Study of Tsunami-Induced Fluid and Debris Load on Bridges using the Material Point Method

Thesis (Ph.D.)--University of Washington, 2016-08

Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.

地震波散射非线性反演问题不动点理论及方法的研究

How to create a new method to solve the problem or reduce the influence of that the result of the seismic waves scattering nonlinear inversion is not uniqueness is a main purpose of this research work in the paper. On the background of research into the seismic inversion, new progress of the nonlinear inversion is introduced at the first chapter in this paper. Especially, the development, basic theories and assumptions on some major theories of seismic inversion are analyzed, discussed and summarized in mathematics and physics. Also, the problems faced by the mathematical basis of investigations of the seismic inversion are discussed, and inverse questions of strongly seismic scattering due to strong heterogeneous media in the Earth interior are analyzed and viewed. What the kernel of paper is that gathers all our attention making a new nonlinear inversion method of seismic scattering. The paper provides a theory and method of how to introduce the fixed-point theory into the nonlinear seismic scattering inversion and how to obtain the solution, and gives the actually method to create a serials of contractive mappings of velocity parameter's in the mapping space of wave. Therefore, the results testify the existence of fixed point of velocity parameter and give the method the find it. Further, the paper proves the conclusion that the value obtained by taking the fixed point of velocity parameter into wave equation is the fixed point of the wave of the contractive mapping. Thence, the fixed point is the global minima since the stabilities quality of the fixed point. Based on the new theory, in the chapter three, many inverse results are obtained in the numerical value test. By analysis the results one could find a basic facts that all the results, which are inversed by the different initial model, are tended to the true value in theoretical true model. In other words, the new method mostly eliminates the non-uniqueness that which is existed in seismic waves scattering nonlinear inversion in degree. But, since the test results are quite finite now, more test is need here to positive our theory. As a new theoretical method, it must be existed many weaken in it. The chapter four points out all the questions which is bother us. We hope more people to join us to solve the problem together.

Range of validity of the method of averaging

Sufficient conditions are derived for the validity of approximate periodic solutions of a class of second order ordinary nonlinear differential equations. An approximate solution is defined to be valid if an exact solution exists in a neighborhood of the approximation.

Two classes of validity criteria are developed. Existence is obtained using the contraction mapping principle in one case, and the Schauder-Leray fixed point theorem in the other. Both classes of validity criteria make use of symmetry properties of periodic functions, and both classes yield an upper bound on a norm of the difference between the approximate and exact solution. This bound is used in a procedure which establishes sufficient stability conditions for the approximated solution.

Application to a system with piecewise linear restoring force (bilinear system) reveals that the approximate solution obtained by the method of averaging is valid away from regions where the response exhibits vertical tangents. A narrow instability region is obtained near one-half the natural frequency of the equivalent linear system. Sufficient conditions for the validity of resonant solutions are also derived, and two term harmonic balance approximate solutions which exhibit ultraharmonic and subharmonic resonances are studied.

State Dependent Riccati Equation (SDRE) control method applied in cancellation of a parametric resonance in a magnetically levitated body

Here, a simplified dynamical model of a magnetically levitated body is considered. The origin of an inertial Cartesian reference frame is set at the pivot point of the pendulum on the levitated body in its static equilibrium state (ie, the gap between the magnet on the base and the magnet on the body, in this state). The governing equations of motion has been derived and the characteristic feature of the strategy is the exploitation of the nonlinear effect of the inertial force associated, with the motion of a pendulum-type vibration absorber driven, by an appropriate control torque [4]. In the present paper, we analyzed the nonlinear dynamics of problem, discussed the energy transfer between the main system and the pendulum in time, and developed State Dependent Riccati Equation (SDRE) control design to reducing the unstable oscillatory movement of the magnetically levitated body to a stable fixed point. The simulations results showed the effectiveness of the (SDRE) control design. Copyright © 2011 by ASME.

A method for revealing phase space structures of general time dependent dynamical systems

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a “heuristic argument” that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper “side-by-side” comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field (“time averages”) are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

Patient Variability in Sciatic Nerve Branch Point Distance Using Ultrasound Guided Localization

Background and Objectives: Improved ultrasound and needle technology make popliteal sciatic nerve blockade a popular anesthetic technique and imaging to localize the branch point of the common peroneal and posterior tibial components is important because successful blockade techniques vary with respect to injection of the common trunk proximally or separate injections distally. Nerve stimulation, ultrasound, cadaveric and magnetic resonance studies demonstrate variability in distance and discordance between imaging and anatomic examination of the branch point. The popliteal crease and imprecise, inaccessible landmarks render measurement of the branch point variable and inaccurate. The purpose of this study was to use the tibial tuberosity, a fixed bony reference, to measure the distance of the branch point. Method: During popliteal sciatic nerve blockade in the supine position the branch point was identified by ultrasound and the block needle was inserted. The vertical distance from the tibial tuberosity prominence and needle insertion point was measured. Results: In 92 patients the branch point is a mean distance of 12.91 cm proximal to the tibial tuberosity and more proximal in male (13.74 cm) than female patients (12.08 cm). Body height is related to the branch point distance and is more proximal in taller patients. Separation into two nerve branches during local anesthetic injection supports notions of more proximal neural anatomic division. Limitations: Imaging of the sciatic nerve division may not equal its true anatomic separation. Conclusion: Refinements in identification and resolution of the anatomic division of the nerve branch point will determine if more accurate localization is of any clinical significance for successful nerve blockade.

