Upward order-preserving 8-grid-drawings of binary trees
|
|
- Melina Simon
- 5 years ago
- Views:
Transcription
1 CCCG 207, Ottawa, Ontaio, July 26 28, 207 Upwad ode-peseving 8-gid-dawings of binay tees Theese Biedl Abstact This pape concens upwad ode-peseving staightline dawings of binay tees with the additional constaint that all edges must be outed along edges of the 8-gid (i.e., hoizontal, vetical, diagonal) o some subset theeof. We give an algoithm that daws n-node tees with width O(log 2 n), while the pevious best esult wee dawings of width O(n 0.48 ). If only some of the gid-lines ae allowed to be used, then the algoithm gives (with mino modifications) the same uppe bounds fo the {,, }-gid and the {,, }-gid. On the othe hand, in the {,, }-gid sometimes Ω( n/ log n) width is equied. Intoduction Thee ae many algoithms to daw tees, especially ooted tees, because diffeent applications impose diffeent equiements on the dawing. In this pape, the dawing should satisfy the following constaints: It is plana, i.e., no vetices o edges ovelap unless the coesponding gaph-elements do. It is staight-line, i.e., evey edge is dawn as a staight-line segment that connects the coesponding vetices. It is stictly-upwad, i.e., paents have lage y- coodinates that childen. (Fo some of the esults this is elaxed to upwad dawings whee edges may be hoizontal.) It is ode-peseving, i.e., a given left-to-ight ode of childen at each node must be espected in the dawing. Such dawings wee called ideal dawings peviously [3]. All dawings in this pape must be plana and staightline, and this will not always be mentioned. Futhe, vetices must always be placed at gid-points, i.e., with intege coodinates. Any dawing is assumed (afte possible tanslation) to eside within the [, W ] [, H]-gid David R. Cheiton School of Compute Science, Univesity of Wateloo, Wateloo, Ontaio N2L A2, Canada. biedl@uwateloo.ca Suppoted by NSERC. The autho would like to thank Timothy Chan and Stephanie Lee fo inspiing discussions. whee W and H ae the width and height. Column i consists of all gid-points with x-coodinate i; ow j consists of all gid-points with y-coodinate j. Since the height may have to be Ω(n) in a stictly-upwad dawing, the objective of this pape is to find ideal dawings of binay tees that have small width. Thee ae many esults concening how to daw ooted tees; see fo example [5] fo an oveview, [2] fo some ecent esults, and Table fo the esults especially elevant to this pape. This pape focuses on giddawings, which means that the edges must be dawn along the lines of a gid. This is well-studied fo socalled othogonal dawings, whee the gid is the ectangula gid (also called the 4-gid) and hence all edges ae hoizontal o vetical. Ceszenci et al. [4] showed that evey binay n-node tee has an upwad staight-line 4-gid-dawing in an O(log n) O(n)-gid (the dawing need not be ode-peseving). Fo complete binay tees as well as fo Fibonacci tees, they achieve an O( n) O( n)-gid. Fo ode-peseving dawings, significantly moe aea may be needed: Fati [6] showed that Ω(n) width and height is necessay fo some binay tees in an upwad staight-line 4-gid dawing. The focus of this pape is the octagonal gid o 8- gid that has hoizontal, vetical and diagonal lines in both diections. Dawings in the 8-gid could also be called ASCII-dawings, since they could easily be done in ASCII using chaactes / _ \. Ceszenci et al. [4] ague that thei upwad 4-gid-dawings can easily be conveted into stictly-upwad 8-gid-dawings via a downwad shea. This peseves the same width and gives asymptotically the same height, hence any binay tee has an (unodeed) stictly-upwad dawing in an O(log n) O(n)-gid. Fo ode-peseving dawings, only much weake bounds ae known. Chan [3] studied ideal dawings of binay tees (not necessaily with edges along the gid). As he points out, the fist and second of his fou algoithms adapt easily to ceate ASCII-dawings of binay tees. The width of these depends much on the chosen spine (a concept that will be used in Theoem as well); with a suitable choice Chan achieves ideal 8-gid-dawings of width O(n 0.48 ) and height O(n). Results of this pape: In this pape, we show how to ceate ideal 8-gid-dawings of binay tees. The pevious best known bounds hee ae dawings of width 232
