User generator - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:09:31Z http://mathoverflow.net/feeds/user/3647 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47751/what-is-known-about-fractions-in-difference-equations What is known about fractions in difference equations? Generator 2010-11-30T03:49:14Z 2010-11-30T03:49:14Z <p>I'm using the following book as reference:</p> <p>** Walter G. Kelly and Allan C. Peterson <em>Difference Equations</em> Second Edition, Harcourt Academic Press, 2001. **</p> <p>I gather from the book and from research that it's generally difficult to find sums involving fractions with complicated denominators.</p> <p>I'd like to know what is known about these sums. One thing that particularly springs up is what techniques generally work well for finding these sums. The book lists some general properties of indefinite sums, and some properties seem to go well with products, but don't seem to work well with fractions. I wonder if there is some general way to organize complicated fractions so that they are easy to sum.</p> http://mathoverflow.net/questions/42778/piece-of-a-sequence Piece of a sequence Generator 2010-10-19T12:42:29Z 2010-10-20T13:18:14Z <p>Suppose we are given a representation of a finite series of natural numbers:</p> <p>$\sum_{i=0}^N{c_i x^i}$ </p> <p>The representation is essentially an expression that is a rational function of two polynomials.</p> <p>Is it possible to add/subtract this series repeatedly to get a result that contains only part of the series?</p> <blockquote> <p>A simple formulation of the problem</p> </blockquote> <p>We can forget that we have a series, for a moment, and consider it more like a tile. It contains a set of numbers on it that are in a particular order that cannot be changed. It's basically striaght; it just contains this set of numbers on a line.</p> <p>Now we actually have many copies of this same tile, and some are oriented differently. The different orientation comes from the fact that the tile could be on a line into another dimension. For example, one tile may start at coordinates (0,0) and end at coordinates $(0,N)$. Another may start at (0,0) and end at $(N,0)$. All tiles are essentially the same ordered set of numbers that start at one point in $m$ dimensions and end at another point.</p> <p>There is essentially one operation that we can perform. We can take the difference of numbers at a given point. For example, if the sequence is (1,2,3), we can take one tile that starts at (0,0) and proceeds to (0,2). Subtract another tile that starts at (0,0) and proceeds to (2,0). This would cancel out the ones in both tiles, since we subtract the one at (0,0) in the second tile from the one at (0,0) in the first tile.</p> <p>I'm wondering if we can somehow add and subtract these tiles so that only a single number remains. It may be that more than one location contains this same number.</p> <p>There are some rules, so an example and explanations will probably help.</p> <blockquote> <p>Could you give an example?</p> </blockquote> <p>Here's an example. Consider the finite series $10 + 200x$.</p> <p>We can repeatedly add and subtract (scalar multiples of) this series (and a similar series) in the $x,y$ "plane" to end up with $200x$, only part of the series.</p> <p>Here's how:</p> <p>(0) Start with $10+200x$. </p> <p>(1) Subtract $10 + 200y$. This eliminates 10 and we're left with $200x-200y$.</p> <p>(2) Add $(10+200x)\cdot y / x$. This eliminates the $-200y$ in the previous result and adds $10\cdot y / x$, resulting in $200x+10\cdot y / x$.</p> <p>(3) Subtract$(10+200y)\cdot y / x$. This eliminates $10\cdot y / x$, subtracts $200\cdot y^2 / x$, and we're left with $200x - 200\cdot y^2 / x$</p> <p>(4) Add $((10+200x)\cdot y / x)\cdot y / x$. This eliminates the $-200\cdot y^2 / x$ in the previous result and adds $(10\cdot y / x)\cdot y / x$, resulting in $((200x+10)\cdot y / x)\cdot y / x$.</p> <p>...</p> <p>We repeatedly add and subtract pieces of the series this way. It can be shown that the result of this infinite series is $200x$, which is only part of the original series.