Month: March 2011

Romeo & Juliet: Nonlinear Dynamics

“My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me
That I must love a loathèd enemy”.

romeo and juliet

Steven Strogatz first analyzed love affairs via differential equations.

Let’s begin with the linear model for two individuals. An additional constant represents the appeal if positive, the repulsion if negative.

\frac{dR}{dt}= aR+bJ

\frac{dJ}{dt}= cR+dJ

Specifically, R(t) is the love (or hate) of individual 1 (R= Romeo) for individual 2 (J=Juliet), at time t; J(t) is the love (or hate) of individual 2 for individual 1, at time t. The coefficient a and b specify  the “romantic style” of individual 1 ( Romeo), while c and d that of individual 2 ( Juliet). Some “Romantic Styles” :

  1. \frac{dR}{dt}= aR+bJ
  • a =0 (out of touch with own feelings)
  • b =0 (oblivious to other’s feelings)
  • a >0, b >0 (eager beaver)
  • a >0, b <0 (narcissistic nerd)
  • a <0, b >0 (cautious lover)
  • a <0, b <0 (hermit)

2. \frac{dJ}{dt}= cR+dJ

Four subclasses:

  •  c > 0, d > 0 (mutual love or hate)
  •  c > 0, d < 0 (never-ending cycle)
  •  c < 0, d > 0 (never-ending cycle)
  •  c < 0, d < 0 (unrequited love)

Classification:

  1. Eager Beaver: individual 1 is encouraged by his own feelings as well as that of individual 2;
  2. Secure or Cautious  lover: individual 1 retreats from his own feelings but is encouraged by that of individual 2;
  3. Hermit: individual 1 retreats from his own feelings and that of individual 2;
  4. Narcissistic Nerd: individual 1 wants more of what he feels but retreat from the feelings of individual 2.

Model for the dynamics of love affairs

Romeo is in love with Juliet but, in our story, Juliet is a fickle lover.

We can model the oscillating emotions felt by Romeo and his fair Juliet: the more Romeo loves her, the more Juliet wants to run away. When Romeo gets discouraged, Juliet begins to find him attractive.

Let:

R(t)= Romeo’s love/hate for Juliet at time t;

J(t)= Juliet’s love/hate for Romeo at time t;

If R>0, J signifies love; negative values of R signifies hate. A dynamic model of Romeo and Juliet’s Relationship is:

\frac{dR}{dt}=aJ

\frac{dJ}{dt}=-bJ

where a,b>0.

The system has a center at (R,J)=(0,0).

Now consider the general linear system:

\frac{dR}{dt}= aR+bJ

\frac{dJ}{dt}= cR+dJ

where the parameters a,b,c,d may have either sign!

A choice of signs specifies the romantic style.

For example: what happens when two identically cautious lovers get together?

The system is:

\frac{dR}{dt}= aR+bJ

\frac{dJ}{dt}= bR+aJ

where a is the measure of cautiousness (a<0) and b represents responsiveness (b>0).

The corresponding matrix is:

a         b

b        a

which has: \tau=2a<0, \Delta= a^2-b^2, \tau^2- 4\Delta=4b^2>0

The fixed point ( R,J)=(0,0) is a saddle point if a^2 <b^2 and a stable node if a^2>b^2.

The eigenvalues and corresponding eigenvectors are:

\lambda_1= a+b ; v_1= (1,1) ; \lambda_2= a-b ; v_2=(1, -1)

The two cases:

——————————————–

References:

– Steven H. Strogatz. Love affairs and differential equations. Mathematics Magazine, 61:35, 1988

– Steven H. Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Westview Press, 2001

The Math of Ponzi Schemes

A first order linear differential equation is often used to describe the dynamics of an investment fund, a.k.a. Ponzi Scheme. This type of fraud is named after its creator—Carlo Ponzi, who collected $9.8 million from 10.550 people and then paid out $7.8 million in just 8 months in 1920 Boston by offering  profits of 50%  return on investment in postal coupons, every 45 days. A Financial Ponzi scheme is a particular kind of pyramid scheme.

Pyramid Schemes

As in Ponzi schemes, the money collected from newer victims (investors) of the fraud is paid to earlier victims to provide a veneer of legitimacy. The profit is earned, by the sale of new distributorships and it is not created by the success of the business venture but instead is derived fraudulently from the capital contributions of other investors. Emphasis on selling franchises rather than the product leads to a point where the supply of potential investors is exhausted; when the operator flees with all of the proceeds or when a sufficient number of new investors cannot be found to allow the continued payment of “dividends”, the scheme falls apart and the pyramid collapses, because the underlying asset upon which the investment was based either never existed, or was overvalued.

The way that it works is that you create some kind of “investment fund“. You promise some kind of great payoff and convince people to invest. Ponzi schemes promise high financial returns or dividends over a short period of time, while still maintaining some semblance of credibility.

