hackenbush dyadic positions

0000049106 00000 n 0000032164 00000 n 0000015370 00000 n (18 pts) Draw each of the given dyadic positions in Hackenbush: (a) (5/8) (b) (23/32) (c) (-121/64) 8. 0000057787 00000 n Figure 1. Consider an interval (a;b), in which a= 1 and b= 1are allowed. <<7E6FA55DDBF7C14FB2C8C1EE0387C61D>]>> 0000005071 00000 n 0000057810 00000 n In addition to the games listed above, many well-studied positions from normal-play game theory have the dead-ending property: integers are dead ends, and non-integer numbers, all-small games, and all Hackenbush positions are dead-ending. In order for R to win, R must always move to a position that is nonpositive. This means payoffs are symmetric and each player has the same operations based on position on board (in this case structure of drawing) Blue-Red Hackenbush: Each line segment is colored either red or blue. 0000039302 00000 n A family caregive… This website is estimated worth of $ 8.95 and have a daily income of around $ 0.15. 2.Compute the values of the following Blue-Red Hackenbush positions, expressed as dyadic rationals. Dr Hackenbush. 0000029530 00000 n 0000019879 00000 n To explore this issue, we designed a study in which two confederates obtained high or low competence-based status by playing a game together with the participant, while the participant always occupied the middle position of the hierarchy. The strategy for the second player is to copy its opponent’s moves in order to win! ISBN 9780982581346; 176 pages . The mirror strategy is not particular to Hackenbush, it works in any game. 0000010191 00000 n Figure 1 .1Followers of 2 a TFTF2 b , with integer components (from the Terminal Toads Theorem) omitted. A Hackenbush Sprig is a Hackenbush string with the ground edge coloured green and the remaining edges either red or blue. Corollary 3 . We also exhibit positions of value x for each x > 0. 0000002651 00000 n After this, the next simplest surreal numbers are the real numbers that are not integers or dyadic rationals. A dyadic number is a rational number that of the form a b where ais an odd integer and bis a power of 2: b2f1;2;4;8;:::g. 3. answered Dec 16 at 17:30. 0000032440 00000 n are the positions each player can move to. 0000047130 00000 n hackenbush.org Introductory Hackenbush Diagram The following picture shows a Hackenbush Hotchpotch position which Berlekamp often uses in his rst demonstration in introductory lectures of game theory: L R Q G L L L L L L R L R R G G G G G L G G G G G G R If all green branches and everything supporting them are removed, only a BLue-Red House remains. red-blue hackenbush positions: 13. 0000001825 00000 n 0000011254 00000 n 0000047251 00000 n 260 likes. (10 pts) Which dyadic numbers are born on days 4 and 5? Latest news. Dr Hackenbush a covers band playing songs from bands such as Led Zeppelin, AC/DC, ZZ Top, The Who, Black Crowes, Arctic Monkeys, Pulp & much more! 0000010774 00000 n 0000045182 00000 n 0000003843 00000 n 0000032558 00000 n 0000000016 00000 n 0000006696 00000 n 0000047789 00000 n 0000021910 00000 n (a) (b) (c) (d) 3.Suppose x and y are numbers such that, when written as dyadic rationals in lowest terms, y has a larger denominator than x. By juxtaposing a unit vector ˆx,yˆ, or ˆz at the posterior position of the three vector equations given by (7) and summing these equations, we obtain ∇×∇× = G(r,r ′)− k2 = G(r,r) = = Iδ(r− r′) (8) 1 Derivation of Field Quantities From the Dyadic Green’s Function Consider a homogeneous medium bounded by a … 0000046606 00000 n Shop for Vinyl, CDs and more from Dr. Hackenbush at the Discogs Marketplace. The versatile Mechatronics Cylinder series developed by Dyadic Systems is now available from US$295. The additive inverse of a game is found by changing the color of every edge. 6 'O�q#˾6t��E=��g��OpE�w?��/��Q{s5GS_�A/�p�y|f�Ӿ~�� F��d�w�E�$_?��2��RsmErO�?��Ty��?QW u�.