Reetesh Mukul’s Blog

Posted by: Reetesh Mukul | June 25, 2008

Time and Motion

Our senses prompt us about the notion of motion. Movement of stars, automobiles, elements in an animation, wind, water, etc. build up the idea of motion. So central is the observation of motion, that our study of any physical phenomenon necessarily begins with it. In common dialect, we usually talk about time taken by certain thing to move from one place to another one. Quite spontaneously, we assert the thought of “time taken” and “change in position” whenever we describe motion. A mathematically inclined person quickly builds up a mental picture of “Euclidean Geometry” and “Time” when it comes to the description of motion. The visuals, the imaginations about motion are so intermixed with our senses that whole Classical Mechanics appears as an application of Differential Geometry. But doing so, we ignore many questions, answering them is not going to be easier, modeling them may lead to mutual inconsistencies; in fact different entities can lead to paradoxes!

For example, if we are going to describe motion in terms of time, then a question arises — what time is ? And if we are going to describe time in terms of ticks, be it of our local quartz clock or on the basis of vibrations in Caesium atom or in terms of length of shadow of a tower, we are effectively defining ticks in terms of some motion. So, Motion requires Time to describe it, but Time in itself requires some Motion to describe it! Circularity of definitions, descriptions. Probably at this juncture, someone can assert that Time is an absolute quantity, and every change in nature has a time parameter $t$ , associated with it. It will be job of human beings to ascertain this time as precisely as he can. But how ? If he tries to measure time in terms of some motion then there is the problem of circularity and otherwise if he uses some other basis for measurement then the fundamental idea about time will get changed. The later in turn can possibly alter our notion of motion. In Physics significant changes in equations may not happen, approximations can still work, … but the vision, the idea may get sea change.

There are fundamental assumptions right from the beginning of description of motion. When we study geometry of a nearly fixed shape ( like pyramid, automobile’s wheel, etc. ) we begin confidently with a well thought structure. Of course to gain more precision we modify our structures, nevertheless we always have something concrete to begin with or at least we think concretely. But the description of motion begins with experiments. We loosely begin with a mathematical structure/framework, try to find laws using this framework, and we find that — some times new laws are inconsistent with older one, often the framework implicitly takes assumption about behavior of components of the system, some times new laws do not favor the underlying framework; and then we redevelop our framework. Reference frames are the elementary term in the process of identification of Space and Time; however reference frame in themselves presume idea of motion. For example, in Classical Mechanics the basic assumption was that changes are propagated instantly; i.e., if a charge appears at $(x,y,z,t)$ , then it affects the Electric field everywhere at time $t$ itself. Changes happen instantly. That changes occurs instantly and same changes in two different reference frames occur at equivalent time intervals is central idea practiced in Classical Newtonian Physics. But things went wrong ! Things went wrong because there were many presumptions, the idea of “Time” being a primary one.

Does Motion Exist ? Is space quantized or time is quantized ? How we can assert coexistence of two instants in Nature ? What instants are ? Since motions are often defined in terms of ratios of infinitesimal quantities ( $\vec {v} = \frac {d\vec r}{dt}$ ), is there any physical existence attributed to these infinitesimal quantities ? What is the core principle behind motion of a particle when it follows a given path ? Does particle exist ? Does paths exist ? And most importantly, why should Nature follow a Mathematical Law ? How Mathematics is linked to Nature ? Why Nature likes Mathematics ? Many questions ! Answer for each of them is linked to answers of the rest. Many answers are yet to be found. Ideas change in one go. Physics is meant for brave-hearts ( and restless minds ).

Heraclitus ( 540 BC ) said — “Everything flows”. Zeno of Elea ( 490 BC ) said — “all motion is illusion”. In the Greek Era, this debate never ended … for almost 250 years. And the debate is still on. It is the description of “Time and Motion”, that is what constitutes Man’s ultimate mental voyage.

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Posted in Motion, Philosophy, Physics | Tags: Motion, Philosophy, Physics

Posted by: Reetesh Mukul | June 14, 2008

Stirling Numbers of First Kind

Vikalpa, is a Sanskrit/Hindi word, often used in context of choice making. The ancient Indian Scripture, Sthananga Sutra, uses this word as a topic name for a text on Permutation and Combination. For the study of subject matter such as, “Vikalpa“, I will like to use the term “Combinatorics“, mainly because the terms, Permutation, Combination are Mathematical quantities, and the word Combinatorics captures the wider essence of Mathematics of “choice making”. So whenever we deal with problems which involve enumerating possible choices, often conforming certain patterns, we will be doing Combinatorics. Combinatorial problems are the theme of many of the every day calculations, for example, the possible decisions to traverse from one place to other place. And unless some patterns are not discovered, calculations can become pretty tedious. But when patterns are found, and effectively used, things can become fun. A very fine approach for Combinatorial proofs are given in Proofs that really count: The art of Combinatorial proof, by Arthur T. Benjamin and Jennifer J. Quinn.

