Domination and Independence on Square Chessboard


 Francis McCormick
 2 years ago
 Views:
Transcription
1 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq lon.edu.iq Domintion nd Independence on Squre Chessbord Abstrct In this pper, new ide for the problems of independence nd domintion on chessbord is introduced. Two clssicl chessbord problems of independence nd domintion on squre chessbord with squre cells of size n re determined for some cses when two different types of pieces re used together. For independence, fixed number of the first type of pieces is plced on the bord with mximum number of pieces of the second type together. For domintion, fixed number of pieces of first type is plced on the bord with minimum number of pieces of the second type. The pieces which re used together in this pper re: kings with rooks, kings with bishops, nd rooks with bishops. Received on: 0/06/015 Accepted on: /0/017 eywords Domintion, Independence, Squre chessbord, ings, ishops nd Rooks. How to cite this rticle: A.A. Omrn, Domintion nd Independence on Squre Chessbord, Engineering nd Technology Journl, Vol. 5, Prt, No.1, pp , Introduction In the chessbord there re six kinds of pieces. Let "P" be piece of ny kind on the chessbord. There re two clssicl chessbord problems; one of them is by plcing mximum number of pieces of single kind, such tht ech piece does not ttck other pieces. This problem is clled independence problem, nd the number of pieces tht stisfies this criterion is the independence number. Independence number for "P" kind is denoted by β(p). The other problem is by plcing minimum number of single kind P, such tht ll unoccupied positions re under ttck by t lest one of the plced pieces. This problem is clled domintion problem nd the number of pieces tht stisfies this criterion is clled domintion number of "P" nd denoted by γ(p). Previous studies were concerned with domintion nd independence problems with single kind of pieces only, while our current study is concerned in the sme problems but with two kinds of pieces t time. This study cn be used in gme theory or ny similr life problems. For squre chessbord (n n) the independence nd domintion numbers re determined for Rook "R", ishop "" nd ing "". They proved tht γ(r) = n, () = n, γ() = n+, lso, β(r) = n, β() = n nd β() = n+ (see [1], [] nd []). In [], JoeMio nd Willim proved tht γ(r) = min m, n} nd β(r) = min m, n}for m n Toroidl chessbord. In [5], the minimum number of rooks tht cn dominte ll squres of the STC is determined. In [6], the tringulr hexgon bord, in which the cells re hexgons nd the bord is tringle is considered. ishops ttck in stright lines through the vertices of their cells, rooks ttck long stright lines through the centers of the edges of their cells, nd queens hve both ttcks. The only generl upper bound they re ble to give on the independence number of the queens grph is by the rooks bound, which is n+1 for ll n. For n =,, 6, 7, 1, 16, 19, 5, 1, they found tht β = 1, nd for the other n 1, β = n+1. In [7],[8],[9],[10] nd [11] the independence nd domintion in Rhombus chessbord, isosceles tringulr chessbord nd cubic chessbord with squre cells re determined. In this pper, we pply the mening of the two clssicl problems on squre chessbord for two different types of pieces tht hve been chosen. The pieces of first type re plced on the chessbord, nd their number is fixed nd then the domintion or independence number of the other type is determined. Let n P be the number of pieces of the sme kind (P) which re chosen to hve fixed number. N P refers to the number of the cells which re ttcked by one piece (P) in ddition to the cell tht is occupied by the piece (P). n+1 The Chessbord The chessbord in this work is squre chessbord of size n with squre cells. Three types of pieces, rooks R, bishops nd kings re used Copyright 017 by UOT, IRAQ 68
2 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 with their usul moving or ttcking. Let the number of cells (squres) in side be the length of tht side. To simplify the form of our results, the mtrix form is used where r i denote the i th row mesured from bove to down, i = 1,,, n nd let c i denote the j th column mesured from left to right, j = 1,,, n. Let the cell (squre) of i th row nd j th column is denoted by s i,j, i = 1,, n, nd j = 1,, n. Theorem.1 [8] In isosceles tringulr chessbord of size n the domintion number of king pieces (γ()) is given by γ() = (1) n 1 ( n 1 ( n 1 +1) (see Figure 1(); n = 6 ). + ), if n 1 is even, if n 1 is odd } Domintion nd Independence of pieces with fixed number of R pieces. In the squre chessbord in this work let ( n r ) be the number of rook pieces nd let γ(, n r ) nd β(, n r ) be the domintion number nd independence number of king pieces () with fixed number of Rook pieces (R) respectively. Theorem.1. The domintion number of king pieces with fixed pieces( n r ) of rooks in squre chessbord of size n is given by γ(, n r ) = n n r Proof. If the pieces re distributed in the chessbord by plcing the R pieces in the cells s i,i, i = 1,,, n 1 in order to keep the minimum number of domintion (ny other plcing mke prtition on the chessbord) then the mximum of N r of these pieces is gotten, nd squre chessbord of length n n r of