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Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-4, DECEMBER (No. 4) |
| Halaman | : | 755-787 |
| Abstrak | : | The model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities. We describe an efficient adiabatic simulation of any given quantum circuit. This implies that the adiabatic computation model and the standard circuit-based quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows one to state the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of Hamiltonians. |
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
| Pengarang | : | |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-4, DECEMBER (No. 4) |
| Halaman | : | 723-752 |
| Abstrak | : | We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere, and the three-dimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the high-resolution wave-propagation algorithm implemented in clawpack can be used effectively to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grids is below 2 for most of our grid mappings, explicit finite volume methods such as the wave-propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitude-longitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reaction-diffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with MATLAB routines for the various mappings. |
Geometry of Arnold Diffusion
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-4, DECEMBER (No. 4) |
| Halaman | : | 702-720 |
| Abstrak | : | The goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of Arnold diffusion. We choose to speak of a specific model—that of geometric rays in a periodic optical medium. This model is equivalent to that of a particle in a periodic potential in with energy prescribed and to the geodesic flow in a Riemannian metric on . |
Bifurcations in Nonsmooth Dynamical Systems
| Pengarang | : | Mario di Bernardo |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-4, DECEMBER (No. 4) |
| Halaman | : | 629-701 |
| Abstrak | : | A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of “normal form” or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory. |
Modeling and Simulating Chemical Reactions
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 347-368 |
| Abstrak | : | Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described. |
Dissipation-Induced Heteroclinic Orbits in Tippe Tops
| Pengarang | : | Nawaf M. Bou-Rabee |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 325-344 |
| Abstrak | : | This paper demonstrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and noninverted states of a tippe top are determined by a complex version of the equations for a simple harmonic oscillator: the modified Maxwell–Bloch equations. A standard linear analysis reveals that the modified Maxwell–Bloch equations describe the spectral instability of the noninverted state and Lyapunov stability of the inverted state. Standard nonlinear analysis based on the energy momentum method gives necessary and sufficient conditions for the existence of a dissipation-induced connecting orbit between these relative equilibria. |
High-Frequency Oscillations of a Sphere in a Viscous Fluid near a Rigid Plane
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 313-322 |
| Abstrak | : | High-frequency oscillations of a rigid sphere in an incompressible viscous fluid moving normal to a rigid plane are considered when the ratio of minimum clearance to sphere radius is small. Asymptotic expansions are constructed that permit an analytical estimate of the force acting on the sphere as a result of its motion. An inner expansion, valid in the neighborhood of the minimum gap, reflects the dominance of viscous effects and fluid inertia. An outer expansion, valid outside the gap, reflects the dominance of fluid inertia with a correction for an oscillating viscous boundary layer. The results are applied to the hydrodynamics of the tapping mode of an atomic force microscope and to the dynamic calibration of its cantilevers. |
Sensitivity Analysis in Calculus of Variations. Some Applications
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 294-312 |
| Abstrak | : | This paper deals with the problem of sensitivity analysis in calculus of variations. A perturbation technique is applied to derive the boundary value problem and the system of equations that allow us to obtain the partial derivatives (sensitivities) of the objective function value and the primal and dual optimal solutions with respect to all parameters. Two examples of applications, a simple mathematical problem and a slope stability analysis problem, are used to illustrate the proposed method. |
Asymptotic Expansions of Mellin Convolution Integrals
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 275-293 |
| Abstrak | : | We present a new method for deriving asymptotic expansions of for small x. We only require that and have asymptotic expansions at and , respectively. Remarkably, it is a very general technique that unifies a certain set of asymptotic methods. Watson's lemma and other classical methods, Mellin transform techniques, McClure and Wong's distributional approach, and the method of analytic continuation turn out to be simple corollaries of this method. In addition, the most amazing thing about it is that its mathematics are absolutely elemental and do not involve complicated analytical tools as the aforementioned methods do: it consists of simple “sums and subtractions." Many known and new asymptotic expansions of important integral transforms are trivially derived from the approach presented here |
Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions
| Pengarang | : | Robin Pemantle,Martin R. W. Hiebl |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-2, JUNE (No. 1) |
| Halaman | : | 199-272 |
| Abstrak | : | Let be a d-dimensional array of numbers for which the generating function is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples. |