Suborbital graphs for the group Γ 2
|
|
- Jean Chandler
- 5 years ago
- Views:
Transcription
1 Hacettepe Jounal of Mathematics and Statistics Volume , Subobital gaphs fo the goup Γ 2 Bahadı Özgü Güle, Muat Beşenk, Yavuz Kesicioğlu, Ali Hikmet Değe Keywods: Abstact In this pape, we investigate subobital gaphs fomed by the action of Γ 2 which is the goup geneated by the second powes of the elements of the modula goup Γ on ˆQ. Fistly, conditions fo being an edge, self-paied and paied gaphs ae povided, then we give necessay and sufficient conditions fo the subobital gaphs to contain a cicuit and to be a foest. Finally, we examine the connectivity of the subgaph F u, and show that it is connected if and only if 2. Modula goup, Goup action, Subobital gaphs 2000 AMS Classification: 20H05, 05C25 Received 13/11/2013 : Accepted 27/05/2014 Doi : /HJMS Intoduction Let PSL2,R denote the goup of all linea factional tansfomations T : z az + b, whee a, b, c and d ae eal and ad bc = 1. cz + d In tems of matix epesentation, the elements of PSL2,R coespond to the matices a b ± ; a, b, c, d R and ad bc = 1. This is the automophism goup of the uppe half plane H := {z C : Imz > 0}. The modula goup Γ=PSL2, Z, is the subgoup of PSL2,R such that a, b, c and d ae integes. It is geneated by the matices U = ; V = Dept. of Math., Kaadeniz Tech. Uni., Tukey, bogule@ktu.edu.t Coesponding Autho. Dept. of Math., Kaadeniz Tech. Uni., Tukey, mbesenk@ktu.edu.t Dept. of Math., Recep Tayyip Edogan Uni., Tukey, yavuzkesicioglu@yahoo.com Dept. of Math., Kaadeniz Tech. Uni., Tukey, ahikmetd@ktu.edu.t
2 1034 with defining elationships U 2 = V 3 = I, whee I is the identity matix. Γ is a Fuchsian goup of signatue 0; 2, 3,, so it is isomophic to a fee poduct C 2 C 3. Define Γ m as the subgoup of Γ geneated by the m th powes of all elements of Γ. Especially, Γ 2 and Γ 3 have been studied extensively by ewman [13,14,15]. It tuns out that, { } Γ 2 a b = Γ : ab + bc + cd 0 mod 2, by Rankin [Eq , 16]. Fom the equation ab + bc + cd 0 mod 2, we see that at least one of the integes a, b, c, d must be even. Suppose fist that a = 2a 0. Then using the deteminant, we have that b and c ae odd. So, d must be odd as well. Hence, we 2a b get the elements of Γ 2 as the matices a b get the elements of the fom c 2d c will be even. To sum up, Γ 2 has thee types of elements 2a b a 2b a b,, 2 c 2d. Similaly, supposing d = 2d 0, we can. Lastly, if a o d is not even, then both b and whee b, c and d of the fist, a and d of the second and a, b, c of the thid matix ae odd Theoem. [13] The goup Γ 2 is the fee poduct of two cyclic goups of ode 3, and Γ : Γ 2 = 2, Γ = Γ Γ The elements of Γ 2 may be chaacteized by the equiement that the sum of the exponents 0 1 of be divisible by The idea of a subobital gaph has been used mainly by finite goup theoists. In [7], Jones, Singeman and Wicks showed that this idea is also useful in the study of the modula goup, whee they poved that the well-known Faey Gaph is an example of a subobital gaph. Futhemoe, they poved the following esult: Theoem A. The subobital gaph G u,n of Γ contains diected tiangles if and only if u 2 ± u mod n. Moeve they posed the conjectue: G u,n is a foest if and only if it contains no tiangles, that is, if and only if u 2 ± u mod n. Akbas poved in [2] that this conjectue is tue. By simila aguments, we concen with subobital gaphs of Picad goup P, which is the subgoup of PSL2, C with enties coming fom Z[i] in [3]. Since Z[i] is a unique factoization domain with finitely many units, ou expectation was to find simila fomulas. Consequently, theoem A was impoved as Theoem B. The subobital gaph G u, of P contains diected tiangles if and only if ε 2 u 2 εu ± 1 0 mod. In this study, we will continue to investigate the combinatoial popeties of these gaphs fo the goup Γ 2. It is an impotant subgoup of Γ since all the goups Γ m can be expessed in the tems of Γ, Γ 2, Γ 3. The pupose of this pape is to chaacteize all cicuits in the subobital gaph and connectedness fo Γ 2. As it can be seen fom Section 3, we show that the main diffeence is in connectedness of elated gaphs.
