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

General Existence Results for Reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory.Comment: 23 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

Acute Tubulo-interstitial Nephritis with Positive Anti-Neutrophil Cytoplasmic Antibodies

Introduction: Anti-neutrophil cytoplasmic antibody (ANCA) associated vasculitis involving the kidney usually comprises pauci-immune, necrotizing glomerulonephritis with crescents. Mononuclear tubulo-interstitial infiltrates are common in ANCA associated vasculitis, but these have usually been described in conjunction with glomerulitis. Acute tubulo-interstitial nephritis (ATIN) is a common cause of acute kidney injury that is most frequently induced by drugs or infections. Idiopathic ATIN has rarely been reported in association with the presence of a positive ANCA. These two entities seem to share a common immunological basis.Case report: We report a 75 years-old male patient who presented with acute kidney injury and his serum tested positive for p-ANCA by indirect immunofluoresence with a titer of 1/320. Testing by ELISA demonstrated anti-myeloperoxidase (MPO) specificity with a level of 28.8 IU/mL. His kidney biopsy showed features of ATIN with no glomerular involvement. Treatment with corticosteroids led to improvement of his kidney function and serology for ANCA became negative. In this case report ATIN seems to be associated with ANCA positivity, in the absence of other obvious causes for the acute tubulo-interstitial insult.Conclusion: ATIN can be associated with positive ANCA without features of renal-limited vasculitis or systemic vasculitis. This can occur in the absence of drug exposure. The outcome in our case was favorable with corticosteroid therapy.Key words: ANCA; Acute Tubulo-interstitial Nephritis; Pathogenesi