cells which re not ttcked by R pieces. (see Figure (b); n = 9). Since, from [5] the domintion number of king in squre chessbord size n is γ() = n+ wich eequl to n. Hence, γ(, n r ) = n n r b Figure 1 This result is used for determining domintion in the isosceles semi tringulr chessbord of size n. In this chessbord, there re two equl sides nd every i th row contin i cells s shown in Figure 1(b), n = 6 Proposition.. In isosceles semi tringulr chessbord of size n the domintion number of king pieces (γ()) is given by R R R n n r γ() = n ( n + ), if n is even ( n +1), if n is odd } () Figure To prove ech of the following theorems we must refer to (1) The remining P pieces in ny step of the proof which we shll denoted by z equl to the difference between n r nd the mximum number n r of the previous step. 96
3 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 Figure () The blck cells which pper in our figures represent the plces of pieces. Theorem.. For n 1 the independence number of pieces with fixed number n r of R pieces, where 1 n r n is given by β(, n r ) = n (n r + 1), if n 1, ( mod ) } ( n ) (n r ), if n 0, ( mod ) Proof. The independence number of the king piece on squre chessbord is β() = n ccording to [6], where the independence distribution of pieces is s shown in Figure (); where n = 1 (shded cells). The ide is to distribute the pieces of R such tht, it ttck minimum number of pieces to keep mximum number of pieces on the chessbord, nd no piece ttcks ny of R pieces. The column nd row for ny cell of the piece contin n pieces of where n 1, ( mod ), nd n 1 pieces of where n 0, (mod ). For this ide, we hve two cses tht depend on the length n of the squre chessbord nd s follows. (i) If n 1, ( mod ): the suitble cell to plce the first R piece is s,, since the neighborhood is the minimum. In this plce the first R piece does not ttck ny of the pieces, but there re four pieces djcent to it. So we must remove the djcent pieces nd we denote ech cell of these pieces by "x" s shown in Figure (b); for n = 1. To tke dvntge from the first R piece, we plce the second R piece in cell such tht N(R) is shred with the most number of pieces in the neighborhood of the first R piece. Now, if the neighborhood of the second R piece is shred with two pieces from the neighborhood of the first R piece, then the plce of the first nd second R pieces re in the sme column or the sme row nd this men tht the two R pieces re not independent. So the most shred pieces between the neighborhood of the first nd second R piece is one piece, thus suitble cell to plce the second R piece is s,. The second R piece is djcent to other three pieces, so we must remove these pieces nd we denote ech cell of the removble king by "x". Continue to plce other R pieces in the cells s i,i, i = 1,,.., n in order (see Figure (b); n = 1). Then we get β(, n r ) = n (n r + 1). (ii) If n 0, ( mod ): In the sme mnner in the previous cse, the suitble cells to plce the R pieces re s i+,i+1, i = 0,,.., n 1, in order. The first R piece does not ttck ny piece nd djcent to one piece, so this piece must be removed. The second R piece is djcent to other three pieces, so these pieces must be removed (see Figure ()). Continue to this procedure until n r = n. Therefore, β(, n r) = n (n r ). The following exmple illustrting the bove theorem for different vlues of n r. Exmple.. 1) n = 1, n r = 6, implies β(, 1) = 0, (see Figure ). ) n = 1, n r =, implies β(, 8) = 9 (see Figure ).
4 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 Figure Domintion nd Independence of pieces with fixed number of pieces Let the domintion (Independence) number of pieces with fixed number n b of pieces re denoted by γ(, n b ) ( β(, n b )). Theorem.1. The domintion number of pieces with fixed pieces n b in squre chessbord is given s follows. For n >, let m = ( n n b ), if μ 1 = μ = m 1 μ = m ( m +1) ( m + ), if m is even },, if m is odd ( m 1 + ), if m 1 is even ( m 1 +1) m+ 1, if m 1 is odd } ( m ), if m + 1 is even ( m+ 1 +1) μ = nd μ 5 = m m 1 ( m 1 +1), if m + 1 is odd } ( m 1 + ), if m 1 is even }, if m 1 is odd ( m + ), if m is even ( m +1), if m is odd },,, then γ(, n b ) = μ μ 5 + μ + μ μ + μ μ + μ 1 + μ 5., if n nd n b re odd, if n is odd nd n b is even }, if n is even nd n b is odd, if n nd n b re even Proof. In ech cses below 1 n b n1. Cse 1) If n is odd: There re two cses tht depend on n b s follows. (i) If n b is odd, then n b pieces of re plced in the middle of the column c n, since these cells give the mximum of N b, so tht the minimum number of pieces is gotten. These pieces of re distributed on the cells which re not ttcked. These cells from four isosceles tringulr chessbord with sme size ( n n b ) in the squre chessbord s shown in Figure 8(); n = 11. Using equtions (1) nd () in section, the result is gotten. (ii) If n b is even, gin the n b pieces of re plced in the middle of the column c n, nd the cells which re not ttcked by these pieces form four shpes. Two of these shpes form two semi isosceles tringulr chessbord with the size ( n n b ), nd the other two form two isosceles tringulr chessbord with different sizes ( n n b ) nd ( n n b + 1), s shown in Figure 8(b); n = 11. y using equtions (1) nd (), the result is obtined.