3 The action of Γ 2 on ˆQ Evey element of ˆQ can be epesented as a educed faction x with x, y Z and y x, y = 1. This epesentation is not unique, because x = x. We epesent as y y 1 a b = 1. The action of the matix on x 0 0 i a b : x ax + by y cx + dy. Hence, the actions of a matix on x x and on ae identical. If the deteminant of the y y a b matix is 1 and x, y = 1, then ax + by, cx + dy = Tansitive Action Lemma. i The action of Γ 2 on ˆQ is tansitive. ii The stabilize of a point is an infinite cyclic goup. Poof. i Hee we only pove the case that any element of the fom a of ˆQ is sent 2b by an element of Γ 2. The est ae simila. Let a ˆQ, a, 2b = 1. Thee exist integes 2b x 0 and y 0 such that ay 0 2bx 0 = 1 known as Bezout s identity [8]. Hence, we have a x0 that T := Γ. All solutions of the equation ay 2bx = 1 ae x = x 2b y 0 + an 0, y = y 0 + 2bn fo n Z. If x 0 is odd, x would be even by taking n-odd. So, x 0 can be chosen as an even numbe. Hence, T Γ 2 and T = a means that a is in the obit 2b 2b of. ii By i, since the stabilizes of any two points in ˆQ ae conjugate in Γ 2, it is 1 2 sufficient to conside the stabilize Γ 2 of. It is clea that Γ 2 =. 0 1 We emak that Lemma 2.1 i can be poven by using the signatue of Γ 2 as well. Thee is a homomophism θ : Γ C 2 = {e, α} defined by θu = α, and θv = e. The kenel is Γ 2. By the pemutation theoem [19], Γ 2 has signatue 0; 3, 3,. It means that thee is only one obit, so the action is tansitive Impimitive Action. We now discuss the impimitivity of the action of Γ 2 on ˆQ. Fo this, let G, Ω be a tansitive pemutation goup, consisting of a goup G acting on a set Ω tansitively. An equivalence elation on Ω is called G-invaiant if, wheneve α, β Ω satisfy α β, then gα gβ fo all g G. The equivalence classes ae called blocks. We call G, Ω impimitive if Ω admits some G-invaiant equivalence elation diffeent fom i the identity elation, α β if and only if α = β; ii the univesal elation, α β fo all α, β Ω. Othewise, G, Ω is called pimitive. These two elations ae supposed to be tivial elations Lemma. [4] Let G, Ω be a tansitive pemutation goup. G, Ω is pimitive if and only if G α, the stabilize of α Ω, is a maximal subgoup of G fo each α Ω. Fom the above lemma we see that wheneve, fo some α, G α H G, then Ω admits some G-invaiant equivalence elation othe than the tivial one and the univesal one.
4 1036 Because of the tansitivity, evey element of Ω has the fom gα fo some g G. Thus one of the non-tivial G-invaiant equivalence elations on Ω by H is given as follows: gα g α if and only if g gh. The numbe of blocks equivalence classes is the index G : H and the block containing α is just the obit Hα. Let and let Γ 2 0 be defined by { } Γ 2 a b 0 := Γ 2 : c 0 mod. Then Γ 2 0 is a subgoup of Γ 2. It is clea that Γ 2 Γ 2 0 Γ 2 fo and Γ 2 Γ 2 0 Γ 2 fo > Lemma. Γ 0 : Γ 2 0 = 2. In fact, 1 0 Γ 2 0 Γ 2 1 0, Γ 0 = Γ 2 0 Γ 2 1 0, is odd is even Poof. Fist, we suppose that is even. Let s show that Γ a b Γ = Γ 0. We have that T := Γ c d 0 with ad bc = 1. Hee, a and d ae odd. If b is even, T would be an element of Γ 2 0. We x y suppose that b is odd. Hence, it can be witten as T =. 1 c z 1 1 a b x y Then, we have that =. Let s say that + 1 c d c z a + c b + d x y =. a + c + 1 b + d + 1 c z }{{} A As b + d is even, A Γ 2 0. ow, let be odd. In this case, assume that b and c ae even in T. Then a and d ae odd. Hence, T is an element of Γ 2 0. Moeove, it can be witten as T = x y c z. As above, let s say that a b } c a {{ d b } B x y. Since d b is even, B Γ 2 c z 0. In the case: b-even and c-odd, it is clea that B Γ 2 0. If a and d ae even in the equation ad bc = 1, B Γ 2 0 again. Finally if a is odd and d is even o vice vesa, the esult is the same. Consequently, we obtain that Γ 0 : Γ 2 0 = 2. Theefoe, fom the above constucted equivalence elation ", we get Γ 2 -invaiant equivalence elation on ˆQ by Γ 2 0. It is clea that, by Lemma 2.3, Γ 2 acts impimitively on ˆQ. Let v = s and w = x y be elements of ˆQ. Because of the tansitive action, we have that v = g 1 and w = g 2 fo some elements g 1, g 2 Γ 2 of the fom =
