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Unformatted text preview: Dot Product Problem Domino m-l E m EQ!E! ELi Em where g is the MEAN, gy the and TV the relaxation time. UZSO FIBONACCI HEXOMINO, TRIOMINO NUMBER, GOMORY'S THEOREM, POLYOMINO, TETROMINO, PENTOMINO, Doob, J. L. “Topics Amer. Math. 52, J. S. Vomino Recreations. Domino Trans. l DERIVATIVE, also Dot Product The dot product DOT PRODUCT, can be defined TIMES by 1XIIYIcose, (‘1) where 0 is the angle between the vectors. It follows immediately that X - Y = 0 if X is PERPENDICULAR to Y. The dot product is also called the INNER PRODUCT and written (n, b). By writing Recreations.” Madachy’s MatheNew York: Dover, pp. 209-219, 1979. it follows Problem Chains.” l X*Y= Madachy, of Markov 1942. 37-64, Dot The “dot” has several meanings in mathematics, including MULTIPLICATION (a b is pronounced “a times b”), computation of a DOT PRODUCT (a-b is pronounced “a dot b”), or computation of a time DERIVATIVE (h is pronounced “a dot”). domino.html. matical in the Theory Sot. References Dickau, R. M. “Fibonacci Numbers.” http: //www* prairienet.org/-pops/fibboard.html. Ch. 13 in The Scientijic AmerGardner, M. “Polyominoes.” ican Book of Mathematical Puzzles & Diversions. New York: Simon and Schuster, pp. 124-140, 1959. Kraitchik, M. “Dominoes.” $12.1.22 in Mathematical Recreations. New York: W. W. Norton, pp. 298-302, 1942. Lei, A. “Domino.” DEVIATION, References see see STANDARD 489 A, = ACOSBA l3, = BcosBg (2) A, = Asir& By = &in&, (3) that (1) yields see WANG~S CONJECTURE A Donaldson Distinguish Invariants between smooth l B MANIFOLDS in 4-D. Donkin’s Theorem The product of three translations along the sides of a TRIANGLE through twice the lengths sides is the identity. = AB - COS(eA AB(cos - 6~ cos 0~ + sin @Asin 0,) = A cm GAB = A,& directed of these eB) cos 0~ + A sin 8AB sin & + A,B,. (4 So, in general, X*Y =x1y1 +...+x,y,. (5) Donut The dot product see TORUS Doob’s Theorem A theorem proved by Doob (1942) which states that any random process which is both GAUSSIAN and MARKOV has the following forms for its correlation function, spectral density, and probability densities: Cy(7) = ay2e--‘lrr GY(f) = Pi(Y) (242 e-CY-&12/2fl,2 = J27r(l - e-2r/p+y2 x exp { [(y2 (6) ASSOCIATIVE (TX) - Y = T(X - Y), (7) x ' (Y + z) =X*Y+X.Z. The DERIVATIVE I P2(YllY2,7-J X*Y=Y.X, and DISTRIBUTIVE 4T3T,2 +TT-2 = -!Jay” is COMMUTATIVE - y) - e-r/rr(yl - y)12 1 2(1 - e --2r/r, joy2 > of a dot product ~[rl(t)*=2(t)]=Tl(t).~+~*r2(~). (8) of VECTORS is (9) 490 Douady’s is invariant The dot product where Rabbit EINSTEIN The dot product A l Double rotations has been used. is also defined bY see under SUMMATION Double fiactal for TENSORS A and B (11) PRODUCT,~EDGE PRODUCT A&en, G. “Scalar Methods Press, for or Dot Product.” pp. 13-18, Douady’s 3rd Physicists, ed. 51.3 in Mathematical Orlando, FL: Academic 1985. Rabbit Conjecture Fkactal BUBBLE References Haas, J. and Schlafy, Preprint, Contraction This equation see see also SAN MARCO + 0.745i, also known as UZSO DISK FRACTAL Minimize.” Relation is satisfied SPHERICAL contraction by zero trace, symmetric unit TEN- areused todefinethes~~~rt~~~~ HARMONIC TENSOR,TENSOR References A&en, FRACTAL,%EGEL Bubbles A TENSOR t is said to satisfy the double relation when t;*t; = a,,. where the hat denotes SORS. These TENSORS HARMONIC TENSOR. A JULIA SET with c = -0.123 the DRAGON FRACTAL. “Double R. 1995. Double References Integration Two partial SPHERES with a separating boundary (which is planar for equal volumes) separate two volumes of air with less AREA than any other boundary. The planar case was proved true for equal volumes by J. Hass and R. Schlafy in 1995 by reducing the problem to a set of 200,260 integrals which they carried out on an ordinary PC. see also DOUBLE l3 = A”B,. CROSS PRODUCT, INNER PRODUCT, OUTER UZSO Bubble Exponential G. “Alternating Physicists, 3rd ed. Series.” Mathematical Orlando, FL: Academic Methods Press, for pa 140, 1985. References Wagon, S. Mathematics man, p. 176, 1991. Double in Action. New York: W. II. F’ree- Double Cusp see DOUBLE POINT Bubble Double Exponential The planar double bubble (three circular arcs meeting in two points at equal 120” ANGLES) has the minimum PERIMETER for enclosing two equal areas (Foisy 1993, Morgan 1995). see FISHER-TIPPETT BUTION see also APPLE, BUBBLE, DOUBLE BUBBLE TURE,~PHERE-SPHERE INTERSECTION Double CONJEC- References Msg. 68, 321, 1995. Campbell, P. J. (Ed.). R eviews. Math. Foisy, J.; Alfaro, M.; Brock, J.; Hodges, N.; and Zimba, J. “The Standard Double Soap Bubble in R” Uniquely MinPacific J. Math. 159, 47-59, 1993. imizes Perimeter.” Morgan, F. “The Double Bubble Conjecture.” FOCUS 15, 6-7, 1995. Sci. Peterson, I. “Toil and Trouble over Double I3ubbles.” News 148, 101, Aug. 12, 1995. Distribution DISTRIBUTION, Exponential LAPLACE DISTRI- Integration An excellent NUMERICAL INTEGRATION technique used by MupZe V [email protected] (Waterloo Maple Inc.) for numerical computation of integrals. see also INTEGRAL, GRATION INTEGRATION, NUMERICAL INTE- References Davis, P. J. and Rabinowitz, P. Methods of Numerical Inte2nd ed. New York: Academic Press, p. 214, 1984. Di Marco, G.; Favati, P.; Lotti, G.; and Romani, F. “Asymptotic Behaviour of Automatic Quadrature.” J. Cumplexity gration, 10,296-340, Mori, for Numerical tional Cungress York: 1994. M. Developments in the Double Exponential Integration. Proceedings of of Mathematicians, Kyoto Springer-Verlag, pp. 1585-1594, 1991. Formula the Interna1990. New Double Factorial Double Formulas for Mori, M. and Ooura, T. “Double Exponential Fourier Type Integrals with a Divergent Integrand.” In Contributions in Numerical Mathematics (Ed. R. P. Agarwal) . World Scientific Series in Applicable Analysis, Vol. 2, pp. 301-308, 1993. Ooura, T. and Mori, M. “The Double Exponential Formula for Oscillatory Functions over the Half Infinite Interval.” J. Comput. Appl. Math. 38, 353-360, 1991. Takahasi, H, and Mori, M. “Double Exponential Formulas for Numerical Integration.” Pub. RIMS Kyoto Univ. 9, 721-741, 1974. Toda, H. and Ono, I-I. “Some Remarks for Efficient Usage of the Double Exponential Formulas.” Kokyuroku RIMS Kyoto Univ. Double 339, 74-109, l n - 3) l)( l l l l l (1) = (n - l)!! l . (7) . (8) For n EVEN, --n! n!! n(n - l)(n - 2) (2) n(n - 2)(n - 4) q . - (2) l - l l - 3) 944 (2) = (n - I)!! for any n, n! n!! n odd n even n= -l,o. l l relating double factorials to = ( n- n! = n!!(n (1) For n = 0, 1, 2, . . . . the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, . . (Sloane’s AO06882). There are many identities FACTORIALS. Since ( n- l of the usual FAC- n - (n - 2) . , . 5.3.1 72' (n - 2)... 6 4 2 i 1 l 491 n(n - l)(n - 2) (1) n(n - 2)(n - 4) 9 (1) = Therefore, factorial is a generalization n! defined by n!! = n! Jj= = (n - l)(n Factorial Fhc tion For n ODD, 1978. The double TORIAL Gamma The FACTORIAL TIFACTORIAL see (9) - I)!!. (10) may be further FACTORIAL, also l)!! generalized to the MUL- MULTIFACTORIAL References Sloane, N. J. A. Sequence AO06882/M0876 in “An On-Line Version of the Encyclopedia of Integer Sequences.” (2n + 1)!!2”n! = [(2n + 1)(2n - 1)***1][2n][2(n - 1)][2(n Double - 2)1*-*2(l) =[(2n+1)(2n-I)ed][2n(2n-2)(2n-4)..*2] = (2n + 1)(2n)(2n - 1)(2n - 2)(2n - 3)(2n - 4) ‘**2(l) Double-Free = (2n + l)!, it follows (2) that (2n)!! (2n + l)!! = (2n)(2n = w. Since - 2) (2n - 4) . = [2(n)][2(n - 1)][2(n l l 2 - 2)J l w l l 2 = 2”n!, that (h)!! = 2nn!. Set A SET of POSITIVE integers is double-free if, for any integer LC, the SET {x, 22) < S (or equivalently, if II: E S IMPLIES 22 $ S). Define = max{S : S c { 1,2, . , . , n} is double-free}. (3) Then it follows Folium see BIF~LIUM an asymptotic formula 44 (2n - 1)!!2”n! -- [(Zn - 1)(2n - 3) .. .1][2n][2(n = (2n - l)(Zn - 3). . .1][2n(2n = 2n(2n - I)(2 n - 2)(2n -- (2 n >!, it follows Similarly, - 1)][2(n - 2)] - 2)(2n - 4) - 3)(2n l l l 9 ‘2(l) l (4) l)!! = -(2n! ) (5) 2"n!' n l)!! = (- 1) (2n - I)!! = 1989). SET Finch, S. “Favorite Mathematical Constants.” http: //wwu. mathsoft.com/asolve/constant/triple/triple~html~ Wang, E. T. H. “On Double-Free Sets of Integers.” Ars Combin. 28, 97-100, 1989. Double Gamma see DIGAMMA (-l)n2"n! (2 n >! $n + 6(lnn) References - 4) q q 2(l) for n = 0, 1, . . . , (-2n- (Wang - see also TRIPLE-FREE 21 l that (2n- is Since ’ (6) Function FUNCTION Double 492 Double Point A point traced The maximum erate QUARTIC PRINT is called Arnold CURVES Doubly Point out twice as a closed curve is traversed. number of double points for a nondegenCURVE is three. An ORDINARY DOUBLE a NODE. DOUBLE POINT, CONIC DOUBLE CRUNODE, CUSP, ELLIPTIC CONE POINT, DOUBLE POINT THEOREM, NODE (ALGEBRAIC CURVE), ORDINARY DOUBLE POINT, QUADRUPLE POINT RATIONAL DOUBLE POINT, SPINODE, TACNODE, TRIPLE POINT, UNIPLANAR DOUBLE POINT Square over binary QUADRATIC FORMS. If S can be decomposed into a linear sum of products of DIRICHLET LSERIES, it is said to be solvable. The related sums S1 (a, b, c; s) = x (md# (1994) gives pictures of spherical and PLANE with up to five double points, as well as other curves* Magic am2 + bmn + cn2)-’ (-l)“( m) (5) &(a, b, c; s) = x (-l)“(am2 + bmn + cn2)-’ IE ( - 1)m+n(am2 BIPLANAR see also POINT, GAUSS’S (6) +(a, b, c; s) = -f- bmn + cn2)-’ (7) References Aicardi, F. Appendix to “Plane Curves, Their Invariants, Perest roikas, and Classifications .” In Singularities & Bifurcations (V. I. Arnold). Providence, RI: Amer. Math. sot l , pp. 80-91, 1994. Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp, 12-13, 1986. Double Sixes Two sextuples of SKEW LICNES on the general CUBIC SURFACE such that each line of one is SKEW to one LINE in the other set. Discovered by Schkfli. see also BOXCARS, CUBIC SURFACE, SOLOMON’S SEAL LINES References Fischer, G. (Ed.). of Universities Vieweg, p. Mathematical and Museums. Models from the Collections Braunschweig, Germany: 11, 1986. can also be defined, S1(l,O, Identities see also EULER x p=o q=o y-)k,, = m=On=O x x T=o s=o and y = 6’(x - a), is the DELTA FUNCTION. DELTA also i=l XiXj = FUNCTION D. CRC Standard Cuwes and Surfaces. FL: CRC Press, p. 324, 1993. Boca Doubly Even Number An even number N for which N E 0 (mod 4). The first few POSITIVE doubly even numbers are 4, 8, 12, 16, . . (Sloane’s AOO8586). l n xx ion References von Seggern, (2) 72 (8) (1) where FUNCTION, l SUM F’unct Raton, is the FLOOR + 5J29) Glasser, M. L. and Zucker, I. J. “Lattice Sums in Theoretical Theoretical Chemistry: Advances and PerChemistry.” spectives, Vol. 5. New York: Academic Press, 1980. Zucker, I. J. and Robertson, M. M. “A Systematic Approach to the Evaluation of ctm 1n~o 1o,(am2 &nn+cn2)-“.” J. 1976. Phys. A: Math. Gen. 9, 1215-1225, involving G,T-23, 7rln(27 References see = 58; 1) = - A complete table of the principal solutions of all solvable S(a, b, c; s) is given in Glasser and Zucker (1980, pp. 126131)* where’d(x) y>luq,,-, gives rise to such impressive m Doublet Double Sum A nested sum over two variables. double sums include the following: which FORMULAS as ?t2 j=l (x2> ’ (3) see also EVEN FUNCTION, ODD NUMBER, SINGLY EVEN NUMBER References Consider Sloane, N. 3. A. Sequence A008586 in CtAn On-Line of the Encyclopedia of Integer Sequences.” the sum S(a, b,c; s) = x bv4#~W) ( am2 + bmn + cn2)-s (4) Doubly Magic see BIMACIC Square SQUARE Version Dougall-Ramanujan Dowker Identity Dougall-Ramanujan Dougall’s Identity Discovered by Ramanujan (1959, pp. 102-103), around 1910. From Theorem in + l,n, -2, -y, --x [ ~n,a:+n+l,y+n+f,z+n.+l r(x + n + l)r(y + n + l)r(z r(n + l)r(x (x + s + where &(a,b,c, PERGEOMETRIC 1p S r(s + 1)r(x + y + z + u + s + 1) r(x X + s + l)F(y + 2 + u + s + 1) t rI r(x + u + + Y + n. + l)r(y 3 = + n + qr(x + y + z + n + 1) + 2: + n + l)r(x + 2 + n + I) ' x:(n) rI x,y,=*u -- 493 Hardy SF4 X Notation s + 1) (1) d,e; f,g, h, i;z) is a GENERALIZED HYFUNCTION and r(z) is the GAMMA FUNCTION. see also DOWALL-RAMANUJAN IDENTITY, IZED HYPERGEOMETRIC FUNCTION GENERAL- X,'II,Z,U Doughnut where s,1+ F0 =- a(u + 1). v (a + n - 1) (2) a(,) G a(a - 1) . - - (a - n + 1) (3) l P~WHAMMER (here, the @I). This 7 &d SYMBOL can be rewritten is,--z has been - y,-25, -u,x-y++++++s+1 ~s,~+s+1,y+s+l,z+s+l,u+s+l, ;I -X-Y-%-U-S [ X X written as 1* 1 = r(S + 1>qx + y + 2 + u + s + 1) r(x + s + i>r(y + z + u + s + 1) rI r(z+21+s+i) * ,t,=,u %a2,a3,a4,a5&6,a7 ;l h,b2,b&i,b5$6 (a+ l>n(m- (a1 - a2 + l)n(Ul X a2 - u3 - u3 (a1 - a2 - bl - 3-l), +1), (4) - u3 - u4 +l), - u2 - u3 - u4 +l>,' where (a), is the P~CHHAMMER SYMBOL (Petkovgek al. 1996) q The identity is a special case of JACKSON'S (5) et IDENTITY. see also DIXON'S THEOREM, DOUGALL? THEOREM, GENERALIZED HYPERGEOMETRIC FUNCTION, HYPERGEOMETRIC FUNCTION, JACKSON'S IDENTITY, SAALSCH~~TZ'S THEOREM References Dixon, Math. A. C. “Summation Sot. 35, 285-289, Hardy, G. H. Ramanujan: gested by His Life and Theorem If the lines joining corresponding points of two directly similar figures are divided proportionally, then the LOCUS of the points of the division will be a figure directly similar to the given figures. References Eves, H. “Solution 50, 64, 1943. to Problem J. R. “Problem E521.” E52L” Amer. Amer. Math. Monthly Math. MonthZy 49, 335, 1942. Dovetailing Problem see CUBE DOVETAILING Dowker PROBLEM Notation A simple way to describe a knot projection. The advantage of this notation is that it enables a KNOT DIAGRAM to be drawn quickly. 