Supplementary Appendix (not for publication) for: The Value of Network Information
|
|
- Reginald Hensley
- 6 years ago
- Views:
Transcription
1 Suppementary Appendix not for pubication for: The Vaue of Network Information Itay P. Fainmesser and Andrea Gaeotti September 6, 03 This appendix incudes the proof of Proposition from the paper "The Vaue of Network Information." For competeness, we repeat the statement of the Proposition. Proposition. Suppose that γk max < /. Then, for any of the price discrimination schemes considered, there exists a unique finite price schedue p that soves the monopoy s profit maximization probem. We now prove the Proposition. Notation and definitions Reca that γ > 0 is the network externaity coefficient, that out-degrees are indexed by k and in-degrees by. Reca further that P k and H are the fractions of consumers with out-degree k and in-degree respectivey, and that = Ek ] and = E k] capture the conditiona expected out-degree of a consumer with in-degree and conditiona expected in-degree of a consumer with out-degree k. Let K out = k max K in = k max K in/out = k max + k max where k max is maxima degree among a in- and out-degrees of a individuas. Let ϕ out : {0,..., k max } {0,..., K out } ϕ in : {0,..., k max } {0,..., K in } ϕ in/out : {0,..., k max } {0,..., K in/out } Fainmesser: Department of Economics, Brown University, Providence, RI 09 emai: Itay_Fainmesser@Brown.edu; Gaeotti: Department of Economics, University of Essex, Wivenhoe Park, Cochester CO 3S, United Kingdom emai: agaeo@essex.ac.uk.
2 be the bijections defined by ϕ out i = i ϕ in i = i ϕ in/out x, y = k max + x + y and et qk, HP k. The monopoist s profit Π is given by Π out = k P kpkxk Π in = Π in/out = H k H k P k pxk, P k pk, xk, and the negative of Hessian matrices are H out = Π out ps pt s,t {0,...,k max } H in = Π in ps pt s,t {0,...,k max } H in/out = Π in/out p ϕ in/out s p ϕ in/out t s,t {0,...,K in/out } The corresponding first order condition of the monopoy s profit maximization probem are: where 0 = Π out ps = P s ps + γ p out s P kpkk ] H s γ k k ] 0 = Π in ps = Hs ps + γ p in s s Hp γ k 0 = Π in/out = HyP x y px, y + γ p ] in/out x H pk, kγ p in/out P k px, y γ k γ k px, y k p out = H k P k pk p in = Hp p in/out = H P k pk, k
3 and p out ps = P s H s p in ps = Hss p in/out HyP x yy = px, y Intermediate resuts The proof of the Proposition makes use of the foowing three emmas. Lemma. Let K, Π, ϕ {K i, Π i, ϕ i }, where i {out, in, in/out}. p j, b j, c j j {0,...,K} R 3 + K+ such that for every s, t {0,..., K}, Then there exists Γ R + and Π p ϕ s p ϕ t = p sp t Γ b t c s + b s c t ] + p s {s=t}. Lemma. Let G = p s p t Γ b t c s + b s c t ] + p s {s=t} s,t {0,...,K}. Then the determinant of G is given by detg = K i p i + Γ i b i c i p i Γ p i p j b j c i b i c j ]. i<j Lemma 3. Assume that γk max < /. Then H out, H in, and H in/out are positive definite. Proofs Proof of Proposition. It suffices to show that the first and second order conditions are satisfied. The first order conditions are verified in the Appendix to the paper. To see that the second order conditions hod, note that for each price discrimination scheme, the second order partia derivatives are of the form given in Lemma. A necessary and sufficient condition for maximizing profit is that the Hessian matrix is negative definite, or equivaenty that the negative of the Hessian matrix is positive definite. The determinant of this matrix is given by the formua in Lemma. By Lemma 3, we concude positive definiteness and hence that the second order condition hods. We now compete the proof by proving the three emmas above. 3