2 29 th Canadian Confeence on Computational Geomety, 207 Gid-lines Upwad? Ode-peseving width uppe bound width lowe bound {, } upwad no O(log n) [4] Ω(log n) [4] {, } upwad yes O(n) (folkloe) Ω(n) [6] {, } stictly-upwad no O(log n) [4] Ω(log n) [4] {, } stictly upwad yes O(n) (folkloe) Ω(n) (Thm.2) {, } stictly upwad yes O(n) (folkloe) Ω(n) (Thm.2) {,, } stictly upwad yes O(n 0.48 ) [3] Ω(log n) [4] O(log 2 n) (Thm.) {,, } upwad yes O(log 2 n) (Thm.3) Ω(log n) [4] {,, } upwad no O(n) (folkloe) Ω( n/ log n) (Thm.4) Table : Results fo plana, upwad, staight-line gid-dawings of binay tees. Some moe esults can be deived in the obvious way, e.g. the uppe bound fo the {,, }-gid also holds fo the 8-gid. O(n 0.48 ) [3]. This pape impoves this to ceate dawings that have width O(log 2 n). In fact, the width is pw(t ) 2, whee the ooted pathwidth pw(t ) is a lowebound on the width of any upwad dawing of a tee T (even if it need not be ode-peseving o staightline). Since pw(t ) log(n+) [2], ou algoithm can be viewed as an (log(n+))-appoximation algoithm fo the width of ideal 8-dawings. We also study what happens if one set of the paallel gid-lines is emoved; depending on which set is emoved we can eithe achieve the same width-bound o ague that a lowe bound of Ω( n/ log n) holds on the width. See Table. 2 Backgound Let T be a ooted tee with n nodes that is binay, i.e., evey node has at most two childen. Fo any node v, use T v to denote the subtee ooted at v. The ooted pathwidth pw(t ) of a ooted tee is defined as follows [2]: If 3 evey node of T has at most one child, then pw(t ) =. (In othe wods, T 3 is a path fom the oot to the unique 2 3 leaf.) Othewise, set pw(t ) := + min P max T T P pw(t ). Hee the minimum is taken ove all paths P fo which one end is at the oot of T, and the maximum is taken ove all subtees that esult when deleting all nodes of P fom T. A path P whee the minimum is achieved is sometimes called an pw-main-path, though this pape uses the tem spine to mimic the notations of [7]. The figue above shows a tee with the ooted pathwidth indicated fo all nodes; one possible spine is bold. 3 Ideal 8-gid dawings of binay tees Theoem Let T be a ooted binay tee. Then T has an ideal 8-gid-dawing of width at most (pw(t )) Poof. The poof is stongly inspied by the algoithm of Gag and Rusu [7] that give ideal dawings of binay tees of width O(log n). (Thei dawings ae not necessaily in the 8-gid.) Thei key idea was to use dawings that ae stetchable in the sense that fo any given α 0 one can pescibe the contents of the top α ows of the dawing. This then allows to mege dawings of subtees in a ecusive constuction. Fo gid-dawings we need a slightly modified definition as follows: Definition Let T be a ooted binay tee with pw(t ) =, and let α 0 be given. An 8-gid-dawing of T is called a left-α-dawing if within the fist α ows, all points in columns + and futhe to the ight ae unused. Put diffeently, within the top α ows, only the leftmost columns may be used fo placing vetices and edges. Note that (in contast to [7]) this definition of a left-α-dawing makes no estictions whee the oot must be placed (othe than that it must be within the leftmost columns). Define symmetically a ight-αdawing to be one whee within the fist α ows only the ightmost columns may be used. We need two moe types of dawings. Define a leftcone-dawing and a ight-cone-dawing to be a dawing of T whee the oot is at the top-left (top-ight) cone. The main claim, to be poved by induction on pw(t ), is the following: Claim Fix an abitay α 0. Then T has a left-αdawing, a ight-α-dawing, a left-cone-dawing and a ight-cone-dawing, and all fou dawings have width at most (pw(t )) 2. To pove this claim, conside the base case whee pw(t ) =. This implies that T is a path fom the oot to a single leaf. Such a path can easily be dawn Inspection of the constuction given below eveals that the oot is always in column o, but we will not make use of this. 233