1 a4 -tl),(Ul a4 +1)&l Douglas-Neumann Musselman, In a more symmetric form, if 7z = 2ul + 1 = u2 + a3 + 424+ ~5, a6 = 1 + u1/2, u7 = -n, and bi = 1 + al - ui+l for i = 1, 2, . . . , 6, then 7F6 see TORUS of a Certain Series.” Proc. London 1903. Twelve Lectures on Subjects SugWork, 3rd ed. New York: Chelsea, 1959. Petkovgek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43, 126-127, and 183-184, 1996. For an oriented ALTERNATING KNOT with n crossings, begin at an arbitrary crossing and label it 1. Now follow the undergoing strand to the next crossing, and denote it 2. Continue around the knot following the same strand until each crossing has been numbered twice. Each crossing will have one even number and one odd number, with the numbers running from 1 to 2n. Now write out the ODD NUMBERS 1, 3, . . , 2n - 1 in a row, and underneath write the even crossing number corresponding to each number. The Dowker NOTATION is this bottom row of numbers. When the sequence of even numbers can be broken into two permutations of consecutive sequences (such as {4,6,2} {10,12,8}), the knot is composite and is not uniquely determined by the Dowker notation. Otherwise, the knot is prime and the NOTATION uniquely defines a single knot (for amphichiral knots) or corresponds to a single knot or its MIRROR IMAGE (for chiral knots). l For general nonalternating ified slightly by making knots, the procedure is modthe sign of the even numbers Down Arrow 494 Notation Droz- Farny Circles POSITIVE if the crossing is on the top strand, ATIVE if it is on the bottom strand. These data are available from Berkeley’s gopher only for knots, site. and NEG- but not for links, References Adams, to the C. C. The Mathematical Knot Book: Theory An of Elementary Knots. New Arrow An inverse W. H. M. B. “Classification 19-31, 1983. of 16, Notation NOTATION defined by e $4 n = In* n e J.&J. n = In** n, also ARROW wood of times to obtain the NATURAL a value -< e. LOG- NOTATION References Vardi, I. Computational Recreations City, CA: Addison-Wesley, in Mathematics. pp, 12 and 231-232, Red1991. Dozen Dickau, also BAKER'S Dragon Curve DOZEN, R. M. GROSS 6, “Two-Dimensional L-Systems.” York: Dover, http:// pp. 180-181, 5-10, 1995. Dragon Fkactal see DOWADY'S Nonintersecting curves which can be iterated to yield more and more sinuosity. They can be constructed by taking a path around a set of dots, representing a left turn by 1 and a right turn by 0. The firstorder curve is then denoted 1. For higher order curves, add a 1 to the end, then copy the string of digits preceding it to the end but switching its center digit. For example, the second-order curve is generated as follows: (1)l --+ (1)1(O) -+ 110, and the third as: (11O)l -+ (llO)l(lOO) + 1101100. Continuing gives 110110011100100. . (Sloane’s A014577). The OCTAL representation sequence is 1, 6, 154, 66344, . . . (Sloane’s A003460). The dragon curves of orders 1 to 9 are illustrated below. l CURVE Puzzles, Part I: Moving Oriental Dubrovsky, V. “Nesting Towers.” Quantum 6, 53-57 (Jan.) and 49-51 (Feb.), 1996. Dubrovsky, V. “Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster.” Quantum 6, 61-65 (Mar.) and 58-59 (Apr.), 1996. Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 207-209 and 215-220, 1978. Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 4853, 1991. Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, pa 284, 1988. Sloane, N. J. A. Sequences A014577 and A003460/M4300 in “An On-Line Version of the Encyclopedia of Integer Sequences .” Vasilyev, N. and Gutenmacher, V. “Dragon Curves.” Quantum 12. see SYSTEM,PEANO forum.swarthmore.edu/advanced/robertd/lsys2d.htmL Dixon, R. Mathographics. New eJn=lnn see LINDENMAYER also 1991. of the up ARROW where In’ n is the number ARITHM must be iterated see References Preeman, pp. 35-40, 1994. Dowker, C. H. and Thistlethwaite, Knot Projections.” Topol. App2. Down Introduction York: This procedure is equivalent to drawing a RIGHT ANGLE and subsequently replacing each RIGHT ANGLE with another smaller RIGHT ANGLE (Gardner 1978). In fact, the dragon curve can be written as a LINDENMAYER SYSTEM with initial string IIFXUf, STRING REWRITING rules +‘X++ -> “X+YF+“, “Y” -> ‘I-FX-Y”, andangle 90’. RABBIT FRACTAL Draughts see CHECKERS Drinfeld’s Symmetric Space A set of points which do not lie on any of a certain of HYPERPLANES. class References Teitelbaum, Not. Amer. Droz-Farny J. “The Geometry of p-adic Symmetric Math. Sot. 42, 1120-1126, 1995. Circles )I--/0 // /I \\ \\ \\ Spaces.” Droz-Farny Circles Du Bois Raymond Constants 495 Draw a CIRCLE with center H which cuts the lines 0203, 0301, and 0102 (where Oi are the MIDPOINTS) at PI, Q1; P2, Q2; and P3, Q3 respectively, then A1PI = A2P2 = A3P3 = A1&; = Az&2 = A3Q3. Conversely, if equal CIRCLES are drawn about the VERTICES of a TRIANGLE, they cut the lines joining the MIDPOINTS of the corresponding sides in six points. These points lie on a CIRCLE whose center is the ORTHOCENTER. If T is the RADIUS of the equal CIRCLES centered on the vertices Al, AZ, and A3, and Ro is the RADIUS of the CIRCLE about H, then RI2 = 4R2 + r2 - +(a~” -t a22 •I az2)m / ‘- References Goormaghtigh, R. “Droz-Farny’s Theorem.” Scripta Math. 18,268-271,195O. Johnson, R. A. Modern Geometry: An Elementary lkatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 256-258, 1929. / \ /# // \ / \ \ I I I If the circles equal to the CIRC~MCIRCLE are drawn about the VERTICES of a triangle, they cut the lines joining midpoints of the adjacent sides in points of a CIRCLE R2 with center H and RADIUS 2 R2 = 5R2 - $(a” Furthermore, the circles about the midpoints of the sides and passing though H cut the sides in six points lying on another equivalent circle Ry whose center is 0. In summary, the second Droz-Farny circle passes through 12 notable points, two on each of the sides and two on each of the lines joining midpoints of the sides. Drum see ISOSPECTRAL Du Bois MANIFOLDS Raymond Constants 0.6 + az2 + ~3~)~ I 2 The constants 4 Cn defined which are difficult are 6 8 10 by to compute numerically. The first few Cl z 455 It is equivalent to the circle obtained by cles with centers at the feet of the altitudes through the CIRCUMCENTER. These circles responding sides in six points on a circle Rb is H. drawing cirand passing cut the corwhose center c2 ...
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