Consider this question — “How many ways $n$ persons can be seated across a round table having $n$ chairs around it, where only relative order of persons matter?”. This question is a part of the problem statement of the plan for seating people across a round table at dinner, where situation like who should/shouldn’t sit adjacent to a person are often dealt. If we consider a person as an element of set,

$\{P_0,P_1,...,P_{n-1}\}$ , we need all possible arrangements

$(P_{p(0)}, P_{p(1)}, ..., P_{p(n-1)})$ ,

where $p$ is a bijective mapping on and into the set $\{0,1,...,n-1\}$ and arrangements are considered equivalent when,

$p_{1}(i) \equiv p_{2}(i)+c\hspace{9 pt} (modulo\hspace{4pt}n)$ , $c\hspace{4 pt}$ being an integer constant.

This is like finding all arrangements of persons, where a given arrangement should not be circular shift of another one. Since for a given arrangement there are $n$ number of circular shifts, so total number of ways in which $n$ people can be seated across a round table, is $n!/n$ , i.e., $(n-1)!$ . So when on a clear sky night, if we want to sketch straight lines between a given $n$ number of stars in such a way that all stars are on the final sketch and they are crossed only once, then we have $(n-1)!$ number of possible sketches. What if we want to make $k$ , such sketches( each sketch should have atleast one star on it ), which do not have any star in common, when we have $n$ stars? Or, suppose if we have $k$ number of identical round tables, how many ways are there to seat $n$ people across them, in such a way that at least one person should seat across a round table ?

If there are $m_i, 1\leq m_i\leq n-k+1$ , number of stars in $i$ th cycle, total number of solutions is given by,

$\sum \frac {{\binom{n}{m_1}(m_1-1)!}{\binom{n-m_1}{m_2}(m_2-1)!}...{\binom{m_k}{m_k}(m_k-1)!}}{Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\}}$ ,

where the summation ranges on all solutions $(m_1,m_2,...,m_k)$ of the equation $m_1 + m_2 +...+ m_k = n$ . The factor ${\binom {n}{m_1}}{\binom{n-m_1}{m_2}}...{\binom{m_k}{m_k}}$ , is number of ways of choosing groups of $m_1, m_2, ..., m_k$ stars out of $n$ stars. The factor $(m_i-1)!$ , is number of sketches that can be made on $m_i$ number of stars ( the same total number of seating arrangement solution ). Also, since solutions of $m_1 + m_2 +...+m_k = n$ can be permutation of each other, so I provided an expression in denominator. A simple reduction can be acheived by expanding the combinations–

$\sum \frac{n!}{(m_1m_2...m_k)(Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\})}$ .

Like the way there is no explicit formula for factorial, permutation and combination; so is the situation with the above solution, and in literature we use ${n\brack k}$ as their symbolic representation. This quantity, ${n\brack k}$ , is called unsigned Stirling number of first kind. We will now explore falling factorial power, $x^{\underline{n}}$ , to gain more insight into Stirling number of first kind:-

${x^{\underline{n}}} = x(x-1)...(x-n+1)$ .

Unsigned coefficient of $x^k$ in the above product equals:-

Unsigned Coefficient of ${x^{k-1}}$ , in $(x-1)(x-2)...(x-n+1)$

$=\sum (a_1a_2...a_{n-k})$ ,

where $a_1,a_2,...,a_{n-k}$ are $n-k$ distinct number chosen from the set $\{ 1,2,...,n-1 \}$ . The summation thus runs for ${\binom{n-1}{k-1}}$ cases which is equal to total number of solutions of $m_1 +m_2 +...+m_k = n$ , $1 \leq m_i \leq n-k+1$ . Literatures assert that, unsigned coefficient of ${x^k}$ , in $x(x-1)(x-2)...(x-k)$ is ${n\brack k}$ . Does this mean,

$\sum \frac{n!}{(m_1m_2...m_k)(Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\})}$ $= \sum \frac{(n-1)!}{q_1q_2...q_{k-1}}$ ,

where $q_1,q_2,...,q_{k-1}$ are $k-1$ distinct numbers from the set $\{ 1,2,...,n-1 \}$ while the summations on both sides are running for equal number of time ? Remember, $m_i$ ’s can have same values.