</p> <p>I'd like to know if we can add and subtract larger series similarly (but possibly in more dimensions) to end up with only a single coefficient or piece of the series.</p> <p>For example, If we consider the series $1 + 2x + 3x^2$, we may be able to eliminate $1$ and $3x^2$ from this series by a set of careful additions and subtractions.</p> <p>I don't know which branches of mathematics this deals with, but I'm hoping an expert can provide me with some direction.</p> <blockquote> <p>Why are you doing this/ What's your motivation?</p> </blockquote> <p>Results from this "puzzle" would help speed up an algorithm significantly, so I'm interested in publishing this result with whoever helps me. I know MO's position on algorithms, but I believe that this is more of a "tiling" or mathematics question than a question on an algorithm.</p> <p>I'd like to know all branches of mathematics that deal with this question, and where I can go, or who I can go to, that will help me solve the general problem.</p> <p>Some additional notes: The value of $x$ is not allowed to be modified. We can rewrite $x$ as another variable (I rewrote $x$ as $y$ above, for example). We are also allowed to "shift" the series (for example, multiply by $y / x$ as done above). Cancellation is, of course, allowed - and it may be done infinitely many times. These are the only allowable operations.</p> <p>Also, I am particularly interested in the coefficient in the middle of the series.</p> <p>One additional consideration: methods that use fewer variables are preferred.</p> http://mathoverflow.net/questions/20203/where-do-i-turn-for-help-with-generating-functions Where do I turn for help with generating functions? Generator 2010-04-03T02:18:16Z 2010-04-06T00:15:07Z <p>Please forgive me if this is inappropriate for MathOverflow. I've been working/playing with generating functions for a little while and may have stumbled upon a new technique or methodology.</p> <p>The problem is that it's incomplete, and I don't have a lot to show to someone to prove its effectiveness. I believe I need some help fine tuning this method to get it to work. I'm unsure if I can solve it, at least easily, on my own. I've tried contacting a few big name professors in the field, but most seemed to busy. I was wondering if there's someone who could spare some time to look into what I'm working on.</p> <p>Are there any suggestions of people that I could try contacting? If it's not advisable to go forward with contacting people at this time, what else should/could I try to help get answers? I realize I haven't described my method here; it's a pretty complicated one, but one that can probably be explained in a few pages.</p> http://mathoverflow.net/questions/15671/which-hadamard-products-of-generating-functions-are-known Which Hadamard Products of Generating Functions Are Known? Generator 2010-02-18T05:44:39Z 2010-02-18T06:59:28Z <p>The <em>Hadamard product</em>, <em>Schur product</em>, or entrywise product of two generating functions is computed as follows:</p> <p>The Hadamard Product, H(x), given two generating functions f(x) and g(x) where</p> <p>$$f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots + c_nx^n + \dots$$ $$g(x) = d_0 + d_1x + d_2x^2 + d_3x^3 + \dots + d_nx^n + \dots$$</p> <p>is defined as</p> <p>$$H(x) = c_0d_0 + c_1d_1x + c_2d_2x^2 + c_3d_3x^3 + \dots + c_nd_nx^n + \dots$$</p> <p>Simply put, it is the result of multiplying individual coefficients of two generating functions.</p> <p>Results detailing certain types of functions are known. I would like to compile a list of results that are known.</p> <p>For example, if we have two generating functions of the form $(1-x)^a$ and $(1-x)^b$, we obtain: ${}_2F_1[-a,-b;1;x]$ where ${}_2F_1$ represents a hypergeometric function of Gauss, according to</p> <blockquote> <p>"Singularity Analysis, Hadamard Products, and Tree Recurrences", by Jim Fill, Philippe Flajolet, and Nevin Kapur In Journal of Computational and Applied Mathematics, volume 174 (February 2005), pages 271--313 (around page 289)</p> </blockquote> <p>Again, which Hadamard products involving generating functions are known or solved? Additionally, solutions to the equations would be greatly appreciated.