3 CCCG 207, Ottawa, Ontaio, July 26 28, 207 with width = 2, and this satisfies the conditions fo all fou dawings. Now assume that := pw(t ) 2. Fom the definition of ooted pathwidth, we know that T has a spine P such that all subtees T of T P satisfy pw(t ) pw(t ). Let the vetices of P be v 0 v... v m whee v 0 is the oot. 2 Fo simplicity of desciption, assume that evey spine-node v i v 0 has a sibling; if it does not then simply add a sibling and delete it in the obtained dawing late. Adding a sibling that is a leaf does not affect the ooted pathwidth since pw(t ) 2, so this does not affect the width-bound. Thus fom now on evey spine-node except v m has a left and a ight child. We explain hee only how to ceate the left-α-dawing and the left-cone-dawing; the othe two dawings ae obtained in a symmetic fashion. Thee ae thee cases, depending on whethe v is the left o ight child of v 0, and which type of dawing is desied. Case : v is the left child of v 0. In this case, the same constuction woks fo both a left-α-dawing and a left-cone dawing (fo the latte, use α := below). Place the oot v 0 at the top-left cone. Let s 0 be the ight child of v 0, and ecusively obtain a left-α - dawing D(T ) of T, whee α = α. Place D(T ), flush left with column 2 and sufficiently fa below such that the -diagonal fom v 0 ends exactly at s 0. Next obtain ecusively a left-cone-dawing D(T v ) of T v, and place it below D(T ), flush left with column. Connect (v 0, v ) vetically (both ae in column ). This finishes the constuction. Note that pw(t ) pw(t ) = by definition of ooted pathwidth and the spine. Theefoe D(T ) uses only the leftmost columns within the fist α ows. So this gives a left-α-dawing with the oot in the top left cone, as desied. As fo the width, D(T ) has width at most ( ) 2 while D(T v ) has width at most 2 ; theefoe the width is at most max{+( ) 2, 2 } = 2 as desied. Case 2: v is the ight child of v 0, and we want a left-cone-dawing. Let s 0 be the left child of v. Recusively find a left-cone-dawing D(T ) of T, say that it has height H. D(T ) has width at most ( ) 2 since pw(t ) < pw(t ). Recusively find a ight-h - dawing D(T v ) of T v of width 2. (If its width is smalle than 2, then pad it with empty columns on the left.) Thus within the topmost H ows of D(T v ), the leftmost 2 > ( ) 2 columns ae empty. D(T ) fits within this empty space; place it flush left with column. Finally place v 0 vetically above s 0 (i.e., in column 2 The notation hee is the same as in [7], though thei spine is chosen diffeently as to always use the heaviest child, athe than the one that has the lagest ooted pathwidth. ) and high enough so that the -diagonal fom v 0 ends exactly at v. This gives a left-cone-dawing of width 2 as desied. v D(T ) D(Tv ) α D(T ) D(Tv ) Figue : The constuction in Case and Case 2. Case 3: v is the ight child of v 0, and we want an α- dawing. This is the most complicated case whee a longe section of the spine may get dawn befoe ecusing. Figue 2 illustates the constuction. Recall that evey spine-vetex v i v 0 has a sibling by assumption; as in [7] let s i be the sibling of v i. Let k be the smallest intege such that v k is eithe v m o s k is the left child of v k. Fist place vetices v,..., v k of the spine; vetex v 0 will be added late. Thus, place v in column. Now epeat fo i k : ecusively find a left-conedawing D(T si ) of T si, place it flush left with column + and one ow below v i, then place v i+ in column and in the last ow used by D(T si ). This ends with vetex v k having been placed in column. Extend a -diagonal fom v k ; this will late be used to complete edge (v k, v k+ ). Next, ecusively obtain a left-conedawing D(T sk ) of T sk, and place it, flush left with column and ( ) 2 ows below the ow of v k. Note that D(T sk ) has width at most ( ) 2. Theefoe (in the dawing of width 2 that is being ceated) thee ae 2 ( ) 2 ( ) = columns fee to the ight of D(T sk ). These will be used fo T vk+ late. Also note that in the topmost ow of D(T sk ), the -diagonal fom v k is within the ightmost ows, and hence it does not intefee with D(T sk ). Let H be the total numbe of ows that ae used thus fa, i.e., fom the ow of v to the bottommost ow of D(T sk ). Note that columns,..., ae (thus fa) entiely fee. Recusively find a left-α -dawing of T, whee α = H + α. Place it, stating α ows above v and flush left with column. Within the top α ows this uses only columns,..., by pw(t ) < pw(t ), and hence this does not intesect the peviously placed subtees. Place v 0 vetically above v (i.e., in column ) and high enough so that the -diagonal fom v 0 ends exactly v 234