�;��S�ew�2��*ߏ:^o� ��]$�Ĥ��I2��%E�"X�=�m\��] q�8��Of��:�Z*��l�сú��&��G��}v��qO��B��РL��ĥ��-��/}L�H�_ �}:� endstream endobj 170 0 obj 1081 endobj 103 0 obj << /Type /Page /Parent 98 0 R /Resources 104 0 R /Contents [ 127 0 R 139 0 R 141 0 R 158 0 R 160 0 R 162 0 R 164 0 R 167 0 R ] /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 104 0 obj << /ProcSet [ /PDF /Text /ImageB ] /Font << /F1 124 0 R /F2 108 0 R /F3 109 0 R /F4 111 0 R /F5 136 0 R /F6 134 0 R /F7 154 0 R /F8 142 0 R /F9 147 0 R >> /XObject << /Im1 165 0 R >> /ExtGState << /GS1 168 0 R >> /ColorSpace << /Cs5 117 0 R >> >> endobj 105 0 obj << /Filter /FlateDecode /Length 262 >> stream 0000003234 00000 n 0000008256 00000 n We begin by drawing a picture 0000027577 00000 n First consider the red-blue hackenbush game in which both players have identical ﬁgures consisting of all branches of their own color. CG|CGT Surreal numbers In this way we de nesurreal numbers: \decent" pairs of sets of previously de ned surreal numbers: all elements from the left set are smaller than those from the right set. 9. Electric actuators - programmable in 15 minutes! Stack Exchange Network. Exercise: Give a position with value 3/8. Combinatorial game theory (CGT) is a branch of applied mathematics and theoretical computer science that typically studies sequential games with perfect information.Study is largely confined to two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. DALHOUSIE UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled “ON THE STRUCTURE OF 178k 12 204 331. 0000007302 00000 n This preference is consistent for individuals who require care and for those who provide the care (Kellett, 1999a). 0000043257 00000 n By Ginger Mayerson. Recommended strategy for L: Make the move that reduces the value of the game the least. Family care can occur across the life span and refers to unpaid assistance to a patient by family and friends in the patient's place of residence (Archbold et al., 2001). least magnitude, and among other numbers, the simplest is the dyadic rational, J=2K, with smallest nonnegative denominator, K. Several structural theorems in WW provide polynomial-time algorithms for sums of strings, trees, spiders, and several other classes of LR Hackenbush positions… 0000002855 00000 n 0000015871 00000 n G. R. will always be the same. %PDF-1.4 %�� If Left has to move from this position then the resulting positions have values of 0, -1 and -2. Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information.Study has been largely confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. In a partisan game, such as red-blue hackenbush, these options can be diﬀerent. Give a position with value 3/8. 0000046231 00000 n 0000055760 00000 n Green Hackenbush Green Hackenbush is an impartial combinatorial game. Wikipedia . 0000005638 00000 n dyadic number with the lowest denominator (power of 2). 0000039886 00000 n Edit (by request): A game of Red-Blue Hackenbush is a rooted graph G whose edges are either red or blue. 0000060643 00000 n H��U{T��a�]�uR�!3��]kN��Q�͂��+��]��T��1�$�q�&5��=�1�߬w��b�is�?��;�}��q�% G�$I�ߺ�}��{��m�7׾2MdIq��>�| [��/��+�� ܕ�n�I��?��'*Q=��C=oɒ�j��]1ꠈ�(큈D��~u��]��`�?��m|�6�g^�xA(b�1݁� ��G b1I��d�Ņ�M�qX��D�#B�Md81��4� q��d8YG��8��. Your idea is correct. A sample game of Red-Blue Hackenbush. Dr. Hackenbush Gains Perspective AIDS and Class Warfare . For instance, if the next permutation is {a,b,d,c}: {e,a,b,d,c} {a,e,b,d,c} {a,b,e,d,c} {a,b,d,e,c} {a,b,d,c,e} It is clear to see why the algorithm works as intended. 0000014990 00000 n We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in misère-play, as well as in normal-play. Features: Set up many moves, each with individual values for position, speed, accel, etc. Y��)�� z��;-Vk2g��Z"ժ( A��ܠS�b��U��Wo\ߺ�L�=})\��ꩩ��o�]�hV��Bc�5��һ�Fێ�/%��.颕5��j��ۘ�l&/�x|ހ��+��aF��z*��!�U+T�f/G��2�,��¼��p"Q��F�U�b:Ú��-�gHU'N-�(?ZԴo�L^��zܾ@�)��5��﹛�x�^��h5��D�K]>��A�B�2�A(��0ZBgЧGW7scs��p=��V/}>RnV 0000020942 00000 n 0000043235 00000 n Theformofapositionisimportant. H�TQ�n�0��St��L�C��"ѩ�Zdg�ߗ��,��t��v� �{�'h;�"��%Z��;Y��u��7$ �q�~��gX��G� �-:��?��}~q��=� 4 8l�ܼ��jz�c�q��Ϯƃ�1��3�VYzm@��D�&N��6Q��ݳR�� h�oj¹c��-��i��R��*.A>W�ꦟ�t�RS�%��:I�L�U��`��:��a��u� �ܒ>_�S�ge/1R��,iqN��!p,��_ ?D�p Original Hackenbush: All line segments are the same color and may be cut by either player. 0000046913 00000 n This new edition ends with an epilogue that sets the stage for further research on surreal numbers. Answer to 7. dyadic number with the lowest denominator (power of 2). 0000056830 00000 n Every finite stalk has a value that is a dyadic integer, but infinite stacks can have values that are infinite, or that differ from a dyadic integer by an infinitesimal. 0000004169 00000 n The next simplest surreal numbers are rational numbers with denominators that are powers of 2 (dyadic rationals), where the simpler dyadic rationals have smaller denominators. 2.2 Deﬁnition of a vector A vector is deﬁned as a quantity havingmagnitude and direction.1 Vectors are represented graphically with straight or curved arrows. 7. Hackenbush. 0000059808 00000 n Show that f0 j100 g= 1. 0000014175 00000 n 0000001750 00000 n It turns out then that the subgroup of the real numbers corresponding to finite Hackenbush positions is the group of dyadic … English Wikipedia has an article on: Nimber#Hackenbush. 0000030557 00000 n Proof. Less information is available about whether induced social status influences dyadic coordination during motor interactions. Plays, Dyadic rational, Ordinal number. H�TP=o� ��ԁ|5]KN�2\[5��89�� B��Ru �g��h�^Z�=�7�=��(��8�p�A�PZ�=K��:{[�O��8{��|��@ߝB�� [��n��G42��6Wa�Ĉ@#��Hy�X�8��_#�Ƹ��!�:��pª�&Ū24U��*�� An increasing body of research has documented that when a relative requires care due to illness or disability, the preference of most adults is for that care to be provided by family or friends (Cutrona & Cole, 2000). Red-Blue Hackenbush is a two-person game – let’s call these two people Left and Right – best played on a chalkboard. share | cite | improve this answer. 0000031030 00000 n 0000002410 00000 n 0000060012 00000 n By analyzing existing cross correlation between Bluebird Bio and Dyadic International, you can compare the effects of market volatilities on Bluebird Bio and Dyadic International and check how they will diversify away market risk if combined in the same portfolio for a given time horizon. trailer << /Size 171 /Info 99 0 R /Root 102 0 R /Prev 176488 /ID[<8757b9d0283ac30e8be579763048218e><2d733564dc506498af925bb028d315e6>] >> startxref 0 %%EOF 102 0 obj << /Type /Catalog /Pages 98 0 R /Metadata 100 0 R >> endobj 169 0 obj << /S 1200 /Filter /FlateDecode /Length 170 0 R >> stream 0000034985 00000 n 0000020378 00000 n 0000048568 00000 n 0000035767 00000 n 0000007058 00000 n UCDP Dyadic Dataset Codebook Version 20.1 Uppsala Conflict Data Program Department of Peace and Conflict Research Uppsala University Centre for the Study of Civil Wars International Peace Research Institute, Oslo This version compiled and updated by Therese Pettersson (2020) When using this data, please always cite: This is the same permutation with ‘e’ inserted at every possible position in order. Abstract: A Hackenbush Sprig is a Hackenbush String with the ground edge colored green and the remaining edges either red or blue. 0000051961 00000 n (Hint: Use HW 2, #3 and HW 3, #2.) Hackenbush. simplest is the one of least magnitude, and among other numbers, the simplest is the dyadic rational, J{2K, with smallest nonnegative denominator, K. Several structural theorems in WW provide polynomial-time algorithms for sums of strings, trees, spiders, and several other classes of LR Hackenbush positions. Restricted universes are of particular interest to those studying TRUE. The simplest or oldest number in the interval is de ned as follows. Proper noun . 