Background: -catenin mutations that constitutively activate the canonical Wnt signaling have been observed in a subset of hepatocellular carcinomas (HCCs). These mutations are associated with chromosomal stability, low histological grade, low tumor invasion and better patient survival. We hypothesized that canonical Wnt signaling is selectively activated in well-differentiated, but repressed in poorly differentiated HCCs. To this aim, we characterized differentiation status of HCC cell lines and compared their expression status of Wnt pathway genes, and explored their activity of canonical Wnt signaling.

Objectives Coronavirus 2019 disease (COVID‐19) lead to one of the pandemics of the last century. We aimed to predict poor prognosis among severe patients to lead early intervention. Methods The data of 534 hospitalized patients were assessed retrospectively. Risk factors and laboratory tests that might enable the prediction of prognosis defined as being transferred to the intensive care unit and/or exitus have been investigated. Results At the admission, 398 of 534 patients (74.5%) were mild‐moderate ill. It was determined that the male gender, advanced age, and comorbidity were risk factors for severity. To estimate the severity of the disease, receiver operating characteristic analysis revealed that the areas under the curve which were determined based on the optimal cut off values that were calculated for the variables of values of neutrophil to lymphocyte ratio (NLR > 3.69), C‐reactive protein (CRP > 46 mg/L), troponin I ( > 5.3 ng/L), lactate dehydrogenase (LDH > 325 U/L), ferritin ( > 303 ug/L), d‐dimer ( > 574 μg/L), neutrophil NE ( > 4.99 × 109/L), lymphocyte (LE < 1.04 × 109/L), SO2 ( < %92) were 0.762, 0.757,0.742, 0.705, 0.698, 0.694,0.688, 0.678, and 0.66, respectively. To predict mortality, AUC of values for optimal cutoff troponin I ( > 7.4 ng/L), age ( > 62), SO2 ( < %89), urea ( > 40 mg/dL), procalcitonin ( > 0.21 ug/L), CKMB ( > 2.6 ng/L) were 0.715, 0.685, 0.644, 0.632, 0.627, and 0.617, respectively. Conclusions The clinical progress could be severe if the baseline values of NLR, CRP, troponin I, LDH, are above, and LE is below the specified cut‐off point. We found that the troponin I, elder age, and SO2 values could predict mortality.

We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids in porous media. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the L ∞ and W 1,2−a (with 0 < a < 1) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration.

We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.and let Q λ = I − P λ be the orthogonal complement of P λ . Now, given λ > 0 and I h define J = P λ P σ I h and E = I − J.(1.6) Note, although no additional orthogonality or regularity properties other than those appearing in Definition 1.1 have been assumed on I h , the above spectral filtering yields an operator J which is nearly orthogonal and has a range contained in D(A). The downscaling data assimilation algorithm studied in this paper may now be stated as Definition 1.2. Let U be an exact solution of (1.2) which evolves according to dynamics given by the semi-process S. Let t n = t 0 + nδ be a sequence of times for which partial

We study the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media. The media's porosity and coefficients of the Forchheimer equation are functions of the spatial variables. The partial differential equation for the pressure is degenerate in its gradient and can be both singular and degenerate in the spatial variables. Suitable weighted Lebesgue norms for the pressure, its gradient and time derivative are estimated. The continuous dependence on the initial and boundary data is established for the pressure and its gradient with respect to those corresponding norms. Asymptotic estimates are derived even for unbounded boundary data as time tends to infinity.