4 29 th Canadian Confeence on Computational Geomety, 207 at s. Let H be the numbe of ows fom the ow of s k to the bottommost ow of D(T ). Recusively find a ight-h -dawing D(T vk ) of T vk. Place it, flush ight with the ightmost column, and in the ow of s k o below such that the -diagonal extending fom v k exactly meets the point containing v k. Within the topmost H ows, dawing D(T ) uses only the ightmost columns. Recall that columns emained fee next to D(T sk ), and also columns ae fee next to D(T ) since this dawing has width at most ( ) 2. Thus dawing D(T vk ) does not intefee with peviously placed dawings. This gives the desied left-α-dawing of width 2. This ends the constuction fo all cases and poves Theoem. α s 0 v v 0 One can easily ague that the height is at most O(n (pw(t )) 2 ), because (as one can see) any ow without vetex in it intesects a diagonal edge, any such diagonal edge intesects at most pw(t ) 2 ows, and these ows can be assigned to the uppe endpoint of the diagonal edge. If one follows the constuction exactly as descibed, then Ω(n 2 ) height (fo = pw(t )) may esult. (Fo example, conside a tee whee the spine has length Ω(n) and nealy all siblings of spine-vetices ae leaves, but the last few siblings have big enough subtees to foce ooted pathwidth.) Howeve, thee ae some obvious possible impovements to the height. To give just one, in Case 2 the dawing D(T ) could be moved much highe, diectly unde the -diagonal, because due to the stict-upwadness of the dawing, the ith ow of it is empty in column i+ and fathe ight. This alone is not enough to ensue a smalle height, but we suspect that combining this with dawing the spine moe caefully when some siblings have vey small size may lead to a dawing of width O(log 2 n) and height O(n). This emains fo futue wok. s D(T s ) 4 Ideal 6-gid dawings of binay tees H v 2 v k D(T ) s 2 D(T s2 ) s k v k+ D(T sk ) ( ) 2 D(T vk+ ) ( ) 2 + Figue 2: The constuction in Case 3. H Now we tun to the 6-gid, which has gid-lines with angles of 60 between them. Fequently it is easie to think of it instead as a gid that has thee of the fou sets of gid-lines of the 8-gid (e.g., hoizontal, ightwad, and -diagonals). Bachmeie et al. [] studied 6-gid dawings of tees. Thei dawing wee not (necessaily) upwad, and as such, it was ielevant which of the gidlines of the 8-gid ae used fo the 6-gid, since they ae all the same afte 90 otation and/o a shea, and a shea does not affect the asymptotic aea. In contast to this, we study hee upwad dawings of binay tees in the 6-gid, and as befoe, focus on keeping the width small. As will be seen, hee it makes a diffeence exactly which gid-lines ae used to epesent the 6-gid. The following gids will be studied: The {,, }-gid: gid-lines ae vetical o along a 45 diagonal in eithe diection. The {,, }-gid: gid-lines ae hoizontal o along a 45 diagonal in eithe diection. Height-consideation: In most pevious tee-dawing papes, the height is easily shown to be O(n), because all ows (o nealy all ows) intesect at least one vetex. In contast to this, the constuction hee has many ows (e.g. most of the ows,..., 2 in the constuction fo Case 2) that intesect only edges. The {,, }-gid: gid-lines ae hoizontal o vetical o along one of the 45 diagonals. (Fo the {,, }-gid a symmetic set of esults is obtained by using a hoizontal flip.) 235
5 CCCG 207, Ottawa, Ontaio, July 26 28, 207 We have thus fa mostly studied ideal dawings, which must be stictly-upwad and hence hoizontal lines ae disallowed. In paticula, Theoem ceated stictlyupwad dawings in the 8-gid, which hence ae automatically dawings in the {,, }-gid. Theefoe evey binay tee T has an ideal dawing that is an embedding in the {,, }-gid and has width at most (pw(t )) 2. Now we tun to othe types of 6-gids. Again, having an ideal dawing means being stictly-upwad, so no hoizontal lines can be used. We show that then no small width is possible. v α D(T ) D(Tv ) α D(T ) D(Tv ) v Theoem 2 Thee exists a binay tee T such that any ideal dawing of T in the {, }-gid o the {, }-gid equies width and height Ω(n). Poof. Let T consist of a path of length n/2 and attach at each node a left child that is a leaf. Fo an odepeseving and stictly-upwad dawing, the path must be dawn following the -diagonals. This gives a width and height of at least n/2. D(T ) v v D(Tv ) D(Ts2 ) D(T ) Theefoe, the emaining dawing-esults will be in a elaxed model