4 Lemma Proof. We compute the second partia derivatives. Reca that Cs = for any t, s, x, y, z, w {0,..., k max }, H s = ˆs and et Ct = ˆt. Then, Π out pt ps = γ p P s ps + pt γ k s γ P kpkk ] H s γ k k = P s {s=t} γs p γ k pt γ P kpkk H s γ k pt k = P s {s=t} γs γ k P t H t γ γ k P tt H s γ = P s {s=t} P sp t s H t + t H s γ k γ = P s {s=t} P sp t sct + tcs γ k ] Π in pt ps = Hs ps + γ ps s Hp pt γ k ] = Hs {s=t} + γ Hp γ k pt Π in/out pz, w px, y = = Hs = Hs {s=t} + {s=t} + γ γ k γ γ k γ = Hs {s=t} HsHt γ k HyP x y pz, w = qx, y {z,w=x,y} s p pt s s Htt s ] Htt ] s t Htt + sht t Cs + s Ct γ p px, y + γ k x γ γ k x p pz, w ] ] ] pz, w H k = qx, y {z,w=x,y} γ ] ww zγ xqz, qz, w γ k γ k γ = qx, y {z,w=x,y} qx, yqz, w γ k xw + γ γ k γ = qx, y {z,w=x,y} qx, yqz, w xw + zy γ k,k pk, kγ p P k γ k px, y pk, kγ qk, γ k qx, yy zy p px, y To concude: for the case of discrimination on out-degree, we take p i = P i, b i = i, c i = Ci, and Γ = for discrimination on in-degree, we take p i = P i, b i = i, c i = Ci, and Γ = in- and out-degrees, we take p i = qi = qi, i, b i = i, c i = i, and Γ = γ γ k γ k. γ k ; ; and for discrimination on
5 Lemma Proof. Fix k max = K > 0. For {,..., K}, and for any J {0,..., K} with J =, et L = {0,..., } and M J = G st s L t J. Denote S = J L. We wi write J = {j 0, j,..., j } with j 0 < j < < j. Aso denote Q = J \ L = {m i } 0 i< Q and R = L \ J = {n i } 0 i< R. Note that J = S Q and L = S R. Aso, Q = R since J = = L. Moreover, note that any eement of the set Q exceeds any eement of the set S. We caim that M J = κ, Jψ, J where κ, J = S i J L if R = 0 p i n 0+ S if R = n 0+n if R = and + Γ i b ic i p i + Γ p ip j b i b j c i c j b i c j if Q = 0 Γ b n0 c m0 + b m0 c n0 + ψ, J = Γ r S bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r if Q =. Γ b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 if Q = 0 if Q 3 Denote Φ = i=0 p i. We wi prove the caim by induction on. First we wi show that the formua is vaid for =. Next we wi show that, if the formua is vaid for < K, then it is aso vaid for +. Base case = : det G st = p 0 p t Γ b t c 0 + b 0 c t ] + p 0 {0=t} p 0 = Γ b 0c 0 ] + p 0 if t = 0, i.e. Q = 0 p 0 p t Γ b t c 0 + b 0 c t ] if t 0, i.e. Q = p 0 + Γ b 0 c 0 p 0 ] if t = 0, i.e. Q = 0 =. p 0 p t Γ b t c 0 + b 0 c t ] if t 0, i.e. Q = Now assume hods for some < K. Let L = L {} = {0,..., }, and et J = J {j} for some j J. Accordingy, et S = J L, Q = J \ L, and R = L \ J. We need to show that hods for +, i.e., that M J + For two sets X and Y, et X Y be the disjoint union of X and Y. = κ +, J ψ +, J. 5
6 By cofactor expansion aong the ast row, we have M J + = i+ M J \{j i } G ji, 0 i so it suffices to show that i+ M J \{j i } G ji = κ +, J ψ +, J. 0 i We consider the foowing cases. Case 0. R = 0 In this case Q = R = and J = S = {0,..., }, so L \ J \ {} = L \ {} \ J \ {} = L \ J = 0 Simiary, for r S \ {}, L \ J \ {r} = J L {r} C C J = L C {r} = L \ J L {r} = {r} = and in particuar R = {r}. Thus r+ p Φ Φ M J \{r} M J \{} We wi show that M J S p Φ S M J p Φ + + G r = Γ b r c + b c r + Γ j,r p p r Γ b c r + b r c ] G = + Γ b i c i p i + Γ p i p j bi b j c i c j b i c j i< = +Γ i {0,...,} b ic i p i + Γ p ip j b i b j c i c j b i c j = r+ r+ Γ b r c + b c r + Γ r< b c r + b r c b j c j p j b b r p j c j + c c r p j b j j,r p Γ b c ] + p.. Since S = +, b c r + b r c b j c j p j b b r p j c j + c c r p j b j 6
