Fractal functions of exponential type that is generated by the $\mathbf{Q_2^*}$-representation of argument

We consider function $f$ which is depended on the parameters $0<a\in R$, $q_{0n}\in (0;1)$, $n\in N$ and convergent positive series $v_1+v_2+...+v_n+...$, defined by equality $f(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=a^{\varphi(x)}$, where $\alpha_n\in \{0,1\}$, $\varphi(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\alpha_1v_1+...+\alpha_nv_n+...$, $q_{1n}=1-q_{0n}$, $\Delta^{Q_2^*}_{\alpha_1...\alpha_n...}=\alpha_1q_{1-\alpha_1,1}+\sum\limits_{n=2}^{\infty}\big(\alpha_nq_{1-\alpha_n,n}\prod\limits_{i=1}^{n-1}q_{\alpha_i,i}\big)$.In the paper we study structural, variational, integral, differential and fractal properties of the function $f$.

ABOUT ONE CLASS OF FUNCTIONS WITH FRACTAL PROPERTIES

We consider one generalization of functions, which are called as «binary self-similar functi- ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, differential and fractal properties are studied for the functions of this class.

BUBNOV-GALERKIN METHOD FOR THE ELASTIC BUCKLING OF EULER COLUMNS

In this work the Bubnov-Galerkin variational method was applied to determine the critical buckling load for the elastic buckling of columns with fixed-pinned ends. Coordinate shape functions for Euler column with fixed-pinned ends are used in the Bubnov-Galerkin variational integral equation to obtain the unknown parameters. One parameter and two parameter shape functions were used. In each case, the Bubnov-Galerkin method reduced the boundary value problem to an algebraic eigen-value problem. The solution of the characteristic homogeneous equations yielded the buckling loads. One parameter coordinate shape function yielded relative error of 4% compared with the exact solution. Two parameter coordinate shape function gave a relative error of 0.77%, which is negligible.

Polynomial chaos expansion for nonlinear geophysical inverse problems

There are lots of geophysical problems that include computationally expensive functions (forward models). Polynomial chaos (PC) expansion aims to approximate such an expensive equation or system with a polynomial expansion on the basis of orthogonal polynomials. Evaluation of this expansion is extremely fast because it is a polynomial function. This property of the PC expansion is of great importance for stochastic problems, in which an expensive function needs to be evaluated thousands of times. We have developed PC expansion as a novel technique to solve nonlinear geophysical problems. To better evaluate the methodology, we use PC expansion for automating the velocity analysis. For this purpose, we define the optimally picked velocity model as an optimizer of a variational integral in a semblance field. However, because computation of a variational integral with respect to a given velocity model is rather expensive, it is impossible to use stochastic methods to search for the optimal velocity model. Thus, we replace the variational integral with its PC expansion, in which computation of the new function is extremely faster than the original one. This makes it possible to perturb thousands of velocity models in a matter of seconds. We use particle swarm optimization as the stochastic optimization method to find the optimum velocity model. The methodology is tested on synthetic and field data, and in both cases, reasonable results are achieved in a rather short time.

ESTABLISHMENT OF DYNAMIC EQUATIONS FOR DAMPED SYSTEMS BASED ON QUASI-VARIATIONAL PRINCIPLES

In this paper, quasi-variational principles for non-conservative damped systems are studied. A Hamiltontype quasi-variational principle for non-conservative systems in analytical mechanics and a quasi-variational principle of potential energy in non-conservative elastodynamics systems are proposed in simplified forms respectively, by using the direct variational integral method. On the basis of the standard linear solid model for viscoelastic materials, the dynamic equations of exponentially damped systems are established through the proposed quasi-variational principles. A distinction between the internal damping described by exponential damping and the external damping described by viscous one in a vibrating structure is according to different physical mechanisms, which gives some indication of the correct mechanism of damping.

Gurtin-Type Quasi-Variational Principle of Buried Pipelines Dynamics

Because earthquakes may cause severe damage to buried pipelines, it is important to study dynamic response of buried pipelines. For dynamic response problem of buried pipelines, variational method is the suitable method. In the paper, Gurtin-type quasi-variational principle of buried pipelines dynamics is established by convolutional variational integral method, which is theoretical foundation of finite element method for solving buried pipelines dynamics problems.