5 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 b Figure 8 b Figure 9 Cse ) If n is even: gin there re two cses with size ( n depend on the vlue of n b s follows. n b ), nd the other two re isosceles semi tringulr chessbord one of them (iii) If n b is odd, then n b pieces of in the middle of column c n +1, re plced. So, the mximl of N with size ( n b n b ) nd the other with size (n nd the minimum number of pieces re obtined. n b 1) s shown in Figure 9(b). Using the These pieces re distributed on the cells which equtions (1) nd () the result is obtined. re not ttcked by n b pieces of. The cells which The following exmple illustrting the bove re not ttcked by n b pieces of form four theorem for different vlues of n nd n b. isosceles tringulr chessbords, two of them re of size ( n n b + 1) nd the others re of size ( n n b ) s shown in Figure 9(). y using the equtions (1) nd (), we get the result. (iv) If n b is even, gin we plce the n b pieces of in the middle of the column c n +1, the cells not ttcked by these pieces form four shpes. Two of these shpes re isosceles tringulr chessbord Exmple.. 1) n = 11, n b =, then γ(, ) = 16, (see Figure 8()) ) n = 11, n b =, then γ(, ) = 10, (see Figure 8(b)) ) n = 10, n b =, then γ(, ) = 1, (see Figure 9()) ) n = 10, n b =, then γ(, ) = 8, (see Figure 9(b))
6 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 Theorem. The independence of pieces with fixed number n b of pieces is given by the following: (i) If n is odd, then β(, n b ) = n (n b + 1), if 1 n r n n (n b + ), if n < n r n 1 } (ii) If n is even, then β(, n b ) = ( n ) n b, if 1 n r n ( n ) (n b + 1), if n < n r n 1 } Proof. There re two cses tht depend on the vlue of n s follows. (i) If n is odd: two successive cses tht depend on n b re obtined s follows. () If 1 n b n, the distribution of the pieces is s before in Section, nd cell is looked for plesing one piece such tht the minimum number of pieces is ttcked by it. The suitble cells of this ide re one of s n,j ; j = 1,,, n, where no pieces re ttcked. For the first piece there re two djcent of pieces to this piece, nd there is one piece djcent to the second piece. So, the djcent pieces must be removed. Continue with the sme mnner for the other pieces until reching cell s n, n s shown in Figure 10(); n = 11. According to [6], where β() = n, hence, (n b + 1). β(, n b ) = n (b) n < n r n 1, by plcing the remining z pieces in the cells s 1,j ; j = 1,,, n in order (see Figure 10; n = 11), s in (i), we get β(, n b ) = n (n b + ). (ii) If n is even: Agin we hve two successive cses depend on n b s follows: () If 1 n b n, the distribution of the pieces is s before in Section. Now, the next step is looking for cell to plce one piece such tht the minimum number of pieces is ttcked by it. The suitble cell of this ide is one of the cells s 1,j ; j = 1,,, n in order. There is one piece djcent to the first piece; therefore removing of this piece is necessrily. y continuing with other bishop pieces s the sme mnner (see in Figure 10(b); n = 1). Thus, β(, n b ) = n n b. (b) If n < n b n 1, by plcing the remining z pieces in the cells s n,+j ; j = 0,1,, n in order. There re two pieces djcent to the first piece, so we must remove these pieces. There is one djcent piece to the second piece, so we must remove this piece. We continue with the sme mnner for the other pieces until reching the cell s n,n 1 s shown in Figure 10(b); n = 1. Hence, we get β(, n b ) = n (n b + 1). b Figure 10 Note.. When n 1 < n r n, it is not esy to find generl formul for β(, n b ). The following exmple illustrting the bove theorem for different vlues of n nd n b.