7 p p r Γ b c r + b r c ] Γ i< = + AΓ + BΓ + CΓ 3. b i c i p i + Γ i,j< It suffices to show A = i {0,...,} b ic i p i, B = A = p b c + i< b ic i p i. Next note that B = p p r b c r + b r c ] b r c + b c r r< + p b c ] b i c i p i i< + p i p j bi b j c i c j b i c j i,j< = p p i b c i + b i c ] b i c + b c i + b i c i p i b c p = = = + + i< i,j< p i p j bi b j c i c j b i c j p p i b c i + b i c i b c + b i c ] + bi c i p i b c p i< i,j< i< i< p i p j bi b j c i c j b i c j p p i b c i + b i c i b c b i c ] + p i p bi b c i c b i c + = p i p j bi b j c i c j b i c j. To see that C = 0, note that C = p r< j,r p i p j bi b j c i c j b i c j p Γ b c ] + p ip j b i b j c i c j b i c j, and C = 0. Ceary we have i,j< p p j b b j c c j b c j j< p i p j bi b j c i c j b i c j + i,j< b c r + b r c b j c j p j b b r p j c j + c c r p j b j pr b c r + b r c ] + p i p j bi b j c i c j b i c j b c ] i,j< = i,j< p i p j bi b j c i c j b i c j b c i + b i c b j c j p j b b i p j c j + c c i p j b j pi b c i + b i c ] + p i p j bi b j c i c j b i c j b c ] 7
8 = p i p j b c i + b i c b j c j b b i c j + c c i b j b c i + b i c ] + ] b i b j c i c j b i c j b c ] i,j< Note that for every i, j, the coefficient of p i p j is zero: b c i + b i c b j c j b b i c j + c c i b j b c i + b i c ] ] + b i b j c i c j b i c j b c ] b c j + b j c b i c i b b j c i + c c j b i b c j + b j c ] ] + b i b j c i c j b jc i b c ] = b c i b j c j b i c jc i + c j b i c i b j c ] i c j c b i b j c j c i b jb i + b j b i c i c j b ] i b j = 0. b c + b c b i b j c i c j ] ci b i b j c j b i c j c i b j b i c j + cj b j b i c i c i b j b i c j b j c i ] Case. R = Case a. S In this case, R = L \ J =. Note that M J + = i M J \{j i } G ji j i J = i M J \{j i } G ji j i S Q = i M J \{} G + First, note anaogousy to Case 0 that j i S\{} i M J \{j i } G ji + i M J \{j i } G ji. j i Q L \ J \ {} = L \ J Next consider r S \{}. Note that r J and hence r J C. Since R, we have L \J = R = R \{} = L \ J, so =. L \ J \ {r} = J L {r} C C J = L C {r} = L \ J L {r} = L \ J {r} 8
9 = L \ J {r} = + =. Now consider j Q. Note that j {0,..., }. As before R = R \ {}, so Thus n0+r p m0 p Φ M J \{r} L \ J \ {j} = L J {j} C C J = L C {j} = L {} C J C L {j} = L {} C J C G r = L \ J = L \ {} \ J = L \ J \ {} = L \ J =. = Γ b n0 b m0 c r c b n0 b c r c m0 b r b m0 c n0 c + b r b c n0 c m0 p p r Γ b c r + b r c ] n0+ M J \{} G p m0 Φ = Γ b n0 c m0 + b m0 c n0 + Γ bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r p Γ b c ] + p r S \{} n0+ M J \{m 0 } G m0 p Φ = Γ b n0 c + b c n0 + Γ b c n0 + b n0 c b r c r p r b b n0 p r c r + c c n0 p r b r p p m0 Γ b c m0 + b m0 c ] We wi show that S M J p m0 p Φ + r S \{} = Γ b n0 c m0 + b m0 c n0 + Γ r S bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r. First note that S M J p m0 p Φ + 9
10 = S \{} i+ n 0+i M J \{j i } G j p m0 p Φ i j i S \{} + + n 0+ M J \{} G + + n 0+ = n 0+ S \{} p m0 p Φ j i S \{} = n 0+ AΓ + BΓ + CΓ 3. M J \{j i } G j i + It suffices to show that A = b n0 c m0 + b m0 c n0, B = and C = 0. First, it is easy to see that and hence A = b n0 c m0 + b m0 c n0. Next, note that M J \{} M J \{m 0 } G m 0 G M J \{m 0 } G m 0 r S bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r, S p m0 p Φ A = S p m0 Φ p b n0 c m0 + b m0 c n0, S p m0 p Φ B = + S p m0 Φ p bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r r S \{} + S p m0 Φ p b c ] b n0 c m0 + b m0 c n0 S p Φ p p m0 b c m0 + b m0 c ] b n0 c + b c n0 B = + bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r r S \{} + p b c ] b n0 c m0 + b m0 c n0 p b c m0 + b m0 c ] b n0 c + b c n0 = + bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r r S \{} + p b n0 c m0 b c + b m0 c n0 b c p b c m0 b n0 c + b c m0 b c n0 + b m0 c b n0 c + b m0 c b c n0 ] = + bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r + = r S \{} bm0 c n0 + b n0 c m0 b c p b m0 b n0 p c + c m 0 c n0 p b bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r r S To see that C = 0, note that C = r S \{} b n 0 b m0 c r c b n0 b c r c m0 b r b m0 c n0 c + b r b c n0 c m0 p p r b c r + b r c ] 0