of ideal dawings whee hoizontal edges ae allowed, hence the dawing is upwad athe than stictly-upwad. Call these weakly-ideal dawings. (As befoe all dawings must be plana, staight-line and ode-peseving.) Theoem 3 Evey binay tee T has a weakly-ideal dawing that is an embedding in the {,, }-gid and has width at most (pw(t )) 2. Poof. The poof is vey simila to the poof of Theoem. As befoe, define (left/ight) cone-dawings and (left/ight) α-dawings. Additionally now demand fo all these dawings that in the topmost ow no point to the ight of the oot is occupied (we say that the oot is ight-fee). Ceate left-cone-dawings and left-α-dawings almost exactly as befoe. The only diffeence is that at the places whee a -diagonal was used befoe, we now use a hoizontal edges instead; this is feasible because the oot of coesponding subtee is ight-fee. Fo ightcone and ight-α-dawings, the constuctions ae not entiely symmetic anymoe, but again, by using hoizontal edges athe than diagonal ones, dawings can be constucted. Figue 3 illustates the constuctions in all cases; the details ae left to the eade. Finally conside the {,, }-gid, which is the same as the 8-gid whee no vetical edges ae allowed. Theoem 2 showed that ideal dawings have to have lage width. We show hee that even weakly-ideal dawings may equie lage with. In fact, the following bound holds fo any plana staight-line dawing in the α D(Tv ) v v2 vk sk D(T ) s D(Ts ) s2 D(Ts2 ) D(Tsk ) ( ) 2 D(Tvk+ ) vk+ α vk+ D(Ts ) s v D(Ts2 ) s2 v2 D(Tsk ) ( ) 2 D(Tvk+ ) vk sk D(T ) Figue 3: The constuction fo the {,, }-gid. {,, }-gid, even if it is not upwad o not odepeseving. Theoem 4 The complete binay tee must have width O( n/ log n) in any staight-line dawing in the {,, }-gid. Poof. The poof is vey simila to the simplest method fo obtaining a width-lowe-bound fo weaklyideal dawings of the complete tenay tee, see [8]. We 236
6 29 th Canadian Confeence on Computational Geomety, 207 epeat the agument hee fo completeness. Fix an abitay staight-line dawing of the complete binay tee in the {,, }-gid, say it has w columns. So any edge spans a hoizontal distance of at most w. Since only hoizontal and diagonal edges ae allowed, theefoe any edge spans a vetical distance of at most w. Obseve that T has height h := log ( ) n+ 2, i.e., the path fom the oot to each leaf contains h edges. In consequence, any node has vetical distance at most h(w ) fom the oot. Theefoe the entie dawing is contained within a ectangle that has w columns and up to h(w ) ows above and below the oot, hence 2h(w )+ ows in total. Theefoe the dawing esides in a gid with at most 2wh(w )+w gid points. Since all n nodes ae placed on these gid-points, necessaily n 2wh(w ) + w O(w 2 log n), which implies w Ω( n/ log n). We stongly suspect that a lowe bound of Ω( n) on the width holds, but this emains fo futue wok. Fo the complete binay tee, it is easy to find a constuction that has width and height O( n), and in fact, no hoizontal edges ae used. Theoem 5 (based on [4]) The complete binay tee has an ideal dawing in the {, }-gid of gid-size O( n) O( n). Poof. Cescenzi et al. gave a simple ecusive constuction that daws the complete binay tee in a 4- gid of size O( n) O( n) [4]. Moeove, all edges go ightwad o downwad. Scale this dawing by 2 and then otate it by 45 clockwise. Due to the scaling, this maps all vetices to gid-points, and all edges ae now diagonal and downwad as desied. 5 Remaks This pape developed algoithms fo weakly-ideal 8- gid-dawings of binay tees, i.e., plana upwad staight-line ode-peseving dawings with edges dawn along gid-lines fo the 8-gid (o some subset theefoe). We gave constuctions of width O(log 2 n) fo a numbe of such gids. The height is athe lage (O(n log 2 n)), and impoving this emains an open poblem. We also showed that width O( n/ log n) is equied fo the gid whee no vetical lines ae allowed. A natual question is whethe simila bounds could be poved fo tenay tees. Fo unodeed dawings, Bachmeie et al. [] gave simple ecusive constuctions that achieve width O(n log 3 2 ) O(n 0.63 ). In wok done simultaneously with the cuent pape, Lee studied odeed dawings of tenay tees and poved that evey tenay tee has such a weakly-ideal 8-gid dawing of width Ω(n 0.68 ) [8]. Futhemoe, the complete tenay tee equies width Ω(n 0.4 ) in any upwad