</p> http://mathoverflow.net/questions/13467/please-help-z-transforms-with-possible-radius-of-convergence-0 Please help - z transforms; with possible radius of convergence 0 Generator 2010-01-30T09:27:23Z 2010-01-31T03:35:41Z <p>Is it always allowed to perform a z transform with radius of convergence 0? I'm looking for a way to use limits of a generating function or related (as the power approaches a certain value, using only the overall function description without relying on a function for individual coefficients) in recursive equations. One possibility seems to be to use a specific value as a dummy variable. However, I would prefer to use a z-transform.</p> <p>More background: I am attempting to extract the coefficient that is the middle in a series. Not much information is known about the series; I am working in the general case and its radius of convergence could be zero. My approach was to attempt to use an integral and differentiation similar to the z-transform. This should allow one to extract coefficients using differences of manipulated generating functions, one at-a-time. However, it seems that I can't use a contour integral if the r.o.c. is zero with garaunteed correctness.</p> <p>I guess I'm really asking if there is any method that would allow one to extract coefficients, like taking the limit as the function approaches a given coefficient. I'd like to be able to handle this procedure in a recursive fashion, and it seems that this is impossible.</p> <p>I don't want to waste anyone's time, but any help would be greatly appreciated. Email is welcomed at mgroff100@hotmail.com</p> http://mathoverflow.net/questions/47751/what-is-known-about-fractions-in-difference-equations Comment by Generator Generator 2010-11-30T06:11:33Z 2010-11-30T06:11:33Z ...We have the understanding that we'll later divide by this same product. We then only have to divide each term by its denominator. I think that this may have an extension to the general problem. http://mathoverflow.net/questions/47751/what-is-known-about-fractions-in-difference-equations Comment by Generator Generator 2010-11-30T06:10:07Z 2010-11-30T06:10:07Z @Gerry: Here's my example...take $\sum_{i=1}^n{1/i}$ = $\left(\sum_{i=1}^n{(\prod_{j=1}^n{j})/i}\right) / \sum_{j=1}^n{j}$. I take the product of all denominators and multiply each term by this product. So I essentially establish a common divisor, which is the product of all denominators. Thusly, I divide each term by its divisor, after having multiplied by the common divisor. When I'm done, I divide by this common divisor, thus finding the correct sum. I believe that this can be generalized. We multiply each term by the product of all denominators. http://mathoverflow.net/questions/47751/what-is-known-about-fractions-in-difference-equations Comment by Generator Generator 2010-11-30T03:59:21Z 2010-11-30T03:59:21Z I'm wondering about this now because I'm examining the possibility of combining discrete multiplicative integrals with these fractional sums. It seems that using the discrete multiplicative techniques may allow one to calculate complicated fractions with ease, by finding a common denominator. http://mathoverflow.net/questions/42778/piece-of-a-sequence/42862#42862 Comment by Generator Generator 2010-10-20T05:09:34Z 2010-10-20T05:09:34Z Please forgive me. I neglected a major point: We don't know the coefficients. We have an expression for a rational generating function instead. http://mathoverflow.net/questions/42778/piece-of-a-sequence Comment by Generator Generator 2010-10-19T18:54:55Z 2010-10-19T18:54:55Z @Scott:Yes, scalar multiples are allowed. It may be problematic, however, if the definition is too complex. http://mathoverflow.net/questions/42778/piece-of-a-sequence Comment by Generator Generator 2010-10-19T13:46:48Z 2010-10-19T13:46:48Z @JBL: Thanks for the LaTeX help. There are really only two operations allowed, and I've edited the question to reflect this. http://mathoverflow.net/questions/42778/piece-of-a-sequence Comment by Generator Generator 2010-10-19T13:14:48Z 2010-10-19T13:14:48Z Sorry, but you can't change the value of x. You can only rewrite the variable x as another variable (y, for