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On Eigen(patch(ix))values - (RWLA,MCT,GT,AM Part VIII)

March 16th, 2011 No comments

So in the continuation of this series, I have been thinking long and hard about the curious property of the existence of eigen(patch(ix))values that I have talked about in a previous post.  I began to question whether such eigen(patch(ix))values are limited to a finite set (much as in finite matrices) or whether there was some other fundamental insight, like, if 1 is an eigen(patch(ix))value, then all elements of  \mathbb{R} are too (or all of  \mathbb{R} minus a finite set).  In my latest attempt to understand this, the question comes down to, using the "star" operator, whether

 a(x) \star p(x,y) = \lambda a(x)

has discrete values of  \lambda or, "what values can lambda take for the equation to be true," in direct analogy with eigenvalues when we're dealing with discrete matrices.  I am not using yet "integral transform notation" because this development seemed more intuitive to me, and thus I'm also limiting the treatment to "surfaces" that are smooth and defined on  [0,1] \times [0,1] , like I first thought of them. Thus, the above equation translates to:

 \int_0^1 a(1-y) p(x,y) dy = \lambda a(x)

and, if we recall our construction of the patch (or patchix if we relax the assumption that integrating with respect to x is 1)  p(x,y) = f_1(x) g_1(y) + f_2(x) g_2(y) :

 \begin{array}{ccc} \lambda a(x) & = &\int_0^1 a(1-y) \left(f_1(x) g_1(y) + f_2(x) g_2(y) \right) dy \\ & = & f_1(x) \int_0^1 a(1-y) g_1(y) dy + f_2(x) \int_0^1 a(1-y) g_2(y) dy \\ & = & B_1 f_1(x) + B_2 f_2(x) \end{array}

where  B_1, B_2 are constants.  It is very tempting to divide  \lambda as

 a(x) = \frac{B_1}{\lambda} f_1(x) + \frac{B_2}{\lambda} f_2(x)

must hold provided  \lambda \neq 0 .  So we have excluded an eigen(patch(ix))value right from the start, which is interesting.

We can systematically write the derivatives of  a(x) , as we're going to need them if we follow the algorithm I delineated in one of my previous posts (NB: we assume a finite number of derivatives or periodic ones, or infinite derivatives such that the subsequent sums we'll write are convergent):

 \begin{array}{ccc} a(x) & = & \frac{B_1}{\lambda} f_1(x) + \frac{B_2}{\lambda} f_2(x) \\ a'(x) & = & \frac{B_1}{\lambda} f'_1(x) + \frac{B_2}{\lambda} f'_2(x) \\ a''(x) & = & \frac{B_1}{\lambda} f''_1(x) + \frac{B_2}{\lambda} f''_2(x) \\ \vdots & \vdots & \vdots \\ a^k(x) & = & \frac{B_1}{\lambda} f^k_1(x) + \frac{B_2}{\lambda} f^k_2(x) \\ \vdots & \vdots & \vdots \end{array}

provided, as before,  \lambda \neq 0 .  We want to calculate the constants  B_1, B_2 , to see if they are restricted in some way by a formula, and we do this by integrating by parts as we did in a previous post to obtain the cool "pasquali series." Thus, we have that if  B_1 = \int_0^1 a(1-y) g_1(y) dy , the tabular method gives:

 \begin{array}{ccccc} \vert & Derivatives & \vert & Integrals & \vert \\ \vert & a(1-y) & \vert & g_1(y) & \vert \\ \vert & -a'(1-y) & \vert & G_1^1(y) & \vert \\ \vert & a''(1-y) & \vert & G_1^2(y) & \vert \\ \vert & \vdots & \vert & \vdots & \vert \end{array}

and so,

 \begin{array}{ccc} B_1 & = & \int_0^1 a(1-y) g_1(y) dy \\ & = & a(1-y) G_1^1(y) \vert_0^1 + a'(1-y) G_1^2(y) \vert_0^1 + \ldots \\ & = & \sum_{i = 0}^\infty a^i(1-y) G_1^{i + 1} \vert_0^1 \end{array}

if we remember the alternating sign of the multiplications, and we are allowed some leeway in notation.  Ultimately, this last bit means:  \sum_{i=0}^\infty a^i(0) G_1^{i+1}(1) - \sum_{i=0}^\infty a^i(1) G_1^{i+1}(0) .

Since we have already explicitly written the derivatives of  a(x) , the  a^i(0), a^i(1) derivatives can be written as  \frac{B_1}{\lambda} f_1^i(0) + \frac{B_2}{\lambda} f_2^i(0) and  \frac{B_1}{\lambda} f_1^i(1) + \frac{B_2}{\lambda} f_2^i(1) respectively.

We have then:

 B_1 = \sum_{i=0}^\infty \left( \frac{B_1}{\lambda} f_1^i(0) + \frac{B_2}{\lambda} f_2^i(0) \right) G_1^{i+1}(1) - \sum_{i=0}^\infty \left( \frac{B_1}{\lambda} f_1^i(1) + \frac{B_2}{\lambda} f_2^i(1) \right) G_1^{i+1}(0)

Since we aim to solve for  B_1 , multiplying by  \lambda makes things easier, and also we must rearrange all elements with  B_1 in them, so we get:

 \lambda B_1 = B_1 \sum_{i=0}^\infty \left( f_1^i(0) G_1^{i+1}(1) - f_1^i(1) G_1^{i+1}(0) \right) + B_2 \sum_{i=0}^\infty \left( f_2^i(0) G_1^{i+1}(1) - f_2^i(1) G_1^{i+1}(0) \right)

Subtracting both sides the common term and factoring the constant we endeavor to solve for, we get:

