User debapriyay - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:38:47Z http://mathoverflow.net/feeds/user/9586 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself A mapping from a lattice to itself debapriyay 2010-09-29T08:39:32Z 2010-10-01T14:58:04Z <p>Consider $\mathbb{Z}^{n}$ for $n = 2^r$ where $r \geq 1$ . Look at the iterates of the following function $T$ from $\mathbb{Z}^n$ to itself. $T((a_1, a_2, \ldots, a_n)) = (|a_1 - a_n|, |a_2 - a_1|, |a_3 - a_2|, \ldots, |a_n - a_{n-1}|)$.</p> <p>It has been proved that when $n = 2^{r}$, then for every $(a_1, a_2, \ldots, a_n) \in \mathbb{Z}^n - \{0\}$, there exists some $i \geq 1$, such that $T^{i}((a_1, a_2, \ldots, a_n)) = 0.$ This does not hold for other values of $n$. Note that, if $T^{i}((a_1, a_2, \ldots, a_n)) = 0$, then $T^{j}((a_1, a_2, \ldots, a_n)) = 0$ for all $j > i.$</p> <p>Findings so far are the following.</p> <p>(i) $T(k(a_1, a_2, \ldots, a_n)) = k T((a_1, a_2, \ldots, a_n))$ for all $k \in \mathbb{Z}$.</p> <p>(ii) $T(k + (a_1, a_2, \ldots, a_n)) = T((a_1, a_2, \ldots, a_n))$ for all $k \in \mathbb{Z}$, where $k + (a_1, a_2, \ldots, a_n) = (k + a_1, k + a_2, \ldots, k + a_n).$</p> <p>(iii) Let, $S_{i} = \{(a_1, a_2, \ldots, a_n) \in Z^{n} : T^{i}((a_1, a_2, \ldots, a_n)) = 0 \text{ and } T^{i-1}((a_1, a_2, \ldots, a_n)) \neq 0\}$ for $i \geq 1$. Note that $S_i$ s are disjoint also their union is equal to $\mathbb{Z}^n$.</p> <p>The questions that I have are the following.</p> <p>(i) What's the maximum value of $i$ such that $S_{i}$ is not empty? Putting it in other words, what's the maximum number of times the function T needs to be applied to a vector so that it gets mapped to $0$ vector.</p> <p>(ii) Since the function $T$ is homogeneous, notions from projective space can be borrowed. How could projective geometry be applied here?</p> http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself/40730#40730 Answer by debapriyay for A mapping from a lattice to itself debapriyay 2010-10-01T09:48:42Z 2010-10-01T09:48:42Z <p>This is not an answer though. I am trying still to solve the problem. I would like to get comments on the approach that I am following. The approach in nutshell is described below. I am trying to characterize the sets $S_{i}$. Note that, any $a \in S_{1}$ can be derived from the single vector $(1, 1, \ldots, 1)$, by doing either $k + (1, 1, \ldots, 1)$ or $k (1, 1, \ldots, 1).$</p> <p>Similarly, any $a \in S_{2}$ can be derived from any of the following vectors by applying $k +$ and $k.$. The vectors are (i) $(0, 1, 0, 1, \ldots, 0, 1)$, (ii) $(1, 0, 1, 0, \ldots, 1, 0)$ and all the vectors that can be formed from the vector $(1, 0, 1, 2, 1, 0, 1, 2, \ldots, 1, 0, 1, 2)$ by shifting the elements of the vectors by one position in the right.</p> <p>This can be verified. Can we now iterate in some way this to come to a value of $i$ such that no such vectors can be formed and then we stop. $i-1$ then is the answer.</p> <p>Is it a viable approach? Or am I missing something?</p> http://mathoverflow.net/questions/40454/practical-applications-of-eigenvalues-and-eigenvectors/40460#40460 Answer by debapriyay for practical applications of eigenvalues and eigenvectors debapriyay 2010-09-29T12:18:03Z 2010-09-29T12:18:03Z <p>Principal Component Analysis is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. It is very difficult to visualize data in high dimensional space, but PCA can be used their to analyze data. From the data set covariance matrix is formed and then eigen values and eigen vectors of that covariance matrix are found. These eigne values and eigen vectors then can be compared to figure out the contribution of a particular feature in the data set. Thus PCA can be successfully applied to reduce dimension of the data.