octagonalgid-dawing [8]. Both the constuctions and the lowe bounds in Lee s thesis ae significantly moe complicated than the ones given hee, and will be published sepaately. As fo open poblems, the obvious one is to close the gap between the width O(log 2 n) achieved with ou algoithm and the lowe bound of Ω(log n) fo the complete binay tee. Ae thee binay tees that equie ω(log n) width in ideal 8-gid dawings? The othe emaining gap concens dawings in the {,, }-gid-gid. Can we achieve a width of O( n) not just fo complete binay tees but fo all tees? Refeences [] Chistian Bachmaie, Fanz-Josef Bandenbug, Wolfgang Bunne, Andeas Hofmeie, Maco Matzede, and Thomas Unfied. Tee dawings on the hexagonal gid. In Ioannis G. Tollis and Mauizio Patignani, editos, Gaph Dawing (GD 2008), volume 547 of Lectue Notes in Compute Science, pages Spinge, [2] Theese Biedl. Ideal tee-dawings of appoximately optimal width (and small height). Jounal of Gaph Algoithms and Applications, 2(4):63 648, 207. [3] Timothy M. Chan. A nea-linea aea bound fo dawing binay tees. Algoithmica, 34(): 3, [4] Pieluigi Cescenzi, Giuseppe Di Battista, and Adolfo Pipeno. A note on optimal aea algoithms fo upwad dawings of binay tees. Comput. Geom., 2:87 200, 992. [5] Giuseppe Di Battista and Fabizio Fati. A suvey on small-aea plana gaph dawing, 204. CoRR epot [6] Fabizio Fati. Staight-line othogonal dawings of binay and tenay tees. In Seok-hee Hong, Takao Nishizeki and Wu Quan, editos, Gaph Dawing (GD 2007), volume 4875 of Lectue Notes in Compute Science, pages 76 87, Spinge, [7] Ashim Gag and Adian Rusu. Aea-efficient odepeseving plana staight-line dawings of odeed tees. Int. J. Comput. Geomety Appl., 3(6): , [8] Stephanie Lee. Upwad octagonal dawings of tenay tees. Maste s thesis, Univesity of Wateloo, August 206. (Supevisos: T. Biedl and T. Chan.) Available at 237
A Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationSupplementary information Efficient Enumeration of Monocyclic Chemical Graphs with Given Path Frequencies
Supplementay infomation Efficient Enumeation of Monocyclic Chemical Gaphs with Given Path Fequencies Masaki Suzuki, Hioshi Nagamochi Gaduate School of Infomatics, Kyoto Univesity {m suzuki,nag}@amp.i.kyoto-u.ac.jp
More informationA solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane
A solution to a poblem of Günbaum and Motzkin and of Edős and Pudy about bichomatic configuations of points in the plane Rom Pinchasi July 29, 2012 Abstact Let P be a set of n blue points in the plane,
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationGraphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
More informationCOLLAPSING WALLS THEOREM
COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationAnalysis of simple branching trees with TI-92
Analysis of simple banching tees with TI-9 Dušan Pagon, Univesity of Maibo, Slovenia Abstact. In the complex plane we stat at the cente of the coodinate system with a vetical segment of the length one
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationRelating Branching Program Size and. Formula Size over the Full Binary Basis. FB Informatik, LS II, Univ. Dortmund, Dortmund, Germany
Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationgr0 GRAPHS Hanan Samet
g0 GRPHS Hanan Samet ompute Science epatment and ente fo utomation Reseach and Institute fo dvanced ompute Studies Univesity of Mayland ollege Pak, Mayland 074 e-mail: hjs@umiacs.umd.edu opyight 1997 Hanan
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationFractional Zero Forcing via Three-color Forcing Games
Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationMatrix Colorings of P 4 -sparse Graphs
Diplomabeit Matix Coloings of P 4 -spase Gaphs Chistoph Hannnebaue Januay 23, 2010 Beteue: Pof. D. Winfied Hochstättle FenUnivesität in Hagen Fakultät fü Mathematik und Infomatik Contents Intoduction iii
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationFall 2014 Randomized Algorithms Oct 8, Lecture 3
Fall 204 Randomized Algoithms Oct 8, 204 Lectue 3 Pof. Fiedich Eisenband Scibes: Floian Tamè In this lectue we will be concened with linea pogamming, in paticula Clakson s Las Vegas algoithm []. The main
More informationDuality between Statical and Kinematical Engineering Systems
Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland.