11 S \{} p b c bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r r S \{} + S \{} p b c m0 + b m0 c ] b c n0 + b n0 c b r c r p r b b n0 p r c r + c c n0 p r b r r S \{} Note that each term in the sum is zero: for each r, b n 0 b m0 c r c b n0 b c r c m0 b r b m0 c n0 c + b r b c n0 c m0 b c r + b r c ] + b c bm0 c n0 + b n0 c m0 b r c r b m0 b n0 c r + c m0 c n0 b r b c m0 + b m0 c ] b c n0 + b n0 c b r c r b b n0 c r + c c n0 b r b m0 c n0 c b c n0 c m0 c b c c m0 c n0 + b c m0 + b m0 c c c n0 = b r + c r b n 0 b m0 c b n0 b c m0 b b c b m0 b n0 + b c m0 + b m0 c b b n0 + b r c r c b n0 b m0 c b n0 b c m0 c b m0 c n0 c b + b c n0 c m0 b + b r c r b m0 c n0 b c + b n0 c m0 b c b r c r b c m0 b c n0 + b c m0 b n0 c + b m0 c b c n0 + b m0 c b n0 c ] = 0. Case b. R First consider j S. Since R = L \ J, we have J. In this case, = R = Q = J \ L. Since j S, we have j L, so J \ {j} \ L = J \ {j} L \ {} = J \ L \ {} = J \ {} \ L \ {} = J \ L Note that R = {}, and hence S = {0,..., }. This impies that for j S, so j+ p m0 Φ M J \{j} = = Γ b j c m0 + b m0 c j + Γ r S \{j} bm0 c j + b j c m0 b r c r p r b m0 b j p r c r + c m0 c j p r b r. Now et j Q. Reca that in this case, L \ J =. Using the facts that j L and J, we have L \ J \ {j} L J = C {j} = L J C L {j}
12 Then for j Q, we have Thus Φ M J \{j} S M J p m0 p Φ = S p m0 p Φ = Γ b ji c m0 + b m0 c ji + Γ j i S p ji Γ b c ji + b ji c ] + + Γ i = L J C = L \ J = L \ {} \ J = L \ J \ {} = R = = 0. = + Γ b i c i p i + Γ p i p j bi b j c i c j b i c j. i L i,j L i+ i+ M J \{j i } G j i + + M J \{m 0 } j i S b i c i p i + Γ = + AΓ + BΓ + CΓ 3 r S \{j i } p i p j bi b j c i c j b i c j It suffices to show that A = b c m0 + b m0 c, B = and C = 0. First, it is easy to see that A = b c m0 + b m0 c. Next, note that bm0 c ji + b ji c m0 b r c r p r b m0 b ji p r c r + c m0 c ji p r b r Γ b c m0 + b m0 c ] r S bm0 c + b c m0 b r c r p r b m0 b p r c r + c m0 c p r b r, B = b ji c m0 + b m0 c ji p ji b c ji + b ji c ] + b i c i p i b c m0 + b m0 c ] j i S i = b ic m0 + b m0 c i p i b c i + b i c ] + b i c i p i b c m0 + b m0 c ] i< = p i b ic m0 b c i + b i c m0 b i c + b m0 c i b c i + b m0 c i b i c + b i c i b c m0 + b i c i b m0 c i< = p i b i c m0 b c i + b m0 c i b i c b i c m0 b i c b m0 c i b c i i< = p i cm0 b b i c i + b m0 c b i c i c m0 c b i b m0 b c i i<
13 = i< bm0 c + b c m0 b i c i p i b m0 b p i c i + c m0 c p i b i, from which the caim foows since S = {0,..., }. To see that C = 0, note that C = j i S r S \{j i } + bm0 c ji + b ji c m0 b r c r p r b m0 b ji p r c r + c m0 c ji p r b r pji b c ji + b ji c ] p i p j bi b j c i c j b i c j b c m0 + b m0 c ] = bm0 c i + b i c m0 b j c j p j b m0 b i p j c j + c m0 c i p j b j pi b c i + b i c ] + p i p j bi b j c i c j b i c j b c m0 + b m0 c ] ] = p i p j bm0 c i + b i c m0 b j c j b m0 b i p j c j + c m0 c i p j b j b c i + b i c ] + b i b j c i c j b i c j b c m0 + b m0 c ] ] We wi show that, for each i, j with i j, the coefficient of p i p j is zero: b m0 c i + b i c m0 b j c j b m0 b i c j + c m0 c i b j b c i + b i c ] + b i b j c i c j b i c j b c m0 + b m0 c ] b m0 c j + b j c m0 b i c