0000015551 00000 n First consider the red-blue hackenbush game in which both players have identical ﬁgures consisting of all branches of their own color. As no active threats were reported recently by users, hackenbush.org is SAFE to browse. Clash Royale CLAN TAG #URR8PPP 5 1 I'm reading Berlekamp/Conway/Guy's Winning Ways for Your Mathematical Plays. 0000034074 00000 n To enter editing 0000004344 00000 n xref ... L must always move to a position that is non-negative. the blog. 0000038320 00000 n We call it the mirror strategy. If you are playing a game, you are in playing mode and cannot edit the position. 96 56 0000028222 00000 n Mis`ere-play Hackenbush Sprigs Neil A. McKay, ... of star and a dyadic rational) ... A position of the form ∗:H for some H is called star-based. 0000034635 00000 n 0000033529 00000 n It’s 1984 and Hackenbush’s broken heart is on the mend as she assists the very roguish, but devastatingly charming theater director, Monte Vista, in his last and greatest production. A Hackenbush Sprig is a Hackenbush String with the ground edge colored green and the remaining edges either red or blue. 0000049996 00000 n It turns out then that the subgroup of the real numbers corresponding to finite Hackenbush positions is the group of dyadic rationals. Original Hackenbush: All line segments are the same color and may be cut by either player. the convention Hackenbush uses is to always assume that a winnable position is winnable by Blue. 0000034334 00000 n the dyadic relationship in lmx is quizlet 0000056808 00000 n 0000003017 00000 n For the next permutation, we reverse the positions where we insert the ‘e’. 0000027191 00000 n 0000031052 00000 n (a) If the interval contains at least one integer, the simplest number in the interval is the integer with smallest absolute value. 4 .34For any integers a ≥ b, there is a Toads and Frogs position with value {a|b}. Exercise: Show that f0 j100 g= 1. 0000036840 00000 n A Short Guide to Hackenbush Padraic Bartlett – VIGRE REU 2006 August 12, 2006 1 Let’s Play A Game! Let Gbe a position of Hackenbush. 0000045913 00000 n 0000013939 00000 n Let G be a Red-Blue Hackenbush position or any game. 0000002779 00000 n Explore releases from Dr. Hackenbush at Discogs. The set of all dead-ending games is thus a meaningful (and large) universe to consider. ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in mis\`ere-play, as well as in normal-play. 0000047772 00000 n This means payoffs are symmetric and each player has the same operations based on position on board (in this case structure of drawing) Blue-Red Hackenbush: Each line segment is colored either red or blue. This means payoffs are symmetric and each player has the same operations based on position on board (in this case structure of drawing) Blue-Red Hackenbush: Each line segment is colored either red or blue. If L moves, he can only chop o the bottom branch, leaving 0 branches, or the endgame. Player x plays with group A and player y plays with group B. x makes the move first. A mathematical game for two players, involving the turn-by-turn removal of line segments. Hackenbush is a two-player mathematical game that may be played on any configuration of colored line segments connected to one another by their endpoints and to the ground. �H@ej��f��q��ױ�\�$SӞqCm��N6.�` �&�r x�b```f``=�� @1v�(`�/�Lv��L��+9'�xqg��q�r. I will call these green line segments "edges." The addition operation is simply the disjoint union with roots identified. 0000001416 00000 n 9. 0000003506 00000 n 101 0 obj << /Linearized 1 /O 103 /H [ 1825 1192 ] /L 178638 /E 60873 /N 10 /T 176499 >> endobj xref 101 70 0000000016 00000 n For example, the vectors depicted below are directed to the right, left, up, down, out from the page, into the page, and inclined at 45 , respectively. 