation of our methodology, we investigated the decomposition reaction dynamics of the above molecules and the calculated dynamic behaviors indicate that the oxidation of the 2,3,4,4′,5‐PeCB, 3,3′,4,4′,5‐PeCB, and 2,3,7,8‐TCDD proceeds through an epoxide intermediate, which is in good agreement with the previous experimental reports and consistent with our static density functional theory calculations. These results proved that our new tight‐binding quantum chemical molecular dynamics method with first‐principles parameterization is an effective tool to clarify the chemical reaction dynamics at reaction temperatures. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Comparative analysis of the underground parts ofGentiana species by HPLC with diode-array and mass spectrometric detection

Summary The aim of this work was to compare the chemical composition of the underground parts (roots and rhizomes) ofGentiana cruciata L.,Gentiana pneumonanthe L., andGentiana asclepiadea L.— the three gentians native to Hungary—with that of the widely used stomachicGentiana lutea L., to determine which of the three Hungarian species could be used as a substitute forGentiana lutea in pharmaceutical preparations. The four gentians were compared by means of RPHPLC with diode-array detection (DAD) and electrospray ionization-mass spectrometric detection (ESI-MSD). The quantities of the lead compounds, the secoiridoid-glycosides, in 220 samples of the underground parts of gentians originating from several locations in Hungary, were determined by a more economical RPHPLC-DAD method. The occurrence of the characteristic compounds investigated—bitter principles and xanthones—in the underground parts of the speciesGentiana asclepiadea L. suggest it might be a potential replacement forGentiana lutea L. in pharmaceutical products. Presented at Balaton Symposium '01 on High-Performance Separation Methods, Siófok, Hungary, September 2–4, 2001     

过渡金属原子X (X=Mn、Fe、Co)掺杂单层Janus WSSe的第一性原理研究

二维Janus WSSe作为一种新兴的过渡金属硫族化合物(TMDs)材料,由于其打破了面外镜像对称性,且具有内在垂直压电和强Rashba自旋轨道耦合效应等丰富的物理特性,在自旋电子器件中具有巨大的应用潜力。本文基于密度泛函理论的第一性原理平面波赝势方法计算了过渡金属原子X(X=Mn、Fe、Co)掺杂单层Janus WSSe的电子结构、磁性和光学性质。结果表明：在Chalcogen-rich(硫族元素为多数元素)条件下的掺杂比在W-rich(钨元素为多数元素)条件下的掺杂展现出更高的稳定性,且掺杂后所有体系均表现出磁性。值得一提的是,对该体系进行Mn掺杂后,自旋向上通道出现杂质能级,WSSe由原来的非磁性半导体转变成磁矩为1.043μ_B的铁磁性半金属。而Fe、Co的掺杂使得自旋向上通道和自旋向下通道均出现杂质能级,呈现出磁矩分别为1.584μ_B、2.739μ_B的金属性。此外,掺杂体系的静态介电常数都显著增加,极化程度增强,且介电函数虚部和光吸收峰都发生了红移,说明掺杂有利于对可见光的吸收。     

Hybrid and multi-field variational principles for geometrically exact three-dimensional beams

This paper addresses the development of several alternative novel hybrid/multi-field variational formulations of the geometrically exact three-dimensional elastostatic beam boundary-value problem. In the framework of the complementary energy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of stresses only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value problem. Both weak and linearized weak forms of the principles are presented. The main features of the principles are highlighted, giving special emphasis to their relationships from both theoretical and computational standpoints.