Super and Hyper Products of Super Relations

If R is a relation on X to Y, U is a relation on P (X) to Y, and V is a relation on P (X) to P (Y), then we say that R is an ordinary relation, U is a super relation, and V is a hyper relation on X to Y. Motivated by an ingenious idea of Emilia Przemska on a unified treatment of open- and closed-like sets, we shall introduce and investigate here four reasonable notions of product relations for super relations. In particular, for any two super relations U and V on X, we define two super relations U * V and U * V, and two hyper relations U ★ V and U * V on X such that : ( U * V ) ( A ) = ( A ∪ U ( A ) ) ∩ V ( A ) , ( U * V ) ( A ) = ( A ∩ U ( A ) ) ∪ U ( A ) \begin{array}{*{20}{l}} {(U*V)(A) = (A\mathop \cup \nolimits^ U(A))\mathop \cap \nolimits^ V(A),}\\ {(U*V)(A) = (A\mathop \cap \nolimits^ U(A))\mathop \cup \nolimits^ U(A)} \end{array} and ( U ★ V ) ( A ) = { B ⊆ X : ( U * V ) ( A ) ⊆ B ⊆ ( U * V ) ( A ) } , ( U * V ) ( A ) = { B ⊆ X : ( U ∩ V ) ( A ) ⊆ B ⊆ ( U ∪ V ) ( A ) } \begin{array}{*{20}{l}} {(UV)(A) = \{ B \subseteq X:\,(U*V)(A) \subseteq B \subseteq (U*V)(A)\} ,}\\ {(U*V)(A) = \{ B \subseteq X:\,(U\mathop \cap \nolimits^ V)(A) \subseteq B \subseteq (U\mathop \cup \nolimits^ V)(A)\} } \end{array} for all A ⊆ X. By using the distributivity of the operation ∩ over ∪, we can at once see that U * V ⊆ U * V. Moreover, if U ⊆ V, then we can also see that U * V = U * V. The most simple case is when U is an interior relation on X and V is the associated closure relation defined such that V (A) = U (Ac ) c for all A ⊆ X.

Uncertainty Quantification of Deep Neural Network-Based Turbulence Model for Reactor Transient Analysis

Deep neural networks (DNNs) have demonstrated good performance in learning highly non-linear relationships in large datasets, thus have been considered as a promising surrogate modeling tool for parametric partial differential equations (PDEs). On the other hand, quantifying the predictive uncertainty in DNNs is still a challenging problem. The Bayesian neural network (BNN), a sophisticated method assuming the weights of the DNNs follow certain uncertainty distributions, is considered as a state-of-the-art method for the UQ of DNNs. However, the method is too computationally expensive to be used in complicated DNN architectures. In this work, we utilized two more methods for the UQ of complicated DNNs, i.e. Monte Carlo dropout and deep ensemble. Both methods are computationally efficient and scalable compared to BNN. We applied these two methods to a densely connected convolutional network, which is developed and trained as a coarse-mesh turbulence closure relation for reactor safety analysis. In comparison, the corresponding BNN with the same architecture is also developed and trained. The computational cost and uncertainty evaluation performance of these three UQ methods are comprehensively investigated. It is found that the deep ensemble method is able to produce reasonable uncertainty estimates with good scalability and relatively low computational cost compared to BNN.

New MicroRNAs Candidates to Treat Human Colorectal Cancer; Molecular Dynamic Simulations Found a Major Down-Regulator of CLCA4 Tumor Suppressor Gene