Instead of investing the funds of victims, the con artist (a con artist is an individual who is skilled and experienced at devising and executing scams and other fraudulent schemes) pays “dividends” to initial investors, using the funds of subsequent investors. As you get new investors, you use their investments to pay off the previous investors. The regular payment of dividends induces investors to bring friends, family members, or business colleagues (commonly targeted victims) into the scheme and to put up additional funds themselves now that they are convinced of its veracity.

 Several distinctions between Ponzi schemes and pyramid selling schemes: Pyramid schemes require active participants who will bring in more participants. Ponzi schemes require passive investors without any responsibility to promote the opportunity.

A LOT OF MONEY… and MATH!

We assume that the fund starts at time t=0

  • c(t): cash in initial deposit at time t \in [0,\infty).
  • \lambda(t): cash influx rate at time t \in [0,\infty). The amount of cash that flows in the deposit scam in an infinitesimal time interval [t, t + dt] is \lambda(t)dt.
  • T: lock-up period (T>0). The con artist promises to return the money to the investors after a lock-up for T units of time.
  • R: promised rate on investment (R>0). It’s called Ponzi rate.

__________

The Law of Conservation of Investment Value

 Money doesn’t evaporate, when it surprisingly dissapears.  It follows the same laws of conservation of energy. It may just transform: be it coffee or book or iphone.

If you spend few minutes in the day, you’ll check what you’ve spent.  So, at the end of month, you could see where your money was transformed. An advanced method is to identify your sources of money inflow and track how it flows out, that way you can determine if your inflow matches up with your outflow.
We will assume that the com artist does not deposit the cash in a bank to earn interest on it.

A first order linear differential equation is used to describe the dynamics of an investment fund.  The following differential equation:

\displaystyle\dot{c}(t) = \displaystyle \lambda (t) - (1+R) \lambda (t-T),

in which c: [0,\infty) \rightarrow \mathbb{R} is the function to be determined, and \lambda: [0,\infty) \rightarrow \mathbb{R} is the forcing function.

The rate at which the cash flows (Cash flow is the movement of cash into or out of a business, financial product) out is higher than the rate at which money flows in. After a large cash influx: since the cash influx cannot grow without bound, the cash influx will be insufficient to pay off the debt.

Solving the differential equation:

If we denote the initial amount of cash by c(0) = c_0 and solve the differential equation above, we obtain:

c(t) = c_0 +\displaystyle\int_{0}^{t} \lambda(\tau - T)d\tau - (1+R)\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau

Fort \in [0,T):

\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = 0,

and for t>T is:

\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = \displaystyle\int_{0}^{t -T} \lambda (\tau ) d\tau.

Thus:

\displaystyle\int_{0}^{t} \lambda (\tau - T) d\tau = \displaystyle\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau,

and the general solution can be written:

c(t) = c_0 +\displaystyle\int_{0}^{t} \lambda(\tau - T)d\tau - (1+R)\displaystyle\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau.

The amount of cash at time t will be given by initial amount of cash, plus the amount of cash his creditors lent him, minus the amount of cash the con artist had to pay to his creditors (which is “amplified” by R) over time interval (t-T, t). The con artist starts returning money to the investors at time t=T.

The general solution can also be written as

c(t) = c_0 +\int_{\max(t-T,0)}^{t } \lambda (\tau)  d\tau -R\int_{0}^{\max(t-T,0)} \lambda (\tau) d\tau .

Example: Cash influx at t=0

\displaystyle \lambda(t) = \displaystyle \lambda_0 \delta(t)

In this case, an amount of cash equal to \lambda_0 is credited to the con artist’s pockets at time t=0. The solution will be:

\displaystyle c(t) = \displaystyle c_0 + \lambda_0 u(t) - (1+R) \lambda_0 u(t-T),

which is the Ponzi scheme’s impulse response (impulse response is response of system when input is provided to it is impulse; yet another explanation: the impulse response function is the inverse Laplace transform of the system transfer function H(s)). Note that c(t) = c_0 - R \lambda_0 for t \in (0, T), while c(t) = c_0 - R \lambda_0 for t>T.

__________

Example: constant cash influx rate

\displaystyle \lambda(t) = \displaystyle \lambda_0 u(t)

A constant influx of cash flows to the con artist’s pockets. He will have to repay this cash with interest T units of time later. The solution of the differential equation is:

c(t) = \displaystyle c_0 + \lambda_0 s(t) - (1+R) \lambda_0 s(t-T),

which is the Ponzi scheme’s step response (the step response is defined as the response of a system to a step input; look here: http://www.atp.rub.de/DynLAB/dynlabmodules/Examples/TimeResponse/TimeResponse02.html ). The cash will increase linearly with time for t \in (0,T), a maximum is reached at t=T (maximum: c(T) = c_0 + \lambda_0 T), and then the debt-paying time starts and the cash will decrease linearly with slope - R \lambda_0 until the amount of cash reaches zero at  t = t_B > T. Making  c(t_B) = 0, we have:

c_0 +\lambda_0 t_B - (1+R)\lambda_0 (t_b - T) = 0,

t_B = \frac{c_0}{R \lambda_0} + (1 +\frac{1}{R}) T,

which is the “bankruptcy time” ( bankruptcy is a legal status of a person or an organization that cannot repay the debts it owes to its creditors). For t > t_B, the con artist will have to go into debt to pay off to his investors.
The bankruptcy proceedings:

