Group Irregular Labelings of Disconnected Graphs

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group \gr of order $s$, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels at every vertex are distinct. We give the exact values and bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we present some results for the \textit{modular edge gracefulness} $k(G)$, i.e. the smallest value of $s$ such that there exists a function f:E(G)\rightarrow \zet_s such that the sums of edge labels at every vertex are distinct

Direct evidence for the magnetic ordering of Nd ions in NdMn2_2Si2_2 and NdMn2_2Ge2_2 by high resolution inelastic neutron scattering

We have investigated the low energy nuclear spin excitations in NdMn$_2$Si$_2$ and NdMn$_2$Ge$_2$ by high resolution inelastic neutron scattering. Previous neutron diffraction investigations gave ambiguous results about Nd magnetic ordering at low temperatures. The present element-specific technique gave direct evidence for the magnetic ordering of Nd ions. We found considerable difference in the process of the Nd magnetic ordering at low temperature in NdMn$_2$Si$_2$ and NdMn$_2$Ge$_2$. Our results are consistent with those of magnetization and recent neutron diffraction measurements

Anomalous thermal expansion and strong damping of the thermal conductivity of NdMnO3_3 and TbMnO3_3 due to 4f crystal-field excitations

We present measurements of the thermal conductivity $\kappa$ and the thermal expansion $\alpha$ of NdMnO$_3$ and TbMnO$_3$. In both compounds a splitting of the $4f$ multiplet of the $R^{3+}$ ion causes Schottky contributions to $\alpha$. In TbMnO$_3$ this contribution arises from a crystal-field splitting, while in NdMnO$_3$ it is due to the Nd-Mn exchange coupling. Another consequence of this coupling is a strongly enhanced canting of the Mn moments. The thermal conductivity is greatly suppressed in both compounds. The main scattering process at low temperatures is resonant scattering of phonons between different energy levels of the $4f$ multiplets, whereas the complex 3d magnetism of the Mn ions is of minor importance.Comment: 9 pages including 6 figure

A novel heavy-fermion state in CaCu_3Ru_4O12

We have measured susceptibility, specific heat, resistivity, and thermopower of CaCu$_3$Ti$_{4-x}$Ru$_x$O$_{12}$ and CaCu$_{3-y}$Mn$_y$Ru$_4$O$_{12}$, and have found that CaCu$_3$Ru$_4$O$_{12}$ can be regarded as a heavy-fermion oxide in d-electron systems. The Kondo temperature is near 200 K, and the susceptibility (1.4$\times10^{-3}$ emu/Cu mol) and the electron specific heat coefficient (28 mJ/Cu molK$^2$) are moderately enhanced. The resistivity is proportional to $T^2$ at low temperatures, and satisfies the Kadowaki-Woods relation. The heavy-fermion state comes from the interaction between the localized moment of Cu 3d and the conduction electron of Ru 4d. An insulator-metal transition occurs between $x=1.5$ and 4 in CaCu$_3$Ti$_{4-x}$Ru$_x$O$_{12}$, which can be regarded as a transition from magnetic insulator to heavy-fermion metal.Comment: 4 pages, 5 figures, submitted to J. Phys. Soc. Jp

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions