Robert G de Boer's blog
https://imechanica.org/blog/27062
enComplementary numerals
https://imechanica.org/node/12369
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/7420">instrumentation</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">
<span>Sometimes a change of notation can advance thought. In today's blog the tradition of numerals 6,7,8, and 9 will be challenged. </span>
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<span>Suppose the five letters a, b, c, d, and e are used to represent numbers satisfying</span>
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<span>a + 1 = 0, b + 2 = 0, c + 3 = 0, d + 4 = 0, e + 5 = 0.</span>
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<span>Then each numeral in {1,2,3,4,5} has a complementary numeral in {a,b,c,d,e}. </span><span>If n is a numeral, a x n is the complementary numeral. For instance, a x a = 1. This equality is an instance of the "rule of signs", the others being a x 1 = a, and 1 x 1 = 1. The rule of signs and the principle of placeholder value are powerful principles of notation that enable modified decimal systems.</span>
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<span>With modified numeration, the number five (5) can be written 10 + e = 1e, where e is in the ones place of a two-digit number. Six is written 1d, seven is 1c, eight 1b, and nine 1a. The old numerals for these numbers are no longer needed. Note that the decimal .5 is written 1.e so that if the value right of the decimal is lost (truncated), the number remaining is 1. , which is the round-off value of .5 . This property of truncation yielding round-off makes e-decimal notation of numbers more attractive than the usual decimal notation.</span>
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<span>Suppose that {a,b,c,d} is the set of complementary numerals and 5 is used instead of 1e for five. In such a d-decimal system .5 rounds down to 0, unlike the usual rounding. Nevertheless, 1.d, 1.c, 1.b, and 1.a all round up to 1. on truncation. Without e, one has a x 5 = a5 = -10 + 5. To multiply a number of several digits by a, change every numeral to its complement, and when 5 is encountered, leave it 5 and add a to the column to the left.<span> </span>Attentive students, such as read iMechanica, will take as an exercise the description of multiplication by a in the e-decimal system where 5 is not used. Evidently one of the costs of using complementary numerals is the extra complication in changing the sign of a multi-digit number.</span>
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<span>Since four is twice two, and five is half of ten, products by two express those of four and five. Multiplication in the d-decimal system is reduced to two and three, once the rule of signs and products by a are in hand. The idea of numerals representing negative quantities is implicit in Roman numeral construction such as IX and XL representing 9 and 40 respectively. In 1726 John Colson published a description of arithmetic with “small numbers”, essentially the d-decimal system described above. In 1840 Augustin Cauchy illustrated use of this arithmetic by comparing the square of 11 with that of 1a, and the square of 12 with that of 1b. He also illustrated the repeating decimals of 1/7 and 1/73 using complementary numerals. For more references see the Wikipedia article <a href="http://www.en.wikipedia.org/wiki/Signed-digit%20representation">Signed-digit representation</a>.</span>
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<span>Check that these physical constants agree with the ordinary decimal expressions:</span>
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<span>c = 3.0a x 10^1b m/s (speed of light in vacuum)</span>
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<span>G = 1c.cc x 10^aa m^3/kg s (gravitational constant)</span>
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<span>N = 1d.02 x10^23 mol^a (Avagadro’s number)</span>
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<span>R = 1b.31 J/mol K (gas constant)</span>
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<span>eps= 1a.b5 x 10^ab C^2/N m^2 (permittivity of free space)</span>
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<span>mp/me =2b40 (ratio of mass of proton to mass of electron)</span>
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<span>g = 10.b1cd5 m/s^2 (acceleration of gravity at Earth’s surface)</span>
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<span>Re = 1d.4c4 x 10^1d m (radius of the Earth)</span>
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<span>Ms = 2.0aa x 10^30 kg (mass of the Sun)</span>
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</div></div></div>Mon, 30 Apr 2012 21:45:23 +0000Robert G de Boer12369 at https://imechanica.orghttps://imechanica.org/node/12369#commentshttps://imechanica.org/crss/node/12369Corner flow
https://imechanica.org/node/10543
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/656">education</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<span>Today we examine a planar flow in the quadrant |y| < x . The boundary lines form the asymptotes of hyperbolas which are the streamlines and the flow will correspond to the usual central angle with |θ| < π/4 . A constant ρ corresponds to a streamline {(ρ cosh <em>a</em> , ρ sinh <em>a</em>) : <em>a</em> in R } where cosh and sinh are hyperbolic cosine and sine functions with parameter <a href="http://www.en.wikipedia.org/wiki/hyperbolic%20angle">hyperbolic angle</a> <em>a</em>. Note that for ρ = 1, the hyperbolic angle is twice the area of the <a href="http://www.en.wikipedia.org/wiki/hyperbolic%20sector">hyperbolic sector</a> corresponding to the angle. Therefore the area of the sector between angles <em>a</em> and <em>b</em> is the same as that between angles (<em>a</em> + <em>c</em>) and (<em>b</em> + <em>c</em>), meaning that the transformation by addition of<span> <em>c</em></span><span> </span>to angles leaves sector areas constant.</span><span> </span>
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<span>Suppose there is a constant <a href="http://www.en.wikipedia.org/wiki/hyperbolic%20rotation">hyperbolic rotation</a> with da/dt = κ > 0. This rate also holds for dθ/dt when θ = 0. Otherwise the hyperbolic rotation corresponds to a somewhat slower turning of the Euclidean angle θ. To obtain the exact rate of change of θ one must use differential calculus on the following structural equation of this model: tanh <em>a</em> = tan θ. </span><span>Differentiating gives </span>
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<span>sech^2 <em>a</em> (d<em>a</em>/dt) = sec^2 θ (dθ/dt),</span> <span>and sec^2 θ = 1 + tanh^2 <em>a</em> then implies dθ/dt = κ/ cosh 2a . So for large <em>a</em> the rate of change of θ is negligible and θ is confined to an interval while <em>a</em> is unbounded.</span><span> </span>
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<span>Planar cross-sections of laminar flows of <span> </span>incompressible fluids correspond to equi-areal <span> </span>mapping. Rather than present a velocity potential with complex number based functions, the above flow suggests the simple multiplication by a <a href="http://www.en.wikipedia.org/wiki/split-complex%20number">split-complex number.</a></span><a href="http://www.en.wikipedia.org/wiki/split-complex%20number"><span> </span> </a>
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<span>In the early days of mathematical physics a professor of astronomy, Egon R. von Oppolzer wrote that in electron theory "everything is a consequence of hydrodynamical theory". Evidently he had in mind the corner flow model discussed above. That model of fluid motion was too good to leave at the water wheel. It was Hermann Minkowski who invented "proper time" for the relativistic age: the above quadrant was re-identified with the future having Now at the corner and the ρ parameter representing proper time. The relativists saw θ as an arbitrary direction of motion into the future determined by an observer’s velocity <em>v</em> < c through Now satisfying tan θ = <em>v</em>/c .The quadrant (future) is a plenum, subject to sudden shifts according to change of observer. Instead of a steady flow model of hydrodynamics, the relativistic picture of spacetime depends on Lorentz transformations to re-set the evolution of time. </span><span>Egon von Oppolzer was quoted by Lewis Peyson in his article "The relativity revolution in </span><span>Germany"</span><span> (page 73) which in contained in <em>Comparative Reception of Relativity</em></span><span> (1987) by Thomas F. Glick (pages 59 to 111). In the context of relativity studies, the hyperbolic angle parameter <em>a</em> is called <a href="http://www.en.wikipedia.org/wiki/rapidity">rapidity</a>.</span><span> </span>
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<p><span><span>For comprehension, the corner flow can be viewed in the context of two other planar linear flows: concentric circular flow and shear flow. These other flows have been cited in connection with the development of the concepts of <a href="http://www.en.wikipedia.org/wiki/viscosity">viscosity </a>and Reynold’s number. For instance, in 1928 Emil Hatschek composed this sketch: </span><span>"In 1890 Couette took up the system of concentric cylinders, the velocity distribution for which had already been given by Stokes, who suggested it might be studied experimentally by observing 'motes in the liquid'. Couette calculated the moment exerted by the outer cylinder on the inner one, and constructed an apparatus in which this moment could be measured and the coefficient of viscosity deduced from it; it was found to agree with Poiseuille’s values. … He showed that the velocities at which turbulence set in were approximately proportional to the kinematic viscosity."(<em>Viscosity of</em> <em>Liquids</em>, page 13)</span> </span></p>
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<span>The laminar flow of a viscous liquid along a wall is called shear flow and corresponds to the linear transformation called <a href="http://www.en.wikipedia.org/wiki/shear_mapping">shear mapping</a>.<span> </span>The physical conditions of corner flow, concentric circular flow, and shear flow differ dramatically, yet</span> <span><a href="http://www.en.wikipedia.org/wiki/2_%C3%97_2_real_matrices">2 x 2 real matrices</a>, with determinant one, provide a single context for their mathematical models. In this mathematical context the corner flow is represented by squeeze mapping, where rectangles of the same area are permuted.</span><span> </span>
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<span><strong>Common Ground:</strong> </span>