i b m0 b j c i + c m0 c j b i b c j + b j c ] + b i b j c i c j b jc i b c m0 + b m0 c ] = b ci bm0 c i + b i c m0 b j c j b m0 b i c j + c m0 c i b j + cm0 bi b j c i c j b i c ] j + c bi bm0 c i + b i c m0 b j c j b m0 b i c j + c m0 c i b j + bm0 bi b j c i c j b i c ] j + b cj bm0 c j + b j c m0 b i c i b m0 b j c i + c m0 c j b i + cm0 bi b j c i c j b jc ] i + c bj bm0 c j + b j c m0 b i c i b m0 b j c i + c m0 c j b i + bm0 bi b j c i c j b jc ] i = 0. Hence C = 0. Case. R = Case a. S Reca that M J + = i M J \{j i } G ji j i J = i M J \{j i } G ji j i S Q = i M J \{} G + j i S \{} i M J \{j i } G ji + i M J \{j i } G ji. j i Q 3
14 Consider r S \ {}. Note as in Case a that This impies M J \{r} = 0 for r S \ {}, so M J + L \ J \ {r} = L \ J {r} = + = 3. = i M J \{} G + i M J \{j i } G ji j i Q Consider j Q {}. Note aso as in Case a that L \ J \ {j} = L \ J =. Denote Q = {m 0, m } and R \ {} = R = {n 0, n }. By, M J \{m } n 0+n S Φ r J \S p r Γ G m = p p m0 b n0 b c n c m0 b n0 b m0 c n c b n b c n0 c m0 + b n b m0 c n0 c p p m Γ b c m + b m c ] M J \{m 0 } n 0+n S Φ r J \S p r Γ G m0 = p p m b n0 b c n c m b n0 b m c n c b n b c n0 c m + b n b m c n0 c p p m0 Γ b c m0 + b m0 c ] M J \{} n 0+n S Φ r J \S p r Γ G = p m0 p m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p Γ b c ] + p The sum of these three terms reduces to n 0+n S Φ r J \S p r Γ j i Q {} i+ M J \{j i } G ji = + p p m0 b n0 b c n c m0 b n0 b m0 c n c b n b c n0 c m0 + b n b m0 c n0 c p p m Γ b c m + b m c ] + + p p m b n0 b c n c m b n0 b m c n c b n b c n0 c m + b n b m c n0 c p p m0 Γ b c m0 + b m0 c ] + + p m0 p m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p Γ b c ] + p = p p m0 b n0 b c n c m0 b n0 b m0 c n c b n b c n0 c m0 + b n b m0 c n0 c p p m Γ b c m + b m c ] p p m b n0 b c n c m b n0 b m c n c b n b c n0 c m + b n b m c n0 c p p m0 Γ b c m0 + b m0 c ] + p m0 p m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p Γ b c ] + p = b c m b n0 b c n c m0 b n0 b m0 c n c b n b c n0 c m0 + b n b m0 c n0 c p p m 0 p m Γ
15 b c m0 b n0 b c n c m b n0 b m c n c b n b c n0 c m + b n b m c n0 c p p m 0 p m Γ + b c b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p p m 0 p m Γ + b m c b n0 b c n c m0 b n0 b m0 c n c b n b c n0 c m0 + b n b m0 c n0 c p p m 0 p m Γ b m0 c b n0 b c n c m b n0 b m c n c b n b c n0 c m + b n b m c n0 c p p m 0 p m Γ + b c b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p p m 0 p m Γ + b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 p = p b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0. Hence M J = j i Q {} = S Φ p r Γ r J \S = S p Φ i M J \{j i } G ji j i Q {} i M J \{j i } G ji r J \S p r Γ b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0. Case b. R First consider j S. Note as in Case b that Now et j Q. As in case b, we have Thus r J \S p r J \ {j} \ L = J \ L =. L \ J \ {j} = L \ J \ {} = R = =. M J = i M J \{j i } G ji j i J = S Φ i+ n0+i Γ b n0 b m0 c ji c m j i J \Q b n0 b m c ji c m0 b ji b m0 c n0 c m + b ji b m c n0 c m0 G j i + S Φ + n 0+ Γ b n0 c m + b m c n0 r J \S \m 0 p r 5
