The effect of seepage on the stress–strain state of rock near a borehole

The effect of the strain-strength and seepage properties of rock and the compressibility of the percolating fluid on the dimensions of the rock fracture zones, which occur in oil and gas boreholes when the bottom hole pressure is reduced, is investigated. The seepage is considered basing on the stationary formulation of the problem, which enables the general case to be investigated. It is shown that in the case of unsteady flow, the stresses on the boundary of the rock fracture zone and, as a consequence, on its dimensions, are independent of the nature of the pressure distribution in the stratum, and are determined solely by the pressure of the percolating fluid on the boundary of this zone. It is established that an increase in the compressibility of the percolating fluid leads to an increase in the dimensions of the rock fracture zone.     

The stress–strain state and disintegration of a meteoroid moving through the atmosphere

The solution of the problem of the stress–strain state of an elastic body of spherical shape when it enters the Earth's atmosphere at a superorbital velocity is obtained in the form of a series in Legendre polynomials, in the case of a viscous gas at hypersonic velocity, in the quasi-stationary formulation. Using the Hubert–von Mises–Hencky criterion the limit stresses corresponding to the start of disintegration are obtained. This enables the nature of the disintegration of the body to be judged and also enables the heights at which destruction of known meteoroids begins to be estimated.     

Modelling the static stress–strain state around the fan-structure in the shear rupture head

The mathematical model of an equilibrium fan-structure in the interface between two elastic blocks, simulating the shear rupture head in a hard rock under high confining pressure, is constructed. The stress–strain state far from the fan-structure is analyzed with the help of a solution of the problem on edge dislocation. The fan length is estimated using this solution. The model of formation of two oppositely directed fans due to the localized action of tangential stress, which pushes two edge dislocations with antiparallel Burgers vectors, is proposed. In complete formulation, the problem on an equilibrium fan-structure in the interface between infinite elastic half-planes is analyzed by means of original method of superposition of dislocations, leading to two nonlinear integral equations in the fan zone. To solve them numerically, the method of successive approximations is applied. Based on this method, fields of stresses and displacements around the equilibrium fan modelling of a deep-seated shear rupture in the seismogenic zone of the Earth’s crust are computed. Such fields can be used, when setting the initial data in the analysis of dynamics of the fan-shaped mechanism.     

An asymptotic expression of the Schrödinger equation

The problem of solving the time–independent Schr?dinger equation for the motion of an electron of mass μ and charge –e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equations to the study of the Schr?dinger equation under certain parametric conditions, and obtain an asymptotic formula. The work has been presented at the International Conference on Quantum Computation and Quantum Technology, Texas A&M University, College Station, Texas, November 13-16, 2005. The author would like to thank the organizer Professor Goong Chen for his generous support. This work is also partly supported by UTPA Faculty Research Council Grant 119100.     

An asymptotic of the negative discrete spectrum of the Schrödinger operator

The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L₂(R^m) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = V_o + V₁; V_o < 0, ¦V_o =o (¦V_o¦^3/2) as ¦x¦ → ∞; σ (t/2, V_o) ?cσ (t. V_o) and V₁∈L_m/2,loc, σ(t, V₁) =o (σ (t, V_o)), where σ (t,f)= mes {x:¦f (x) ¦ > t).     

Expansion in the eigenfunctions of the schrödinger equation with a potential having a periodic asymptotic behavior

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = ?d ^z/dx ²+q(x)(?∞<x<∞)with a potential q(x), satisfying the condition $$\int\limits_0^{ + \infty } {(1 + x^2 )|q(x) - q_ \pm (x)|dx< \infty ,} $$ where q±(x) are periodic functions.     

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

By using the steepest descent method for Riemann–Hilbert problems introduced by Deift–Zhou (Ann Math 137:295–370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial $L^{(\alpha)}_n(\nu z)$ as n→∞, where ν=4n+2α+2. One expansion holds uniformly in a right half-plane $\text{Re}\; z\geq \delta_1, 0<\delta_1<1$ , which contains the critical point z=1; the other expansion holds uniformly in a left half-plane $\text{Re}\; z\leq 1-\delta_2, 0<\delta_2<1-\delta_1$ , which contains the other critical point z=0. The two half-planes together cover the entire complex z-plane. The critical points z=1 and z=0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by $L^{(\alpha)}_n(\nu z)$ .     

An analysis of single-index model with monotonic link function

The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h（.）. Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.     