5 g 1 := s x, g 2 := y Fom the elation we get v w if and only if g 1 1 g2 Γ2 0, v w if and only if y sx 0 mod. By ou geneal discussion of impimitivity, the numbe of blocks unde is given by Ψ = Γ 2 : Γ 2 0. So the block of is obtained as [ ] := { } x y ˆQ y 0 mod Lemma. Ψ = p dividing whee the poduct is ove the distinct pimes p p Poof. To calculate Ψ we use two decomposition of the index Γ : Γ 2 0 as the following Γ : Γ 2 Γ 2 : Γ 2 0 = Γ : Γ 0 Γ 0 : Γ 2 0. Hee, Γ : Γ 2 = 2 and Γ : Γ 0 = p ae well-known by [13,16] and p [16,17] espectively. We pove that the index of Γ 0 : Γ 2 0 is equal to 2 in Lemma 2.3. Witing these values in above equation, the esult is obvious. 3. Subobital Gaphs fo Γ 2 on ˆQ In[18], Sims intoduced the idea of the subobital gaphs of a pemutation goup G acting on a set, these ae gaphs with vetex-set, on which G induces automophisms. We summaise Sims theoy as follows: Let G, be tansitive pemutation goup. Then G acts on by gα, β = gα, gβg G, α, β. The obits of this action ae called subobitals of G. The obit containing α, β is denoted by Oα, β. Fom Oα, β we can fom a subobital gaph Gα, β : its vetices ae the elements of, and thee is a diected edge fom γ to δ if γ, δ Oα, β. A diected edge fom γ to δ is denoted by γ δ. If γ, δ Oα, β, then we will say that thee exists an edge γ δ in Gα, β. In this pape ou calculation concens Γ 2, so we can daw this edge as a hypebolic geodesic in the uppe half-plane H, that is, as euclidean semi-cicles o half-lines pependicula to the eal line. The obit Oβ, α is also a subobital gaph and it is eithe equal to o disjoint fom Oα, β. In the latte case Gβ, α is just Gα, β with the aows evesed and we call, in this case, Gα, β and Gβ, α paied subobital gaphs. In the fome case Gα, β = Gβ, α and the gaph consists of pais of oppositely diected edges; it is convenient to eplace each such pai by a single undiected edge, so that we have an undiected gaph which we call self paied Definition. By a diected cicuit in a gaph we mean a sequence v 1, v 2,..., v m of diffeent vetices such that v 1 v 2... v m v 1, whee m 3. If m = 3, then the cicuit, diected o not, is called a tiangle. If m = 2, then we will say the configuation v 1 v 2 v 1 is self paied.
6 1038 A gaph which contains no cicuit is called a foest. The above ideas ae also descibed in a pape by eumann [12] and in books by Tsuzuku [20] and by Biggs and White [4], the emphasis being on applications to finite goups. The eade is efeeed to [1, 2, 3, 6, 7, 9, 10, 11] fo some elevant pevious wok on subobital gaphs. If α = β, then Oα, α = {γ, γ γ } is the diagonal of. The coesponding subobital gaph Gα, α, called the tivial subobital gaph, is self-paied: it consists of a loop based at each vetex γ. We shall be mainly inteested in the emaining non-tivial subobital gaphs. Since Γ 2 acts tansitively on ˆQ, each subobital contains a pai, v fo some v ˆQ; witing v = u, u, = 1 and 0. We denote this subobital by Ou, and the coesponding subobital gaph by G u, Gaph G u,. If v =, we would have the simplest subobital gaph, namely G 1,0 = G 1,0. Theefoe, we can take v Q. Let v = u Q. The necessay and sufficient condition fo O, v = O, v is that v and v ae in the same obit of Γ 2. Since Γ 2 is geneated by z : v v + 2, then z u = u+2 = u. Theefoe, we have that = and u u mod 2. Hence, G u, = G u, if and only if = and u u mod 2. Consequently, fo a fixed thee ae 2ϕ distinct subobital gaphs G u, whee ϕ is Eule s phi function Theoem. x Gu, if and only if i If is even, then x ±u mod, y ±us mod, y ±us mod 2 and y sx =. ii If s is even, then x ±u mod 2, y ±us mod and y sx =. iii If and s ae odd, then x ±u mod, y ±us mod 2 and y sx =. Poof. i Let be even. By the tansitivity of Γ 2, without loss of geneality, we assume that < x whee all lettes ae positive integes. Thus, we have that y sx < 0. Since a b x Gu,, thee exist some T = Γ 2 such that T 1, u 0 =, x. a b 1 u x A sx < 0, the multiplication of is equal to o 0 x. If the fist case is valid, we have that a =, c = s, au + b = x and s y cu + d = y. That is, x u mod and y us mod. Since is even, then a is also even. To have T Γ 2, d must be odd. Fom us + d = y, we have that y ±us mod 2. ii Suppose s is even. In a simila way, we see that b and c must be even because T 1 0 = a b 1 u x = a. As in i, we may assume that =. s c 0 Hence, we have that a =, c = s, au+b = x, cu+d = y and y sx =. That is, u + b = x and us + d = y. Since b is even, we have that x u mod 2 and y us mod. iii Let and s be odd. With simila agument, it can be seen that d must be even. Fom the same matix equation in ii, we obtain that x u mod and y us mod 2.
