Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

Estimation of quantitative criteria of carbonate reservoirs of triassic deposits

This paper discusses the boundary values of the reservoir properties of carbonate rocks of the Triassic sediments of South Mangyshlak, which are important for the interpretation of production geophysical data and for perforating and blasting operations. In terms of lithological composition, Triassic deposits are represented by two types of commercial reservoirs terrigenous and carbonate. Carbonate reservoirs are localized in the volcanic-dolomite and volcanic-limestone strata of the Middle Triassic. These rocks are characterized by a complex type of reservoir: porous-fractured, porous-cavernous and fractured. Sediments of the Upper Triassic occur with erosion on the Middle Triassic sedimentary complex and are represented by alternating tuffaceous, silt-sandy and mudstone rocks. Polymictic sandstones are oil-saturated to varying degrees; oil deposits are confined to them. To substantiate the quantitative criteria of the reservoir, the results obtained during special laboratory studies of the core were used. Filtration studies were carried out, where physical and hydrodynamic characteristics were determined when oil was displaced by displacing reagents. The obtained parameters were used to construct correlations collector non-collector. Using the relationships between the reservoir properties of the reservoir, the dependence of the porosity and permeability on the residual water content, as well as open porosity and permeability on the dynamic porosity, the boundary values were determined.

Tissue Segmentation from Whole-Slide Images Using Lightweight Neural Networks

Pathology slides of malignancies are segmented using lightweight convolutional neural networks (CNNs) that may be deployed on mobile devices. This is made possible by preprocessing candidate images to make CNN analysis tractable and also to exclude regions unlikely to be diagnostically relevant. In a training phase, labeled whole-slide histopathology images are first downsampled and decomposed into square tiles. Tiles corresponding to diseased regions are analyzed to determine boundary values of a visual criterion, image entropy. A lightweight CNN is then trained to distinguish tiles of diseased and non-diseased tissue, and if more than one disease type is present, to discriminate among these as well. A segmentation is generated by downsampling and tiling a candidate image, and retaining only those tiles with values of the visual criterion falling within the previously established extrema. The sifted tiles, which now exclude much of the non-diseased image content, are efficiently and accurately classified by the trained CNN. Tiles classified as diseased tissue ¾ or in the case of multiple possible subtypes, as the dominant subtype in the tile set ¾ are combined, either as a simple union or at a pixel level, to produce a segmentation mask or map. This approach was applied successfully to two very different datasets of large whole-slide images, one (PAIP2020) involving multiple subtypes of colorectal cancer and the other (CAMELYON16) single-type breast-cancer metastases. Scored using standard similarity metrics, the segmentations exhibited notably high recall, even when tiles were large relative to tumor features.

On the Geometric Approach to the Boundary Problem in Supergravity

We review the geometric superspace approach to the boundary problem in supergravity, retracing the geometric construction of four-dimensional supergravity Lagrangians in the presence of a non-trivial boundary of spacetime. We first focus on pure N=1 and N=2 theories with negative cosmological constant. Here, the supersymmetry invariance of the action requires the addition of topological (boundary) contributions which generalize at the supersymmetric level the Euler-Gauss-Bonnet term. Moreover, one finds that the boundary values of the super field-strengths are dynamically fixed to constant values, corresponding to the vanishing of the OSp(N|4)-covariant supercurvatures at the boundary. We then consider the case of vanishing cosmological constant where, in the presence of a non-trivial boundary, the inclusion of boundary terms involving additional fields, which behave as auxiliary fields for the bulk theory, allows to restore supersymmetry. In all the cases listed above, the full, supersymmetric Lagrangian can be recast in a MacDowell-Mansouri(-like) form. We then report on the application of the results to specific problems regarding cases where the boundary is located asymptotically, relevant for a holographic analysis.

