The Mathematics of the Movie Gifted by Marvin L. Bittinger Recent Films with Math Context: d p 2 édu du du du ù - + mÑ u + Fx = r ê + u + v + w ú, (1) Good Will Hunting, Stand and Deliver, d x ë dt d x d y d z û A Beautiful Mind, The Man Who Knew Infinity, Hidden d p 2 édv dv dv dv ù - + mÑ v + Fy = r ê + u + v + w ú, (2) Figures, The Imitation Game, Gifted, d y ë dt d x d y d z û d p 2 éd w d w d w d wù - + mÑ w + Fz = r ê + u + v + w ú, (3) Trachtenberg Method-Speed Math: d z ë dt d x d y d z û du dv d w References: + + = 0, Incompressibility (4) -www.TrachtenbergSpeedMath.com d x d y d z -U-Tube or Wiki: “Trachtenberg Method” -Book: The Trachtenberg Speed System of Basic Is there a proof as to whether solutions exist, are they Mathematics by Ann Cutler & Rudolph McShane. unique, and do all the graphs have derivatives with no bumps? Millennium Prize Problems: 1) Poincare’ Conjecture: Every simply connected, closed 3- Solutions often pertain to turbulence. General solutions manifold is homeomorphic to the 3-sphere. Can’t get from a remain one of the greatest unsolved problems in sphere to a donut without cutting or deforming it such physics, despite its immense importance in science and as pinching a hole. PROVED by Grigori Perelman in engineering, in particular aerodynamics. Engineers at 2003. Boeing are driven crazy trying to deal with NS equations. 2) Yang-Mills Existence and Mass Gap: Establish rigorously the existence of the quantum Yang-Mills theory and Mass gap. Has to do with behavior of 7) The Riemann Hypothesis: The Zeta Function restricted subatomic particles. to real numbers is defined: 3) Birch & Swinnerton-Dyer Conjecture: Elliptic curves are usually described by an equation with combinations of 2nd & 3rd power variables. The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. Let’s make some substitutions: 4) P vs NP: Is it easy to check that a solution to a problem is correct, is it also easy to solve the problem? 5) Hodge Conjecture: Every harmonic differential form (of a certain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles. Refs: See U-Tube Videos by David Metzler. 6) Navier-Stokes Existence & Smoothness (Occurs in our movie, GIFTED): Navier-Stokes equations govern the Even though x=1 is not part of definition, let’s flow of fluids such as water, air, and blood and involve substitute 1: the following system of nonlinear partial differential equations which is divergent. \1 not in dom 1 1 - x2 f (x)= e The “actual” Riemann Zeta Function described in 2p terms of ANALYTIC CONTINUATION is: ì ¥ 1 ï , Â(z) >1, (1) The area under the curve over the x-axis is a å nz ï n=1 modification of Mary’s statistical problem: ï ¥ n+1 ¥ ï 1 æ u - nö - x2/s 2 e dx = 2p s Letting s = 1, and dividing by 2p , we get z (z) = í 1+ - zå z+1 du, 0 < Â(z) £ 1, (2) ò This area is ò ç ÷ -¥ z -1 n=1 n è u ø ï ¥ 1 2 ï e- x dx =1 z z p (1- z) ò ï 2 p cos G(1- z)z (1- z), Â(z) £ 0, (3) -¥ 2p ï 2 î 1. The trivial zeros at the negative, even integers. Riemann Hypothesis: All the non-trivial zeros of the Riemann Zeta Function are on the line z = ½. æ 1 ö æ 1 ö x +14.134725i = 0, x + 25.0108575i = 0, èç 2 ø÷ èç 2 ø÷ For example, æ 1 ö x ç +158.849985171i÷ = 0, è 2 ø æ 1 ö x + 267,653,395,648.847523127i = 0 Without the minus sign, the area is infinite (see right) ç 2 ÷ è ø All trivial zeros are on the negative x-axis, at negative even Hint: First show that ¥ ¥ 2 2 2 numbers. -(x + y )2s 2 e dx dy = 2ps A proof of RH would have far-reaching implications for ò-¥ ò-¥ the distribution of prime numbers. RH has been shown The proof involves a couple of “tricks” one of which is true for 10 Trillion or 10,000 billion roots. the use of a polar coordinate substitution, found in probability and stats books involving calculus. For Following is a single expression for z : example, Meyer, Paul L., Introductory Probability & nd ¥ n Statistical Applications, 2 ed, 1970, Addison-Wesley 1 1 k æ n ö - z z z = -1 k +1 Pub Co. ( ) 1-z å n+1 å( ) ç ÷ ( ) 1- 2 n=0 2 k=0 è k ø …………………………………………. Refs: U-tube videos under “Riemann Hypotheses, especially by 3Blue1Brown, Numberphile, and Randall Heymann. Book: Riemann’s Zeta Function, by H. M. Edwards The Missing Minus Sign: Problem as posed to Mary: Show that ∞ 푥2/휎2 ∫−∞ 푒 푑푥 = √2휋휎 ¥ ¥ (x2+ y2 )2s 2 2 Hint: First show e dxdy = 2ps ò-¥ ò-¥ A simplistic explanation: 1) First consider the graph of 2 .