A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449--474] and the data-driven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452--493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen--Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation

Dynamical symmetries of two-dimensional systems in relativistic quantum mechanics

The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum $L$. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the harmonic oscillator potential possesses an SU(2) symmetry. The generators of the symmetric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits are also discussed.Comment: 3 pages, Accepted by Annals of Physics (New York

Deletion within the Src homology domain 3 of Bruton's tyrosine kinase resulting in X-linked agammaglobulinemia (XLA).

The gene responsible for X-linked agammaglobulinemia (XLA) has been recently identified to code for a cytoplasmic tyrosine kinase (Bruton's agammaglobulinemia tyrosine kinase, BTK), required for normal B cell development. BTK, like many other cytoplasmic tyrosine kinases, contains Src homology domains (SH2 and SH3), and catalytic kinase domain. SH3 domains are important for the targeting of signaling molecules to specific subcellular locations. We have identified a family with XLA whose affected members have a point mutation (g-->a) at the 5' splice site of intron 8, resulting in the skipping of coding exon 8 and loss of 21 amino acids forming the COOH-terminal portion of the BTK SH3 domain. The study of three generations within this kinship, using restriction fragment length polymorphism and DNA analysis, allowed identification of the mutant X chromosome responsible for XLA and the carrier status in this family. BTK mRNA was present in normal amounts in Epstein-Barr virus-induced B lymphoblastoid cell lines established from affected family members. Although the SH3 deletion did not alter BTK protein stability and kinase activity of the truncated BTK protein was normal, the affected patients nevertheless have a severe B cell defect characteristic for XLA. The mutant protein was modeled using the normal BTK SH3 domain. The deletion results in loss of two COOH-terminal beta strands containing several residues critical for the formation of the putative SH3 ligand-binding pocket. We predict that, as a result, one or more crucial SH3 binding proteins fail to interact with BTK, interrupting the cytoplasmic signal transduction process required for B cell differentiation