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The Axisymmetric Antidynamo Theorem Revisited
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| Pengarang | : | Ralf Kaiser, Andreas Tilgner |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 74 (No. 2) |
| Halaman | : | 571-597 |
| Abstrak | : | The axisymmetric kinematic dynamo problem is reconsidered, and a number of open questions are answered. Apart from axisymmetry and smoothness of data and solution we deal with this problem under quite general conditions; i.e., we assume a compressible fluid of variable (in space and time) conductivity moving in an arbitrary (axisymmetric) domain. We prove unconditional, pointwise, and exponential decay of the magnetic field and electric current to zero. The decay rate of the external (meridional) magnetic field can become very small (compared to free decay) for special flow fields and large magnetic Reynolds numbers, as we demonstrate with an example. On the other hand, we show for fluids with weak variation of mass density and conductivity that the meridional and azimuthal decay rates do not drop significantly below those of free decay. |
Counting Binomial Coefficients Divisible by a Prime Power
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 6) |
| Halaman | : | 531-540 |
| Abstrak | : | We define a new class of integer partitions generalizing hyperbinary partitions. We then utilize these partitions to provide a new, relatively simple formula for the number of binomial coefficients in a particular row of Pascal’s triangle that are divisible by a fixed power of a prime. Our formula specializes to a famous result due to N. J. Fine from 1947. Furthermore, we demonstrate a connection between the binomial coefficients congruent to zero modulo a fixed prime power and the so-called digital dominance orders on the natural numbers. |
Hilbert’s Proof of His Irreducibility Theorem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 6) |
| Halaman | : | 513-530 |
| Abstrak | : | Hilbert’s irreducibility theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and combinatorics. We try to expose the motivations that led Hilbert to this synthesis. His famous cube lemma anchored the proof but without the analytical foundation and framework it would have had no purpose. We also assess this lemma as a precursor of Ramsey theory. |
The n-Cube is Rupert
| Pengarang | : | Greg Huber |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 6) |
| Halaman | : | 505-512 |
| Abstrak | : | An oval in is called Rupert if a straight tunnel can be made in it through which a second congruent oval can be passed. We show that the n-cube is Rupert for each n ? 3. |
Study of a Model Equation in Detonation Theory
| Pengarang | : | Luiz M. Faria, Aslan R. Kasimov, Rodolfo R. Rosales |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 74 (No. 2) |
| Halaman | : | 547-570 |
| Abstrak | : | Here we analyze properties of an equation that we previously proposed to model the dynamics of unstable detonation waves [A. R. Kasimov, L. M. Faria, and R. R. Rosales, Model for shock wave chaos, Phys. Rev. Lett., 110 (2013), 104104]. The equation is ???????? +12?(????2−??????????(0−,????))???? =?????(????,?????(0−,????)), ???? ≤0, ???? >0. It describes a detonation shock at ???? =0 with the reaction zone in ???? <0. We investigate the nature of the steady-state solutions of this nonlocal hyperbolic balance law, the linear stability of these solutions, and the nonlinear dynamics. We establish the existence of instability followed by a cascade of period-doubling bifurcations leading to chaos. |
Rupert Property of Archimedean Solids
| Pengarang | : | Makito Miyazaki |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 6) |
| Halaman | : | 497-504 |
| Abstrak | : | We say that a polytope has the Rupert property if we can make a hole large enough in to permit another copy of to pass through. In this article, we show that among the 13 Archimedean solids, 8 have this property, namely, the cuboctahedron, the truncated octahedron, the truncated cube, the rhombicuboctahedron, the icosidodecahedron, the truncated cuboctahedron, the truncated icosahedron, and the truncated dodecahedron. |
Using Dynamical Systems to Construct Infinitely Many Primes
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 6) |
| Halaman | : | 483-496 |
| Abstrak | : | Euclid’s proof can be reworked to construct infinitely many primes, in many different ways, using ideas from arithmetic dynamics. |
A Macroscopic Model Including Membrane Exchange for Diffusion MRI
| Pengarang | : | Julien Coatléven, Houssem Haddar, Jing-Rebecca Li |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 74 (No. 2) |
| Halaman | : | 516-546 |
| Abstrak | : | Diffusion Magnetic Resonance Imaging is a promising tool to obtain useful information on the microscopic structure and has been extensively applied to biological tissues. We establish a new macroscopic model from homogenization theory for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of intermediate water exchange across biological cellular membranes. Based on a particular scaling of the permeability condition modeling cellular membranes, this macroscopic model reproduces the memory effects often observed in experiments. Explicit formulae given by homogenization for the coefficients of this model emphasize their link to the relevant physiological quantities. In addition, we explicitly solve the macroscopic model to obtain an ODE model for the dMRI signal. This ODE model is numerically easy to invert, and the inverse problem of retrieving model coefficients from synthetic diffusion MRI (dMRI) signal data is considered. |
Discrete Models of Fluids: Spatial Averaging, Closure, and Model Reduction
| Pengarang | : | Alexander Panchenko, Alexandre Tartakovsky, Kevin Cooper |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 74 (No. 2) |
| Halaman | : | 477-515 |
| Abstrak | : | We consider semidiscrete ODE models of single-phase fluids and two-fluid mixtures. In the presence of multiple fine-scale heterogeneities, the size of these ODE systems can be very large. Spatial averaging is then a useful tool for reducing computational complexity of the problem. The averages satisfy exact balance equations of mass, momentum, and energy. These equations do not form a satisfactory continuum model because evaluation of stress and heat flux requires solving the underlying ODEs. To produce continuum equations that can be simulated without resolving microscale dynamics, we recently proposed a closure method based on the use of regularized deconvolution. Here we continue the investigation of deconvolution closure with the long term objective of developing consistent computational upscaling for multiphase particle methods. The structure of the fine-scale particle solvers is reminiscent of molecular dynamics. For this reason we use nonlinear averaging introduced for atomistic systems by Noll, Hardy, and Murdoch-Bedeaux. We also consider a simpler linear averaging originally developed in large eddy simulation of turbulence. We present several simple but representative examples of spatially averaged ODEs, where the closure error can be analyzed. Based on this analysis we suggest a general strategy for reducing the relative error of approximate closure. For problems with periodic highly oscillatory material parameters we propose a spectral boosting technique that augments the standard deconvolution and helps to correctly account for dispersion effects. We also conduct several numerical experiments, one of which is a complete mesoscale simulation of a stratified two-fluid flow in a channel. In this simulation, the operation count per coarse time step scales sublinearly with the number of particles. |
Peer Feedback for Learning Mathematics
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 125 (No. 7) |
| Halaman | : | 653-658 |
| Abstrak | : | This article describes Peer-Assisted Reflection (PAR), a cycle for engaging students with rich mathematical problems. PAR gives students an opportunity to receive peer feedback and revise their work, supporting sustained mathematical engagement. PAR can be incorporated into nearly any course to address logistical constraints that limit how much feedback an instructor can provide. Strategies for using PAR and sample problems are provided. |