Repeating earthquakes,episodic tremor and slip: Emerging patterns in complex earthquake cycles?

The lack of regularity in earthquake cycles continues to be a confounding issue in earthquake science. Lately, observations of episodic nonvolcanic tremor and slip (ETS) along a few well‐instrumented tectonic plate boundaries are intriguing: these features recur together with predictable time intervals. Data now trace recurring ETS back to 1990 and no significant earthquake ever followed an ETS episode. This observation and the fact that stress drops associated with episodic slips are low, only on the order of 0.01 MPa, suggests that repeated ETS has little cumulative effects in priming the fault for the next large earthquake. Another known regularity in seismic activity is the so‐called repeating earthquakes that rupture the same patch of fault repetitively. Current hypotheses for repeating earthquakes point to the interaction of continual, aseismic fault slip with locked, seismogenic patches of the fault. Interestingly, ETS also recur near where the transition between brittle faulting and plastic flow is expected, although it is not clear how and why regularities in space and time are interconnected. New data show that ETS recurs throughout the entire length of the Cascadia subduction zone, thus ruling out any special, local factors as necessary conditions for ETS. Recurrence intervals of ETS vary along the Cascadia. Such variations are not governed by the rate of plate motion that ultimately drives the earthquake process, but they do coincide with variations in the geology of the overriding plate that can influence the rheology along the plate interface. To this end, we call attention to the Portevin‐Le Chatelier effect (PLC, or jerky flow) as a potential analog to the earthquake process. The dynamics of the PLC has been extensively studied and shows many intriguing features as the system goes from chaotic to self‐organized critical regimes as strain rate increases. In particular, the PLC exhibits not only stick‐slip behavior (stress serration) over time but also spatial interactions over extended regions—features that are necessary to account for complex spatio‐temporal variations associated with earthquake activities. © 2007 Wiley Periodicals, Inc. Complexity 12: 33–43, 2007     

Dynamic fractal unifying interaction confirmed with magnetospheric behavior and orbital data

The solar wind makes the magnetosphere to expand and contract as indicated by the expansions and contractions of the auroral oval due to balancing of the dynamic pressure of the ambient space plasma at inner and outer magnetic lines. This self‐similar magnetospheric behavior elucidates the controversial magnetic storm‐substorm relationship and reveals the 3D‐spiral structure of magnetic interaction. The found self‐similarly evolving structure of one seen as fundamental interaction suggests dynamic fractal unifying interaction that builds a firework universe having 3D‐spiral code. The unifying interaction is described with equation drawn in new fundamental dynamic fractal framework. The equation of unifying interaction converges to the inverse square laws and the principle of uncertainty at laboratory scales. The dynamic fractal fundamental framework is made of one 3D‐spirally‐faster‐inward contracting and expanding, oscillating, basic matter. It simply accounts for observed constant speed of light and for the creation of bright and dark bands on a screen behind a tiny slit. The dynamic fractal framework is quantitatively confirmed with the orbital data for the Milky Way Galaxy, the Sun, the Earth, and the triple asteroid system 87 Sylvia. Many testable predictions are also made. The presented new fundamental dynamic fractal framework allows qualitative and quantitative modeling and simplification. © 2007 Wiley Periodicals, Inc. Complexity 12: 61–76, 2007     

Exponential attractor for a planar shear‐thinning flow

We study the dynamics of an incompressible, homogeneous fluid of a power‐law type, with the stress tensor T = ν(1 + µ|Dv|)^p?2Dv, where Dv is a symmetric velocity gradient. We consider the two‐dimensional problem with periodic boundary conditions and p ∈ (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so‐called method of ??‐trajectories. Copyright © 2007 John Wiley & Sons, Ltd.     

Least‐squares problems for Michaelis–Menten kinetics

The Michaelis–Menten kinetics is fundamental in chemical and physiological reaction theory. The problem of parameter identification, which is not well posed for arbitrary data, is shown to be closely related to the Chebyshev sum inequality. This inequality yields sufficient conditions for existence of feasible solutions both for nonlinear and for linear least‐squares problems. The conditions are natural and practical as they are satisfied if the data show the expected monotone and concave behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

Interaction of modulated pulses in nonlinear oscillator chains

We formally derive and rigorously justify the modulation equations of lowest order for the interaction of two modulated pulses on a one-dimensional nonlinear oscillator chain. We show that solutions with the initial form of the assumed ansatz preserve this form over time intervals with positive macroscopic length, and we show a bound on the possible shift of the envelope caused by the interaction. Thus, we rigorously justify and quantify the statement that under the given conditions there is almost no interaction of the modulated pulses.     

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

We derive an asymptotic solution of the vacuum Einstein equations that describe the propagation and diffraction of a localized, large‐amplitude, rapidly varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations. © 2007 Wiley Periodicals, Inc.     

Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.     

Analytical solutions for vibrating fractal composite rods and beams

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.     

高阶差分方程Lidstone BVP的特征值问题

应用锥上的不动点定理和锥理论,我们在对非线性项的不同假设下分别得到了一类2m阶离散Lidstone BVP特征值的存在区间．     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Existence Results on Semi-Infinite Intervals for Nonlocal Evolutions Equations and Inclusions via Semigroup Theory

In this paper, we prove existence and controllability results on semi-infinite intervals for first- and second-order evolutions equations and inclusions in Banach spaces with nonlocal conditions. Our theory makes use of semigroups, cosine families, nonlinear alternative of Leray–Schauder, and a diagonalization argument.     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

MULTIPLE POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEMS OF DELAY DIFFERENTIAL EQUATIONS WITH DENUMBERABLE SINGULARITIES ON INFINITE INTERVAL

This paper deals with the existence of denumberable positive solutions to boundary value problems of delay differential equations with denumberable singularities on infinite intervals. By the fixed-point index theory and a new fixed-point theorem in cones, the existence of denumberable positive solutions is obtained under some suitable growth conditions imposed on the nonlinear term.     

Neumann problems for nonlinear hemivariational inequalities

In this paper we study nonlinear Neumann problems driven by the p ‐Laplacian and having a nonsmooth potential. Using techniques from the nonsmooth critical point theory, we prove two existence theorems and a multiplicity result. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decay of the energy for the Benjamin–Bona–Mahony equation posed on bounded intervals and on a half‐line

This paper concerns nonlinear dissipative initial boundary value problems for the Benjamin–Bona–Mahony equation posed on a half‐line and on bounded intervals. We prove the existence and uniqueness of global solutions and decay of the energy as time tends to infinity as well as convergence of solutions on bounded intervals to a solution on a half‐line. Copyright © 2012 John Wiley & Sons, Ltd.     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

Heat transfer in composite materials with Stefan–Boltzmann interface conditions

In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan–Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one‐dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.