Closed Forms by Borwein and Crandall - Essay Example

Closed Forms by Borwein and Crandall 1. Introduction In this paper, a brief report on the article named “Closed forms: What they are and why we care” (Borwein and Crandall 2010, pp. 1-30) is presented. There are six major sections in this article. The first section explains various kinds of definitions of closed form. The second section provides us with a summary of reasons that why scientists and mathematicians are investigating closed forms. Third section deals with detailed examples on closed forms. Next, recent examples of advanced research on closed forms are discussed. Then there is the fifth section titled “profound curiosities” (Borwein and Crandall 2010, p. 24) followed by the concluding section of the article. In the concluding section, several open questions have been discussed. The first section of this paper is particularly important because it explains the very significance of this article. In this section, the authors attempt to furnish a definition of closed form. But in doing this, the authors revisit a basic concept of mathematics, that is the concept of rigorous proof. The authors wish to furnish a rigorous definition of closed forms with the help of the concept of rigorous proof. However, the problem is that the general notion of rigorous proof is a kind of “community-varying and epoch dependent” concept (Borwein and Crandall 2010, p. 1). Consequently, even a potential rigorous definition of closed forms is likely to provide an exhaustive treatment to the matter. 2. Discussion The authors have adopted seven different approaches to define a closed form. The first three approaches are very basic and theoretical in nature. The fourth approach chiefly utilises set algebra with particular focus on exponential and logarithmic functions. Using this approach, Chow (1999) remarks that the term closed form must imply explicit in the sense that the expression in closed form is meaningful, clearly open to all calculations and standard mathematical operators can be applied (Borwein and Crandall 2010, section 1.0.4). Although most algebraic functions do not have a simple explicit expression, scientists and mathematicians are trying to introduce concepts like hyperclosure and superclosure. The fifth approach is again elementary in nature with emphasis on theory rather than correlative analysis with respect to sufficiently complicated equations and identities (Borwein and Crandall 2010). In discussing the sixth approach, the authors have put their own input to refine the understanding of this concept as deduced from previous research works of experts like Bailey, Borwein, and Crandall (2008). First, the Borwein and Crandall (2010) consider any convergent sum given by the following expression: x = ∑cnzn (where x is a member of the set X) … … … (1) Explaining the different variables and operators that are seen in (1), we must mention that c0 is rational; z is algebraic; and n ≥ 0. Furthermore, for n > 0 we have: , where B and A are integer polynomials such that deg B ≥ deg A. Also, the set X contains generalised hypergeometric evaluations as established by the authors (Borwein and Crandall 2010, section 1.2.2) as a part of the ring of hyperclosure denoted by H (which is begot from all generalised hypergeometric evaluations). Now according to the authors: “Under these conditions the expansion for x converges absolutely on the open disk |z| < 1. However, we also allow x to be any finite analytic-continuation value of such a series; moreover, when z lies on a branch cut we presume both branch limits to be elements of X. (See ensuing examples for some clarification.) It is important to note that our set X is closed under rational multiplication, due to freedom of choice for c0. ” (Borwein and Crandall 2010, section 1.2.2) The merit of this approach is that it introduces us to the concept of hyperclosure. The operation of hyperclosure is capable of establishing at least a tentative but workable relationship between the generalised hypergeometric evaluations and the complex number system. Furthermore, several fundamental numbers have been proved to be hyperclosed. For example, ew can be proved to be a hyperclosed function where e is the Euler’s number and w is any arbitrary algebraic number. Analysing the theoretical aspects of the seventh approach, the definition of closed forms appears to need an increased algebraic typology-based environment. In this kind of setting, “it might make sense to define closed forms to be those arising as periods – that is as integrals of rational functions (with integer parameters) in n variables over domains defined by algebraic equations” (Borwein and Crandall 2010, section 1.3). These propositions have their roots in the theory of elliptic and abelian integrals, which have been deeply studied by Kontsevich and Zagier (2001). Periods can form an algebra and they can also capture several constants. Unfortunately, Borwein and Crandall (2010) admit in this section (that is