The fundamental cycle of concept construction underlying various theoretical frameworks

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout mathematical learning

Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, M\"obius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(\Lambda,L_y \times L_x,q,v,w)=\sum_{d=0}^{L_y} \tilde c^{(d)} Tr[(T_{Z,\Lambda,L_y,d})^m]$, where $\Lambda$ denotes the lattice type, $\tilde c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,\Lambda,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $\Lambda=sq, tri (hc)$, respectively. An analogous formula is given for M\"obius strips, while only $T_{Z,\Lambda,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,\Lambda,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,\Lambda,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,\Lambda,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.Comment: Reference added to a relevant paper by F. Y. W

Substructures in lens galaxies: PG1115+080 and B1555+375, two fold configurations

We study the anomalous flux ratio which is observed in some four-image lens systems, where the source lies close to a fold caustic. In this case two of the images are close to the critical curve and their flux ratio should be equal to unity, instead in several cases the observed value differs significantly. The most plausible solution is to invoke the presence of substructures, as for instance predicted by the Cold Dark Matter scenario, located near the two images. In particular, we analyze the two fold lens systems PG1115+080 and B1555+375, for which there are not yet satisfactory models which explain the observed anomalous flux ratios. We add to a smooth lens model, which reproduces well the positions of the images but not the anomalous fluxes, one or two substructures described as singular isothermal spheres. For PG1115+080 we consider a smooth model with the influence of the group of galaxies described by a SIS and a substructure with mass $\sim 10^{5} M_{\odot}$ as well as a smooth model with an external shear and one substructure with mass $\sim 10^{8} M_{\odot}$ . For B1555+375 either a strong external shear or two substructures with mass $\sim 10^{7} M_{\odot}$ reproduce the data quite well.Comment: 26 pages, updated bibliography, Accepted for publication in Astrophysics & Space Scienc

4f spin density in the reentrant ferromagnet SmMn2Ge2

The spin contribution to the magnetic moment in SmMn2Ge2 has been measured by magnetic Compton scattering in both the low and high temperature ferromagnetic phases. At low temperature, the Sm site is shown to possess a large 4f spin moment of 3.4 +/- 0.1 Bohr magnetons, aligned antiparallel to the total magnetic moment. At high temperature, the data show conclusively that ordered magnetic moments are present on the samarium site.Comment: 5 pages, 2 figures, transferred from PRL to PRB (Rapid Comm.

Potts model on recursive lattices: some new exact results

We compute the partition function of the Potts model with arbitrary values of $q$ and temperature on some strip lattices. We consider strips of width $L_y=2$, for three different lattices: square, diced and `shortest-path' (to be defined in the text). We also get the exact solution for strips of the Kagome lattice for widths $L_y=2,3,4,5$. As further examples we consider two lattices with different type of regular symmetry: a strip with alternating layers of width $L_y=3$ and $L_y=m+2$, and a strip with variable width. Finally we make some remarks on the Fisher zeros for the Kagome lattice and their large q-limit.Comment: 17 pages, 19 figures. v2 typos corrected, title changed and references, acknowledgements and two further original examples added. v3 one further example added. v4 final versio

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

Multi-ancestry GWAS of the electrocardiographic PR interval identifies 202 loci underlying cardiac conduction

The electrocardiographic PR interval reflects atrioventricular conduction, and is associated with conduction abnormalities, pacemaker implantation, atrial fibrillation (AF), and cardiovascular mortality. Here we report a multi-ancestry (N = 293,051) genome-wide association meta-analysis for the PR interval, discovering 202 loci of which 141 have not previously been reported. Variants at identified loci increase the percentage of heritability explained, from 33.5% to 62.6%. We observe enrichment for cardiac muscle developmental/contractile and cytoskeletal genes, highlighting key regulation processes for atrioventricular conduction. Additionally, 8 loci not previously reported harbor genes underlying inherited arrhythmic syndromes and/or cardiomyopathies suggesting a role for these genes in cardiovascular pathology in the general population. We show that polygenic predisposition to PR interval duration is an endophenotype for cardiovascular disease, including distal conduction disease, AF, and atrioventricular pre-excitation. These findings advance our understanding of the polygenic basis of cardiac conduction, and the genetic relationship between PR interval duration and cardiovascular disease

FELIX: construction and testing of a facility to study electromagnetic effects for first wall, blanket, and shield systems

An experimental test facility for the study of electromagnetic effects in the FWBS systems of fusion reactors has been constructed over the past 1-1/2 years at Argonne National Laboratory (ANL). In a test volume of 0.76 m/sup 3/ a vertical pulsed 0.5 T dipole field (B < 50 T/s) is perpendicular to a 1 T solenoid field. Power supplies of 2.75 MW and 5.5 MW and a solid state switch rated 13 kV, 13.1 kA (170 MW) control the pulsed magnetic fields. The total stored energy in the coils is 2.13 MJ. The coils are designed for a future upgrade to 4 T or the solenoid and 1 T for the dipole field (a total of 23.7 MJ). This paper describes the design and construction features of the facility. These include the power supplies, the solid state switches, winding and impregnation of large dipole saddle coils, control of the magnetic forces, computer control of FELIX and of experimental data acquisition and analysis, and an initial experimental test setup to analyze the eddy current distribution in a flat disk