More informationCentripetal Force OBJECTIVE INTRODUCTION APPARATUS THEORY
Centipetal Foce OBJECTIVE To veify that a mass moving in cicula motion expeiences a foce diected towad the cente of its cicula path. To detemine how the mass, velocity, and adius affect a paticle's centipetal
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 9 Solutions
Math 451: Euclidean and Non-Euclidean Geomety MWF 3pm, Gasson 04 Homewok 9 Solutions Execises fom Chapte 3: 3.3, 3.8, 3.15, 3.19, 3.0, 5.11, 5.1, 5.13 Execise 3.3. Suppose that C and C ae two cicles with
More informationThe Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr
The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationThe Millikan Experiment: Determining the Elementary Charge
LAB EXERCISE 7.5.1 7.5 The Elementay Chage (p. 374) Can you think of a method that could be used to suggest that an elementay chage exists? Figue 1 Robet Millikan (1868 1953) m + q V b The Millikan Expeiment:
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More informationLiquid gas interface under hydrostatic pressure
Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,
More informationMotion in One Dimension
Motion in One Dimension Intoduction: In this lab, you will investigate the motion of a olling cat as it tavels in a staight line. Although this setup may seem ovesimplified, you will soon see that a detailed
More informationThe Chromatic Villainy of Complete Multipartite Graphs
Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at:
More informationEQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS
EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS ATHULA GUNAWARDENA AND ROBERT R MEYER Abstact A d-dimensional gid gaph G is the gaph on a finite subset in the intege lattice Z d in
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationForest-Like Abstract Voronoi Diagrams in Linear Time
Foest-Like Abstact Voonoi Diagams in Linea Time Cecilia Bohle, Rolf Klein, and Chih-Hung Liu Abstact Voonoi diagams ae a well-studied data stuctue of poximity infomation, and although most cases equie
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationEncapsulation theory: radial encapsulation. Edmund Kirwan *
Encapsulation theoy: adial encapsulation. Edmund Kiwan * www.edmundkiwan.com Abstact This pape intoduces the concept of adial encapsulation, wheeby dependencies ae constained to act fom subsets towads
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationEuclidean Figures and Solids without Incircles or Inspheres
Foum Geometicoum Volume 16 (2016) 291 298. FOUM GEOM ISSN 1534-1178 Euclidean Figues and Solids without Incicles o Insphees Dimitis M. Chistodoulou bstact. ll classical convex plana Euclidean figues that
More informationProbablistically Checkable Proofs
Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol
More informationAnalytical time-optimal trajectories for an omni-directional vehicle
Analytical time-optimal tajectoies fo an omni-diectional vehicle Weifu Wang and Devin J. Balkcom Abstact We pesent the fist analytical solution method fo finding a time-optimal tajectoy between any given
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationSplay Trees Handout. Last time we discussed amortized analysis of data structures
Spla Tees Handout Amotied Analsis Last time we discussed amotied analsis of data stuctues A wa of epessing that even though the wost-case pefomance of an opeation can be bad, the total pefomance of a sequence
More informationTHE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX. Jaejin Lee
Koean J. Math. 23 (2015), No. 3, pp. 427 438 http://dx.doi.og/10.11568/kjm.2015.23.3.427 THE JEU DE TAQUIN ON THE SHIFTED RIM HOOK TABLEAUX Jaejin Lee Abstact. The Schensted algoithm fist descibed by Robinson
More informationRECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationSUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER
Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More informationβ β β β β B B B B o (i) (ii) (iii)
Output-Sensitive Algoithms fo Unifom Patitions of Points Pankaj K. Agawal y Binay K. Bhattachaya z Sandeep Sen x Octobe 19, 1999 Abstact We conside the following one and two-dimensional bucketing poblems:
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationCERFACS 42 av. Gaspard Coriolis, Toulouse, Cedex 1, France. Available at Date: April 2, 2008.
ON THE BLOCK TRIANGULAR FORM OF SYMMETRIC MATRICES IAIN S. DUFF and BORA UÇAR Technical Repot: No: TR/PA/08/26 CERFACS 42 av. Gaspad Coiolis, 31057 Toulouse, Cedex 1, Fance. Available at http://www.cefacs.f/algo/epots/
More informationLecture 18: Graph Isomorphisms
INFR11102: Computational Complexity 22/11/2018 Lectue: Heng Guo Lectue 18: Gaph Isomophisms 1 An Athu-Melin potocol fo GNI Last time we gave a simple inteactive potocol fo GNI with pivate coins. We will
More informationOn the Structure of Linear Programs with Overlapping Cardinality Constraints
On the Stuctue of Linea Pogams with Ovelapping Cadinality Constaints Tobias Fische and Mac E. Pfetsch Depatment of Mathematics, TU Damstadt, Gemany tfische,pfetsch}@mathematik.tu-damstadt.de Januay 25,
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationContact impedance of grounded and capacitive electrodes
Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can
More informationRADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION
RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION JINGYI CHEN AND YUXIANG LI Abstact. We show that a smooth adially symmetic solution u to the gaphic Willmoe suface equation is eithe
