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.

Petrophysical evaluation of sandstone gas reservoir using integrated well logs and core data for the Lower Cretaceous Kharita formation, Western Desert, Egypt

This study aims to evaluate Kharita gas reservoir to enhance the production. The increase in water-cut ratio reduces the left hydrocarbons’ amount behind pipe. Accurate determination of pore throats, pores connectivity and fluid distribution are central elements in improved reservoir description. The integration of core and logging data responses is often used to draw inferences about lithology, depositional sequences, facies, and fluid content. These inferences are based on petrophysical models utilizing correlations among tools’ responses as well as rocks and fluids properties. Upper Kharita Formation produces gas and condensate from the clastic sandstone in Badr-3 field, western desert of Egypt. It consists mainly of sandstone with shale intercalations. It is subdivided into three sub-units Kharita A, Kharita B and Kharita C that are in pressure communication. Hence, a new further investigation and review for the previously calculated GIIP (gas initially in place) was initiated. The results of this study yielded that the main uncertainty in the volumetric calculations was the petrophysical evaluation; subsequently, a new unconventional petrophysical evaluation approach was performed. The sands thickness in Upper Kharita Formation varies between more than 9 up and more than 61 m with average porosity values range between 0.08 and 0.17 PU while the average permeability values range between 1.89 and 696.66 mD. The average hydrocarbon saturation values range between 46 and 97%. The sands thickness in Upper Kharita Formation varies between more than 9 up and more than 61 m with average porosity values range between 0.08 and 0.17 PU while the average permeability values range between 1.89 and 696.66 mD. The average hydrocarbon saturation values range between 46 and 97%. Reservoir shale cutoff of 55% by using cross-plot between shale volume and porosity (Toby Darling concept) was utilized to discriminate the reservoir from non-reservoir sections. The porosity model was used to calculate reservoir porosity, using the density log. The Archie and saturation/height function models were used to calculate the water saturation and used to calibrate the water saturation in the transition zone. The porosity–permeability (POR-PERM) transform equation was used to estimate the reservoir connectivity (absolute permeability) for the four petrophysical facies (High Quality Reservoir, Moderate Quality Reservoir, Low Quality Reservoir and Highly Shale Reservoir). Core data have shown variations in reservoir quality parameters (porosity and permeability) from one well to the other. Integration of all the reservoir pressures indicated that there are different fluid types (oil, gas and water) in the Upper Kharita Formation level. The saturation/height function model was used to calibrate the saturation in the transition zone. The integration of geological core and geophysical log data helped to conduct a comprehensive petrophysical assessment of Upper Kharita Formation for a better estimation of the reservoir and to achieve a better understanding of the water encroachment in the Upper Kharita reservoir. The big challenge is the determination of the most correct model for calculating porosity, permeability and water saturation in this reservoir of different quality sand. The new petrophysical evaluation resulted in doubling the volumes in Upper Kharita reservoir and so a perforation campaign was performed to confirm the new volumetric calculations, which showed a good match with the model results. Hence, a new well was drilled targeting the low quality sand and found them with high pressure almost near virgin pressure.

A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

On the frequency of height values

We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height $${\mathcal {H}}$$ H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to $${\mathcal {H}}$$ H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $$k \in \{0,d\}$$ k ∈ { 0 , d } or $$\gcd (k,d) = 1$$ gcd ( k , d ) = 1 . We therefore study the behaviour in the case where $$0< k < d$$ 0 < k < d and $$\gcd (k,d) > 1$$ gcd ( k , d ) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.

On Sinaĭ Billiards on Flat Surfaces with Horns

We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

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.