7 1039 In the opposite diection, we shall pove i fo minus sign. Suppose that is even, x u mod, y us mod, y us mod 2 and y sx =. In this case, thee exist integes b, d such that x = u b, y = us d. So, it is b clea that Γ 2 which means s d x Gu,. Because = y sx = us d s u b. This implies d + sb = 1. As is even, d must be even. Othewise, it contadicts ou hypothesis. With simila agument, we obtain the elements 2b b of Γ 2 which ae and fo ii and iii espectively. s d s 2d 3.3. Theoem. All subobital gaphs fo Γ 2 on ˆQ ae paied. Poof. Because of the tansitivity of Γ 2, it is sufficient to show that G, u G u,. It means that thee is no T Γ 2 which sends the pai, u to the pai u,. On the contay, assume that T = u and T u =. By compaing the deteminants, we have that a b 1 u u 1 a b 1 u u 1 = o = In the fist case, we obtain a = u, c =, au + b = 1 and cu + d = 0. That u b is, d = u and u 2 = 1 + b. Taking T = we see that the only case fo u T to be an element of Γ 2 is that and b must be even. Since u 2 = 1 + b, then u 2 1 mod b. As and b ae even, u 2 1 mod 4 which has no solution. Fo u b the second case, taking T =, simila contadiction is obtained. u 3.4. Coollay. Thee ae no self-paied subobital gaphs fo Γ 2 on ˆQ.. In section 2 we intoduced, fo each intege, a Γ 2 -invaiant equivalence elation on ˆQ, with x if and only if y sx 0 mod. If x in Gu,, then Theoem s y 3.2 implies that y sx = ±, so x. Thus, each connected component of Gu, lies s y in a single block fo, of which thee ae Ψ, so we have: 3.5. Coollay. The gaph G u, is a disjoint union of Ψ subgaphs Subgaph F u,. We epesent the subgaph of G u, whose vetices fom the block [ ] = {x/y ˆQ y 0 mod } by F u, Coollay. The gaph G u, consists of Ψ disjoint copies of F u,. Poof. The vetices of each subgaph fom a single block with espect to the Γ 2 -invaiant equivalence elation defined by y sx 0 mod. Theefoe, if x 1 x 2 is an edge in the subgaph F u,, T x 1 T x 2 is also an edge in any othe subgaph with T Γ 2 because of the tansitivity of Γ 2 on ˆQ. ow, Theoem 3.2 immediately gives: 3.7. Theoem. x Fu, if and only if i If is even, then x ±u mod, y ±us mod, y ±us mod 2 and y sx =. ii If s is even, then x ±u mod 2, y ±us mod and y sx =. iii If and s ae odd, then x ±u mod, y ±us mod 2 and y sx =.
8 Theoem. Γ 2 0 pemutes the vetices and the edges of F u, tansitively. Poof. Let v, w be any vetices of F u,. Since Γ 2 acts on ˆQ tansitively, thee exist a b T Γ 2 such that T v = w. Taking T =, v = k 1 and w = k 2 l 1 l 2 we see that c. It means that Γ 2 0 pemutes the vetices of F u,. Let x e 1 1 y 1 b1 and x e 2 2 y 2 b2 be any edges of F u,. We can give following diagam: 1, u T2 0 x2, y 2 b2 T 1 x1 y 1, b1 T 2 T 1 1 x1 x2 By this epesentation, we have T 1 = and T y 1 2 =. Since y 2 T 2 T 1 1 has the fom fo some intege k, then T := T k 2 T 1 1 Γ 2 0. It is clea that T x1 y 1 = x 2 y 2 and T b1 = b2. Since T is an element of a goup of hypebolic isometies of H, geodesics ae sent to geodesics unde its action. So, T tansfom the edges e 1 to e 2. Consequently, Γ 2 0 pemutes the edges of F u, Lemma. Thee is an isomophism F u, F u, given by v v. Poof. It is clea that v v is one-to-one and onto. Let s show that the stuctue is peseved. Hee, it means that if a b F u,, then a b F u,. Suppose that x Fu, and is even. By Theoem 3.7i, taking < x, we have that x u mod, y us mod, y us mod 2 and y sx =. Since < x, then > x. Taking x u mod, y us mod, y us mod 2 and y + sx =, we have that x F u,. Fo othe conditions, the est ae simila Lemma. If M, then thee is a homomophism F u, F u,m given by v v/m., s < x y x Poof. We suppose that ae adjacent vetices in Fu, and < x and y that is witten as Fu,. If is even, then x u mod, y s us mod, y us mod 2 and y sx = 1. Since M, x u mod M, ym usm mod M, ym us mod 2M. y sx = 1 is also tue fo M. Fo othe conditions, the est ae simila Theoem. F u, contains diected tiangles if and only if u 2 u+1 0 mod. Poof. Suppose that F u, contains a diected tiangle. Because of the tansitive action, the fom of diected tiangle can be taken as u < fo some intege. Fist, let u be even. Fom the second edge, we have u = 1 and u 2 mod by Theoem 3.2. So, we obtain u 2 + u mod. Similaly, if u >, then we see that u 2 u mod. ow, is even. By applying Theoem 3.2 to the second edge, we have u = 1 and u 2 mod 2, giving u 2 + u mod 2. It is impossible, because thee is no solution fo this equivalence. Finally, suppose that u, ae odd. Again, fom the second edge, we have u = 1 and u 2 mod, giving u 2 + u mod. If u >, it would be u2 u mod. Combining all of the equivalences, we obtain u 2 u mod. Convesely, if u 2 u mod, we see that eithe u + 1 u 2 mod o u + 1 u 2 mod. Theoem 3.2. implies that thee is an edge u u+1 with
