Form factor expansion of the row and diagonal correlation functions of the two dimensional Ising model

We derive and prove exponential and form factor expansions of the row correlation function and the diagonal correlation function of the two dimensional Ising model

The diagonal Ising susceptibility

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions $\chi_{d}^{(1)}$ and $\chi_{d}^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${\chi}^{(5)}_d(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $n$-particle contributions $\chi_{d}^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $f^{(n)}$. We show that the $n^{th}$ particle contributions $\chi_{d}^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh2E_v/kT \sinh 2E_h/kT|=1$ as $n\to \infty$.Comment: 18 page

Singularities of nn-fold integrals of the Ising class and the theory of elliptic curves

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for $n=1, 2, 3, 4$ and only modulo some primes for $n=5$ and $6$, thus providing a large set of (possible) new singularities of the $\chi^{(n)}$. We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to $n= 6$) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a {\em finite number of one dimensional integrals}. Among the singularities found, we underline the fact that the quadratic polynomial condition $1+3 w +4 w^2 = 0$, that occurs in the linear differential equation of $\chi^{(3)}$, actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.Comment: 39 pages, 7 figure

Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

We evaluate the virial coefficients B_k for k<=10 for hard spheres in dimensions D=2,...,8. Virial coefficients with k even are found to be negative when D>=5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D>=5. Further analysis provides evidence that negative virial coefficients will be seen for some k>10 for D=4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D=3.Comment: 33 pages, 12 figure

Development of an international survey attitude scale: measurement equivalence, reliability, and predictive validity

Declining response rates worldwide have stimulated interest in understanding what may be influencing this decline and how it varies across countries and survey populations. In this paper, we describe the development and validation of a short 9-item survey attitude scale that measures three important constructs, thought by many scholars to be related to decisions to participate in surveys, that is, survey enjoyment, survey value, and survey burden. The survey attitude scale is based on a literature review of earlier work by multiple authors. Our overarching goal with this study is to develop and validate a concise and effective measure of how individuals feel about responding to surveys that can be implemented in surveys and panels to understand the willingness to participate in surveys and improve survey effectiveness. The research questions relate to factor structure, measurement equivalence, reliability, and predictive validity of the survey attitude scale. The data came from three probability-based panels: the German GESIS and PPSM panels and the Dutch LISS panel. The survey attitude scale proved to have a replicable three-dimensional factor structure (survey enjoyment, survey value, and survey burden). Partial scalar measurement equivalence was established across three panels that employed two languages (German and Dutch) and three measurement modes (web, telephone, and paper mail). For all three dimensions of the survey attitude scale, the reliability of the corresponding subscales (enjoyment, value, and burden) was satisfactory. Furthermore, the scales correlated with survey response in the expected directions, indicating predictive validity