Proceed with care, because some infinities are larger than others

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

SOME ASSOCIATIVITY RESULTS FOR RANDOM VARIABLES IN STRANGE GROUPS

Let ̃z be a bare-foot dependent path. Recently, there has been muchinterest in the derivation of isomorphisms on strange groups that come inthe way of ~z. Let there exist a countable infinity of such groups, [[G]]. Weshow that [[G]] has one or more isomorphisms to a Dojo D.Keywords; Bare-foot paths, Measure Theory, Higher Operators, StrangeGroups, Techno Gibberish, model theory, Dojo Theory

Solitary Waves in 1:N Dimer Chains

In the present work we report the discovery of new families of solitary waves in a 1:N (N>1) granular dimers (a heavy bead followed and preceded by N light beads) wherein the Hertzian interaction law governs the interaction between spherical beads. We consider the dimer chain with zero precompression. The dynamics of such a dimer chain is governed by two system parameters, the stiffness ratio and the mass ratio between the light and the heavy beads. In particular we study in detail the solitary waves in 1:2 dimer chains [11]. The solitary waves in a 1:2 dimer are contrastingly different from that in a homogeneous chain and 1:1 dimer chain. Solitary waves realized in homogeneous and 1:1 dimer chains possess symmetric velocity waveforms. In contrast, in a 1:2 dimer chain we realize solitary waves that have symmetric velocity waveforms on the heavy beads, whereas that on the light beads is non-symmetric. The existence of families of solitary waves in these systems is attributed to the dynamical phenomenon to ‘anti-resonance’. This leads to the complete elimination of radiating waves in the trail of the propagating pulse. Anti-resonances are associated with certain symmetries of the velocity waveforms of the beads of the dimer. We conjecture that a countable infinity of family of solitary waves can be realized in 1:2 dimer chains. Interestingly, solitary waves in a general 1:N (N>2) dimer chain are far more difficult to realize. For the case of 1:2 dimers, we can vary the two parameters to satisfy conditions such that the oscillatory tails in the trail of the primary pulse of the two light beads decay to zero. In contrast, for a 1:N (N>2) dimer chain, we have the same two parameters but need to satisfy the decaying conditions on N light beads simultaneously. This leads to a mathematically ill-posed problem and as such rigorously no solitary waves can be realized in general.

Resonance and Anti-Resonance Phenomenon in Granular Dimer Chains With No Pre-Compression

It is a well known fact that many interesting phenomena in the theory of waves in nonlinear lattices, e.g., the significant reduction of the amplitude of a propagating primary pulse or the essential growth of the phase velocity, may be explained in terms of various resonant mechanisms existing in the system (e.g. Frankel-Kontorova model). Recently, we have demonstrated analytically and numerically that similar resonant mechanisms also exist in periodically disordered granular chains with no pre-compression. Moreover, these mechanisms are responsible for the aforementioned phenomena of intensive pulse attenuation as well as speeding up of solitary waves in periodic granular chains. In our studies we have considered regular dimer chains consisting of pairs of ‘heavy’ and ‘light’ beads with no pre compression and with elastic Hertzian interaction between beads. A new family of solitary waves was discovered for these systems. These solitary waves may be considered analogous to the solitary wave of a homogeneous chain studied by Nesterenko [1], in the sense that they do not involve separations between beads, but rather satisfy special symmetries or, equivalently resonances in the dynamics. We show that these solitary waves arise from a countable infinity (we conjecture) of nonlinear anti-resonances in the dimer chains. Moreover, solitary waves in the dimers propagate faster than solitary waves in the homogeneous granular chain obtained in the limit of no mass mismatch (i.e., composed of only ‘heavy’ beads). This finding, which might seem to be counter intuitive, indicates that under certain conditions nonlinear anti-resonances can increase the speed of disturbance propagation in disordered granular media, by generating new ways of transferring energy to the far field in these media. Finally, we discuss a contrasting resonance mechanism that leads to the opposite effect, that is, very efficient shock attenuation in the dimer chain. Indeed, we show that under a certain nonlinear resonance condition a granular dimer chain can greatly reduce the amplitude of propagating pulses, through effective scattering of the energy of the pulse to higher frequencies and excitation of alternative intrinsic dynamics of the dimer. This resonance condition may be theoretically predicted and explained, and a very fair correspondence is observed between the analytical solutions and direct numerical simulations. From a practical point of view, these results can have interesting implications in applications where granular media are employed as shock transmitters or attenuators.

ON THE INFINITE IN MEREOLOGY WITH PLURAL QUANTIFICATION

In “Mathematics is megethology,” Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if—as Lewis maintains—MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects.

Parametric Identification of Chaotic Systems Via a Long-Period Harmonic Balance

Hyperbolic chaotic sets are composed of a countable infinity of unstable periodic orbits (UPOs). Symbol dynamics reveals that any long chaotic segment can be approximated by a UPO, which is a periodic solution to an ideal model of the system. Treated as such, the harmonic balance method is applied to the long chaotic segments to identify model parameters. Ultimately, this becomes a frequency domain identification method applied to chaotic systems.

Locating oscillatory orbits of the parametrically-excited pendulum

A method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

Static spherically symmetric solutions of a Yang–Mills field coupled to a dilaton

We give a detailed analytical study of static spherically symmetric solutions for an SU (2) Yang–Mills field coupled to a scalar graviton (or dilaton). We show by a ‘shooting’ argument that there are a countable infinity of such solutions satisfying the relevant boundary conditions, there being at least one for each given number of local maxima and minima for the Yang-Mills potential.