 \left( \lambda - \sum_{i=0}^\infty \left( f_1^i(0) G_1^{i+1}(1) - f_1(1) G_1^{i+1}(0) \right) \right) B_1 = B_2 \sum_{i=0}^\infty \left(f_2^i(0) G_1^{i+1}(1) - f_2^i(1) G_1^{i+1}(0) \right)

or

 B_1 = \frac{B_2 \sum_{i=0}^\infty f_2^i(1-y) G_1^{i+1}(y) \vert_0^1}{\lambda - \sum_{i=0}^\infty f_1^i(1-y) G_1^{i+1}(y) \vert_0^1} = \frac{B_2 D}{\lambda - C}

A similar argument for  B_2 suggests

 B_2 = \frac{B_1 \sum_{i=0}^\infty f_1^i(1-y) G_2^{i+1}(y) \vert_0^1}{\lambda - \sum_{i=0}^\infty f_2^i(1-y) G_2^{i+1}(y) \vert_0^1} = \frac{B_1 E}{\lambda - F}

where the new constants introduced emphasizes the expectation that the sums converge.  Plugging in the one into the other we get:

 B_1 = \frac{\left( \frac{B_1 E}{\lambda - F} \right) D}{\lambda - C} = \frac{B_1 E D}{(\lambda - F) (\lambda - C)}

and now we seem to have additional restrictions on lambda:  \lambda \neq F and  \lambda \neq C .  Furthermore, the constant  B_1 drops out of the equation, suggesting these constants can be anything we can imagine (all of  \mathbb{R} without restriction), but then we have the constraint:

 (\lambda - F)(\lambda - C) = ED

which is extraordinarily similar to its analogue in finite matrix or linear algebra contexts.  Expanding suggests:

 \lambda^2 - (F + C) \lambda + (CF - ED) = 0

which we can solve by the quadratic equation of course, as:

 \lambda_{1,2} = \frac{(F + C) \pm \sqrt{(F-C)^2 + 4ED} }{2}

So not only is  \lambda not equal to a few values, it is incredibly restricted to two of them.

So here's a sort of conjecture, and a plan for the proof.  The allowable values of  \lambda is equal to the number of x terms  a(x) (or  p(x,y) ) carries.  We have already shown the base case, we need only show the induction step, that it works for  k and  k+1 terms.

On Patch(ix)es as Kernels of Integral Transforms (RWLA,MCT,GT,AM Part VII)

February 7th, 2011 No comments

[This post is ongoing, as I think of a few things I will write them down too]

So just a couple of days ago I was asked by a student to give a class on DEs using Laplace transforms, and it was in my research that I realized that what I've been describing by converting a probability distribution on [0,1] to another is in effect a transform (minus the transform pair, which was unclear to me how to obtain, corresponding perhaps to inverting the patch(ix)).  The general form of integral transforms is, according to my book Advanced Engineering, 2nd ed., by Michael Greenberg p. 247:

 F(s) = \int_a^b f(t) K(t,s) dt , where  K(t,s) is called the kernel of the transform, and looks an awful lot like a function by patch(ix) "multiplication," which I described as:

 b(x) = \int_0^1 a(1-y) p(x,y) dy you may recall.  In the former context  p(x,y) looks like a kernel, but here  a(1-y) is a function of  y than of  x , and I sum across  y .  To rewrite patch(ix)-multiplication as an integral transform, it would seem we need to rethink the patch position on the xy plane, but it seems easy to do (and we do in number 1 below!).

In this post I want to (eventually be able to):

1. Formally rewrite my function-by-patch(ix) multiplication as a "Pasquali" integral transform.

If we are to modify patch multiplication to match the integral transform guideline, simply think of  p(t,s) as oriented a bit differently, yielding the fact that  \int_0^1 p(t,s) ds = 1 for any choice of  t .  Then, for a probability distribution  b(t) in [0,1], the integral transform is  B(s) = \int_0^1 b(t) p(t,s) dt .  Now  p(t,s) is indeed then a kernel.

2. Extend a function-by-patch multiplication to probability distributions and patches on all  \mathbb{R} and  \mathbb{R}^2 , respectively.

When I began thinking about probability distributions, I restricted them to the interval [0,1] and a patch on  [0,1] \times [0,1] , to try to obtain a strict analogy of (continuous) matrices with discrete matrices.  I had been thinking for a while that this need not be the case, but when I glanced at the discussion of integral transforms on my Greenberg book, and particularly the one on the Laplace transform, I realized I could have done it right away.  Thus, we can redefine patch multiplication as

 B(s) = \int_{-\infty}^{\infty} b(t) p(t,s) dt

with

 \int_{-\infty}^{\infty} p(t,s) ds = 1

3. Explore the possibility of an inverse-patch via studying inverse-transforms.

3a. Write the patch-inverse-patch relation as a transform pair.

4. Take a hint from the Laplace and Fourier transforms to see what new insights can be obtained on patch(ix)es (or vice-versa).

Vice-versa: Well one of the things we realize first and foremost, is that integral transforms are really an extension of the concept of matrix multiplication: if we create a matrix "surface" and multiply it by a "function" vector we obtain another "function," and the kernel (truly a continuous matrix) is exactly our path connecting the two.  Can we not think now of discrete matrices (finite, infinite) as "samplings" of such surfaces?  I think so.  We can also combine kernels with kernels (as I have done in previous posts) much as we can combine matrices with matrices.  I haven't really seen a discussion exploring this in books, which is perhaps a bit surprising.  At any rate, recasting this "combination" shouldn't be much of a problem, and the theorems I proved in previous posts should still hold, because the new notation represents rigid motions of the kernel, yielding new kernel spaces that are isomorphic to the original.

On Patch Stationariness (RWLA,MCT,GT,AM Part VI)

January 16th, 2011 No comments

In my previous posts, I have been discussing how we can extend functional analysis a little bit by "inventing" continuous matrices (surfaces) which contain all the information we may want on how to transform, in a special case, probability distributions from one to another, and I have tried, by reason of analogy, to extend Markov theory as well.  In this special case, I have been talking about how a surface "continuous collection of distributions" can reach steady-state: by self-combining these surfaces over and over; I even showed how to obtain a couple steady-states empirically by calculating patch powers specifically and then attempting to infer the time evolution, quite successfully in one case. The usual Markov treatment suggests another way to obtain the steady-state (the limiting transition probability matrix), by finding a stationary distribution so that left multiplying the vector  \mathbf{\widehat p}  by the transition probability matrix  P gives us  \mathbf{\widehat p} .  Within the discrete transition probability matrix context, a vector  \mathbf{\widehat p} with this property is also a (left) eigenvector of  P with eigenvalue 1.  See for example Schaum's series Probability, Random Variables, and Random Processes p. 169, as well as Laurie Snell's chapter 11 on Markov Chains on his online Probability book. An important theorem says that the limiting transition probability matrix  \lim_{n \rightarrow \infty} P^n = \mathbf{\widehat P} is a matrix whose rows are identical and equal to the stationary distribution  \mathbf{\widehat p} .  To calculate the stationary distribution (and the limiting transition probability matrix) one would usually solve a system of equations. For example, if:

 P = \left[ \begin{array}{cc} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{2} \end{array} \right]

the stationary distribution

 \mathbf{\widehat p} P = \mathbf{\widehat p}

looks explicitly like:

 \left[ \begin{array}{cc} p_1 & p_2 \end{array} \right] \left[ \begin{array}{cc} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{2} \end{array} \right] = \left[ \begin{array}{cc} p_1 & p_2 \end{array} \right]

in other words, the system:

 \frac{3}{4} p_1 + \frac{1}{2} p_2 = p_1

 \frac{1}{4} p_1 + \frac{1}{2} p_2 = p_2

each of which gives  p_1 = 2 p_2 and is solvable if we notice that  p_1 + p_2 = 1 , yielding  \mathbf{\widehat p} = \left[ \begin{array}{cc} \frac{2}{3} & \frac{1}{3} \end{array} \right] , and

 \mathbf{\widehat P} = \left[ \begin{array}{c} \mathbf{\widehat p} \\ \mathbf{\widehat p} \end{array} \right]

In this post, I want to set up an algorithm to calculate the stationary surface (steady-state) of patches as I've defined them, following in analogy the above argument.  To do so, I revisit both of my previous examples, now calculating the steady state from this vantage point.  The fact that we can define such an algorithm in the first place has ginormous implications, in the sense that we can define stationary function distributions that would seem therefore to be eigen(patch(ix))vectors (corresponding to eigen(patch(ix))values) of surface distributions, and we can seemingly also solve a continuously infinite quantity of independent equations, however strange this actually sounds.

Example 1, calculating the stationary patch  p_{\infty}(x,y) when  p_1(x,y) = 2 x - \frac{2 x y^3}{3} + x^2 y^3 .

I have already shown that  p_1(x,y) is indeed a patch because  \int_0^1 p_1(x,y) dx = 1 , for any choice of  y .

Suppose there exists a distribution defined as always on  x \in [0,1] , say  a(x) , so that

 a(x) \star p(x,y) = a(x) .  Explicitly,  \int_0^1 a(1 - y) \cdot \left( 2 x - \frac{2 x y^3}{3} + x^2 y^3 \right) dy = a(x)  We can break up the integral as:

 2 x \int_0^1 a(1-y) dy - \frac{2 x}{3} \int_0^1 y^3 a(1-y) dy + x^2 \int_0^1 y^3 a(1-y) dy = a(x)

The first part, we've seen many times, adds up to one because  a(x) is a probability distribution, so let's rewrite the whole thing as:

 2 x - \left( \frac{2 x}{3} - x^2 \right) \int_0^1 y^3 a(1 - y) dy = a(x)

The integral is in reality just a constant, so we have that a(x) looks something like:

 2 x - \left( \frac{2 x}{3} - x^2 \right) B = a(x) if we let

 B = \int_0^1 y^3 a(1 - y) dy

Now this integral in  y , though it is a constant, is seemingly impossible to solve without more knowledge of  a(1-y) ; but the truth of the matter is we have everything we need because we have a specification of  a(x) .  The crucial thing to notice is that derivatives of  a(x) do not exist "eternally," because  a(x) is a polynomial of maximal degree 2.  Thus we can attempt integration by parts and try to see where this takes us.  The tabular method gives us an organized way to write this out:

 \begin{array}{ccccc} | & Derivatives & | & Integrals & | \\ | & a(1-y) & | & y^3 & | \\ | & -a'(1-y) & | & \frac{y^4}{4} & | \\ | & a''(1-y) & | & \frac{y^5}{20} & | \\ | & 0 & | & \frac{y^6}{120} & | \\ | & \vdots & | & \vdots & | \end{array}

and, remembering the alternating sign when we multiply, we get the series:

 \frac{a(1-y) y^4}{4} + \frac{a'(1-y) y^5}{20} + \frac{a''(1-y) y^6}{120} + 0 + \ldots \arrowvert_0^1

The zeroth substitution of the lower limit of the integral gives us all zeroes, but the one-substitution gives us the interesting "pasquali series":

 \frac{a(0) }{4} + \frac{a'(0)}{20} + \frac{a''(0)}{120} + 0 + \ldots

which asks of us to evaluate  a(x) and its derivatives (until just before it vanishes) at zero:

 \begin{array}{ccc} a(x) & = & 2 x - \left( \frac{2 x}{3} - x^2 \right) B \\ a'(x) & = & 2 - \frac{2 B}{3} + 2 B x \\ a''(x) & = & 2 B \end{array}

 \begin{array}{ccc} a(0) & = & 0 \\ a'(0) & = & 2 - \frac{2 B}{3} = \frac{6 - 2 B}{3}\\ a''(0) & = & 2 B \end{array}

All that's left now is to substitute back into the series:

 B = \frac{\frac{6 - 2 B}{3}}{20} + \frac{2 B}{120} = \frac{1}{10} - \frac{B}{60} solves to  B = \frac{6}{61} which is what we want (I tested the following code with Wolfram Alpha: "integrate [[2(1-y) - (.0983607)(((2(1-y))/3) - (1-y)^2)]*[2x - (2 x y^3)/3 + x^2 y^3]] dy from y = 0 to 1" and obtained the same numeric decimal value at the output).

We have therefore that  a(x) = x - \left( \frac{2 x}{3} - x^2 \right) \frac{6}{61} is a stationary distribution, and the steady-state patch would seem to be  p_\infty(x,y) = a(x) .  I personally think this is very cool, because it validates several propositions: that we can find steady-state patches analytically (even when we may think we have a (continuously) infinite system to solve, it will reduce essentially to a (countable!) series estimable provided the "pasquali" series converges) by a means other than finding the patch powers and attempting to see a pattern, prove perhaps by induction, and then take the limit as patch powers go to infinity, much as I did in my previous post.  It also validates the "crazy" idea that (certain?) special surfaces like patches have eigen(patch(ix))vectors, as arguing  a(x) would suggest exist, and which we would have to obtain, in discrete matrixes, by solving a finite system of equations (and which we did here, again, by solving the "pasquali" series).

Example 2.  In my second example, take the patch  1 - cos(2 \pi x) cos( 2 \pi y) . Again we are looking at a patch because  \int_0^1 p(x,y) dx = 1 for any value of  y .  To establish the steady-state surface, or  p_\infty(x,y) , we proceed as before and write

 \int_0^1 a(1-y) \left( 1 - cos(2 \pi x) cos(2 \pi y) \right)dy = a(x) , or, explicitly,

 \int_0^1 a(1-y) dy - cos(2 \pi x) \int_0^1 a(1-y) cos(2 \pi y) dy = a(x)

The first integral adds up to 1 by hypothesis, where the second one is zero after integrating by parts:

 \begin{array}{ccccc} | & Derivatives & | & Integrals & | \\ | & a(1-y) & | & cos(2 \pi y) & | \\ | & -a'(1-y) & | & \frac{sin(2 \pi y)}{2 \pi} & | \\ | & a''(1-y) & | & \frac{-cos(2 \pi y)}{4 \pi} & | \\ | & -a'''(1-y) & | & \frac{-sin(2 \pi y)}{8 \pi} & | \\ | & \vdots & | & \vdots & | \end{array}

so we have:

 \frac{a(1-y) sin(2 \pi y)}{2 \pi} - \frac{a'(1-y) cos(2 \pi y)}{4 \pi} - \frac{a''(1-y) sin (2 \pi y)}{8 \pi} + \ldots \vert_0^1 and the awesome-slash-interesting "pasquali series"

 -\frac{a'(0)}{4 \pi} + \frac{a'''(0)}{16 \pi} - \frac{a^v (0)}{64 \pi} + \ldots from which we must subtract by the Fundamental Theorem of Calculus

 -\frac{a'(1)}{4 \pi} + \frac{a'''(1)}{16 \pi} - \frac{a^v(1)}{64 \pi} + \ldots

So we are left with  B = \frac{a'(1)}{4 \pi} - \frac{a'(0)}{4 \pi} + \frac{a'''(0)}{16 \pi} - \frac{a'''(1)}{16 \pi} + \frac{a^v(1)}{64 \pi} - \frac{a^v(0)}{64 \pi} + \ldots and also with

 \begin{array}{ccc} a(x) & = & 1 - cos(2 \pi x) B \\ a'(x) & = & 2 \pi sin(2 \pi x) B \\ a''(x) & = & 4 \pi cos(2 \pi x) B \\ a'''(x) & = & -8 \pi sin(2 \pi x) B \\ \vdots & \vdots & \vdots \end{array}

To show this thoroughly, we should prove by induction that every odd derivative of  a(x) contains a  sin term (or we can attempt an argument by periodicity of the derivative, as we do), and so evaluating such at 0 and at 1 literally causes the term to vanish, and leaving us with the fact that  B = 0 and that  a(x) = 1 .  Therefore, as before,  p_\infty(x, y) = a(x) = 1 , and this is consistent with my derivation in the previous post, too.

On Patchix by Patchix Products – Tying Up Loose Ends - (RWLA,MCT,GT,AM Part V)

October 17th, 2010 No comments

In this post I want to "tie up a few lose ends."  For example, in my last post I stated that the patchix pattern

 \begin{array}{ccc} p_1(x,y) & = & 1 - cos(2 \pi x) cos(2 \pi y) \\ p_2(x,y) & = & 1 + \frac{cos(2 \pi x) cos(2 \pi y)}{2} \\ p_3(x,y) & = & 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{4} \\ p_2(x,y) & = & 1 + \frac{cos(2 \pi x) cos(2 \pi y)}{8} \\ \vdots \\ p_t(x,y) & = & 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{t-1}} \end{array}

for  t \in \mathbb{Z^+} , but I didn't prove it.  It's simple to do by induction: by the inductive hypothesis,

 p_1(x,y) = 1 - cos(2 \pi x) cos(2 \pi y) = 1 - \frac{cos(2 \pi x) cos(2 \pi t)}{(-2)^{1-1}}

By the inductive step, assume

 p_k(x,y) = 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{k-1}}

Then,

 \begin{array}{ccc} p_{k+1}(x,t) & = & \int_0^1 p_1(1-y,t) \cdot p_k(x,y) dy \\ & = & \int_0^1 \left( 1 - cos(2 \pi (1-y))cos(2 \pi t) \right) \cdot \left( 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{k-1}} \right) dy \end{array}

Now, if one dislikes shortcuts one can expand the product and integrate term by term to one's heart's content.  The "shorter" version is to relate the story: notice the product of 1 with itself is 1, and such will integrate to 1 in the unit interval.  So we save it.  The integrals  \int_0^1 cos(2 \pi y) dy and  \int_0^1 cos(2 \pi - 2\pi y) dy both evaluate to zero, so we are left only with the task of evaluating the crossterm:

 \begin{array}{ccc} && \int_0^1 cos(2 \pi (1-y))cos(2 \pi t) \cdot \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{k-1}} dy \\ & = & \frac{cos(2 \pi t) cos (2 \pi x)}{(-2)^{k-1}} \int_0^1 cos(2 \pi - 2 \pi y) cos(2 \pi y) dy \\ & = & \frac{cos(2 \pi t) cos (2 \pi x)}{(-2)^{k-1}} \int_0^1 cos^2(2 \pi y) dy \\ & = & \frac{cos(2 \pi t) cos (2 \pi x)}{(-2)^{k-1}} \cdot \frac{1}{2} \\ & = & -\frac{cos(2 \pi t) cos (2 \pi x)}{(-2)^{k}} \end{array}

Let's not forget the 1 we had saved, so:

 p_{k+1}(x,t) = 1 - \frac{cos(2 \pi x) cos(2 \pi t)}{(-2)^{k}} \rightsquigarrow 1 - \frac{cos(2 \pi x) cos(2 \pi y)}{(-2)^{k}} = p_{k+1}(x,y)

as we wanted to show.

So finally notice that, of course, if we take the limit as  t approaches infinity, the patch evolution tendency is to become 1, the uniform distribution:

 \lim_{t \rightarrow \infty} p_t(x,y) = 1 = u(x,y)

From here on out, I want to set up the operative framework of patchixes, in analogy with discrete matrices.  I want to show that in general, patchix products are non-commutative.  This is easily done by counterexample:

We want to show that  p(x,y) \star q(x,y) \neq q(x,y) \star p(x,y) . So suppose the patchixes  p(x,y) = x and  q(x,y) = y . Then

 p(x,y) \star q(x,y) = \int_0^1 p(1-y,t) \cdot q(x,y) dy = \int_0^1 (1-y) y dy = \int_0^1 y - y^2 dy = \frac{1}{6}

and

 q(x,y) \star p(x,y) = \int_0^1 q(1-y,t) \cdot p(x,y) dy = \int_0^1 (t \cdot x) dy = t \cdot x \rightsquigarrow x \cdot y

are clearly not-equal.  It would be great to say that, because patchixes are non-commutative, patches are too, but we don't know that patches as a whole subset of patchixes commute, so let's disprove it.  Now suppose the patches  p(x,y) = x + \frac{1}{2} and  q(x,y) = 1 + xy - \frac{y}{2} .  Then

 \begin{array}{ccc} p(x,y) \star q(x,y) & = & \int_0^1 p(1-y,t) \cdot q(x,y) dy \\ & = & \int_0^1 \left( \frac{3}{2} - y \right) \cdot \left( 1 + xy - \frac{y}{2} \right) dy \\ & = & \frac{5x}{12} + \frac{19}{24} \end{array}

where

 \begin{array}{ccc} q(x,y) \star p(x,y) & = & \int_0^1 q(1-y,t) \cdot p(x,y) dy \\ & = & \int_0^1 q(1-y,t) \cdot p(x) dy \\ & = & p(x) \int_0^1 q(1-y,t) dy \\ & = & p(x) \cdot u(t) = p(x) \\ & = & x + \frac{1}{2} \end{array}

By refraining from calculating this last bit explicitly, we have (serendipitously) proved that any patch by a patch that is solely a function of  x returns the last patch, a result which reminds us of the analogous distribution by patch result I have shown in my previous post (a distribution on [0,1] times a patch that is solely a function of  x returns the patch, that viewed from the point of view of functions is a distribution on [0,1]).  A quick note: the integral  \int_0^1 q(1-y,t) dy is the unit distribution because  \int_0^1 q(x,y) dx = u(y) and  x \rightsquigarrow (1-y) and  dx \rightsquigarrow -dy .

The end result of these observations is that patches are also, in general, non-commutative.

Next, I want to show that patchixes in general are associative.  This is a bit tricky because of the "after integral" transformations we have to do, but it is doable if we keep careful track of our accounting.  We want to show that  [p(x,y) \star q(x,y)] \star r(x,y) = p(x,y) \star [q(x,y) \star r(x,y)] .  Let's begin with the left hand side.

 \begin{array}{ccc} [p(x,y) \star q(x,y)] \star r(x,y) & \rightsquigarrow & [p(x,w) \star q(x,w)] \star r(x,y) \\ & = & \left( \int_0^1 p(1-w, y) \cdot q(x, w) dw \right) \star r(x, y) \\ & = & \int_0^1 \left( \int_0^1 p(1-w, t) \cdot q(1-y, w) dw \right) \cdot r(x, y) dy \\ & = & \int_0^1 \int_0^1 p(1-w, t) \cdot q(1-y, w) \cdot r(x, y) dw dy \\ & = & s(x,t) \rightsquigarrow s(x,y) \end{array}

Now the right hand side

 \begin{array}{ccc} p(x,y) \star [q(x,y) \star r(x,y)] & \rightsquigarrow & p(x,w) \star \left( \int_0^1 q(1-y, w) \cdot r(x,y) dy \right) \\ & = & \int_0^1 p(1-w, t) \cdot \left( \int_0^1 q(1-y, w) \cdot r(x,y) dy \right ) dw \\ & = & \int_0^1 \int_0^1 p(1-w,t) \cdot q(1-y, w) \cdot r(x,y) dy dw \\ & = & s(x,t) \rightsquigarrow s(x,y) \end{array}

The two sides are equal when we can apply the Fubini theorem to exchange the order of integration.

Of course, patches, being a subset of patchixes, inherit associativity.

Defining a patchix left and right identity is extremely difficult, in the sense that, if we take a hint from discrete matrices, we'd be looking at a very special function on the  xy plane, so that  i(1-y,y) = i(x,1-x) = 1 and  0 everywhere else.  Because there is no "pretty" way to define this as a function of  x and  y both, showing that when we multiply a patchix by this function on either the right or the left requires elaborate explication. Unless we take it as axiomatic high ground, postulating the existence of an identity function  i(x,y) so that  i(x,y) \star p(x,y) = p(x,y) = p(x,y) \star i(x,y) to make the framework work, there is no easy way out.  Let's give it a shot then.

Left identity:

 i(x,y) \star p(x,y) = \int_0^1 i(1-y,t) \cdot p(x,y) dy

Now  i(1-y,t) = 1 only for values where  t = y , as we've defined it, otherwise the integral is zero and there is nothing to solve.  So then we've got

 \int_0^1 i(1-t,t) \cdot p(x,t) dy = \int_0^1 (1) \cdot p(x,t) dy = p(x,t) \rightsquigarrow p(x,y)

which is essentially the argument I make for the zero patch power in my informal paper on continuous Markov transition matrices or patches (however, there's a problem with this definition on patches, more of this below).  There's the question of why we didn't force the change of  dy \rightsquigarrow dt , and this is because the only way to obtain a function of both  x and  t is to force the patchix to the  x t plane and let the integral be taken in the  x y plane.  If this argument is unsatisfactory, consider this one:  at  t = 0 = y the patchix takes the values  p(x, 0) which is a function of  x alone.  Thus,

 \int_0^1 i(1,0) \cdot p(x,0) dy = p(x,0) \int_0^1 (1) dy = p(x,0)

if we do this for all  t \in [0,1] , we are certainly left with  p(x,t) .  We may raise the objection that, if we create a mental picture of the situation, at  t = 0 ,  i takes a value of 1 only at  y = 0 , so that, on the  x y plane, all values of  p(x, y) are zeroed except those at  y = 0 .  Thinking about it this way creates the difficulty of the integral with respect to  y : it evaluates to zero (there is no "area" in the  x y plane anymore, only a filament or fiber at  y=0 ), and we would be left with the zero patchix.  There is no way to resolve this except two ways: to send the patchix  p(x,y) to  p(x,t) before we take the integral in the  x,y plane, and then toss the integral out the window (or take it on the uniform distribution), or, to think of the filament  p(x,0) = p_0(x) as  p_0(x) \times [0,1] = p_0(x,y) and then integrate in the  x y plane to obtain  p_0(x) \rightsquigarrow p(x,0) and do this for all  t to get  p(x,t) .  Hence yes, the difficulty of defining the identity function on "surface" matrices (because it is not smooth like they are and because it is defined piece-wise).

Right identity:

 p(x,y) \star i(x,y) = \int_0^1 p(1-y,t) \cdot i(x,y) dy

Here we remind ourselves that  i(x,1-x) = 1 and zero otherwise, so that we can make the substitution

 \int_0^1 p(x,t) \cdot i(x,1-x) dy = \int_0^1 p(x,t) \cdot (1) dy = p(x,t) \rightsquigarrow p(x,y)

We of course have issues: it may seem redundant to send  x \rightsquigarrow 1-y \rightsquigarrow x , sending  x back to itself, but again this is the only way to remain consistent and get back the original function.  Again there's an issue of why we didn't send the integral  dy \rightsquigarrow -dx , but this has to remain in the  x y plane for the mechanics to work.  Other objections are likewise not easily resolved; but the argument would work out algebraically if we concede on a few things: otherwise we cannot but shrug at the fact that it is, indeed, a little bit of hocus pocus, and we return to our suggestion to postulate the identity function as an axiom. Perhaps maybe these issues can be resolved or elucidated a little later, I don't lose hope.

Defining inverse patchixes will also present a great difficulty, particularly because they have to produce the identity function when we "patchix multiply" two mutually inverse patchixes  together.  I was thinking that we could perhaps determine whether a particular patchix has one, by extending Sarrus's rule (for determinants) to be continuous, which would involve, I'm sure, multiple integrations.  This will be a topic of further investigation for me. The cool thing is, if we can elucidate how this "continuous version" of the determinant works, many different results from Linear Algebra could follow.  I am also trying to figure out how two inverse patchixes would look like, and if I can produce an example (at all), virtually from thin air.  If I can, then perhaps we're on our way to constructing patchix groups of all flavors.

Unfortunately, patches can't inherit the identity as we've defined it: the integral with respect to  x of  i(x,y) is zero for all  y .  Thus  i(x,y) is not a patch.

This problem makes us want to think of the uniform distribution  u(x,y) as another possible candidate for the identity for patchixes all, and it might just work if we agree that, when we don't have a function of  t or of  x after doing the setup-transformations for the integral, we send whatever function remains there before taking the integral.

Left identity:

 u(x,y) \star p(x,y) = \int_0^1 u(1-y,t) \cdot p(x,y) dy \rightsquigarrow \int_0^1 (1) \cdot p(x,t) dy = p(x,t) \rightsquigarrow p(x,y)

Right identity:

 p(x,y) \star u(x,y) = \int_0^1 p(1-y,t) \cdot u(x,y) dy \rightsquigarrow \int_0^1 p(x,t) \cdot (1) dy = p(x,t) \rightsquigarrow p(x,y)

This has several happy consequences: we avoid dealing with a piece-wise defined function  i(x,y) which is zero everywhere except on  y = 1-x , the uniform distribution is smooth, we can now more easily define inverses (by finding multiplicative inverse functions, more on this below), and, specifically regarding patches,  \int_0^1 u(x,y) dx = u(y) = 1 so the uniform distribution is indeed a patch.

In my mental picture, the "patchix product" of the uniform distribution with a patchix (and vice versa) doesn't "add up" (pun intended), but the algebraic trickery would seem to be the same even when using the alternative  i(x,y) .  So.  At this point I sort of have to convince myself into accepting this for now.

On Patchixes and Patches - or Pasqualian Matrixes - (RWLA,MCT,GT,AM Part II)

October 10th, 2010 3 comments

For the last few months I have been thinking about several very curious properties of patchixes and patches (mentioned here); in particular, having studied patch behavior in a "continuous Markov chain" context, and, at having been drinking a bowl of cereal and  observing the interesting movement of the remaining milk, it hit me: a patch could certainly describe milk movement at particular time steps.  It is my hope to try to elucidate this concept a little better here today.  In particular, I think I have discovered a new way to describe waves and oscillations, or rather, "cumulative movement where the amount of liquid is constant" in general, but, in my honest belief, I think this new way and the old way converge in limit (this based on my studies, here and here, or discrete Markov chains at the limit of tiny time steps, so that time is continuous), although it is a little bit unclear to me how at the moment.  It is my hope that this new way not only paves the way for a new and rich field of research, but I foresee it clarifying studies in, for example, turbulence, and, maybe one day, Navier-Stokes related concepts.  This last part may sound a little lofty and ambitious, but an approach in which, for example, vector fields of force or velocity need to be described for every particle and position of space, with overcomplicated second and third order partial derivatives, is in itself somewhat ambitious and lofty, and often prohibitive for finding exact solutions;  perhaps studying particle accumulations through a method of approximation, rather than individual particles, is the answer.

I want to attempt to describe the roadmap that led me to the concept of a patchix (pasqualian matrix) in the first place; it was in the context of discrete Markov chains.  Specifically, I thought that, as we study linear algebra, for a function or transformation  T(\textbf{v}) , with  \textbf{v} is an n-vector with  n entries (finite), we have  T can be described succinctly by an  n \times n matrix.  Such a matrix then, converts  \textbf{v} into another n-dimensional vector, say  \textbf{w} .  This field is very well studied of course: in particular, invertible transformations are very useful, and many matrixes can be used to describe symmetries, so that they underlie Group Theory:

 \textbf{v} \underrightarrow{T} \textbf{w}

Another useful transformation concept resides in  l_2 , the space of sequences whose lengths squared (dot product with itself) converge, that was used, for example by Heisenberg, in quantum mechanics, as I understand it.  For example, the sequence  x_1 + x_2 + \ldots can be transported to another  y_1 + y_2 + \ldots via  T , as by  T(x_1 + x_2 + \ldots) = y_1 + y_2 + \ldots .  Key here then was the fact that  x_1^2 + x_2^2 + \ldots converged, so that  \sqrt{x_1^2 + x_2^2 + \ldots} , the norm, is defined.  Also the dot product  x_1 y_1 + x_2 y_2 + \ldots converges (why?).  Importantly, however, this information points in the direction that a transformation matrix could be created for  T to facilitate computation, with an infinite number of entries, so that indeed a sequence is taken into another in this space in a manner that is easy and convenient.  I think this concept was used by Kolmogorov in extending Markov matrices as well, but I freely admit I am not very versed in mathematical history.  Help in this regard is muchly appreciated.

In function space such as  C^{\infty}[0,1] , the inner product of, say, f(x) with g(x) is also defined, as  \langle f(x), g(x) \rangle = \int_0^{1} f(x) \cdot g(x) dx , point-wise continuous multiplications of the functions summed absolutely convergently (which results from the integral).  Then the norm of  f(x) is  \sqrt{\langle f(x), f(x) \rangle} = \sqrt{\int_0^{1} f(x)^2 dx} .  The problem is of course no convenient "continuous matrix" that results in the transform  T(f(x)) = g(x) , although transforms of a kind can be achieved through a discrete matrix, if its coefficients represent, say, the coefficients of a (finite) polynomial.  Thus, we can transform polynomials into other polynomials, but this is limiting in scope in many ways.

The idea is that we transform a function to another by point-wise reassignment: continuously.  Thus the concept of a patchix (pasqualian matrix) emerges, we need only mimic the mechanical motions we go through when conveniently calculating any other matrix product.  Take a function  f(x) defined continuously on  [0,1] , send  x \rightsquigarrow 1-y so that  f(1-y) is now aligned with the y-axis. From the another viewpoint, consider  f(1-y) as  f(1-y,t) so that, at any value of  t , the cross-section looks like  f .  Define a patchix  p(x,y) on  [0,1] \times [0,1] .  Now "multiply" the function (actually a patchix itself from the different viewpoint) with the patchix as  \int_{0}^{1} f(1-y) \cdot p(x,y) dy = g(x) to obtain  g(x) .  The patchix has transformed  f(x) \rightsquigarrow g(x) as we wanted.  I think there are profound implications from this simple observation; one may now consider, for example, inverse patchixes (or how to get  g(x) back to  f(x) , identity patchixes, and along with these one must consider what it may mean, as crazy as it sounds, to solve an infinite (dense) system of equations; powers of patchixes and what they represent; eigenpatchixvalues and eigenfunctionvectors; group theoretical concepts such as symmetry groups the patchixes may give rise to, etc.

As much as that is extremely interesting to me, and I plan on continuing with my own investigations, my previous post and informal paper considered the implications of multiplying functions by functions, functions by patchixes, and patchixes by patchixes.  Actually I considered special kinds of patchixes  p(x,y) , those having the property that for any specific value  y_c \in [0,1] , then  \int_0^1 p(x,y_c) dx = 1 .  Such special patchixes I dubbed patches (pasqualian special matrixes), and I went on to attempt an extension of a Markov matrix and its concept into a Continuous Markov Patch, along with the logical extension of the Chapman-Kolmogorov equation by first defining patch (discrete) powers (this basically means "patchix multiplying" a patch with itself).  The post can be found here.

So today what I want to do is continue the characterization of patches that I started.  First of all, emulating some properties of the Markov treatment, I want to show how we can multiply a probability distribution (function) "vector" by a patch to obtain another probability distribution function vector. Now this probability distribution is special, in the sense that it doesn't live in all of  \mathbb{R} but in  [0,1] .  A beta distribution, such as  B(2,2) = 6(x)(1-x) , is the type that I'm specifically thinking about. So suppose we have a function  b(x) , which we must convert first to  b(1-y) in preparation to multiply by the patch.  Suppose then the patch is  p(x,y) with the property that, for any specific  y_c , then  \int_0^1 p(x,y_c) dx = 1 .  Now, the "patchix multiplication" is done by

 \int_0^1 b(1-y) \cdot p(x,y) dy

and is a function of  x .  We can show that this is indeed a probability distribution function vector by taking the integral for every infinitesimal change in  x , and see if it adds up to one, like this:

 \int_0^1 \int_0^1 b(1-y) \cdot p(x,y) dy dx

If there is no issue with absolute convergence of the integrals, there is no issue with the order of integration by the Fubini theorem, so we have:

 \int_0^1 \int_0^1 b(1-y) \cdot p(x,y) dx dy = \int_0^1 b(1-y) \int_0^1 p(x,y) dx dy

Now for the inner integral,  p(x,y) adds up to 1 for any choice of  y , so the whole artifact it is in effect a uniform distribution in  [0,1] with value 1 (i.e., for any choice of  y \in [0,1] , the value of the integral is 1).  Thus we have, in effect,

 \int_0^1 b(1-y) \int_0^1 p(x,y) dx dy = \int_0^1 b(1-y) \cdot u(y) dy = \int_0^1 b(1-y) (1) dy

for any choice of  y in  [0,1] , and that last part we know is 1 by hypothesis.

Here's a specific example:  Let's declare  b(x) = 6(x)(1-x) and  p(x,y) = x + \frac{1}{2} .  Of course, as required,  \int_0^1 p(x,y) dx = \int_0^1 x + \frac{1}{2} dx = (\frac{x^2}{2} + \frac{x}{2}) \vert^1_0 = 1 .  So then  b(1-y) = 6(1-y)(y) , and by "patchix multiplication"

 \int_0^1 b(1-y) \cdot p(x,y) dy = \int_0^1 6(1-y)(y) \cdot \left(x + \frac{1}{2} \right) dy = x + \frac{1}{2}

Thus, via this particular patch, the function of  b(x) = 6(x)(1-x) \rightsquigarrow c(x) = x + \frac{1}{2} , point by point.  Which brings me to my next point.

If  p(x,y) is really solely a function of  x , then it follows that  b(x) \rightsquigarrow p(x) any initial probability distribution becomes the patch function distribution (from the viewpoint of a single dimension, than two).  Here's why:

 \int_0^1 b(1-y) \cdot p(x,y) dy = \int_0^1 b(1-y) \cdot p(x) dy = p(x) \int_0^1 b(1-y) dy = p(x)

I think, of course, a lot more interesting are patches that are in fact functions of both  x and of  y .  There arises a problem in constructing them.  For example, let's assume that we can split  p(x,y) = f(x) + g(y) .  Forcing our requirement that  \int_0^1 p(x,y) dx = 1 for any  y \in [0,1] means:

 \int_0^1 p(x,y) dx = \int_0^1 f(x) dx + g(y) \int_0^1 dx = \int_0^1 f(x) dx + g(y) = 1

which implies certainly that   g(y) = 1 - \int_0^1 f(x) dx is a constant since the integral is a constant.  Thus it follows that  p(x,y) = p(x) is a function of  x alone.  Then we may try  p(x,y) = f(x) \cdot g(y) .  Forcing our requirement again,

 \int_0^1 p(x,y) dx = \int_0^1 f(x) \cdot g(y) dx = g(y) \int_0^1 f(x) dx = 1

means that  g(y) = \frac{1}{\int_0^1 f(x) dx} , again, a constant, and  p(x,y) = p(x) once more.  Clearly the function interactions should be more complex, let's say something like:  p(x,y) = f_1(x) \cdot g_1(y) + f_2(x) \cdot g_2(y) .

 \int_0^1 p(x,y) dx = g_1(y) \int_0^1 f_1(x) dx + g_2(y) \int_0^1 f_2(x) dx = 1

so that, determining three of the functions determines the last one, say

 g_2(y) = \frac{1-g_1(y) \int_0^1 f_1(x) dx}{\int_0^1 f_2(x) dx} is in fact, a function of  y .

Let's construct a patch in this manner and see its effect on a  B(2,2) .  Let  f_1(x) = x^2 , and  g_1(y) = y^3 , and  f_2(x) = x , so that

 g_2(y) = \frac{1 - g_1(y) \int_0^1 f_1(x) dx}{\int_0^1 f_2(x) dx} = \frac{1 - y^3 \int_0^1 x^2 dx}{\int_0^1 x dx} = \frac{1 - \frac{y^3}{3}}{\frac{1}{2}} = 2 - \frac{2y^3}{3}

and  p(x,y) = x^2 y^3 + x \left(2 - \frac{2y^3}{3} \right) .

So now the "patchix product" is

 \int_0^1 6(1-y)(y) \cdot \left(x^2 y^3 + x \left(2 - \frac{2y^3}{3} \right) \right) dy = \frac{x^2}{5} + \frac{28x}{15} which is a probability distribution on the interval  [0,1] and, as a matter of check, we can integrate with respect to  x to obtain 1.  Thus the probability distribution function  6(x)(1-x) is carried, point by point, as  6(x)(1-x) \rightsquigarrow \frac{x^2}{5} + \frac{28x}{15} which, quite frankly, is very amusing to me!

From an analytical point of view, it may be interesting or useful to see what happens to the uniform distribution on  [0,1] when it's "patchix multiplied" by the patch above.  We would have:

 \int_0^1 u(y) \cdot \left(x^2 y^3 + x \left(2 - \frac{2y^3}{3} \right) \right) dy = \int_0^1 (1) \cdot \left(x^2 y^3 + x \left(2 - \frac{2y^3}{3} \right) \right) dy = \frac{x^2}{4} + \frac{11x}{12}

so that  u(x) \rightsquigarrow \frac{x^2}{4} + \frac{11x}{12} .

In my next post, I want to talk about more in detail about "patchix multiplication" of, not a probability distribution on [0,1] vectors by a patch, but of a patch by a patch, which is the basis of (self) patch powers: with this I want to begin a discussion on how we can map oscillations and movement in a different way, so that perhaps we can trace my cereal milk movement in time.