9 1041 u < u+1 in Fu, o u u 1 with u > u+1 in Fu,. tiangle u u±1 in Fu,. Consequently, thee is a diected 1 Let us give some examples. Fo u, -odd, o is a diected tiangle in F3,13. Fo u-even and -odd, o is a diected tiangle in F2,7. Fo -even, we know that thee is no tiangle. Obsevation. We know that thee is no tiangle in F u,20 fo -even by Theoem Because of the elationships between elliptic elements with cicuits, ou expectation is u 2b that thee is no elliptic element of ode 3 of the fom Γ 2. Indeed, being 2 0 d an elliptic element of ode 3, it is well-known that u + d = ±1. Taking deteminant 1 d 2b of, we have d d 2 4b 2 0 d 0 = 1. It is clea that thee is no solution fo d d 2 1 mod 4. On the othe hand, we know that the subobital gaph fo modula goup is a foest if and only if it contains no tiangles [2]. Using this fact, we can give the following esult; Coollay. The gaph G u, is a foest if and only if u 2 ± u mod Connectedness. In this last section, we examine the connectedness of F u, Definition. A subgaph K of G u, is called connected if any pai of its vetices can be joined by a path in K Theoem. The subgaphs F 0,1 and F 1,1 ae connected. Poof. Hee, to see the situation bette, we wite the edge conditions fo F 0,1 and F 1,1 by Theoem 3.2 explicitly. Case F 0,1: x F0,1 if and only if i If -even, then y-odd and y sx = 1. ii If s-even, then x-even and y sx = 1. iii If, s-odd, then y-even and y sx = 1. We will show that each vetex a of F0,1 can be joined to by a path in F0,1. It is clea b fo b = 1. Since a, b = 1, we can wite the equation ad bc = 1 by Bezout s identity. Fo this pai c, d satisfying the equation we claim that a can be joined with c by above b d edge condition. Subcase1. Suppose a-even. By the equation we have that b, c must be odd and thee ae two possibilities fo d. If d-odd, then a i c means that we have c a by i. If b d d b d-even, then c ii a. d b Subcase2. Let b-even. By the equation we have that a, d must be odd and thee ae two possibilities fo c. If c-odd, then c iii a. If d-even, then a ii c. d b b d Subcase3. Assume that a-odd and b-odd. By the equation it is impossible that c, d ae odd o even at once, so thee ae two possibilities. If c-odd and d-even, then a iii c. If b d c-even and d-odd, then c i a. d b Consequently F 0,1 is connected. Case F 1,1: x F1,1 if and only if i If -even, then y-even and y sx = 1.
10 1042 ii If s-even, then x-odd and y sx = 1. iii If, s-odd, then y-odd and y sx = 1. Taking a vetex a in F1,1, thee exists the equation ad bc = 1 by Bezout s identity. b We shall show that a is adjacent to vetex c in F1,1. b d Subcase1. Suppose a-even. By the equation we have that b, c must be odd and thee ae two possibilities fo d. If d-odd, then c iii a. If d-even, then a i c. d b b d Subcase2. Let b-even. By the equation we have that a, d must be odd and thee ae two possibilities fo c. If c-odd, then a ii c. If c-even, then c i a. b d d b Subcase3. Assume that a-odd and b-odd. By the equation it is impossible that c, d ae odd o even at once, so thee ae two possibilities. If c-odd and d-even, then c ii a. If d b c-even and d-odd, then a iii c. b d Consequently, F 1,1 is connected Theoem. The subgaphs F 1,2 and F 3,2 ae connected. Poof. We shall show that each vetex v = a b 1 of F1,2 is joined to by a path 2b in F 1,2. Since the patten is peiodic with peiod 2, we can show by induction on b. If b = 1, then v = a can be joined with. If a = 1, it is clea that 1 1. If a = 1, then because 1 3 mod 4 and = 2. If a = 5, then 1 5. The same holds fo the est peiodically. So we can assume that b 2. To complete the poof, we show that v is adjacent to a vetex w = a 2b 1 with b 1 < b. It means that, w is connected by a path to, and hence so is v. As a, b = 1, thee exist integes c, d such that ad bc = 1. Fo some k Z, eplacing c and d by c + ka and d + kb, we can suppose 0 < d < b. i If c is odd, then w = c can be joined with a. Indeed, a > c gives that 2d 2b 2b 2d a 2d c 2b = 2 and c a mod 4. If c a mod 4, taking c a mod 4 we obtain a < c by 2bc 2ad = 2. Hence, if c is odd, a c is adjacent to in F1,2. 2b 2d 2b 2d ii If c is even, then a c is odd. As 0 < b d < b, we can take w = a c, 2b d adjacent to a because 2bc cd = 2. Hee, if 2a c 0 mod 4, then we have that 2b a c a mod 4 and 2ad bc = 2. Consequently, F 1,2 is connected. By the isomophism F 1,2 F 1,2 = F v 3,2, F 3,2 is v also connected Coollay. All gaphs F u,2 ae connected Coollay. The gaph G u,2 has 2 ψ2 = 6 connected components. Its blocks ae [ ], [1], [0]. The connected components of [ ] ae F 1,2 and F 3, Theoem. The subgaphs F 1,3, F 2,3, F 4,3 and F 5,3 ae not connected. Poof. It is sufficient to study with F 1,3 and F 2,3. Because thee is an isomophism fom F 1,3F 2,3 to F 5,3F 4,3 espectively. Case F 1,3: If F 1,3 is connected, then each vetex v = a would be joined to. We shall 3b show that no vetices of F 1,3 whee 1 < v < 2 ae adjacent to. Futhe, we asset that thee is no such a vetex v adjacent to vetices outside this inteval. Of couse, thee is at least some vetex of F 1,3 in this stip. Suppose 2 c < 1 < a < 2. Then we have 3 3d 3b c < 3 < a. This is impossible because cd ad = 1. Similaly, if 1 < k < f 7, then d b 3l 3e 3 k < 4 < f contadicts ke lf = 1. It means that no vetices of F1,3 between 1 and 2 l e ae adjacent to and that F 1,3 is not connected.