Impact of Cold Weather on Setup Errors in Radiotherapy

Objective. To investigate the influence of cold weather on setup errors of patients with chest and pelvic disease in radiotherapy. Methods. The image-guided data of the patients were collected from the Radiotherapy Center of Cancer Hospital Affiliated to Guangxi Medical University from October 2020 to February 2021. During this period, the cold weather days were December 15, 16, and 17, 2020, and January 7 and 8, 2021. For body fixation in radiotherapy, an integrated plate and a thermoplastic mold were employed in 18 patients with chest disease, while an integrated plate and a vacuum pad were applied in 19 patients with pelvic disease. All patients underwent cone beam computed tomography (CBCT) scans in the first five treatments and once a week thereafter. The obtained data were registered to the planning CT image to get the setup errors of the patient in the translational direction including X, Y, and Z axes and rotational direction including RX, RY, and RZ. Then, the Mann–Whitney U test was performed. The expansion boundary values of the chest and pelvis were calculated according to the formula M PTV = 2.5 ∑ + 0.7 δ . Results. A total of 286 eligible results of CBCT scans were collected. There were 138 chest CBCT scans, including 26 taken in cold weather and 112 in usual weather, and 148 pelvic CBCT scans, including 33 taken in cold weather and 115 in usual weather. The X-, Y-, and Z-axis translational setup errors of patients with chest disease in the cold weather group were 0.16 (0.06, 0.32) cm, 0.25 (0.17, 0.52) cm, and 0.35 (0.21, 0.47) cm, respectively, and those in the usual weather group were 0.14 (0.08, 0.29) cm, 0.23 (0.13, 0.37) cm, and 0.18 (0.1, 0.35) cm, respectively. The results indicated that there was a statistical difference in the Z-axis translational error between the cold weather group and the usual weather group (U = 935.5; p = 0.005 < 0.05 ), while there was no statistical difference in the rotational error between the two groups. The external boundary values of X, Y, and Z axes in the cold weather group were 0.57 cm, 0.92 cm, and 0.99 cm, respectively, and those in the usual weather group were 0.57 cm, 0.78 cm, and 0.68 cm, respectively. There was no significant difference in the translational and rotational errors of patients with pelvic disease between the cold weather group and the usual weather group ( p < 0.05 ). The external boundary values of X, Y, and Z axes were 0.63 cm, 0.79 cm, and 0.68 cm in the cold weather group and 0.61 cm, 0.79 cm, and 0.61 cm in the usual weather group, respectively. Conclusion. The setup error of patients undergoing radiotherapy with their bodies fixed by an integrated plate and a thermoplastic mold was greater in cold weather than in usual weather, especially in the ventrodorsal direction.

On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of α0,λ* and λ* such that this problem has a unique solution when 0<α<α0 and λ>0, and has three solutions when α>α0 and λ*<λ<λ*. The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that 4.0686722336<α0<4.0686722344. This result improves the existing result for α0≈4.069 and increases the accuracy of α0 to 10−8. We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of λ for this problem to have three solutions for given values of α is also computed with accuracy up to 10−14.

NUMERICAL ESTIMATION OF THERMAL STABILITY BOUNDARIES OF FIXED BED BIOMASS COMBUSTION USING LOCAL CHEMICAL EQUILIBRIUM APPROXIMATION

В статье исследуется зависимость стационарной температуры слоевого горения биомассы от скорости подачи и удельного расхода воздушного дутья. Для этого стационарное уравнение теплового баланса вместе с простейшим уравнением кинетики для химической реакции решается в широком диапазоне параметров. Для численного решения вводится ряд допущений (узкая зона реакции, преимущественный отвод теплоты путем лучистой теплопроводности, равновесный состав продуктов окисления). Результаты расчетов дают граничные значения расходных и стехиометрических параметров, при которых возможно устойчивое горение. The article investigates the dependence of the stationary combustion temperature of layer combustion of biomass on the feed rate and specific consumption of air blast. For this, the stationary heat balance equation, together with the simplest kinetic equation for a chemical reaction, is solved over a wide range of parameters. For a numerical solution, a number of assumptions are introduced (a narrow reaction zone, preferential heat removal by radiant heat conduction, equilibrium composition of oxidation products). The calculation results give the boundary values of the consumption and stoichiometric parameters at which stable combustion is possible.

PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

Systematization and mathematical modeling of classification features of forest rakes

With proper maintenance of many territories a technological operation of collecting and raking up chopped residues of undesirable tree and shrub vegetation is often performed. At the same time, despite the variety of mechanization tools used, their classification is currently difficult. The purpose of the research was to develop a classification of technical means for collecting and raking chopped residues with the justification of their name and rational definition. Based on the analysis of more than 500 models the authors of the research recommended to designate the considered technical means as “Forest Rake” and a corresponding definition has been formulated. The authors of the carried out the classification of forest rakes, a system of equations has been developed that describes the position of the boundaries of the characteristic zones of the distribution of the working width of the forest rake, the boundary values of a number of parameters are shown, the criteria for the density of the teeth and the specific width of the rake are proposed, the values of these criteria are given. The results of the work can be used by manufacturing enterprises engaged in the maintenance and operation of transport infrastructure facilities.