section 1.3, paragraph 2) that a deeper study of closed form involving periods is beyond their area of expertise. In continuation with this discussion, the authors have also tried to explain that why closed forms are being investigated. From a purely theoretical and inquisitive point of view, closed forms are of course a kind of intellectual challenge and hence highly thought provoking. However, understanding of closed forms has utility in several real world applications. For example, researchers like Heston (1993, p. 332) have attempted to obtain closed form solutions to explore the drawbacks of Black-Scholes formula that can be used to study “bond options, currency options, and other extensions.” However, in the article under discussion, Borwein and Crandall (2010) have used a very simple example of the working of pendulum to explain real world application of closed forms. 3. Focus on the Equations 3.1 Detailed Examples To begin with the intricacies of closed forms, Borwein and Crandall (2010) have utilised detailed citations from The Siam 100-Digit Challenge: A Study in High-accuracy Numerical Computing (see Bornemann et al 2004). Three problems originating in this book have been discussed by the authors. The first problem (that is problem # 2) presented by Borwein and Crandall (2010, Example 3.1) involves complicated interval arithmetic requiring “high-precision interval computation.” The second problem (that is problem # 9) presented by the authors in Example 3.2 involves advanced level integral calculus aimed at calculating maxima and minima of trigonometric functions. In finding a satisfactory solution to this problem, the authors further suggest that Meijer-G function can be used to solve the problem, but the issue is that the Meijer-G function gives very complex values and the concept of superclosure must be applied to utilise them. The second problem (that is problem # 10) presented by the authors in Example 3.3. And this problem appears to be rather interesting with respect to the very purpose and scope of closed functions (and not merely defining them). The statement of the problem is: “A particle at the center of a 10 X 1 rectangle undergoes Brownian motion (i.e., 2-D random walk with in infinitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?” [See Borwein and Crandall (2010), Example 3.3; also see Bornemann et al (2004)] In solving the problem as stated above, Bornemann et al (2004) have adopted a deterministic approach. They used Laplace’s Equation and then implemented sparse Cholesky solver. In this way, the problem was finally solved. But Borwein and Crandall (2010) choose to proceed further from this point. The authors explore different methods for solving this problem, but the method of conformal mapping appears to be most thought provoking. This method yields the following equation for any rectangle with dimensions 2a X 2b: … … … (2) Where p (a, b) denotes the probability under investigation, K is the first kind complete elliptic integral, and ρ := a/b. Now, according to the problem stated in Example 3.3, a/b =10, where a=5 and b=0.5. Putting these values in Equation (2), we get p = (2/π) arcsin (k100), where k100 := ((3-2√2)(2+√5)(-3+√10)(-√2+4√5)2)2 Most interestingly, this value is completely in closed form. And now the question is: “Where does this come from?” (See Borwein and Crandall 2010, p. 16) Actually, k100 is k(a/b)2 (or kρ2), which is parameterised with the help of theta functions. In this way, the authors first obtained the following equation: … … … (3) [Where q := e-πρ] Then manipulating the underlying partial differential equation (with relation to the Brownian motion as mentioned in the problem statement of Example 3.3), the technique of separation of variables helped Borwein and Crandall (2010) to arrive at a converging probability distribution that was finally summed up inside a closed form! In the subsequent examples (those are Example 3.4 and Example 3.5) Borwein and Crandall (2010) have presented some real world examples where the concept of closed form is necessary. But the authors have confined themselves only to a theoretical treatment of these topics (which are related to formation of mirage, as in Example 3.4 and Lane-Emden Equation, as in Example 3.5). 3.2 Recent Examples Borwein and Crandall (2010) have furthered their research in the field of exploring possible varieties of closed forms with the help of recent research borrowing ideas from the physical world. In Example 4.1, the authors have shown some very complex forms of integral equations. They have presented Cn integrals that are related to quantum mechanics and Dn and En integrals that are related to the Ising Model. According to the authors, the Cn integrals are highly complex yet they can be calculated and the values obtain can be deemed in closed form as per almost all existing definitions. But Dn and En integrals are far more complicated and their values cannot be obtained by standard means. Even if these values are obtained, then they cannot