Global Schauder Estimates for the $$\mathbf {p}$$-Laplace System

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $$\text {BMO}$$ BMO and $${{\,\mathrm{VMO}\,}}$$ VMO spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $$p=2$$ p = 2 , and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

INTERACTION OF LARGE PILES WITH A MULTILAYER SOIL MASS, TAKING INTO ACCOUNT HARDENING AND SOFTENING

This article discusses the formulation and solution of the problem of the interaction of a long pile with thesurrounding multilayer and underlying soils, taking into account the rheological properties of the surrounding soil mass. The creep process is considered taking into account hardening and softening. The problem was considered in a linear setting. The solution is presented by analytical method. To describe the creep process, the rheological parameters of hardening and softening were used. An expression is obtained for finding the reduced shear modulus for a multilayer soil mass. A dependence is obtained for determining the force on the pile heel on time, taking into account the rheological parameters of hardening and softening. Analytical solutions in the article are supported by a graphical part. The graphs of the dependence of the settlement of the pile, the force on the heel of the pile cutting through alternating layers, on time for various parameters of viscosity, as well as for variable parameters of hardening and softening are given. The solutions obtained can be used for preliminary determination of the movement of long piles with the surrounding multilayer and underlying soils.

Effect of Water Dilution on Full-Arch Gypsum Implant Master Casts

The objective of this study was to assess if the setting expansion, compressive strength, compressive modulus and Knoop hardness is reduced in Type III, IV and V gypsum products when excess water is added to recommended water/powder (W/P) ratios for full-arch implant master casts. Four properties were measured: linear setting expansion, compressive strength, compressive modulus, and Knoop hardness. Three dental stones were evaluated: ISO Type III, Type IV, and Type V. Water was added to recommended W/P ratios as follows: (1) 0%, the manufacturer’s recommend W/P ratio, (2) 50%, 1.5 times the manufacturer’s recommendation of water, (3) 100%, three times the manufacturer’s recommendation of water, (4) 150%, 2.5 times the manufacturer’s recommendation of water, and (5) 200%, three times the manufacturer’s recommendation of water. For all three gypsum types, 50% water additions minimally affected the setting expansion and compressive modulus (p > 0.05), but lowered compressive strength by 26% to 52% (p < 0.05). Knoop hardness was lowered by 24% and 48% for Types V and III (p < 0.05), whereas Type IV was not significantly affected (p > 0.05). With few exceptions, the physical properties for the three gypsum types reached minimum values at 100% dilution, with few changes noted at higher dilutions. Increased W/P ratios did not adequately reduce the setting expansion and adversely affected other physical properties. Of the gypsums tested, Type IV is still recommended due to its lower setting expansion for full-arch implant master casts.

Abnormal stresses in underground pipeline due to static and dynamic reversal of several foundation blocks

The article discusses the issues of forecasting the strength of underground pipelines laid in seismically active areas through sections composed of relatively rigid moving blocks. In such dangerous areas, in addition to the normal pressure load of the transported product, the pipe is subjected to additional effects from the movements of the fragments of the block foundation. As the literature data show, the problems of the influence of the interaction of faults on the stress state of the pipeline have not yet been studied. The aim of the study is to develop a model for the analysis of abnormal stresses in the underground pipeline on a damaged foundation caused by static or time-harmonic reciprocal turns of the blocks around the axis of the pipe on both sides of several faults. Static equilibrium and harmonic oscillations of the pipeline are investigated in a linear setting, modelling it with a rod with an annular cross section. The inertia of the transported product is not taken into account. To consider the issues of the ultimate equilibrium of the pipe, the momentless theory of shells and the energy theory of strength are used. The soil backfill is considered as Winkler’s elastic layer. Multiple damages to the solid foundation are presented in the form of several faults on which there is a rupture of the angle of rotation around the axis of the pipe. We formulated boundary value problems for differential equations of static torsion and torsional harmonic oscillations with discontinuous right-hand sides. Based on the analytical solutions of these problems for the cases of antisymmetric and symmetrical reversal of the foundation blocks, the distributions of the torsion angle and equivalent stress in the pipe, depending on the distance between faults and the frequency of forced oscillations of the system, are investigated.

On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others.