16 + Γ r S bm c n0 + b n0 c m b r c r p r b m b n0 p r c r + c m c n0 p r b r ] G m 0 + S Φ r J \S \m p r + n 0+ Γ b n0 c m0 + b m0 c n0 + Γ r S bm0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r ] G m = +n 0 S Φ r J \S p r b n0 b m0 c ji c m b n0 b m c ji c m0 j i J \Q Γ b ji b m0 c n0 c m + b ji b m c n0 c m0 p p ji Γ b c ji + b ji c ] + +n 0 S Φ Γ b n0 c m + b m c n0 + Γ b m c n0 + b n0 c m b r c r p r r S r J \S \m 0 p r b m b n0 p r c r + c m c n0 p r b r ] p p m0 Γ b c m0 + b m0 c ] +n 0 S Φ r J \S \m p r Γ b n0 c m0 + b m0 c n0 + Γ r S b m0 c n0 + b n0 c m0 b r c r p r b m0 b n0 p r c r + c m0 c n0 p r b r ] p p m Γ b c m + b m c ] = AΓ 3 + +n 0 S p Φ p r BΓ r J \S First we wi show that B = b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 : B = b n0 c m + b m c n0 b c m0 + b m0 c ] b n0 c m0 + b m0 c n0 b c m + b m c ] = b n0 c m b c m0 + b n0 c m b m0 c + b m c n0 b c m0 + b m c n0 b m0 c b n0 c m0 b c m + b n0 c m0 b m c + b m0 c n0 b c m + b m0 c n0 b m c = b n0 b m0 c c m b n0 b m c c m0 b b m0 c n0 c m + b b m c n0 c m0 Now we wi show that A = 0: S p Φ r Q p A = b n0 b m0 c r c m b n0 b m c r c m0 b r b m0 c n0 c m + b r b m c n0 c m0 b c r + b r c ] r r S bm c n0 + b n0 c m b r c r b m b n0 c r + c m c n0 b r b c m0 + b m0 c ] r S + r S bm0 c n0 + b n0 c m0 b r c r b m0 b n0 c r + c m0 c n0 b r b c m + b m c ] It suffices to show that each term is zero. Indeed for every r we have b n0 b m0 c r c m b n0 b m c r c m0 b r b m0 c n0 c m + b r b m c n0 c m0 b c r + b r c ] 6
17 b m c n0 + b n0 c m b r c r b m b n0 c r + c m c n0 b r b c m0 + b m0 c ] + b m0 c n0 + b n0 c m0 b r c r b m0 b n0 c r + c m0 c n0 b r b c m + b m c ] = b n0 b m0 c r c m b n0 b m c r c m0 b r b m0 c n0 c m + b r b m c n0 c m0 b c r ] + b n0 b m0 c r c m b n0 b m c r c m0 b r b m0 c n0 c m + b r b m c n0 c m0 b r c ] b m c n0 + b n0 c m b r c r b c m0 + b m0 c ] + b m b n0 c r + c m c n0 b r b c m0 + b m0 c ] + b m0 c n0 + b n0 c m0 b r c r b c m + b m c ] b m0 b n0 c r + c m0 c n0 b r b c m + b m c ] ] ] = c r b n0 b m0 c m b b n0 b m c m0 b + b m b n0 b c m0 + b m0 c b m0 b n0 b c m + b m c ] ] + b r b m0 c n0 c m c + b m c n0 c m0 c + c m c n0 b c m0 + b m0 c c m0 c n0 b c m + b m c + b r c r b m0 c n0 c m b + b m c n0 c m0 b + b n0 b m0 c m c b n0 b m c m0 c = 0. ] ] b m c n0 + b n0 c m b c m0 + b m0 c + b m0 c n0 + b n0 c m0 b c m + b m c Case 3. R = 3 Case 3a. S For any j J, we have L \ J \ {j} = L \ J {j} C = L J {j} C C J C {j} = L Since S J, we have L \ J = L \ J and hence By, this impies M J \{j} = L J C L {j} L \ J \ {j} L J C = L \ J = L \ J = 3. = 0 for every j J and hence M J + = 0. 7
18 Case 3b. R First consider j S. Note that R impies J. Since j S L, we have This impies M J \{j} = 0 for j S. Hence Now et j Q. As in case b, we have J \ {j} \ L = J \ {j} L M J + = J \ L = J \ {} \ L = J \ {} L = J \ L = 3. = M j i Q i J \{j i } G ji. L \ J \ {j} = L \ J \ {} = R = 3 =. Denote R \ {} = {n 0, n } and Q = {m 0, m, m } with n 0 < n and m 0 < m < m. By, we have M J \{m 0 } S Φ r S p r Γ G m0 = p m p m b n0 b m c n c m b n0 b m c n c m b n b m c n0 b m + b n b m c n0 c m p p m0 Γ b c m0 + b m0 c ] M J \{m } S Φ r S p r Γ G m = p m0 p m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 b m + b n b m c n0 c m0 p p m Γ b c m + b m c ] M J \{m } S Φ r S p r Γ G m = p m0 p m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 b m + b n b m c n0 c m0 p p m Γ b c m + b m c ] Thus p m0 p m p m S Φ r S p r Γ i+ M J \{j i } G ji j i Q = b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 b c m + b m c b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 b c m + b m c + b n0 b m c n c m b n0 b m c n c m b n b m c n0 c m + b n b m c n0 c m b c m0 + b m0 c = b c m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 b c m b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 8