0000048224 00000 n %PDF-1.2 %�� 0000058932 00000 n 0000044176 00000 n 0000048319 00000 n trailer We show that in canonical form … 0000060034 00000 n 0000033188 00000 n If all the components are purely red or blue, we can assign a value of +nto a componentwith n edges if it is … 0000036195 00000 n 8. 0000049327 00000 n 0000045836 00000 n We show that in canonical form a Sprig is a star-based number (the ordinal sum of star and a dyadic rational) in misère-play, as well as in normal-play. 0 (42 pts) Find a dyadic position equivalent to the following given positions: (a) A 3 6 board in Cut-Cake.ⅹ (b) The Domineering position (c) The sum of the position in (a), (b), and (-5/4).⦁ 10. 0000014594 00000 n Clash Royale CLAN TAG #URR8PPP 5 1 I'm reading Berlekamp/Conway/Guy's Winning Ways for Your Mathematical Plays. For instance, the following is a valid position in Hackenbush: Alice and Bob have opted out of being used as example names on Complex Projective 4-Space , so I’ll instead have to resort to using names of people who particularly enjoy the site. Chapter. 0000006012 00000 n 0000035003 00000 n 0000049580 00000 n (j) It is not the case that every one-stalk game of Red-Blue Hackenbush has a value that is a real number. 0000032954 00000 n 0000002812 00000 n Each player has the same number of possible moves. A dyadic number is a rational number that of the form a b where ais an odd integer ... below), Cut-Cake, Hackenbush. 0000048050 00000 n 0000047930 00000 n 0000019971 00000 n We choose to let "$1$" be a position with $1$-move advantage for Left, so that we can loosely think of the number associate to a position as "the number of moves that Left is ahead by". SPEDIZIONE GRATUITA su ordini idonei In gure b, if R moves rst, he can only chop o the top branch, leaving 1 branch for L and a game with a value of 1 since L now has a one move advantage. 0000038994 00000 n hackenbush.org is 1 decade 8 years old. In a partisan game, such as red-blue hackenbush, these options can be diﬀerent. Download Citation | On May 8, 2018, Elwyn R. Berlekamp and others published Hackenbush | Find, read and cite all the research you need on ResearchGate. So the game is denoted as G = ff;j;gj1g. English . From any position, exactly the same moves are legal for each of th e two playe rs. 0000049701 00000 n A Hackenbush Sprig is a Hackenbush String with the ground edge colored green and the remaining edges either red or blue. (cable sold separately) This servo actuator system is unique in that the motor, encoder, drive and the program memory are one integral package. 0000033732 00000 n 0000040312 00000 n 0000048797 00000 n 0000007178 00000 n It's helpful to think of the surreal value of the game G as the number of moves by which Blue is ahead. 0000011496 00000 n Consider the position G¯ where all the colors are changed (bLue becomes Red, and Red becomes bLue), then the game G+ G¯ is a zero game. Remember we are strictly looking at removing blue values only right now and determining what the game value would be after removing each blue edge. We show that for any given dyadic rational x, there is a unique TOPPLING DOMINOES position G equal to x, and that G is necessarily a palindrome. 0000013642 00000 n G. L. and. 0000027749 00000 n 0000011384 00000 n SemAlg Knuth and Conway Donald E.(Ervin) KnuthJohn H.(Horton) Conway 1938, US 1937, UK !US NP; KMP Co1, Co2, Co3 TEX Doomsday algoritme change-ringing; 3:16 game of Life; Angel problem The Art of Computer Winning Ways for your Programming Mathematical Plays Surreal … 0000026832 00000 n 9. There are some piles of numbers. 