7 R Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 Exmple.5. 1) n = 11, n b = 10, then β(, 10) =, (see Figure 10()) ) n = 1, n b = 11, then β(, 11) =, (see Figure 10(b)) R n n r b Figure 11 5 Domintion nd Independence of pieces with fixed number of R pieces. We denote the domintion (Independence) number of pieces with fixed number n r of R pieces by γ(, n r ) ( β(, n r )). Theorem 5.1. The domintion (independence) of pieces γ(, n r ) (β(, n r )) with fixed number n r of R pieces is given by the following: (i) γ(, n r ) = n n r (ii) β(, n r ) = (n n r ) Proof. The ide is to plce n r pieces of R nd then distributing the pieces to get the domintion (independence) number of the pieces together with fixed number n r. (i) We look for cell to plce the first R piece such tht we obtin the mximum of N R. Therefore we cn distribute minimum number of pieces such tht they re not ttcked by the first R piece. The suitble cells for (n 1) R pieces distribution re the min digonl of the chessbord s i,i, i = 1,.., r in order. The vcuum cells which re not ttcked by R pieces form squre chessbord of length n n r s shown in Figure 11(); n =10. We know tht γ() = n (see [6]), where n is the size of squre chessbord. Therefore by our distribution we get γ(, n r ) = n n r. (ii) Similrly s in (i) we plce the R pieces in the min digonl of squre chessbord s i,i, i = 1,.., r in order. The cells which re not ttcked by these pieces form s squre chessbord of length n n r. We know tht β() = n n n r (see [6]) where n is the size of the squre chessbord. So we distribute (n n r )  in the chessbord of size n n r, but we must remove the piece from the min digonl, since it is ttcked by the R pieces (see Figure 11 (b), 11(c); n = 10). Thus we get β(, n r ) = (n n r ). The following exmple illustrting the bove theorem for different vlues of n r. Exmple 5.. 1) n = 10, n r = 1, then γ(,1 ) = 9, (see Figure 11()). ) n = 10, n r = 1, then β(, 1) = 15, (see Figure 11(b)). Open problems for two different types of pieces Find the generl formul of ech of the following numbers γ(r, n k ), γ(, n k ), γ(, n r ), β(r, n k ), β(, n k ) nd β(, n r ). ReferencesNo.6, [1] J. J. Wtkins, C. Ricci," ings Domintion on Torus ", Colrdo College, Colrdo Spring, Co [] O. Fvron," From Irredundnce to Annihiltion : A rief Overview of Some Domintion Prmeters of Grphs", Sber, Universidd de Oriente, Venezuel,1(), 669, 000. [] R.Lskr, C.Wllis,"Chessbord Grphs, Relted Designs, nd Domintion Prmeters", Journl of Sttisticl Plnning nd Inference, (76), 859,1999. [] J. DeMio nd W. Fust, "Domintion nd Independence on the Rectngulr Torus by Rooks nd ishops ", Deprtment of Mthemtics nd Sttistics ennesw Stte University, ennesw, Georgi, 01, USA [5] H. Chen, T. Hob,"The Rook Problem on Swtoothed Chessbords", Applied Mthemtics Letters,( 1),1 17,008. [6] H.Hrborth, V. ultn,. Nyrdyov, Z. Spendelov, "Independence on tringulr
8 Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 hexgon ords", in: Proceedings of the ThirtyFourt Southestern Interntionl Conference on Combintorics, Grph Theory nd Computing,160, 15, 00. [7] A. A.Omrn,et l., " Independence in Isosceles Tringle Chessbord ", Applied Mthemticl Sciences, 6 (11),65165, (01). [8] A. A. Omrn et l., "Domintion in Isosceles Tringle Chessbord", Mthemticl Theory nd Modeling Journl, 7(),7180, 01. [9] A. A. Omrn, et l., "Independence in Rhombus Chessbord ", Archive Des Sciences Journl,(66), 858, 01. [10] A. A. Omrn, et l., "Domintion in Rhombus Chessbord ", Journl of Asin Scientific Reserch, (5), 859, 01. [11] A. A. Omrn, Domintion nd Independence in Cubic Chessbord, E Eng. nd Tech. Journl, (1),Prt (), 659, 016. Author biogrphy Ahmed Abed Ali Omrn Deprtment of Mthemtics, Collegeof Eduction for Pure Science, University of bylon, bylon, Irq Prgrph He published previously in the field of combintorics. He is member of hwrizmi legue.