19 so M J + + b c m0 b n0 b m c n c m b n0 b m c n c m b n b m c n0 c m + b n b m c n0 c m = 0, = 0. + b m c b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 b m c b n0 b m0 c n c m b n0 b m c n c m0 b n b m0 c n0 c m + b n b m c n0 c m0 + b m0 c b n0 b m c n c m b n0 b m c n c m b n b m c n0 c m + b n b m c n0 c m Case. R > 3 In this case, R = L \ J = L J C. Since we have By, this impies M J \{j} L \ J \ {j} = L \ J {j} C = L J {j} C C = L J C {j} = L J C L {j}, L \ J \ {j} L J C L J C = 3. = 0 for every j J, and hence M J + = 0. Lemma 3 Proof. First we wi consider the case of discrimination on in-degree. The case of discrimination on out-degree is identica. Using the determinant formua given in Lemma, it suffices to show that Γ i<j p i p j b j c i b i c j ] < + Γ i b ip i c i. Reca from the proof of Lemma that p i = P i, b i = i, c i = Ci, and Γ = γ k. Since γkmax < by assumption, we have γ k <. This impies γ k > and hence γ k <, so Γ = γ γ k < k max = k max. Reca that for any random variabe X, we have EX ] = x px x max xpx = x max EX. Thus i<j p i p j b j c i b i c j ] = i<j pi p j i c j ijc i c j + j c i pi p j i c j ijc i c j = i i j i pi p j i c j ijc i c j j 9
20 pi p j i c j + p i p j ijc i c j i j i j i p i p j c j + p i ic i i j i = E k ] E + k max p i c i i = E k ] ] E + k max Ec] = E ] k ] E + k max = E ] k ] E ] + k E max k max. Since 0 < Γ < k max, we have Γ pi p j b j c i b i c j ] < pi k max p j b j c i b i c j ] i<j < < + Γ i i<j b i p i c i as caimed. Now we wi consider the case of discrimination on in- and out-degrees. Again by Lemma it suffices to show that Γ i j p i p j b j c i b i c j ] < + Γ i b ip i c i. As before, we have <. This impies γ k Γ = γ γ k < k max = k max. Reca that = i,i i qi, i, ˆ = i,i i qi, i, and ˆ = due to consistency. Note that i j i j i j + i j < i j k max + i j k max, Using the parameters p i = qi = qi, i, b i = i, c i = i, and Γ = have i j qiqj i j i j ] < qiqj k max i j + j i i j < k max i,j γ obtained in the proof of Lemma, we γ k i j qiqj 0
21 = k max i qi j qj = k max ˆ = k max. i,i j,j Since 0 < Γ <, we have kmax Γ i j qiqj i j i j ] < k max qiqj i j i j ] i j < < + Γ i i i qi. This shows that det H > 0 where the size of the matrix K is arbitrary. Note that the K K upper-eft sub-matrix of H is defined exacty by the same formua as H. Then, since the resut hods for the sub-matrix of any size K, we concude that H is positive definite.
Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications
Pricing Mutipe Products with the Mutinomia Logit and Nested Logit Modes: Concavity and Impications Hongmin Li Woonghee Tim Huh WP Carey Schoo of Business Arizona State University Tempe Arizona 85287 USA
More informationHomework 5 Solutions
Stat 310B/Math 230B Theory of Probabiity Homework 5 Soutions Andrea Montanari Due on 2/19/2014 Exercise [5.3.20] 1. We caim that n 2 [ E[h F n ] = 2 n i=1 A i,n h(u)du ] I Ai,n (t). (1) Indeed, integrabiity
More information2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationu(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0
Bocconi University PhD in Economics - Microeconomics I Prof M Messner Probem Set 4 - Soution Probem : If an individua has an endowment instead of a monetary income his weath depends on price eves In particuar,
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationAkaike Information Criterion for ANOVA Model with a Simple Order Restriction
Akaike Information Criterion for ANOVA Mode with a Simpe Order Restriction Yu Inatsu * Department of Mathematics, Graduate Schoo of Science, Hiroshima University ABSTRACT In this paper, we consider Akaike
More informationODE Homework 2. Since M y N x, the equation is not exact. 2. Determine whether the following equation is exact. If it is exact, M y N x 1 x.