0000044198 00000 n 0000047504 00000 n 96 0 obj <> endobj 0000004008 00000 n 0000019533 00000 n It is a domain having org extension. endstream endobj 111 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 122 /Widths [ 671 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 575 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 575 575 575 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 383 882 756 724 383 900 383 383 383 383 1092 900 383 786 383 383 639 383 383 869 1189 383 383 383 383 383 383 383 383 383 559 639 511 639 527 351 575 639 319 351 607 319 958 639 575 639 607 474 454 447 639 607 831 607 607 511 ] /Encoding 113 0 R /BaseFont /BPOANC+CMBX10 /FontDescriptor 112 0 R /ToUnicode 110 0 R >> endobj 112 0 obj << /Type /FontDescriptor /Ascent 698 /CapHeight 679 /Descent -202 /Flags 262150 /FontBBox [ -301 -250 1164 946 ] /FontName /BPOANC+CMBX10 /ItalicAngle 0 /StemV 114 /XHeight 447 /StemH 47 /CharSet (/g/e/o/hyphen/m/t/r/i/c/a/l/M/u/p/n/f/v/w/h/s/V/d/y/D/ff/x/z/j/k/S/q/E/W\ /zero/circumflex/N/b/P/F/H/slash/one) /FontFile3 114 0 R >> endobj 113 0 obj << /Type /Encoding /Differences [ 1 /ff 26 /circumflex 45 /hyphen 47 /slash /zero /one 68 /D /E /F 72 /H 77 /M /N 80 /P 83 /S 86 /V /W 97 /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z ] >> endobj 114 0 obj << /Filter /FlateDecode /Length 4101 /Subtype /Type1C >> stream %%EOF By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers (so named by Donald Knuth) that includes at the same time the real numbers and the ordinal numbers. 0000006058 00000 n • Many Hackenbush positions can be broken up into components where the components have no edges or vertices in common except possibly for the vertex on the ground. H��[h�VǏ�d�R,�v㻭n�l�C)��:k�dM��i��&�_"'͚t�{`�1�ԗA�P�R�0��=�б�m�=��i��aB�~�_�s� b P@ �@��5H �>�|�0�#�7�_��`i��S ��|�˻w��|�|�FV9d��(�D�f��&7*pT��z8�:�t�W.��J��,ϳV�ym�R��GB��%"f診(��mF��c��4R��F�i3��:��^jĲk�y��l%/F��}�)T��J3#rM��(Ϛ-&kc~mAAӌf@EPZ�X�H��j��٪��6�6s��'��x�� 0000034493 00000 n Jump to navigation Jump to search. 2 .32For any dyadic rational number q, there is a Toads and Frogs position with value q. Corollary 3 . 1.Show that every position in Blue-Red Hackenbush is a number. Here are some examples of using the simplicity theorem. Definition from Wiktionary, the free dictionary. (18 pts) Draw each of the given dyadic positions in Hackenbush: (a) (5/8)⦁ (b) (23/32)⦁ (c) (-121/64) ⦁ 8. ��P�Z�!��@�,4��۝./��ZFM#�A��p�±�؎��8�� &A�H�\��j��p�C9��Mj\��`�n��5M��|�k"�V9�N)4r�uE��ⶦ��+��>�D�}�� 0000032061 00000 n Eric Wofsey. endstream endobj 106 0 obj << /Type /FontDescriptor /Ascent 699 /CapHeight 679 /Descent -203 /Flags 262212 /FontBBox [ -29 -250 1274 754 ] /FontName /BPOANB+CMBXTI10 /ItalicAngle -14.03999 /StemV 107 /XHeight 455 /StemH 47 /CharSet (/v/e/c/t/o/r/s/d/y/a/i/p/l/n/f/w/m/g/u/b/z/D/q/j/V/x/h/hyphen/C/A) /FontFile3 115 0 R >> endobj 107 0 obj << /Type /Encoding /Differences [ 45 /hyphen 65 /A 67 /C /D 86 /V 97 /a /b /c /d /e /f /g /h /i /j 108 /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z ] >> endobj 108 0 obj << /Type /Font /Subtype /Type1 /FirstChar 1 /LastChar 148 /Widths [ 833 583 833 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 389 389 333 778 278 333 278 500 500 500 500 500 500 500 500 500 500 500 278 333 333 778 333 333 333 750 333 722 764 681 653 785 750 361 514 778 333 917 750 778 681 778 736 556 722 750 750 1028 333 333 333 278 333 278 333 333 333 500 556 444 556 444 306 500 556 278 306 528 278 833 556 500 556 528 392 394 389 556 528 722 528 528 444 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 500 500 333 278 333 333 556 556 ] /Encoding 125 0 R /BaseFont /BPOAMG+CMR10 /FontDescriptor 121 0 R /ToUnicode 123 0 R >> endobj 109 0 obj << /Type /Font /Subtype /Type1 /FirstChar 45 /LastChar 122 /Widths [ 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 866 414 827 876 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 866 414 414 414 414 414 414 414 414 414 414 591 532 532 591 532 400 532 591 356 356 414 297 944 650 591 591 532 502 487 385 621 532 768 561 562 491 ] /Encoding 107 0 R /BaseFont /BPOANB+CMBXTI10 /FontDescriptor 106 0 R /ToUnicode 105 0 R >> endobj 110 0 obj << /Filter /FlateDecode /Length 299 >> stream
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