Spiral Tilings with Ccurves
Spirl Tilings with curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More informationLECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY
LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd
More informationMATH 118 PROBLEM SET 6
MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the
More informationSection 16.3 Double Integrals over General Regions
Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is
More informationMAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES
MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine Emil: rt_science@hotmil.com 1 INTRODUCTION During the considertion
More informationSeven Sisters. Visit for video tutorials
Seven Sisters This imge is from www.quiltstudy.org. Plese visit this website for more informtion on Seven Sisters quilt ptterns. Visit www.blocloc.com for video tutorils 1 The Seven Sisters design cn be
More informationGeneral Augmented Rook Boards & Product Formulas
Forml Power Series nd Algebric Combintorics Séries Formelles et Combintoire Algébriue Sn Diego, Cliforni 006 Generl Augmented Rook Bords & Product Formuls Brin K Miceli Abstrct There re number of soclled
More information10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides  Solving Generl Tringles  Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More informationPolar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xyplne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationStudent Book SERIES. Fractions. Name
D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types
More informationLecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More informationExercise 11. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 11 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationCongruences for Stirling Numbers of the Second Kind Modulo 5
Southest Asin Bulletin of Mthemtics (2013 37: 795 800 Southest Asin Bulletin of Mthemtics c SEAMS. 2013 Congruences for Stirling Numbers of the Second Kind Modulo 5 Jinrong Zho School of Economic Mthemtics,
More informationMETHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin
METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or
More informationStudy on SLT calibration method of 2port waveguide DUT
Interntionl Conference on Advnced Electronic cience nd Technology (AET 206) tudy on LT clibrtion method of 2port wveguide DUT Wenqing Luo, Anyong Hu, Ki Liu nd Xi Chen chool of Electronics nd Informtion
More informationAbacabaDabacaba! by Michael Naylor Western Washington University
AbcbDbcb! by Michel Nylor Western Wshington University The Abcb structure shows up in n mzing vriety of plces. This rticle explores 10 surprising ides which ll shre this pttern, pth tht will tke us through
More informationCS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates
Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the
More informationSection 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.
Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line
More informationConvolutional Networks. Lecture slides for Chapter 9 of Deep Learning Ian Goodfellow
Convolutionl Networks Lecture slides for Chpter 9 of Deep Lerning In Goodfellow 20160912 Convolutionl Networks Scle up neurl networks to process very lrge imges / video sequences Sprse connections Prmeter
More informationGeometric quantities for polar curves
Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions
More informationCHAPTER 2 LITERATURE STUDY
CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationMesh and Node Equations: More Circuits Containing Dependent Sources
Mesh nd Node Equtions: More Circuits Contining Dependent Sources Introduction The circuits in this set of problems ech contin single dependent source. These circuits cn be nlyzed using mesh eqution or
More informationEE Controls Lab #2: Implementing StateTransition Logic on a PLC
Objective: EE 44  Controls Lb #2: Implementing Stternsition Logic on PLC ssuming tht speed is not of essence, PLC's cn be used to implement stte trnsition logic. he dvntge of using PLC over using hrdwre
More informationBoolean Linear Dynamical System (Topological Markov Chain)
/ 23 Boolen Liner Dynmicl System (Topologicl Mrkov Chin) Yokun Wu (~ˆn) Shnghi Jio Tong University (þ ÏŒA) S ŒAŽA AAH 7 ÊcÊ ÔF(.õ&F) Dynmicl system sed on one digrph 2 / 23 2 5 3 4 Figure: A digrph, representing,
More informationREVIEW, pages
REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:
More informationSequential Logic (2) Synchronous vs Asynchronous Sequential Circuit. Clock Signal. Synchronous Sequential Circuits. FSM Overview 9/10/12
9//2 Sequentil (2) ENGG5 st Semester, 22 Dr. Hden So Deprtment of Electricl nd Electronic Engineering http://www.eee.hku.hk/~engg5 Snchronous vs Asnchronous Sequentil Circuit This Course snchronous Sequentil
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationAlgorithms for Memory Hierarchies Lecture 14
Algorithms for emory Hierrchies Lecture 4 Lecturer: Nodri Sitchinv Scribe: ichel Hmnn Prllelism nd Cche Obliviousness The combintion of prllelism nd cche obliviousness is n ongoing topic of reserch, in
More informationSection 10.2 Graphing Polar Equations
Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The
More informationShuli s Math Problem Solving Column
Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Edited d Authored by Shuli Sog Colordo Sprigs, Colordo shuli_sog@yhoocom Cotets Mth Trick: Metl Clcultio: b cd Mth Competitio Skill: Divisibility by
More information& Y Connected resistors, Light emitting diode.