  1. free you from overwhelming debts so you can make a fresh start, subject to some restrictions;
  2. make sure your assets are shared out fairly among your creditors;

An individual can be made bankrupt either in one of three ways:

  • Voluntarily – By the debtor themselves;
  • Involuntarily – By the creditor owed money;
  • The supervisor or anyone bound by an IVA.

________

The development of the atomic models

 Dalton model (Billiard Ball Model)

John Dalton proposed a modern atomic model based on experimentation. The Law of Multiple Proportions states that when elements combine, they do so in the ratio of small whole numbers. Carbon and Oxygen can form either CO ( a 1:1 ratio) or CO2 (a 1:2 ratio). In his theory, there are six basic ideas:

–          All matter is composed of atoms

–          Atoms cannot be made or destroyed

–          All atoms of the same element are identical

–          Different elements have different types of atoms

–          Chemical reactions occur when atoms are rearranged

–          Compounds are formed from atoms of the constituent elements.

THE KELVIN’S-THOMSON’S ATOMIC MODEL -THE PLUM CAKE: Physics and radioactivity after the discovery of polonium and radium.

 When the phenomena of the radioactivity was discovered, the Dalton’s theory of the indivisible atom fell down. Basing on the researches on polonium and radium ( discovered by Marie e Pierre Curie), lord Kelvin formulated a model, assuming that every particle is more or less but radioactive.

The “plum pudding model” was J.J. Thomson’s theory of atomic structure that he developed in 1904. In this model the electrons and protons are uniformly mixed throughout the atom.

In the years 1909-1911 Ernest Rutheford and his students – Hans Geiger (1882-1945) and Ernest Marsden conducted some experiments, testing Thomson’s hypothesis by devising his “gold foil” experiment.

The experiments caused the creation of the new model of atom – the “planetary” model. An alpha particle ( a heavy, positively charged particle) is a helium nucleus released by radioactive substances. Polonium was put into a lead box that sent out alpha particles to a thin sheet of gold foil. The foil was then surrounded by a luminescent zinc sulphide screen that served as a backdrop for the alpha particles to appear on.  A microscope was placed above the screen so they could observe any contact made between the alpha particles and the screen. In order to prove Thomson’s “plum pudding model”, the alpha particles were supposed to go straight through the foil. Shockingly, although most alpha particles passed straight through the gold foil, some did not.

 

Most alpha particles indeed went through the foil with only small deflections. But a few were scattered by large angles or bounced directly back at source. Thomson’s “plum pudding model” was proven incorrect. Thomson’s theory said that an atom was made up of empty space, but that couldn’t be correct if the particles had bounced back because they had to have hit something. The way and angles in which the alpha particles bounced off the foil indicated that the majority of the mass of an atom was concentrated in one small region, that Rutherford later called the nucleus. He reasoned that the nucleus held all the positive charge, while electrons occupied most of the atom’s space.

The Bohr’s Model

In 1913 Niels Bohr published a theory about the structure of the atom based on an earlier theory of Rutherford’s. Rutherford had shown that the atom consisted of a positively charged nucleus, with negatively charged electrons in orbit around it.

Bohr expanded upon this theory by proposing that electrons travel only in certain successively larger orbits. Bohr also described the way atoms emit radiation by suggesting that when an electron jumps from an outer orbit to an inner one, that it emits light. The Bohr’s Model is a planetary model in which the negatively-charged electrons orbit a small, positively-charged nucleus similar to the planets orbiting the Sun. The gravitational force of the solar system is mathematically akin to the Coulomb (electrical) force between the positively-charged nucleus and the negatively-charged electrons.

There were problems with the Bohr Model:

  • It violated the Heisenberg Uncertainty Principle because it considered electrons to have both   a known radius and orbit.
  • The Bohr Model provided an incorrect value for the ground state orbital angular momentum.
  • It made poor predictions regarding the spectra of larger atoms.
  • It did not predict the relative intensities of spectral lines.
  • The Bohr Model did not explain fine structure and hyperfine structure in spectral lines.
  • It did not explain the Zeeman Effect.

The Zeeman effect describes the splitting of spectral lines in the presence of a magnetic field. In the absence of a magnetic field, emission is observed as a single spectral line and is dependent only on the principal quantum numbers of the initial and final states. In the presence of an external magnetic field, the principal quantum number of each state is split into different substates.