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<span>When the relativity of Lorentz, Einstein, and Minkowski (LEM) was studied for foundations, it was found necessary to formulate the presumptions of earlier kinematics such as the space transformation x' = x + v t on a plane (x,t) expressing absolute space and time. <a href="http://www.en.wikipedia.org/wiki/Isaak_Yaglom">Isaak Yaglom</a> (1921 to 1988) investigated the geometric algebra associated with this classical spacetime and that associated with the LEM spacetime. However, for fundamental studies the notion of space is attenuated to a single real dimension so that the spacetimes are planar geometries. There are so many possibilities to consider algebraically for a four-dimensional spacetime that reasonable steps into spacetime algebra work first in the plane. Yaglom (1979) noted that the shear mappings of the classical view leave area invariant in the planar spacetime. Remarkably, the Lorentz transformations also leave area in spacetime invariant. These facts follow from the nature of such mappings in the group of equi-areal mappings in the plane, which consists of three types, two of which correspond to classical and LEM relativity. Such correspondence is common ground between classical and LEM physics; rather than accentuate linear differences in models for each, readers of this announcement in iMechanica can use the mutual currency to advance coherent communication.</span>
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<span>The common physical unit is meter-second (not to be confused with meters per second). Consider the poise = kilograms per meter-second, most commonly used to express viscosity. Here the meter-second is used to provide a "density" for the massy unit, kilogram. In this blog, the unit of poise will be used to measure <strong>kinsity</strong>. Readers may exercise a bit of <a href="http://en.wikipedia.org/wiki/dimensional_analysis">dimensional analysis</a> to show that the geometric mean of a force and a mass density results in a kinsity.</span><span> </span><span> </span>
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</div></div></div>Sat, 09 Jul 2011 20:54:52 +0000Robert G de Boer10543 at https://imechanica.orghttps://imechanica.org/node/10543#commentshttps://imechanica.org/crss/node/10543Adventures in 9-space
https://imechanica.org/node/10518
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1774">matrix</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<span>The three by three matrix is a workhorse for mechanicians, yet the study of this tool is seldom addressed mathematically. One reason is that the space of such matrices is nine-dimensional, quite beyond diagrams and<span> </span>vision. The advantage of a mathematical investigation of the matrix space is clear in the study of two by two real matricies where rotations, shears, and squeezes correspond to area-preserving transformations of the plane. For transformations of space that preserve volume there is a much richer variety of phenomena that may correspond to mechanical events.</span>
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<span>This blog will consider the 3 x 3 real matrices as a <a href="http://www.wikipedia.org/wiki/hypercomplex_number">hypercomplex number</a> system M(3,R) endowed with certain <a href="http://www.en.wikipedia.org/wiki/nilpotent">nilpotents.</a> These nilpotents will be used to build the whole algebra from four three-dimensional subalgebras that arise from second order nilpotents. Let Z denote the zero matrix and Z1[(n,m)] denote the matrix with all zeroes except a one at row n and column m of the matrix. For instance, the identity is Z1[(1,1),(2,2),(3,3)].</span><span>Let p = Z1[(1,2),(2,3)]. Check that p x p = Z1[(1,3)] and that p x p x p = Z. Then p is a nilpotent of the second order.</span>
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<span>The nine-space will be profiled in terms of 3-dimensional subspaces like</span>
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<span>S = { x + y p + z pp : x, y, z<span> </span>in R }. An important curve is this subspace is</span>
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<span>{exp(ap) = 1 + ap + (aa/2) pp : a in R}. This curve is a parabola in the plane of S given by x = 1. In the terminology of Sophus Lie it is a one-parameter subgroup of the multiplicative group of non-singular matrices. The curve is a conic section; the parabola in this case corresponds to the circle found in the complex plane, the unit hyperbola in the split-complex plane, or the line x = 1 in the dual number plane. </span>
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<span><span>The subspace S of M(3,R) is spanned by p, pp, and the identity matrix. As this subspace is closed under products, it is a subalgebra. To more fully comprehend M(3,R) it is useful to consider p as one of four matrices that generate subalgebras isomorphic to S.</span><span> </span></span>
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<span><span>Let q be the same as p except that the entry at (2,3) is changed to negative one.</span><span>Let r = Z1[(2,1),(3,2)]. Finally, let s be the same as r except that the entry at (3,2) is changed to minus one.</span><span> </span></span>
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<span><span>Readers at iMechanica should be able to confirm these matrix identities that arise from <a href="http://www.en.wikipedia.org/wiki/matrix_multiplication">matrix multiplication</a>: pp = – qq , rr = – ss, and </span><span>pr = qs, ps = qr, rp = sq, sp = rq. </span></span>
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<p><span><span>Another exercise that will confirm your engagement with M(3,R) will be the verification that 13 pr + 5 rp – ps – 7sp is twelve times the identity matrix. </span><span>See if you can tell how the identity matrix can be obtained more generally. For instance, can you find a linear combination of three nilpotent products that gives the identity ?</span> </span><span></span></p>
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</div></div></div>Tue, 05 Jul 2011 21:10:56 +0000Robert G de Boer10518 at https://imechanica.orghttps://imechanica.org/node/10518#commentshttps://imechanica.org/crss/node/10518