IntroductionColorectal cancer (CRC) is one of the most common malignancies worldwide. The expression of CLCA4, a tumor suppressor gene, decreases significantly in cancer cells of CRC. In this study, we identified miRNAs target the mRNA of the CLCA4 gene. ObjectiveThe aim of this study was the identification of miRNAs involved in CRC.Material and methodsWe predicted miRNA(s) that target CLCA4 mRNA applying TargetScan v.7. Then through analysis of Gene Expression Omnibus (GEO) datasets, among them, miRNA(s) over-expressed in CRC cells were determined. To identify miRNAs with the highest potential to down-regulate CLCA4 through binding, we calculated the binding free energies of the candidate miRNA- mRNA complexes using the molecular mechanics energies combined with several solvation models: The Poisson–Boltzmann (MM/PBSA), the generalized Born (MM/GBSA), and the three-dimensional reference interaction site model with Kovalenko–Hirata closure relation (3D-RISM-KH). ResultsOur TargetScan analysis predicted that 106 miRNAs could bind to CLCA4 3' UTR mRNA. Hsa-miR-934, hsa-miR-574-5p, hsa-miR-377-3p, hsa-miR-5580-3p, hsa-miR-4775, hsa-miR-590-3p and hsa-miR-501-5p showed increased expression in CRC samples compared to normal cells. MD results found the lowest free energy changes in three hsa-miR-377-3p, hsa-miR-574-5p and hsa-miR-501-5p miRNAs. ConclusionThis research beside introducing a new fast and low cost plan to find best candidate of miRNAs to bind their targets, suggested miR-501-5p as a biomarker for early diagnosis of CRC. As well, preventing of down regulation of the CLCA4 expression through interrupting in the expression of miR-574-5p and miR-377-3p and more effectively miR-501-5p probably treat or slow down the development of colorectal cancer.

Radiatively-driven black-hole winds revisited

We examine general relativistic radiatively-driven spherical winds, using the basic equations for relativistic radiation hydrodynamics under the moment formalism. Moment equations are often closed, using the equilibrium diffusion approximation, which has an acausal problem, and furthermore, gives nodal-type critical points. Instead, we use the nonequilibrium diffusion approximation with a closure relation of a variable Eddington factor, f(τ, β), where τ is the optical depth and β is the flow speed normalized by the speed of light. We then analyse the critical properties in detail for several parameters, and found that there appear saddle-type critical points as well as nodal type and spiral one. The most suitable type is the saddle one, which appears in a region close to a black hole. We also calculate transonic solutions with typical parameters, and show that the luminosity is almost comparable to the Eddington luminosity, the gas is quickly accelerated in the vicinity of the black hole, and wind terminal speeds are on the order of 0.1–0.3 c. These results of radiatively-driven black hole winds can be applied, e.g. to ultra-fast outflows (UFOs), which are supposed to be fast outflows from the vicinity of super massive black holes.

THE SLICE BALANCE APPROACH USING AN ADAPTIVE-WEIGHTED CLOSURE

In this paper, we present a formulation of the slice balance approach using a nonlinear closure relation derived analogously from the adaptive-weighted diamond-difference form of the weighted diamond-difference method for Cartesian grids. The method yields strictly positive solutions that reduce to a standard diamond closure with fine-enough mesh granularity. It can be efficiently solved using Newton-like nonlinear iterative methods with diffusion preconditioning.

Radial Distribution Function for a Hard Sphere Liquid: A Modified Percus-Yevick and Hypernetted-Chain Closure Relations

Establishment of the radial distribution function by solving the Ornstein-Zernike equation is still an important problem, even more than a hundred years after the original paper publication. New strategies and approximations are common in the literature. A crucial step in this process consists in defining a closure relation which retrieves correlation functions in agreement with experiments or molecular simulations. In this paper, the functional Taylor expansion, as proposed by J. K. Percus, is applied to introduce two new closure relations: one that modifies the Percus‑Yevick closure relation and another one modifying the Hypernetted-Chain approximation. These new approximations will be applied to a hard sphere system. An improvement for the radial distribution function is observed in both cases. For some densities a greater accuracy, by a factor of five times compared to the original approximations, was obtained.