11 1043 Figue 1. The subgaph F 1,3 Case F 2,3: As above, let s show that no vetices of F 2,3 between 3 and 2 ae adjacent to 2 vetices outside this inteval. Suppose that 1 x < 3 < a < 2 and x < a F2,3. 3y 2 3b 3y 3b Then we have that x < 9 < a and xb ay = 1. By [7], we obtain that x = 4, y = 1, y 2 b a = 5 and b = 1. But 4 5 is not in F2,3. If 2 < x < 2 < a < 8 and x < a F2,3, y 3b 3 3y 3b then we would have x < 6 < a and xb ay = 1. It is impossible because of well-known y b Faey sequence. Consequently, F 2,3 is not connected Coollay. All gaphs F u,3 ae not connected. Figue 2. The subgaph F 2, Theoem. The subgaphs F 1,4, F 3,4, F 5,4 and F 7,4 ae not connected. Poof. As emaked in the poof of Theoem 3.18, it is sufficient to study with F 1,4 and F 3,4. Case F 1,4: We will show that no vetices in F 1,3 between 1 and 1 ae adjacent to vetices 2 outside this inteval. Suppose 1 a < 1 < x < 1. Then we have a < 2 < x. This is 4 4b 2 4y b y
12 1044 impossible because ay bx = 1. Similaly, if a contadiction. So F 1,4 is not connected. a 4b < 1 < x 4y 7 4, then a b < 4 < x y < 7 is Case F 3,4: As above, it is seen that no vetices of F 3,4 between 1 and 2 ae adjacent to vetices outside this inteval. Consequently, F 3,4 is not connected Theoem. The subgaph F u, is connected if and only if 2. Poof. If F u, is connected, we know that 4 by [7]. Fo = 3, 4, we poved that F u, is not connected by Theoem 3.18 and Convesely, if 2, the esult immediately follows fom Theoem 3.14 and Refeences [1] Akbaş, M. and Başkan, T. Subobital gaphs fo the nomalize of Γ 0, T. J. of Mathematics, 20, , [2] Akbaş, M. On subobital gaphs fo the modula goup, Bull. London Math. Soc., 33, , [3] Beşenk, M. et al. Cicuit lengths of gaphs fo the Picad goup, J. Inequal. Appl., 2013:106, 8 pp., [4] Bigg,.L. and White, A.T. Pemutation goups and combinatoial stuctues, London Mathematical Society Lectue ote Seies, 33, CUP, Cambidge, 140 pp.,1979. [5] Dixon, J. D. and Motime, B. Pemutation Goups, Gaduate Texts in Mathematics 163, Spinge-Velag, [6] Güle, B.Ö. et al. Elliptic elements and cicuits in subobital gaphs, Hacet. J. Math. Stat., 40 2, , [7] Jones, G.A., Singeman, D. and Wicks, K. The modula goup and genealized Faey gaphs, London Mathematical Society Lectue ote Seies, 160, CUP, Cambidge, , [8] Jones, G.A. and Jones, J.M. Elementay umbe Theoy, Spinge Undegaduate Mathematics Seies, Spinge-Velag, [9] Kade, S., Güle, B. Ö. and Değe, A. H. Subobital gaphs fo a special subgoup of the nomalize, IJST. Tans A., 34 A4, , [10] Kade, S. and Güle, B. Ö. On subobital gaphs fo the extended Modula goup ˆΓ, Gaphs and Combinatoics, 29, no. 6, , [11] Keskin, R. Subobital gaphs fo the nomalize of Γ 0 m, Euopean J. Combin., 27, no. 2, , [12] eumann, P.M. Finite Pemutation Goups, Edge-Coloued Gaphs and Matices, Topics in Goup Theoy and Computation, Ed. M. P. J. Cuan, Academic Pess, [13] ewman, M. The Stuctue of some subgoups of the modula goup, Illinois J. Math., 6, , [14] ewman, M. Fee subgoups and nomal subgoups of the modula goup, Illinois J. Math., 8, , [15] ewman, M. Classification of nomal subgoups of the modula goup, Tansactions of the Ameican Math. Society, Vol.126, 2, , [16] Rankin, R. A. Modula Foms and Functions, Cambidge Univesity Pess, [17] Schoenebeg, B. Elliptic modula functions, Spinge Velag, [18] Sims, C.C. Gaphs and finite pemutation goups, Math. Z., 95, 76-86, [19] Singeman, D. Subgoups of Fuchsian goups and finite pemutation goups, Bull. London Math. Soc., 2, , [20] Tsuzuku, T. Finite Goups and Finite Geometies, Cambidge Univesity Pess, Cambidge, 1982.
