A new theoretical error estimate of the method of fundamental solutions applied to reduced wave problems in the exterior region of a disk

AbstractIn this paper, we present a mathematical study of the method of fundamental solutions (MFS) applied to reduced wave problems with Dirichlet boundary conditions in the exterior domain of a disk. A theorem in this paper shows that the MFS with N source points in equi-distantly equally phased arrangement with assignment parameter q(0<q<1), which characterizes the position of the source points and the collocation points, gives an approximate solution with error of O(qN) if the Fourier coefficients of the boundary data decay exponentially. This error estimate is an extension of the results of the previous studies. Numerical examples make good agreements with the results of the theoretical study

Chronic Inflammatory Demyelinating Polyneuropathy With Concurrent Membranous Nephropathy: An Anti-paranode and Podocyte Protein Antibody Study and Literature Survey

Background: Several case reports have described the concurrence of chronic inflammatory demyelinating polyneuropathy (CIDP) and membranous nephropathy (MN). The presence of autoantibodies against podocyte antigens phospholipase A2 receptor (PLA2R) and thrombospondin type 1 domain containing 7A (THSD7A) in MN suggests an autoimmune mechanism. Some CIDP patients also harbor autoantibodies against paranodal proteins such as neurofascin 155 (NF155) and contactin-1 (CNTN1). We investigated the relationship between CIDP and MN by assaying autoantibodies against paranodal and podocyte antigens in a CIDP patient with MN, and by a literature survey on the clinical features of CIDP with MN.Methods: Anti-CNTN1 and NF155 antibodies were measured by flow cytometry using HEK293 cell lines stably expressing human CNTN1 or NF155. Binding capacity of antibodies was validated by immunostaining mouse teased sciatic nerve fibers. Anti-PLA2R antibodies were measured by enzyme-linked sorbent assay and anti-THSD7A antibodies by indirect immunofluorescence assay. Clinical features between 14 CIDP with MN cases including two with anti-CNTN1 antibodies and 20 anti-CNTN1 antibody-positive CIDP cases were compared.Results: A patient whose ages was in the late 70 s complained of progressive weakness and superficial and deep sensory impairment in four extremities over 6 months. Nerve conduction studies showed prominent demyelination patterns. The patient presented with nephrotic syndrome. Renal biopsy disclosed basement membrane thickening with local subepithelial projections and glomerular deposits of IgG4, compatible with MN. Autoantibody assays revealed the presence of IgG4 and IgG1 anti-CNTN1 antibodies, but an absence of anti-NF155, anti-PLA2R, and anti-THSD7A antibodies. The patient's serum stained paranodes of teased sciatic nerves. CIDP with MN and anti-CNTN1 antibody-positive CIDP commonly showed male preponderance, relatively higher age of onset, acute to subacute onset in 35–50% of cases, distal dominant sensorimotor neuropathy, proprioceptive impairment leading to sensory ataxia, and very high cerebrospinal fluid protein levels. However, 11 of 13 CIDP patients with MN had a favorable response to mono- or combined immunotherapies whereas anti-CNTN1 antibody-positive CIDP was frequently refractory to corticosteroids and intravenous immunoglobulin administration.Conclusion: CIDP with MN and anti-CNTN1 antibody-positive CIDP show considerable overlap but are not identical. CIDP with MN is probably heterogeneous and some cases harbor anti-CNTN1 antibodies