An asymptotic formula for the primitive of Hardy’s function

Let Z(t) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term O(T ^1/6log?T) is given for the integral of Z(t) over the interval [0,T], with special attention paid to the critical cases when the fractional part of \(\sqrt{T/2\pi }\) is close to \(\frac{1}{4}\) or \(\frac{3}{4}\).     

On the asymptotic distribution of the Koenker–Bassett estimator for a parameter of the nonlinear model of regression with strongly dependent noise

We prove that, under certain regularity conditions, the asymptotic distribution of the Koenker – Bassett estimator coincides with the asymptotic distribution of the integral of indicator process generated by a random noise weighted by the gradient of the regression function.     

Mathematical analysis of a population model with an age–weight structured two-stage life history: asymptotic behavior of solutions

In this paper, we propose a model for the dynamics of a physiologically structured population of individuals whose life cycle is divided into two stages: the first stage is structured by the weight, while the second one is structured by the age, the exit from the first stage occurring when a threshold weight is attained. The model originates in a complex one dealing with a fish population and covers a large class of situations encompassing two-stage life histories with a different structuring variable for each state, one of its key features being that the maturation process is determined in terms of a weight threshold to be reached by individuals in the first stage. Mathematically, the model is based on the classical Lotka–MacKendrick linear model, which is reduced to a delayed renewal equation including a constant delay that can be viewed as the time spent by individuals in the first stage to reach the weight threshold. The influence of the growth rate and the maturation threshold on the long-term behavior of solutions is analyzed using Laplace transform methods.     

Graphlike Manifolds with the Type of the 1—skeleton of a 4—simplex

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An analysis of a ‘Sunday Supplement’ problem

A problem posed in an influential textbook is analysed in more detail than is given there. The textbook answer to the problem has been represented as counterintuitive, as is probably the case; however, it is here shown that it depends critically on an assumption. If this is relaxed, then a wide variety of possible answers is available. All the same, if the special assumption made is replaced by one of a number of more realistic ones, then the order of magnitude of the approximation given by the author of the textbook is preserved.     

New exact solutions with functional parameters of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity

We use the Zakharov—Manakov δ-dressing method to construct new classes of exact solutions with functional parameters of the hyperbolic and elliptic versions of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity. We show that the constructed solutions contain classes of multisoliton solutions, which at a fixed time are exact potentials of the perturbed telegraph equation (the perturbed string equation) and the two-dimensional stationary Schrödinger equation. We interpret the stationary states of a microparticle in soliton-type potential fields physically in accordance with the constructed exact wave functions for the two-dimensional stationary Schrödinger equation.     

Objective Bayesian analysis of the Frechet stress–strength model

Several reference priors and a general form of matching priors are derived for a stress–strength system, and it is concluded that none of the reference priors is a matching prior. The study shows that the matching prior performs better than Jeffreys prior and reference priors in meeting the target coverage probabilities.     

Transferring a system with an unknown disturbance under the optimal control to a state of dynamic balance and to an ε-vicinity of the final state

A problem of transferring a linear system to a state of dynamic balance under a simultaneous action of an unknown disturbance and time-optimal control is considered. The optimal control is calculated along the phase trajectory, and it is periodically updated for discrete phase coordinate values. It is proved that the phase trajectory comes to the dynamic equilibrium point and makes undamped periodic motions (a stable limit cycle). The location of the dynamic equilibrium point and the limit cycle form are considered as functions of various parameters. With the disturbance calculated in the process of control, the accuracy of transferring to the required final state increases. A method for estimating attainable accuracy is presented. Results of simulation and numerical calculations are given.     

Qualitative analysis of a variable yield turbidostat model with impulsive state feedback control

A turbidostat is an apparatus with feedback control system used to continuously culturing microorganisms. The dilution rate of the turbidostat can be regulated by the control system when the concentration of microorganism, detected by photoelectricity system or other devices, reaches a preset value. Based on the design ideas of the turbidostat, a differential equation with impulsive state feedback control is proposed for a kind of turbidostat system in this paper. By the existence criteria of periodic solution of a general planar impulsive autonomous system, the conditions for the existence of periodic solution of order one are obtained according to the preset value and the types of the positive equilibrium of the corresponding system without impulsive control. Furthermore, it is pointed out that the system either tends to a stable state or has a periodic solution. Finally, the theoretical results are verified by numerical simulations.