Proof problems with diagrams: An opportunity for experiencing proofs and refutations

It has become gradually accepted that proof and proving are essential at all grades of mathematical learning. Among the various aspects of proof and proving, this study addresses proofs and refutations described by Lakatos, in particular a part of increasing content by deductive guessing, to introduce an authentic process into mathematics classrooms. This paper analyzes proof problems with diagrams as an appropriate task for this mathematical action, and illustrates, with a sequence of three lessons in the eighth grade, that a certain type of such problems can provide students with an opportunity to engage in this action.ArticleFor the Learning of Mathematics. 34(1):36-42 (2014)journal articl

Thermal Conductivity in the Bose-Einstein Condensed State of Triplons in the Bond-Alternating Spin-Chain System Pb2V3O9

In order to clarify the origin of the enhancement of the thermal conductivity in the Bose-Einstein Condensed (BEC) state of field-induced triplons, we have measured the thermal conductivity along the [101] direction parallel to spin-chains, $kappa_{\|[101]}$, and perpendicular to spin-chains, $kappa_{\perp[101]}$, of the S=1/2 bond-alternating spin-chain system Pb2V3O9 in magnetic fields up to 14 T. With increasing field at 3 K, it has been found that both $kappa_{\|[101]}$ and $kappa_{\perp[101]}$ are suppressed in the gapped normal state in low fields. In the BEC state of field-induced triplons in high fields, on the other hand, $kappa_{\|[101]}$ is enhanced with increasing field, while $kappa_{\perp[101]}$ is suppressed. That is, the thermal conductivity along the direction, where the magnetic interaction is strong, is markedly enhanced in the BEC state. Accordingly, our results suggest that the enhancement of $kappa_{\|[101]}$ in the BEC state is caused by the enhancement of the thermal conductivity due to triplons on the basis of the two-fluid model, as in the case of the superfluid state of liquid 4He.Comment: 5 pages, 3 figure

Pseudogap Phase Boundary in Overdoped Bi_2Sr_2CaCu_2O_8 Studied by Measuring Out-of-plane Resistivity under the Magnetic Fields

The characteristic pseudogap temperature T* in Bi2Sr2CaCu2O8 system has been systematically evaluated as a function of doping, especially focusing on its overdoped region, by measuring the out-of-plane resistivity under the magnetic fields. Overdoped samples have been prepared by annealing TSFZ-grown Bi2Sr2CaCu2O8 single crystals under the high oxygen pressures (990 kgf/cm2). At a zero field, the out-of-plane resistivity showed a metallic behavior down to Tc (= 62 K), while under the magnetic fields of over 3 T,it showed typical upturn behavior from around 65 K upon decreasing temperature. This result suggests that the pseudogap and superconductivity are different phenomena.Comment: 2 pages, 2 figures. Final version accepted for the Proceedings of the M2S-IX Conference (Tokyo, September 2009

Design and Performance of Remote Controlled ^3He Refrigerator in Hybrid Magnet(Part II. Several Instruments and Techniques Developed in HFLSM)

A remote controlled ^3He refrigerator for use in a hybrid magnet is described. Temperatures between 1.3 K and 0.43 K can be remote controlled by ^3He vapor pressure within the temperature drift of 0.01 K in magnetic fields up to 23 T. Using this refrigerator, several kinds of physical measurements, such as electrical resistance, capacitance and ultrasonic attenuation, have been carried out without being disturbed by noise peculiar to the hybrid magnet system. Additionally, magnetic field dependence of capacitance thermometers at liquid ^3He temperatures is discussed

Na依存性PiトランスポーターNpt2cは、KlothoノックアウトマウスPi恒常性において成長期と成熟期では異なる作用を有する

SLC34A3/NPT2c/NaPi-2c/Npt2c is a growth-related NaPi cotransporter that mediates the uptake of renal sodium-dependent phosphate (Pi). Mutation of human NPT2c causes hereditary hypophosphatemic rickets with hypercalciuria. Mice with Npt2c knockout, however, exhibit normal Pi metabolism. To investigate the role of Npt2c in Pi homeostasis, we generated α-klotho−/−/Npt2c−/− (KL2cDKO) mice and analyzed Pi homeostasis. α-Klotho−/− (KLKO) mice exhibit hyperphosphatemia and markedly increased kidney Npt2c protein levels. Genetic disruption of Npt2c extended the lifespan of KLKO mice similar to that of α-Klotho−/−/Npt2a−/− mice. Adult KL2cDKO mice had hyperphosphatemia, but analysis of Pi metabolism revealed significantly decreased intestinal and renal Pi (re)absorption compared with KLKO mice. The 1,25-dihydroxy vitamin D3 concentration was not reduced in KL2cDKO mice compared with that in KLKO mice. The KL2cDKO mice had less severe soft tissue and vascular calcification compared with KLKO mice. Juvenile KL2cDKO mice had significantly reduced plasma Pi levels, but Pi metabolism was not changed. In Npt2cKO mice, plasma Pi levels began to decrease around the age of 15 days and significant hypophosphatemia developed within 21 days. The findings of the present study suggest that Npt2c contributes to regulating plasma Pi levels in the juvenile stage and affects Pi retention in the soft and vascular tissues in KLKO mice