Face recognition using fractal codes

In this paper we propose a new method for face recognition using fractal codes. Fractal codes represent local contractive, affine transformations which when iteratively applied to range-domain pairs in an arbitrary initial image result in a fixed point close to a given image. The transformation parameters such as brightness offset, contrast factor, orientation and the address of the corresponding domain for each range are used directly as features in our method. Features of an unknown face image are compared with those pre-computed for images in a database. There is no need to iterate, use fractal neighbor distances or fractal dimensions for comparison in the proposed method. This method is robust to scale change, frame size change and rotations as well as to some noise, facial expressions and blur distortion in the image

Security Analysis of Randomize-Hash-then-Sign Digital Signatures

At CRYPTO 2006, Halevi and Krawczyk proposed two randomized hash function modes and analyzed the security of digital signature algorithms based on these constructions. They showed that the security of signature schemes based on the two randomized hash function modes relies on properties similar to the second preimage resistance rather than on the collision resistance property of the hash functions. One of the randomized hash function modes was named the RMX hash function mode and was recommended for practical purposes. The National Institute of Standards and Technology (NIST), USA standardized a variant of the RMX hash function mode and published this standard in the Special Publication (SP) 800-106. In this article, we first discuss a generic online birthday existential forgery attack of Dang and Perlner on the RMX-hash-then-sign schemes. We show that a variant of this attack can be applied to forge the other randomize-hash-then-sign schemes. We point out practical limitations of the generic forgery attack on the RMX-hash-then-sign schemes. We then show that these limitations can be overcome for the RMX-hash-then-sign schemes if it is easy to find fixed points for the underlying compression functions, such as for the Davies-Meyer construction used in the popular hash functions such as MD5 designed by Rivest and the SHA family of hash functions designed by the National Security Agency (NSA), USA and published by NIST in the Federal Information Processing Standards (FIPS). We show an online birthday forgery attack on this class of signatures by using a variant of Dean’s method of finding fixed point expandable messages for hash functions based on the Davies-Meyer construction. This forgery attack is also applicable to signature schemes based on the variant of RMX standardized by NIST in SP 800-106. We discuss some important applications of our attacks and discuss their applicability on signature schemes based on hash functions with ‘built-in’ randomization. Finally, we compare our attacks on randomize-hash-then-sign schemes with the generic forgery attacks on the standard hash-based message authentication code (HMAC).

Effect of inter-edge Coulomb interactions on transport through a point contact in nu=5/2 quantum Hall state

We study transport across a point contact separating two line junctions in a nu = 5/2 quantum Hall system. We analyze the effect of inter-edge Coulomb interactions between the chiral bosonic edge modes of the half-filled Landau level (assuming a Pfaffian wave function for the half-filled state) and of the two fully filled Landau levels. In the presence of inter-edge Coulomb interactions between all the six edges participating in the line junction, we show that the stable fixed point corresponds to a point contact that is neither fully opaque nor fully transparent. Remarkably, this fixed point represents a situation where the half-filled level is fully transmitting, while the two filled levels are completely backscattered; hence the fixed point Hall conductance is given by G(H) = 1/2e(2)/h. We predict the non-universal temperature power laws by which the system approaches the stable fixed point from the two unstable fixed points corresponding to the fully connected case (G(H) = 5/2e(2)/h) and the fully disconnected case (G(H) = 0).

{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

Quantum Algorithms with Fixed Points: The Case of Database Search

The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\frac{\pi}{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $\epsilon$ to $\epsilon^{2q+1}$, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.

Cost effective campaigning in social networks

Campaigners are increasingly using online social networking platforms for promoting products, ideas and information. A popular method of promoting a product or even an idea is incentivizing individuals to evangelize the idea vigorously by providing them with referral rewards in the form of discounts, cash backs, or social recognition. Due to budget constraints on scarce resources such as money and manpower, it may not be possible to provide incentives for the entire population, and hence incentives need to be allocated judiciously to appropriate individuals for ensuring the highest possible outreach size. We aim to do the same by formulating and solving an optimization problem using percolation theory. In particular, we compute the set of individuals that are provided incentives for minimizing the expected cost while ensuring a given outreach size. We also solve the problem of computing the set of individuals to be incentivized for maximizing the outreach size for given cost budget. The optimization problem turns out to be non trivial; it involves quantities that need to be computed by numerically solving a fixed point equation. Our primary contribution is, that for a fairly general cost structure, we show that the optimization problems can be solved by solving a simple linear program. We believe that our approach of using percolation theory to formulate an optimization problem is the first of its kind. (C) 2016 Elsevier B.V. All rights reserved.

Experimental-study of free-surface oscillations of a liquid bridge by optical diagnostics

A non-contact optical method, consisting of a projecting grating technique for the relative measurement of a surface, and a technique of absolute measurement at a fixed point on the surface, are applied to measure the free surface vibration in a liquid bridge of half floating zone with small typical scale of a few of mm for emphasizing the thermocapillary effect in comparison with the effect of buoyancy. The radii variations in both longitudinal and azimuthal directions are obtained, and, then, the feature of surface wave could be analyzed in detail. The results show that there are values of principal oscillatory frequencies at different positions of free surface. The amplitudes of surface waves in longitudinal and azimuthal directions are several mum and several tenths of mum in order of magnitude. The phase of two-dimensional surface waves is different at different height for fixed cross section or at different azimuthal angle for fixed height. The wave features are discussed for the cases of typical parameter ranges.