Supply Chain Contracting with Linear Utility Function

The supply chain contracting has traditionally been based on the profit maximization assumption. Recent research has shown that some behavior factors may influence the decision making of supply chain members. The authors utilize a linear utility function to depict such behavior factors and incorporate these into the newsvendor model. The linear utility function provides sufficient flexibility to better capture people's various behavior factors. By supposing the agents are concerned with behavior factors, the authors first investigate how the factors affect the supply chain under wholesale price contract, and find that they do not influence coordination condition, but can adjust the distribution of profits. Then they extend their study to other four common contracts with a similar method and systematically demonstrate that the behavior of agents in such a linear setting has no effect on the conditions of coordinating supply chain.

DEVELOPMENT OF THE THEORY OF COMPUTATION OF PIVOTALLY-CONNECTED BEAMS ON AN ELASTIC FOUNDATION TAKING INTO ACCOUNT THEIR PHYSICAL NONLINEARITY

This work presents a brief review of the literature on the theory and technique of computation of pivotally-connected structures on a linearly-elastic foundation. The authors refer to the works of B.G.Korenev, G.Ya.Popov, I.A.Simvulidi, R.V.Serebryany and A.G.Yuryev, in which investigations for calculating the pivotally-connected beams and slabs on an elastic foundation are performed using different approaches. From the analysis of the scientific and normative literature on the subject under consideration, a conclusion can be made that there is no common approach to solution of this problem, which would hold for any pivotally connected structures being in contact with any elastic foundation model under the action of an arbitrary external load. Besides, when designing the load carrying members of pavements of motor roads of various purposes in the Republic of Belarus, a number of branch-specific normative documents, where the pavements with the load carrying member and interconnection of members over the track length are considered separately in unconnected setting, is used. In this work, a universal approach for computation of pivotally-connected beams on an elastic foundation in the linear setting and taking into account the physical nonlinearity of the beam material is proposed. This approach is based on a mixed method of structural mechanics and implemented in different foundations taking into account the Zhemochkins relations for the functions of influences of an elastic medium. The following hypotheses and assumptions of the linear theory of elasticity and structural mechanics are taken into consideration: only normal stresses act at the contact of the beam with the foundation for beams the hypotheses of the flexural theory the pivot joints are cylindrical and the distribution of the contact stresses over the beam width is uniform. The physical nonlinearity of the beam material is taken into consideration through the variable rigidity of the Zhemochkins areas. Namely: after determining the forces in the Zhemochkins bonds at the contact of every beam with an elastic foundation as a result of the linear computation, the values of bending moments in each section of every beam are determined by the structural mechanics methods. From the calculated values of the moments, the tangential rigidity for each Zhemochkins area on the beam is determined using the formula of the moment-curvature dependence for the beam sections are determines as hyperbolic tangent. In the results of nonlinear computation, the stress-strain behaviour of the system of pivotally-connected beams on an elastic foundation is investigated as it was made earlier in the linear setting: distribution of contact stresses under the beams, internal forces in the beams and pivot joints as well as elastic foundation settlements. The proposed approach is implemented numerically with the use of the Mathematica 10.4 mathematical package. The computation example for three pivotally-connected beams on the Winkler foundation taking into account their physical nonlinearity.В работе приводится краткий обзор литературы по теории и методикам расчета шарнирно-соединенных конструкций на линейно-упругом основании. Авторы ссылаются на работы Б. Г. Коренева, Г. Я. Попова, И. А. Симвулиди, Р. В. Серебряного, А. Г. Юрьева, в которых различными подходами проведены исследования по расчету шарнирно-соединенных балок и плит на упругом основании. Из анализа научной и нормативной литературы по рассматриваемой тематике можно сделать вывод об отсутствии общего подхода к решению этой проблемы, справедливого для любых шарнирно-соединенных конструкций, контактирующих с любой моделью упругого основания под действием произвольной внешней нагрузки. Кроме того, при проектировании несущих элементов дорожных покрытий автомобильных дорог различного назначения в Республике Беларусь используется ряд отраслевых нормативных документов, в которых дорожная одежда с несущим элементом и соединение элементов между собой по длине трассы рассматриваются отдельно, в несвязной постановке. В данной работе предлагается универсальный подход для расчета шарнирно-соединенных балок на упругом основании в линейной постановке и с учетом физической нелинейности материала балок. Этот подход основан на смешанном методе строительной механики и реализуется в разных основаниях с учетом соотношений Жемочкина для функций влияний упругой среды. В расчет принимаются следующие гипотезы и допущения линейной теории упругости и строительной механики: на контакте балки с основанием действуют только нормальные напряжения, для балок справедливы гипотезы теории изгиба, шарниры между балками являются цилиндрическими, распределение контактных напряжений по ширине балок равномерное. Физическая нелинейность материала балок в предлагаемом расчете учитывается через переменную жесткость участков Жемочкина. А именно: после определения усилий в связях Жемочкина на контакте каждой балки с упругим основанием в результате линейного расчета, методами строительной механики определяются величины изгибающих моментов в каждом сечении каждой балки. По вычисленным значениям моментов определяется касательная жесткость для каждого участка Жемочкина на балках по формуле зависимости момент-кривизна для сечений балки в виде гиперболического тангенса. В результатах нелинейного расчета, как и ранее в линейной постановке, исследуется напряженно-деформированное состояние системы из шарнирно-соединенных балок на упругом основании: распределение контактных напряжений под балками, внутренние усилия в балках и шарнирных соединениях, а также осадки упругого основания. Численная реализация предлагаемого подхода выполнена с использованием математического пакета Mathematica 10.4. Приведен пример расчета для трех шарнирно-соединенных балок на основании Винклера с учетом их физической нелинейности.