ON 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 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 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 informationarxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More information9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.
Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this
More informationSPECTRAL SEQUENCES. im(er
SPECTRAL SEQUENCES MATTHEW GREENBERG. Intoduction Definition. Let a. An a-th stage spectal (cohomological) sequence consists of the following data: bigaded objects E = p,q Z Ep,q, a diffeentials d : E
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 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 informationSOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES
italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics
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 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 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 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 informationarxiv: v1 [math.co] 1 Apr 2011
Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and
More informationEnumerating permutation polynomials
Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem
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 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 informationA 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 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 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 informationBounds for Codimensions of Fitting Ideals
Ž. JOUNAL OF ALGEBA 194, 378 382 1997 ATICLE NO. JA966999 Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, 30-059, Kakow, Poland Communicated
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 informationOn the Quasi-inverse of a Non-square Matrix: An Infinite Solution
Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J
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 informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationA NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI
A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI Abstact. Wepove the conjectue that an inteval exchange tansfomation on 3-intevals with coesponding pemutation (1; 2;
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 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 informationSemicanonical basis generators of the cluster algebra of type A (1)
Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:
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 informationGalois points on quartic surfaces
J. Math. Soc. Japan Vol. 53, No. 3, 2001 Galois points on quatic sufaces By Hisao Yoshihaa (Received Nov. 29, 1999) (Revised Ma. 30, 2000) Abstact. Let S be a smooth hypesuface in the pojective thee space
More informationChromatic number and spectral radius
Linea Algeba and its Applications 426 2007) 810 814 www.elsevie.com/locate/laa Chomatic numbe and spectal adius Vladimi Nikifoov Depatment of Mathematical Sciences, Univesity of Memphis, Memphis, TN 38152,
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More informationMeasure Estimates of Nodal Sets of Polyharmonic Functions
Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
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 informationChaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments
Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity,
More informationIntroduction Common Divisors. Discrete Mathematics Andrei Bulatov
Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes
More informationSymmetries of embedded complete bipartite graphs
Digital Commons@ Loyola Maymount Univesity and Loyola Law School Mathematics Faculty Woks Mathematics 1-1-014 Symmeties of embedded complete bipatite gaphs Eica Flapan Nicole Lehle Blake Mello Loyola Maymount
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 informationA Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction
A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić
More informationSOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS
Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal
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 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 informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
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 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 information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationDo Managers Do Good With Other People s Money? Online Appendix
Do Manages Do Good With Othe People s Money? Online Appendix Ing-Haw Cheng Haison Hong Kelly Shue Abstact This is the Online Appendix fo Cheng, Hong and Shue 2013) containing details of the model. Datmouth
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
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 informationAsymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function
"Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationOn a generalization of Eulerian numbers
Notes on Numbe Theoy and Discete Mathematics Pint ISSN 1310 513, Online ISSN 367 875 Vol, 018, No 1, 16 DOI: 10756/nntdm018116- On a genealization of Euleian numbes Claudio Pita-Ruiz Facultad de Ingenieía,
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationConstruction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity
Constuction and Analysis of Boolean Functions of 2t + 1 Vaiables with Maximum Algebaic Immunity Na Li and Wen-Feng Qi Depatment of Applied Mathematics, Zhengzhou Infomation Engineeing Univesity, Zhengzhou,
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 informationAvailable online through ISSN
Intenational eseach Jounal of Pue Algeba -() 01 98-0 Available online though wwwjpainfo ISSN 8 907 SOE ESULTS ON THE GOUP INVESE OF BLOCK ATIX OVE IGHT OE DOAINS Hanyu Zhang* Goup of athematical Jidong
More informationON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS
ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS L. MICU Hoia Hulubei National Institute fo Physics and Nuclea Engineeing, P.O. Box MG-6, RO-0775 Buchaest-Maguele, Romania, E-mail: lmicu@theoy.nipne.o (Received