</p> http://mathoverflow.net/questions/40337/ingenuity-in-mathematics/40426#40426 Answer by debapriyay for Ingenuity in mathematics debapriyay 2010-09-29T05:45:02Z 2010-09-29T05:45:02Z <p>The RSA algorithm publicly described in 1978 by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT is a classic example of "Ingenuinity in Mathematics". Its the famous Public Key Cryptography scheme widely used everywhere and is solidly founded on Mathematics.</p> http://mathoverflow.net/questions/40298/how-to-study-the-behavior-of-a-particular-function-on-a-vector-space How to study the behavior of a particular function on a Vector Space. debapriyay 2010-09-28T11:38:19Z 2010-09-28T12:28:52Z <p>Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that $T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also $T(k + X) = T(X)$ for all $k \in K$ and for all $X \in V$.</p> <p>If $X = (x_1, x_2, \ldots, x_n)$, then $k + X = (k + x_1, k_ + x_2, \ldots, k + x_n)$.</p> <p>I would like to know how to study the behavior of $T$ and its effect on the vector space $V$ be studied. </p> http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself Comment by debapriyay debapriyay 2010-10-18T12:33:05Z 2010-10-18T12:33:05Z I haven't understood the 3rd line of your comment. Whats the definition of e and how &quot;93 is not a power of 2&quot; applies here to conclude that answer may be infinity. http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself/40760#40760 Comment by debapriyay debapriyay 2010-10-18T12:08:25Z 2010-10-18T12:08:25Z I don't think for any $a \in Z$, and for $a &gt; 1$, the relation $a^{3} = a^{2} + a + 1$ holds. Can you provide a real example in support of your claim. http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself Comment by debapriyay debapriyay 2010-09-30T05:31:34Z 2010-09-30T05:31:34Z Answer to the first question is not at all infinity. For example, take $n = 2$.$S_{1}$ is then the set of points with $a_1 = a_2$. The other points of $Z^{2} - {0}$ are in $S_{2}$. And $S_{i}$ for $i &gt;= 3$ is empty. So answer here is 2. Infact, for $n = 2$ this holds for $\mathbb Q$ and $\mathbb R$. http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself Comment by debapriyay debapriyay 2010-09-29T12:21:59Z 2010-09-29T12:21:59Z Whats the notion of homogenity in algebraic geometry then. Can you please elaborate a bit. http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself Comment by debapriyay debapriyay 2010-09-29T09:11:28Z 2010-09-29T09:11:28Z Yes $Z$ is the set of integers. Each $a_{j} \in Z$ for all $j = 1,2, \ldots, n.$ The earlier paper, I referred is the following. A number-theoretic game Prithvi Ramesh Published in Resonance, January 2003, P.84-88 http://mathoverflow.net/questions/40450/a-mapping-from-a-lattice-to-itself Comment by debapriyay debapriyay 2010-09-29T08:46:39Z 2010-09-29T08:46:39Z They mean absolute values. http://mathoverflow.net/questions/40298/how-to-study-the-behavior-of-a-particular-function-on-a-vector-space/40304#40304 Comment by debapriyay debapriyay 2010-09-28T12:56:34Z 2010-09-28T12:56:34Z $X = K'^{n}$, where $K'$ is a field and $K \subset K'$. This can also be the case. Now, you talked about an equivalence relation, one question is if $K'$ is a finite field, then can it be possible to enumerate all the equivalence classes, or the number of equivalence classes. http://mathoverflow.net/questions/40298/how-to-study-the-behavior-of-a-particular-function-on-a-vector-space Comment by debapriyay debapriyay 2010-09-28T12:09:35Z 2010-09-28T12:09:35Z If $X = (x_1, x_2, \ldots, x_n)$, then $k + X = (k + x_1, k_ + x_2, \ldots, k + x_n)$.