Evalutional of the Effect of Physically Magnetized Water on some Properties of Dental Stone

The Aims of Study: to study the effect of interaction of dental stone with tap water and physically magnetized water on two periods of time 24h and 12h to study the change in the physical properties of final product. Materials and Methods: We started prepare no. of control samples and other of samples interacted with physically magnetized water for two periods of time 24h and 12h, then we studied the liner setting expansion and compressive strength. Results: Results obtained from comparing control sample with that treated with physically magnetized water showed an increase in the compressive strength of physically magnetized group and decrease of linear setting expansion in comparison with that of control sample. Conclusion: Reaction of dental stone with physically magnetized water lead to change in some physical properties of dental stone.

Predictive Regressions

Predictive regressions are a widely used econometric environment for assessing the predictability of economic and financial variables using past values of one or more predictors. The nature of the applications considered by practitioners often involve the use of predictors that have highly persistent, smoothly varying dynamics as opposed to the much noisier nature of the variable being predicted. This imbalance tends to affect the accuracy of the estimates of the model parameters and the validity of inferences about them when one uses standard methods that do not explicitly recognize this and related complications. A growing literature aimed at introducing novel techniques specifically designed to produce accurate inferences in such environments ensued. The frequent use of these predictive regressions in applied work has also led practitioners to question the validity of viewing predictability within a linear setting that ignores the possibility that predictability may occasionally be switched off. This in turn has generated a new stream of research aiming at introducing regime-specific behavior within predictive regressions in order to explicitly capture phenomena such as episodic predictability.

Causal Graphical Views of Fixed Effects and Random Effects Models

Despite the long-standing discussion on fixed effects (FE) and random effects (RE) models, how and under which conditions both methods can eliminate unmeasured confounding bias have not yet been widely understood in practice. Using a simple pretest-posttest design in a linear setting, this article translates the conventional algebraic formalization of FE and RE models into causal graphs and provides intuitively accessible graphical explanations about their data-generating and bias-removing processes. The proposed causal graphs highlight that FE and RE models consider different data-generating models. RE models presume a data-generating model that is identical to a randomized controlled trial while FE models allow for unobserved time-invariant treatment-outcome confounding. Augmenting regular causal graphs that describe data-generating processes by adding the computational structures of FE and RE estimators, the article visualizes how FE estimators (gain score and deviation score estimators) and RE estimators (quasi-deviation score estimator) offset unmeasured confounding bias. In contrast to standard regression or matching estimators that reduce confounding bias by blocking non-causal paths via conditioning, FE and RE estimators offset confounding bias by deliberately creating new non-causal paths and associations of opposite sign. Though FE and RE estimators are similar in their bias-offsetting mechanisms, the augmented graphs reveal their subtle differences that can result in different biases in observational studies.