Scaling functions in the square Ising model

We show and give the linear differential operators ${\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $F_{\pm}(r)= \lim_{scaling} {\cal M}_{\pm}^{-2} = \sum_{n} I_{n}(r)$. The integrals $I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\, {\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $r^{1/4}\,\exp(r^2/8)$ is a solution of the Painlev\'e VI equation in the scaling limit. Combinations of the (analytic at $r= 0$) solutions of ${\cal L}^{scal}_q$ sum to $\exp(r^2/8)$. We show that the expression $r^{1/4} \exp(r^2/8)$ is the scaling limit of the correlation function $C(N, N)$ and $C(N, N+1)$. The differential Galois groups of the factors occurring in the operators ${\cal L}^{scal}_q$ are given.Comment: 26 page

Canonical decomposition of linear differential operators with selected differential Galois groups

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.Comment: 40 page

Landau singularities and singularities of holonomic integrals of the Ising class

We consider families of multiple and simple integrals of the ``Ising class'' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODE's and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= \sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {\bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $\chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $\chi^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $2+s+s^2=0$ oustside the unit circle ($| s | > 1$).Comment: 34 pages, 1 figur

Rheological Model for Wood

Wood as the most important natural and renewable building material plays an important role in the construction sector. Nevertheless, its hygroscopic character basically affects all related mechanical properties leading to degradation of material stiffness and strength over the service life. Accordingly, to attain reliable design of the timber structures, the influence of moisture evolution and the role of time- and moisture-dependent behaviors have to be taken into account. For this purpose, in the current study a 3D orthotropic elasto-plastic, visco-elastic, mechano-sorptive constitutive model for wood, with all material constants being defined as a function of moisture content, is presented. The corresponding numerical integration approach, with additive decomposition of the total strain is developed and implemented within the framework of the finite element method (FEM). Moreover to preserve a quadratic rate of asymptotic convergence the consistent tangent operator for the whole model is derived. Functionality and capability of the presented material model are evaluated by performing several numerical verification simulations of wood components under different combinations of mechanical loading and moisture variation. Additionally, the flexibility and universality of the introduced model to predict the mechanical behavior of different species are demonstrated by the analysis of a hybrid wood element. Furthermore, the proposed numerical approach is validated by comparisons of computational evaluations with experimental results.Comment: 37 pages, 13 figures, 10 table

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.Comment: 100 page

Effect of Dual Surface Activation on the Surface Roughness of Titanium Dental Implant

Titanium is the most prevalent material for use in dental implants because of its mechanical properties and intrinsic osteoconductivity. In dental implant, the surface treatment is used to modify surface topography resulting in an improved biocompatibility. In this research surface activation were used for commercial pure Ti alloys manufactured by two different methods; the first method involved the use of commercial pure titanium rod converted to form implant screw by using wire cut machine and lathe. The second method included the use of powder technology for producing the implant screws. Then dual surface treatments were used for samples in two treatment stages. A primary treatment  used to prepare the surface for subsequent treatment which involves acid and alkali etching. The second surface activation treatment process were employed; ulrasonic surface treatment, Nd:YAG laser pulses . The  characterization of samples  before and after surface treatment procedure have been done to examine implant samples in terms of the best surface treatment method which produced the preferable surface properties .The characterization included ; microstructure observation, surface chemical composition analysis(EDS) , surface roughness (AFM) , ion release analysis. From microstructure observation The use of dual chemical treatment (HCl and NaOH etching ) as primary treatment resulted in a change in the surface topography by the formation of sodium titanate hydro gel layer. The surface topography was displayed by Atomic-force microscopy (AFM). From the master group, the powder technology process produced samples with high surface roughness compared with machining process. While there was a large decrease in roughness of samples treated primarily by the acid and alkaline etching. After laser treatment, all samples had the same response to laser irradiation and slight differences in roughness were observed. From the results of ion release analysis, it was found that all samples in all groups had similar ion release behavior when the samples immersed in Hank's solution for seven days. It was observed that the release of Ti ion rose in first three days and after that it began to stabilize. Keywords: Surface Activation, Surface Roughness Dental Implan