instance). http://mathoverflow.net/questions/40665/is-there-an-alternative-to-the-limit-operation-with-generating-functions Comment by Generator Generator 2010-09-30T20:43:32Z 2010-09-30T20:43:32Z Oops - that should be $c_i x^i$ becomes $c_i d^i x^i$. $I$ and $H$ are two intermediate &quot;generating functionoids&quot;, for lack of a better term, that I use to try to achieve the limit algebraicly, which I guess is impossible. I really want to know if there is some workaround, but I lack a good background. http://mathoverflow.net/questions/40665/is-there-an-alternative-to-the-limit-operation-with-generating-functions Comment by Generator Generator 2010-09-30T20:32:21Z 2010-09-30T20:32:21Z I'm very sorry (you shouldn't be); I'm an armchair enthusiast looking for a math expert - and I didn't know how to tag for generating functions (maybe combinatorics). I use $a$ as a second &quot;index&quot; in a &quot;generating function&quot;. $d$ is what I want to multiply by, and generally it will be a root of unity. By $x \to xd$ I meant that $c_i x^i$ becomes $c_i d_i x^i$. In general, I'm attempting to go through a list of numbers and multiply each number in the list by successive powers of d. So for $1, 2x, 3x^2$ I'd like $1, 2x*d, 3x^2*d^2$. http://mathoverflow.net/questions/40665/is-there-an-alternative-to-the-limit-operation-with-generating-functions Comment by Generator Generator 2010-09-30T20:15:04Z 2010-09-30T20:15:04Z Sorry for the confustion. The first f is correct, but I thought that I had it in the second form - and I now guess I should recheck my math. I intended to show that I multiply the ith coefficient by d^i. I am wondering if alternatives exist so that I am able to use recursion on them. I am having trouble using recursion with limits in this fashion. http://mathoverflow.net/questions/30994/extracting-information-from-a-series Comment by Generator Generator 2010-07-08T13:09:37Z 2010-07-08T13:09:37Z @Willie: We're given a set of functions, and one general form is shown in the additional iterations part. Occaisonally we subtract the result from a function of ones. We proceed to multiply each function as described above, in sequence. We stop when the set is exhausted. I hope this helps to clarify. http://mathoverflow.net/questions/30994/extracting-information-from-a-series Comment by Generator Generator 2010-07-08T11:51:15Z 2010-07-08T11:51:15Z To create one function: Take an ordinary power series with binary coefficients and finite length in, say, x. Multiply by another similar series in y. Do $x \mapsto w/v$ and $y \mapsto v$. repeating multiplication and then mapping with another series. Continue with $N$ series. Now it is ready to extract information. I've experimented with the Fourier coefficients, but it seems to be too difficult (inefficient) a calculation in general. http://mathoverflow.net/questions/20203/where-do-i-turn-for-help-with-generating-functions Comment by Generator Generator 2010-04-03T03:12:22Z 2010-04-03T03:12:22Z I actually asked a question regarding something similar, but it was very difficult for me to describe. I'm going to consult with the tips and tricks for authoring here on MO. http://mathoverflow.net/questions/20203/where-do-i-turn-for-help-with-generating-functions/20204#20204 Comment by Generator Generator 2010-04-03T02:50:07Z 2010-04-03T02:50:07Z Thanks for your volunteering and answers, Will. I'm embarassed, I tried to contact Wilf regarding something very simple. I don't want to waste anyone's time. However, maybe someone with more expertise than me, a novice, will be able to point me in the right direction. I'll try to send you some TeX/pdf detailing my results. http://mathoverflow.net/questions/19793/nonstandard-integration-of-generating-functions Comment by Generator Generator 2010-03-30T19:24:27Z 2010-03-30T19:24:27Z Thanks for your response, Jacques. I guess I tagged it wrong, besides having trouble explaining. I think that if we could find a way to add integration of this fashion into the tricks of generating functions, we could answer many more questions of mathematics with ease. I believe a generating function for the Collatz conjecture could be constructed, for example.