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

Density functional theory for collisionless plasmas – equivalence of fluid and kinetic approaches

Density functional theory (DFT) is a powerful theoretical tool widely used in such diverse fields as computational condensed-matter physics, atomic physics and quantum chemistry. DFT establishes that a system of $N$ interacting electrons can be described uniquely by its single-particle density $n(\boldsymbol{r})$ , instead of the $N$ -body wave function, yielding an enormous gain in terms of computational speed and memory storage space. Here, we use time-dependent DFT to show that a classical collisionless plasma can always, in principle, be described by a set of fluid equations for the single-particle density and current. The results of DFT guarantee that an exact closure relation, fully reproducing the Vlasov dynamics, necessarily exists, although it may be complicated (non-local in space and time, for instance) and difficult to obtain in practice. This goes against the common wisdom in plasma physics that the Vlasov and fluid descriptions are mutually incompatible, with the latter inevitably missing some ‘purely kinetic’ effects.

On the spatial and temporal non-locality of dynamo mean-field effects in supersonic interstellar turbulence

ABSTRACT The interstellar medium (ISM) of the Milky Way and nearby disc galaxies harbour large-scale coherent magnetic fields of microgauss strength, that can be explained via the action of a mean-field dynamo. As in our previous work, we aim to quantify dynamo effects that are self-consistently emerging in realistic direct magnetohydrodynamic simulations, but we generalize our approach to the case of a non-local (non-instantaneous) closure relation, described by a convolution integral in space (time). To this end, we leverage our comprehensive simulation framework for the supernova-regulated turbulent multiphase ISM. By introducing spatially (temporally) modulated mean fields, we extend the previously used test-field method to the spectral realm – providing the Fourier representation of the convolution kernels. The resulting spectra of the dynamo mean-field coefficients that we obtain broadly match expectations and allow to rigorously constrain the degree of scale separation in the Galactic dynamo. A surprising result is found for the diamagnetic pumping term, which increases in amplitude when going to smaller scales. Our results amount to the most comprehensive description of dynamo mean-field effects in the Galactic context to date. Surveying the relevant parameter space and quenching behaviour, this will ultimately enable the development of assumption-free subgrid prescriptions for otherwise unresolved global galaxy simulations.

A new hybrid radiative transfer method for massive star formation

Context. Frequency-dependent and hybrid approaches for the treatment of stellar irradiation are of primary importance in numerical simulations of massive star formation. Aims. We seek to compare outflow and accretion mechanisms in star formation simulations. We investigate the accuracy of a hybrid radiative transfer method using the gray M1 closure relation for proto-stellar irradiation and gray flux-limited diffusion (FLD) for photons emitted everywhere else. Methods. We have coupled the FLD module of the adaptive-mesh refinement code RAMSES with RAMSES-RT, which is based on the M1 closure relation and the reduced speed-of-light-approximation. Our hybrid (M1+FLD) method takes an average opacity at the stellar temperature for the M1 module, instead of the local environmental radiation field. Due to their construction, the opacities are consistent with the photon origin. We have tested this approach in radiative transfer tests of disks irradiated by a star for three levels of optical thickness and compared the temperature structure with the radiative transfer codes RADMC-3D and MCFOST. We applied it to a radiation-hydrodynamical simulation of massive star formation. Results. Our tests validate our hybrid approach for determining the temperature structure of an irradiated disk in the optically-thin (2% maximal error) and moderately optically-thick (error smaller than 25%) regimes. The most optically-thick test shows the limitation of our hybrid approach with a maximal error of 65% in the disk mid-plane against 94% with the FLD method. The optically-thick setups highlight the ability of the hybrid method to partially capture the self-shielding in the disk while the FLD alone cannot. The radiative acceleration is ≈100 times greater with the hybrid method than with the FLD. The hybrid method consistently leads to about + 50% more extended and wider-angle radiative outflows in the massive star formation simulation. We obtain a 17.6 M⊙ star at t ≃ 0.7τff, while the accretion phase is still ongoing, with a mean accretion rate of ≃7 × 10−4 M⊙ yr−1. Finally, despite the use of refinement to resolve the radiative cavities, no Rayleigh–Taylor instability appears in our simulations, and we justify their absence by physical arguments based on the entropy gradient.