On distribution densities of algebraic points under different height functions

In the article we consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed degree. The distribution is introduced using a height function. We have obtained universal upper and lower bounds of the distribution density of such points using an arbitrary height function. We have shown how from a given joint density function of coefficients of a random polynomial of degree n, one can construct such a height function H that the polynomials q of degree n uniformly chosen under H[q] ≤1 have the same distribution of zeros as the former random polynomial.

On algebraic points of fixed degree and bounded height

We consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed de- gree and bounded height. In the article the main result of a recent joint work by the author and F. Götze, and D. N. Zaporozhets is extended to the case of arbitrary height functions. We prove an asymptotic formula for the number of such algebraic points lying in a given spatial region. We obtain an explicit expression for the density function of algebraic points under an arbitrary height function.

Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results.

ROCK TYPING AND NOVEL APPROACH FOR FLUID-SATURATION DISTRIBUTION IN TILTED WATER/OIL CONTACT RESERVOIRS

Rock typing in carbonate reservoirs has always represented a difficult challenge due to rock heterogeneity. When interpreting electrical logs, the thick carbonate formation can leave an impression of a homogenous environment; however, looking at core analysis and mercury injection capillary pressure (MICP) data, reservoir heterogeneity can be determined. This complexity of the formation characterization presents challenges in reservoirs that contain tilted water/oil contact (WOC). Tilted WOC discovers hydrocarbon saturation below the free-water level, and different events during geological time can contribute to this specific fluid accumulation. Knowledge of the fluid distribution is needed to understand the mechanisms of oil entrapment, oil volumetrics, and potential recovery mechanisms involved in reservoirs under this wettability and WOC conditions. This case study will describe the workflow used to characterize and model an atypical regime like non-water wet formations in reservoirs with tilted WOC. In this study, a combination of electrical logs, core analysis (lithofacies, poro-perm, MICP), and customized workflow was used to characterize, classify, and map facies. Capillary pressure information and formation tester data were integrated and compiled for each facies. Moving forward, a new method was developed to model saturation height functions representing non-water wet formations and tilted WOC phenomena. Fluid and saturation properties are estimated and assigned to each reservoir point and after reservoir rock types (RRT) were defined. This method has been validated by applying the new approach to actual well data. The drainage capillary pressure (Pc) lab data in the reservoir intervals with established conventional WOC complemented interpretation results derived from acquired logs; however, for the reservoirs zones with identified tilted WOC, correlation and matching Pc lab data with logs was not possible. The new method provides saturation properties in formations with complex fluid-rock interactions and phenomena. This work introduces a novel approach to estimate saturation height functions and saturation distribution for reservoirs with complex fluid-rock interaction and distribution, such as non-water wet formations in tilted WOC conditions.

An All-Mach Number HLLC-Based Scheme for Multi-Phase Flow with Surface Tension

This paper presents an all-Mach method for two-phase inviscid flow in the presence of surface tension. A modified version of the Hartens–Lax–van Leer Contact (HLLC) solver is developed and combined for the first time with a widely used volume-of-fluid (VoF) method: the compressive interface capturing scheme for arbitrary meshes (CICSAM). This novel combination yields a scheme with both HLLC shock capturing as well as accurate liquid–gas interface tracking characteristics. It is achieved by reconstructing non-conservative (primitive) variables in a consistent manner to yield both robustness and accuracy. Liquid–gas interface curvature is computed via height functions and the convolution method. We emphasize the use of VoF in the interest of interface accuracy when modelling surface tension effects. The method is validated using a range of test-cases available in the literature. The results show flow features that are in sensible agreement with previous experimental and numerical work. In particular, the use of the HLLC-VoF combination leads to a sharp volume fraction and energy field with improved accuracy.

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order $$\sqrt{\log N}$$ log N . The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Determinantal Expressions in Multi-Species TASEP

Consider an inhomogeneous multi-species TASEP with drift to the left, and define a height function which equals the maximum species number to the left of a lattice site. For each fixed time, the multi-point distributions of these height functions have a determinantal structure. In the homogeneous case and for certain initial conditions, the fluctuations of the height function converge to Gaussian random variables in the large-time limit. The proof utilizes a coupling between the multi-species TASEP and a coalescing random walk, and previously known results for coalescing random walks.

Multiple positive solutions for singular higher-order semipositone fractional differential equations with p-Laplacian

In this article, together with Leggett–Williams and Guo–Krasnosel’skii fixed point theorems, height functions on special bounded sets are constructed to obtain the existence of at least three positive solutions for some higher-order fractional differential equations with p-Laplacian. The nonlinearity permits singularities both on the time and the space variables, and it also may change its sign.