be deemed to be in closed form until and unless the concept of hyperclosure is applied. In fact for n = 5, En integral appears to be a “conjectured identity” (Borwein and Crandall 2010, equation no. 4.1). Similar problems arise regarding weak coupling oscillation problems, and the authors advocate implementation of high accuracy computation. In general, the authors seem to believe that coupled system problems and their solution give rise to equations, expressions, and values that are in closed form. But the most thought provoking topic in this context is that of box integrals. Suppose there is a definite point inside a hypercube. Now the expected distance of that point from the hypercube walls is to be calculated. Let’s put a random point R at an expected distance of . Furthermore, for random ϵ [0, 1]3 the value of the expected distance can be computed in closed form by (1/4) √3 - (1/24) π + (1/2) log (2+√3). This kind of expected distances or expectations are termed as box integrals which have recently found application in the analysis of brain synapses (in the field of medical biophysics). The moments of the expected distance between two given points inside a hypercube is given as the following box integral: … … … (4) Where ∆d(s) is the notion for the moments obtained in two dimensions. Of late, Steinerberger (2011) has found what happens to this kind of box integrals when number of dimensions (denoted by d) tends to infinity. Accordingly, Borwein and Crandall (2010) present the following equation for any s, p > 0: … … … (5) From this point, the authors have very concisely discussed that how researchers proceed further to calculate the values of box integrals in higher dimensions. For example, the authors mention that even when ϵ [0, 1]5 a box integral can be proved to be explicitly hyperclosed. In the case of Equation (4), such a mathematical reduction would involve putting d = 5 and then going ahead with the double integration process to find the expectation between two given points. 4. Further Research and Conclusion Borwein and Crandall (2010, section 5) have expressed “profound curiosities” regarding different kinds of mathematical functions, equations, expressions, and identities in relation with the definition of closed forms. The authors attempt to focus on the scope of future research in topic areas like Bessel expansion, Meijer-G function, noncomputable real numbers, etc. With relation to these complex areas of mathematical study however, the main difficulty in the way of defining closed forms is inconsistency of approach. While a basic approach is too theoretical, an innovative approach is not all agreeable. Comparing the sixth and seventh approaches for defining closed forms (Borwein and Crandall 2010, sections 1.2.2 and 1.2.3), a strange problem is witnessed. The sixth approach implements hypergeometry and introduces hyperclosure functions. This approach can extend the sphere of closed forms up to several complex numbers and generalised hypergeometric functions. For example, (eπ + πe) is considered to be hyperclosed. But the seventh approach implements algebra of periods. “They are especially well suited to the study of L-series, multi zeta values, polylogarithms and the like, but again will not capture all that we wish” (Borwein and Crandall 2010, p. 8). For example, since e is conjectured not to be a period, a closure function devised with the help of the seventh approach cannot give rise to convergence or exhaustively investigate the nature of the numbers like (eπ + πe). Ultimately, there appears to be several open questions. Borwein and Crandall (2010) have appended these questions in section 6. However, the most crucial question is presumably the fact that the sixth and seventh approaches of defining a closed form have yet not reached a reasoned and practical conciliation or harmonisation. Therefore, the need of continuous research towards exploring closed from is undeniable although lengthy and precise computing is necessary to accomplish these explorations. List of References Bailey, D.H., Borwein, J.M. and Crandall, R.E. 2008. Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics, Experimental Mathematics, 18, pp. 107-116. Bornemann, F., Laurie, D., Wagon, S. and Waldvogel, J. 2004. The Siam 100-Digit Challenge: A Study in High-accuracy Numerical Computing, SIAM, Philadelphia. Borwein, J.M. and Crandall, R.E. 2010. Closed forms: What they are and why we care. In: Notices of American Mathematical Society. Available: http://www.ams.org/notices/201301/rnoti-p50.pdf. Last accessed on 5th October 2013 Chow, T.Y. 1999. What is a closed-form number? American Mathematical Monthly, 106, pp. 440-448. Heston, S.L. 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6, pp. 327-343. Kontsevich, M. and Zagier, D. 2001. Periods. In: Mathematics Unlimited – 2001 and beyond, Berlin: Springer-Verlag, pp. 771-808. Steinerberger, S. 2011. External uniform distribution and random chord lengths, Acta Mathematica Hungarica, 130, pp. 321-339. Read More