Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Genome-Wide Studies Reveal that H3K4me3 Modification in Bivalent Genes Is Dynamically Regulated during the Pluripotent Cell Cycle and Stabilized upon Differentiation

Indexación: Web of Science; Scopus.Stem cell phenotypes are reflected by posttranslational histone modifications, and this chromatin-related memory must be mitotically inherited to maintain cell identity through proliferative expansion. In human embryonic stem cells (hESCs), bivalent genes with both activating (H3K4me3) and repressive (H3K27me3) histone modifications are essential to sustain pluripotency. Yet, the molecular mechanisms by which this epigenetic landscape is transferred to progeny cells remain to be established. By mapping genomic enrichment of H3K4me3/H3K27me3 in pure populations of hESCs in G2, mitotic, and G1 phases of the cell cycle, we found striking variations in the levels of H3K4me3 through the G2-M-G1 transition. Analysis of a representative set of bivalent genes revealed that chromatin modifiers involved in H3K4 methylation/demethylation are recruited to bivalent gene promoters in a cell cycle-dependent fashion. Interestingly, bivalent genes enriched with H3K4me3 exclusively during mitosis undergo the strongest upregulation after induction of differentiation. Furthermore, the histone modification signature of genes that remain bivalent in differentiated cells resolves into a cell cycle-independent pattern after lineage commitment. These results establish a new dimension of chromatin regulation important in the maintenance of pluripotencyhttp://mcb.asm.org/content/36/4/61

Absence of squirt singularities for the multi-phase Muskat problem

In this paper we study the evolution of multiple fluids with different constant densities in porous media. This physical scenario is known as the Muskat and the (multi-phase) Hele-Shaw problems. In this context we prove that the fluids do not develop squirt singularities.Comment: 16 page

Antagonistic effects of transforming growth factor-beta on vitamin D3 enhancement of osteocalcin and osteopontin transcription: reduced interactions of vitamin D receptor/retinoid X receptor complexes with vitamin E response elements

Osteocalcin and osteopontin are noncollagenous proteins secreted by osteoblasts and regulated by a complex interplay of systemic and locally produced factors, including growth factors and steroid hormones. We investigated the mechanism by which transforming growth factor-beta (TGF beta) inhibits 1,25-dihydroxyvitamin D3 (1,25-(OH)2D3)-enhanced expression of the osteocalcin (OC) and osteopontin (OP) genes. ROS 17/2.8 cells, in which both genes are expressed, were transfected with reporter constructs driven by native (i.e. wild-type) rat OC and mouse OP promoters. TGF beta abrogated the 1,25-(OH)2D3 enhanced transcription of both the OC and OP genes. The inhibitory TGF beta response for each requires vitamin D response element (VDRE) sequences, although there are additional contributions from proximal basal regulatory elements. These transcriptional effects were further investigated for contribution of the trans-activating factors, which interact with OC and OP VDREs, involving the vitamin D receptor (VDR) and retinoid X receptor (RXR). Gel mobility shift assays show that TGF beta significantly reduces induction of the heterodimers VDR/RXR complexes in 1,25-(OH)2D3-treated ROS 17/2.8 cells. However, Western blot and ligand binding analysis reveal that TGF beta does not affect nuclear availability of the VDR. We also show that activator protein-1 activity is up-regulated by TGF beta; thus, activator protein-1 binding sites in the OC promoter may potentially contribute to inhibitory effects of TGF beta on basal transcription. Our studies demonstrate that the inhibitory action of TGF beta on the 1,25-(OH)2D3 enhancement of OC and OP transcription in osteoblastic cells results from modulations of protein-DNA interactions at the OC and OP VDRE, which cannot be accounted for by changes in VDR protein levels. As OC and OP participate in bone turnover, our results provide insight into the contributions of TGF beta and 1,25-(OH)2D3 to VDR-mediated gene regulatory mechanism operative in bone formation and/or resorption events

pMHC affinity controls duration of CD8+ T cell-DC interactions and imprints timing of effector differentiation versus expansion.

During adaptive immune responses, CD8(+) T cells with low TCR affinities are released early into the circulation before high-affinity clones become dominant at later time points. How functional avidity maturation is orchestrated in lymphoid tissue and how low-affinity cells contribute to host protection remains unclear. In this study, we used intravital imaging of reactive lymph nodes (LNs) to show that T cells rapidly attached to dendritic cells irrespective of TCR affinity, whereas one day later, the duration of these stable interactions ceased progressively with lowering peptide major histocompatibility complex (pMHC) affinity. This correlated inversely BATF (basic leucine zipper transcription factor, ATF-like) and IRF4 (interferon-regulated factor 4) induction and timing of effector differentiation, as low affinity-primed T cells acquired cytotoxic activity earlier than high affinity-primed ones. After activation, low-affinity effector CD8(+) T cells accumulated at efferent lymphatic vessels for egress, whereas high affinity-stimulated CD8(+) T cells moved to interfollicular regions in a CXCR3-dependent manner for sustained pMHC stimulation and prolonged expansion. The early release of low-affinity effector T cells led to rapid target cell elimination outside reactive LNs. Our data provide a model for affinity-dependent spatiotemporal orchestration of CD8(+) T cell activation inside LNs leading to functional avidity maturation and uncover a role for low-affinity effector T cells during early microbial containment

Occurrence of FVIII Inhibitors in Hemophilia A Patients Following an Institutional Switch to a Third Generation B-Domain-Deleted FVIII

In 2018, Refacto AFR, a B-domain-deleted third-generation FVIII concentrate, became our preferential product. After the introduction, the development of inhibitors was prospectively monitored; retrospectively, we sought for risk factors in the patients who developed a de-novo inhibitor. Over a period of 15 months, 4/19 adult patients with non-severe haemophilia who were treated on demand for surgery, developed high titer antibodies to FVIII after administration of Refacto AFR; 5/52 mostly severe patients on prophylaxis, developed an inhibitor (3 ≥ 0.1 BU; 1 &gt; 0.6 BU, 1 high titre) after they switched to Refacto AFR; all were children &lt;14 years of age and with &gt;100 exposure days, none related to surgery or intensive treatment; all received KovaltryR before. In conclusion: inhibitors were encountered in on demand patients and previously treated prophylaxis patients; this observation might be a coincidental finding, but also risk factors like genotype and surgery and/or that Refacto AFR is more immunogenic should be considered. For the patients on prophylaxis we hypothesize that loss of tolerance by preceding KovaltryR might have contributed to inhibitor development.</p