& Y Connected resistors, Light emitting diode. Experiment # 02 Ojectives: To get some hndson experience with the physicl instruments. To investigte the equivlent resistors, nd Y connected resistors, nd
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 300Q Model the concept of ddition for sums to 10. (QT N 36) Model the concept of sutrction using numers less thn or equl to 10. (QT N 37) Write ddition
More informationc The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?
3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem
More informationEQ: What are the similarities and differences between matrices and real numbers?
Unit 4 Lesson 1 Essentil Question Stndrds Objectives Vocbulry Mtrices Mtrix Opertions Wht re the similrities nd differences between mtrices nd rel numbers? M.ALGII.2.4 Unit 4: Lesson 1 Describe how you
More informationPolar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a
Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.
More informationINTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite
More information(CATALYST GROUP) B"sic Electric"l Engineering
(CATALYST GROUP) B"sic Electric"l Engineering 1. Kirchhoff s current l"w st"tes th"t (") net current flow "t the junction is positive (b) Hebr"ic sum of the currents meeting "t the junction is zero (c)
More informationNUMBER THEORY Amin Witno
WON Series in Discrete Mthemtics nd Modern Algebr Volume 2 NUMBER THEORY Amin Witno Prefce Written t Phildelphi University, Jordn for Mth 313, these notes 1 were used first time in the Fll 2005 semester.
More information5 I. T cu2. T use in modem computing systems, it is desirable to. A Comparison of HalfBridge Resonant Converter Topologies
74 EEE TRANSACTONS ON POER ELECTRONCS, VOL. 3, NO. 2, APRL 988 A Comprison of HlfBridge Resonnt Converter Topologies AbstrctThe hlfbridge seriesresonnt, prllelresonnt, nd combintion seriesprllel
More informationAQA Level 2 Further mathematics Further algebra. Section 3: Inequalities and indices
AQA Level Further mthemtics Further lgebr Sectio : Iequlities d idices Notes d Emples These otes coti subsectios o Iequlities Lier iequlities Qudrtic iequlities Multiplyig epressios The rules of idices
More informationMEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR
Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure
More informationUniversity of North CarolinaCharlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009
Problem 1: Using DC Mchine University o North CrolinChrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding:
More informationFubini for continuous functions over intervals
Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =
More informationPerformance Comparison between Network Coding in Space and Routing in Space
Performnce omprison etween Network oding in Spce nd Routing in Spce Yunqing Ye, Xin Hung, Ting Wen, Jiqing Hung nd lfred Uwitonze eprtment of lectronics nd Informtion ngineering, Huzhong University of
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd
More informationPatterns and Algebra
Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing
More informationUnit 1: Chapter 4 Roots & Powers
Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.
More informationAnalysis of circuits containing active elements by using modified T  graphs
Anlsis of circuits contining ctive elements using modified T  grphs DALBO BOLEK *) nd EA BOLKOA**) Deprtment of Telecommunictions *) dioelectronics **) Brno Universit of Technolog Purknov 8, 6 Brno CECH
More informationFP2 POLAR COORDINATES: PAST QUESTIONS
FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)
More informationModule 9. DC Machines. Version 2 EE IIT, Kharagpur
Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson40) 4 40.1 Gols
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationInterference Cancellation Method without Feedback Amount for Three Users Interference Channel
Open Access Librry Journl 07, Volume, e57 ISSN Online: 97 ISSN Print: 9705 Interference Cncelltion Method without Feedbc Amount for Three Users Interference Chnnel Xini Tin, otin Zhng, Wenie Ji School
More informationApplication Note. Differential Amplifier
Appliction Note AN367 Differentil Amplifier Author: Dve n Ess Associted Project: Yes Associted Prt Fmily: CY8C9x66, CY8C7x43, CY8C4x3A PSoC Designer ersion: 4. SP3 Abstrct For mny sensing pplictions, desirble
More informationSolutions to exercise 1 in ETS052 Computer Communication
Solutions to exercise in TS52 Computer Communiction 23 Septemer, 23 If it occupies millisecond = 3 seconds, then second is occupied y 3 = 3 its = kps. kps If it occupies 2 microseconds = 2 6 seconds, then
More informationSimulation of Transformer Based ZSource Inverter to Obtain High Voltage Boost Ability
Interntionl Journl of cience, Engineering nd Technology Reserch (IJETR), olume 4, Issue 1, October 15 imultion of Trnsformer Bsed Zource Inverter to Obtin High oltge Boost Ability A.hnmugpriy 1, M.Ishwry
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationEnergy Harvesting TwoWay Channels With Decoding and Processing Costs
IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting TwoWy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student
More informationECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design
ECE 27 Digitl Logic Shifters, Comprtors, Counters, Multipliers Digitl Design..7 Digitl Design Chpter : Slides to ccompny the textbook Digitl Design, First Edition, by Frnk Vhid, John Wiley nd Sons Publishers,
More informationVector Calculus. 1 Line Integrals
Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q
More informationExperiment 3: NonIdeal Operational Amplifiers
Experiment 3: NonIdel Opertionl Amplifiers Fll 2009 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output
More informationSynchronous Machine Parameter Measurement
Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions
More information13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.
CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny
More informationABB STOTZKONTAKT. ABB ibus EIB Current Module SM/S Intelligent Installation Systems. User Manual SM/S In = 16 A AC Un = 230 V AC
User Mnul ntelligent nstlltion Systems A B 1 2 3 4 5 6 7 8 30 ma 30 ma n = AC Un = 230 V AC 30 ma 9 10 11 12 C ABB STOTZKONTAKT Appliction Softwre Current Vlue Threshold/1 Contents Pge 1 Device Chrcteristics...
More informationMONOCHRONICLE STRAIGHT
UPDATED 092010 HYDROCARBON Hydrocrbon is ponchostyle cowl in bulkyweight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the
More informationFrancis Gaspalou Second edition of February 10, 2012 (First edition on January 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY?
Frncis Gslou Second edition of Ferury 10, 2012 (First edition on Jnury 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY? ABSTRACT In this er, I enumerte ll the 8x8 imgic sures given y the Trry s ttern. This
More informationY9.ET1.3 Implementation of Secure Energy Management against Cyber/physical Attacks for FREEDM System
Y9.ET1.3 Implementtion of Secure Energy ngement ginst Cyber/physicl Attcks for FREED System Project Leder: Fculty: Students: Dr. Bruce cillin Dr. oyuen Chow Jie Dun 1. Project Gols Develop resilient cyberphysicl
More informationNetwork Theorems. Objectives 9.1 INTRODUCTION 9.2 SUPERPOSITION THEOREM
M09_BOYL3605_13_S_C09.indd Pge 359 24/11/14 1:59 PM f403 /204/PH01893/9780133923605_BOYLSTAD/BOYLSTAD_NTRO_CRCUT_ANALYSS13_S_978013... Network Theorems Ojectives Become fmilir with the superposition theorem
More informationPB735 HD DP. Industrial Line. Automatic punch and bind machine for books and calendars
PB735 HD DP Automtic punch nd bind mchine for books nd clendrs A further step for the utomtion of double loop binding. A clever nd flexible mchine ble to punch nd bind in line up to 9/16. Using the best
More informationDESIGN OF CONTINUOUS LAG COMPENSATORS
DESIGN OF CONTINUOUS LAG COMPENSATORS J. Pulusová, L. Körösi, M. Dúbrvská Institute of Robotics nd Cybernetics, Slovk University of Technology, Fculty of Electricl Engineering nd Informtion Technology
More informationA Development of EarthingResistanceEstimation Instrument
A Development of ErthingResistnceEstimtion Instrument HITOSHI KIJIMA Abstrct:  Whenever erth construction work is done, the implnted number nd depth of electrodes hve to be estimted in order to obtin
More informationRegular languages can be expressed as regular expressions.
Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected
More informationSynchronous Machine Parameter Measurement
Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions
More informationCHAPTER 3 AMPLIFIER DESIGN TECHNIQUES
CHAPTER 3 AMPLIFIER DEIGN TECHNIQUE 3.0 Introduction olidstte microwve mplifiers ply n importnt role in communiction where it hs different pplictions, including low noise, high gin, nd high power mplifiers.
More informationNotes on Spherical Triangles
Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The
More informationIndoor Autonomous Vehicle Navigation A Feasibility Study Based on Infrared Technology
Concept Pper Indoor utonomous Vehicle Nvigtion Fesibility Study Bsed on Infrred Technology RyShine Run ZhiYu Xio * ID Deprtment Electronics Engineering, Ntionl United University, 36003 Mioli, Tiwn; rsrun@nuu.edu.tw
More informationThis is a repository copy of Effect of power state on absorption cross section of personal computer components.