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationOutput-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries
Output-Sensitive Algoithms fo Computing Neaest-Neighbou Decision Boundaies David Bemne 1, Eik Demaine 2, Jeff Eickson 3, John Iacono 4, Stefan Langeman 5, Pat Moin 6, and Godfied Toussaint 7 1 Faculty
More informationDeterministic vs Non-deterministic Graph Property Testing
Deteministic vs Non-deteministic Gaph Popety Testing Lio Gishboline Asaf Shapia Abstact A gaph popety P is said to be testable if one can check whethe a gaph is close o fa fom satisfying P using few andom
More informationSection 26 The Laws of Rotational Motion
Physics 24A Class Notes Section 26 The Laws of otational Motion What do objects do and why do they do it? They otate and we have established the quantities needed to descibe this motion. We now need to
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationConstraint Satisfaction Problems
Constaint Satisfaction Polems Seach and Look-ahead Alet-Ludwigs-Univesität Feiug Stefan Wölfl, Chistian Becke-Asano, and Benhad Neel Noveme 17, 2014 Seach and Look-ahead Enfocing consistency is one way
More informationBounds on the performance of back-to-front airplane boarding policies
Bounds on the pefomance of bac-to-font aiplane boading policies Eitan Bachmat Michael Elin Abstact We povide bounds on the pefomance of bac-to-font aiplane boading policies. In paticula, we show that no
More informationFailure Probability of 2-within-Consecutive-(2, 2)-out-of-(n, m): F System for Special Values of m
Jounal of Mathematics and Statistics 5 (): 0-4, 009 ISSN 549-3644 009 Science Publications Failue Pobability of -within-consecutive-(, )-out-of-(n, m): F System fo Special Values of m E.M.E.. Sayed Depatment
More informationCircular Orbits. and g =
using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is
More informationPermutations and Combinations
Pemutations and Combinations Mach 11, 2005 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication Pinciple
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationLab 10: Newton s Second Law in Rotation
Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have
More informationarxiv: v1 [math.nt] 12 May 2017
SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking
More informationA THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM
A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical
More informationVoltage ( = Electric Potential )
V-1 of 10 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE
THE p-adic VALUATION OF STIRLING NUMBERS ANA BERRIZBEITIA, LUIS A. MEDINA, ALEXANDER C. MOLL, VICTOR H. MOLL, AND LAINE NOBLE Abstact. Let p > 2 be a pime. The p-adic valuation of Stiling numbes of the
More informationApproximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs
Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs A. Kaim Abu-Affash 1 Softwae Engineeing Depatment, Shamoon College of Engineeing Bee-Sheva 84100, Isael abuaa1@sce.ac.il Paz Cami Depatment
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug 4 Complete Section Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a
More informationClassical Worm algorithms (WA)
Classical Wom algoithms (WA) WA was oiginally intoduced fo quantum statistical models by Pokof ev, Svistunov and Tupitsyn (997), and late genealized to classical models by Pokof ev and Svistunov (200).
More informationCALCULUS II Vectors. Paul Dawkins
CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx
More informationChapter 1: Introduction to Polar Coordinates
Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More information! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an
Physics 142 Electostatics 2 Page 1 Electostatics 2 Electicity is just oganized lightning. Geoge Calin A tick that sometimes woks: calculating E fom Gauss s law Gauss s law,! E da = 4πkQ enc, has E unde
More informationOn the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University
Zapiski Nauchn. Seminaov POMI, to appea On the Numbe of Rim Hook Tableaux Segey Fomin* Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239 Theoy of Algoithms Laboatoy SPIIRAN,
More informationLargest and smallest minimal percolating sets in trees
Lagest and smallest minimal pecolating sets in tees Eic Riedl Havad Univesity Depatment of Mathematics ebiedl@math.havad.edu Submitted: Sep 2, 2010; Accepted: Ma 21, 2012; Published: Ma 31, 2012 Abstact
More informationRigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018
Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining
More informationMath Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationMathematisch-Naturwissenschaftliche Fakultät I Humboldt-Universität zu Berlin Institut für Physik Physikalisches Grundpraktikum.
Mathematisch-Natuwissenschaftliche Fakultät I Humboldt-Univesität zu Belin Institut fü Physik Physikalisches Gundpaktikum Vesuchspotokoll Polaisation duch Reflexion (O11) duchgefüht am 10.11.2009 mit Vesuchspatne
More informationFEASIBLE FLOWS IN MULTICOMMODITY GRAPHS. Holly Sue Zullo. B. S., Rensselaer Polytechnic Institute, M. S., University of Colorado at Denver, 1993
FEASIBLE FLOWS IN MULTICOMMODITY GRAPHS by Holly Sue Zullo B. S., Rensselae Polytechnic Institute, 1991 M. S., Univesity of Coloado at Denve, 1993 A thesis submitted to the Faculty of the Gaduate School
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More informationFlux. Area Vector. Flux of Electric Field. Gauss s Law
Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is
More information