More informationSecret Exponent Attacks on RSA-type Schemes with Moduli N = p r q
Secet Exponent Attacks on RSA-type Schemes with Moduli N = p q Alexande May Faculty of Compute Science, Electical Engineeing and Mathematics Univesity of Padebon 33102 Padebon, Gemany alexx@uni-padebon.de
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationTurán Numbers of Vertex-disjoint Cliques in r- Partite Graphs
Univesity of Wyoming Wyoming Scholas Repositoy Honos Theses AY 16/17 Undegaduate Honos Theses Sping 5-1-017 Tuán Numbes of Vetex-disjoint Cliques in - Patite Gaphs Anna Schenfisch Univesity of Wyoming,
More informationarxiv: v1 [math.nt] 28 Oct 2017
ON th COEFFICIENT OF DIVISORS OF x n axiv:70049v [mathnt] 28 Oct 207 SAI TEJA SOMU Abstact Let,n be two natual numbes and let H(,n denote the maximal absolute value of th coefficient of divisos of x n
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 informationConnectedness of Ordered Rings of Fractions of C(X) with the m-topology
Filomat 31:18 (2017), 5685 5693 https://doi.og/10.2298/fil1718685s Published by Faculty o Sciences and Mathematics, Univesity o Niš, Sebia Available at: http://www.pm.ni.ac.s/ilomat Connectedness o Odeed
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
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 informationLecture 16 Root Systems and Root Lattices
1.745 Intoduction to Lie Algebas Novembe 1, 010 Lectue 16 Root Systems and Root Lattices Pof. Victo Kac Scibe: Michael Cossley Recall that a oot system is a pai (V, ), whee V is a finite dimensional Euclidean
More informationAn intersection theorem for four sets
An intesection theoem fo fou sets Dhuv Mubayi Novembe 22, 2006 Abstact Fix integes n, 4 and let F denote a family of -sets of an n-element set Suppose that fo evey fou distinct A, B, C, D F with A B C
More informationMiskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp
Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp Miskolc Mathematical Notes HU
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 informationLacunary I-Convergent Sequences
KYUNGPOOK Math. J. 52(2012), 473-482 http://dx.doi.og/10.5666/kmj.2012.52.4.473 Lacunay I-Convegent Sequences Binod Chanda Tipathy Mathematical Sciences Division, Institute of Advanced Study in Science
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 informationOn Continued Fraction of Order Twelve
Pue Mathematical Sciences, Vol. 1, 2012, no. 4, 197-205 On Continued Faction of Ode Twelve B. N. Dhamenda*, M. R. Rajesh Kanna* and R. Jagadeesh** *Post Gaduate Depatment of Mathematics Mahaani s Science
More informationRELIABILITY is an important concept in the design
Poceedings of the Wold Congess on Engineeing 0 Vol I WCE 0, July -, 0, London, U.K. Reliability Measues in Ciculant Netwok *Inda Rajasingh, Bhaati Rajan, and R. Sundaa Rajan Abstact Reliability and efficiency
More informationarxiv: v2 [math.ag] 4 Jul 2012
SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
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 informationCONSTRUCTION OF EQUIENERGETIC GRAPHS
MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 57 (007) 03-10 ISSN 0340-653 CONSTRUCTION OF EQUIENERGETIC GRAPHS H. S. Ramane 1, H. B. Walika * 1 Depatment
More informationON SPARSELY SCHEMMEL TOTIENT NUMBERS. Colin Defant 1 Department of Mathematics, University of Florida, Gainesville, Florida
#A8 INTEGERS 5 (205) ON SPARSEL SCHEMMEL TOTIENT NUMBERS Colin Defant Depatment of Mathematics, Univesity of Floida, Gainesville, Floida cdefant@ufl.edu Received: 7/30/4, Revised: 2/23/4, Accepted: 4/26/5,
More informationGroup Connectivity of 3-Edge-Connected Chordal Graphs
Gaphs and Combinatoics (2000) 16 : 165±176 Gaphs and Combinatoics ( Spinge-Velag 2000 Goup Connectivity of 3-Edge-Connected Chodal Gaphs Hong-Jian Lai Depatment of Mathematics, West Viginia Univesity,
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 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 informationLecture 1.1: An introduction to groups
Lectue.: An intoduction to goups Matthew Macauley Depatment o Mathematical Sciences Clemson Univesity http://www.math.clemson.edu/~macaule/ Math 85, Abstact Algeba I M. Macauley (Clemson) Lectue.: An intoduction
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationSMT 2013 Team Test Solutions February 2, 2013
1 Let f 1 (n) be the numbe of divisos that n has, and define f k (n) = f 1 (f k 1 (n)) Compute the smallest intege k such that f k (013 013 ) = Answe: 4 Solution: We know that 013 013 = 3 013 11 013 61
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 informationON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS
STUDIA UNIV BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Numbe 4, Decembe 2003 ON LACUNARY INVARIANT SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS VATAN KARAKAYA AND NECIP SIMSEK Abstact The
More information5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS
5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,
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 informationJournal of Number Theory
Jounal of umbe Theoy 3 2 2259 227 Contents lists available at ScienceDiect Jounal of umbe Theoy www.elsevie.com/locate/jnt Sums of poducts of hypegeometic Benoulli numbes Ken Kamano Depatment of Geneal
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 informationTHE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS
THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS ESMERALDA NĂSTASE MATHEMATICS DEPARTMENT XAVIER UNIVERSITY CINCINNATI, OHIO 4507, USA PAPA SISSOKHO MATHEMATICS DEPARTMENT ILLINOIS STATE UNIVERSITY
More informationBrief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis
Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion
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 informationTHE CONE THEOREM JOEL A. TROPP. Abstract. We prove a fixed point theorem for functions which are positive with respect to a cone in a Banach space.
THE ONE THEOEM JOEL A. TOPP Abstact. We pove a fixed point theoem fo functions which ae positive with espect to a cone in a Banach space. 1. Definitions Definition 1. Let X be a eal Banach space. A subset
More information