This is repository copy of Effect of power stte on bsorption cross section of personl computer components. White Rose Reserch Online URL for this pper: http://eprints.whiterose.c.uk/10547/ Version: Accepted
More informationExperiment 3: NonIdeal Operational Amplifiers
Experiment 3: NonIdel Opertionl Amplifiers 9/11/06 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output
More informationInvestigation of Ground Frequency Characteristics
Journl of Electromgnetic Anlysis nd Applictions, 03, 5, 337 http://dx.doi.org/0.436/jem.03.58050 Published Online August 03 (http://www.scirp.org/journl/jem) Mohmed Nyel Electricl Engineering Deprtment,
More informationJoanna Towler, Roading Engineer, Professional Services, NZTA National Office Dave Bates, Operations Manager, NZTA National Office
. TECHNICA MEMOANDM To Cc repred By Endorsed By NZTA Network Mngement Consultnts nd Contrctors NZTA egionl Opertions Mngers nd Are Mngers Dve Btes, Opertions Mnger, NZTA Ntionl Office Jonn Towler, oding
More informationSpherical Geometry. This is an article from my home page:
Spheril Geometry This is n rtile from my home pge: www.olewitthnsen.dk Ole WittHnsen nov. 6 Contents. Geometry on sphere.... Spheril tringles...3. Polr tringles...4 3. The rightngle spheril tringle...6
More informationSynchronous Generator Line Synchronization
Synchronous Genertor Line Synchroniztion 1 Synchronous Genertor Line Synchroniztion Introduction One issue in power genertion is synchronous genertor strting. Typiclly, synchronous genertor is connected
More informationREVIEW QUESTIONS. Figure For Review Question Figure For Review Question Figure For Review Question 10.2.
HAPTE 0 Sinusoidl StedyStte Anlysis 42 EVIEW QUESTIONS 0. The voltge cross the cpcitor in Fig. 0.43 is: () 5 0 V () 7.07 45 V (c) 7.07 45 V (d) 5 45 V Ω 0.5 efer to the circuit in Fig. 0.47 nd oserve
More informationSeries. Teacher. Numbers
Series B Techer Copyright 2009 3P Lerning. All rights reserved. First edition printed 2009 in Austrli. A ctlogue record for this book is vilble from 3P Lerning Ltd. ISBN 9781921860171 Ownership of
More informationOPERATING INSTRUCTION
2 AUTOMATIC POLARIMETER OPERATING INSTRUCTION Plese red through these operting instruction before using MRC.VER.01. CONTENTS I. APPLICATIONS.1 II. III. IV. PERFORMANCE 1 CONSTRUCTION AND PRINCIPLE 2
More information1 tray of toffee 1 bar of toffee. 10 In the decimal number, 0 7, the 7 refers to 7 tenths or
Chpter 3 Deciml Numers Do you know wht DECIMAL is? In chpter, we delt with units, s, 0 s nd 00 s. When you tke single unit nd divide it into (or 0 or 00) its, wht we then hve re deciml frctions of whole
More informationECE 274 Digital Logic Fall 2009 Digital Design
igitl Logic ll igitl esign MW :PM, IL Romn Lysecky, rlysecky@ece.rizon.edu http://www.ece.rizon.edu/~ece hpter : Introduction Slides to ccompny the textbook igitl esign, irst dition, by rnk Vhid, John
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
Hll Ticket No Question Pper Code: AEC009 INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigl, Hyderd  500 043 MODEL QUESTION PAPER Four Yer B.Tech V Semester End Exmintions, Novemer  2018 Regultions:
More informationEET 438a Automatic Control Systems Technology Laboratory 5 Control of a Separately Excited DC Machine
EE 438 Automtic Control Systems echnology bortory 5 Control of Seprtely Excited DC Mchine Objective: Apply proportionl controller to n electromechnicl system nd observe the effects tht feedbck control
More informationA New Algorithm to Compute Alternate Paths in Reliable OSPF (ROSPF)
A New Algorithm to Compute Alternte Pths in Relile OSPF (ROSPF) Jin Pu *, Eric Mnning, Gholmli C. Shoj, Annd Srinivsn ** PANDA Group, Computer Science Deprtment University of Victori Victori, BC, Cnd Astrct
More informationA Novel Back EMF Zero Crossing Detection of Brushless DC Motor Based on PWM
A ovel Bck EMF Zero Crossing Detection of Brushless DC Motor Bsed on PWM Zhu Bopeng Wei Hifeng School of Electricl nd Informtion, Jingsu niversity of Science nd Technology, Zhenjing 1003 Chin) Abstrct:
More informationSeparation Numbers of Chessboard Graphs. Doug Chatham Morehead State University September 29, 2006
Separation Numbers of Chessboard Graphs Doug Chatham Morehead State University September 29, 2006 Acknowledgments Joint work with Doyle, Fricke, Reitmann, Skaggs, and Wolff Research partially supported
More informationRe: PCT Minimum Documentation: Updating of the Inventory of Patent Documents According to PCT Rule 34.1
C. SCIT 2508 00 August 10, 2000 Re: PCT Minimum Documenttion: Updting of the Inventory of Ptent Documents According to PCT Rule 34.1 Sir, Mdm, The current version of the Inventory of Ptent Documents for
More information