ODE Homework.6. Exact Equations and Integrating Factors 1. Determine whether the foowing equation is exact. If it is exact, find the soution pe x sin qdx p3x e x sin qd 0 [.6 #8] So. Let Mpx, q e x sin,
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationBASIC NOTIONS AND RESULTS IN TOPOLOGY. 1. Metric spaces. Sets with finite diameter are called bounded sets. For x X and r > 0 the set
BASIC NOTIONS AND RESULTS IN TOPOLOGY 1. Metric spaces A metric on a set X is a map d : X X R + with the properties: d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, y) d(x, z) + d(z, y), for a
More informationReichenbachian Common Cause Systems
Reichenbachian Common Cause Systems G. Hofer-Szabó Department of Phiosophy Technica University of Budapest e-mai: gszabo@hps.ete.hu Mikós Rédei Department of History and Phiosophy of Science Eötvös University,
More informationGeneralized Bell polynomials and the combinatorics of Poisson central moments
Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43,
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationAnother Class of Admissible Perturbations of Special Expressions
Int. Journa of Math. Anaysis, Vo. 8, 014, no. 1, 1-8 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.31187 Another Cass of Admissibe Perturbations of Specia Expressions Jerico B. Bacani
More informationA CLUSTERING LAW FOR SOME DISCRETE ORDER STATISTICS
J App Prob 40, 226 241 (2003) Printed in Israe Appied Probabiity Trust 2003 A CLUSTERING LAW FOR SOME DISCRETE ORDER STATISTICS SUNDER SETHURAMAN, Iowa State University Abstract Let X 1,X 2,,X n be a sequence
More informationProblem set 6 The Perron Frobenius theorem.
Probem set 6 The Perron Frobenius theorem. Math 22a4 Oct 2 204, Due Oct.28 In a future probem set I want to discuss some criteria which aow us to concude that that the ground state of a sef-adjoint operator
More informationThe arc is the only chainable continuum admitting a mean
The arc is the ony chainabe continuum admitting a mean Aejandro Ianes and Hugo Vianueva September 4, 26 Abstract Let X be a metric continuum. A mean on X is a continuous function : X X! X such that for
More informationAn Extension of Almost Sure Central Limit Theorem for Order Statistics
An Extension of Amost Sure Centra Limit Theorem for Order Statistics T. Bin, P. Zuoxiang & S. Nadarajah First version: 6 December 2007 Research Report No. 9, 2007, Probabiity Statistics Group Schoo of
More informationThe Group Structure on a Smooth Tropical Cubic
The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note,
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationXSAT of linear CNF formulas
XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
More informationStat 155 Game theory, Yuval Peres Fall Lectures 4,5,6
Stat 155 Game theory, Yuva Peres Fa 2004 Lectures 4,5,6 In the ast ecture, we defined N and P positions for a combinatoria game. We wi now show more formay that each starting position in a combinatoria
More information#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG
#A48 INTEGERS 12 (2012) ON A COMBINATORIAL CONJECTURE OF TU AND DENG Guixin Deng Schoo of Mathematica Sciences, Guangxi Teachers Education University, Nanning, P.R.China dengguixin@ive.com Pingzhi Yuan
More informationOn the Goal Value of a Boolean Function
On the Goa Vaue of a Booean Function Eric Bach Dept. of CS University of Wisconsin 1210 W. Dayton St. Madison, WI 53706 Lisa Heerstein Dept of CSE NYU Schoo of Engineering 2 Metrotech Center, 10th Foor
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationPattern Frequency Sequences and Internal Zeros
Advances in Appied Mathematics 28, 395 420 (2002 doi:10.1006/aama.2001.0789, avaiabe onine at http://www.ideaibrary.com on Pattern Frequency Sequences and Interna Zeros Mikós Bóna Department of Mathematics,
More informationA NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC
(January 8, 2003) A NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC DAMIAN CLANCY, University of Liverpoo PHILIP K. POLLETT, University of Queensand Abstract
More informationNOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs
More informationTHE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES
THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES by Michae Neumann Department of Mathematics, University of Connecticut, Storrs, CT 06269 3009 and Ronad J. Stern Department of Mathematics, Concordia
More informationCoupling of LWR and phase transition models at boundary
Couping of LW and phase transition modes at boundary Mauro Garaveo Dipartimento di Matematica e Appicazioni, Università di Miano Bicocca, via. Cozzi 53, 20125 Miano Itay. Benedetto Piccoi Department of
More informationUniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly conp-complete
Uniprocessor Feasibiity of Sporadic Tasks with Constrained Deadines is Strongy conp-compete Pontus Ekberg and Wang Yi Uppsaa University, Sweden Emai: {pontus.ekberg yi}@it.uu.se Abstract Deciding the feasibiity
More informationSrednicki Chapter 51
Srednici Chapter 51 QFT Probems & Soutions A. George September 7, 13 Srednici 51.1. Derive the fermion-oop correction to the scaar proagator by woring through equation 5., and show that it has an extra
More informationMA 201: Partial Differential Equations Lecture - 10
MA 201: Partia Differentia Equations Lecture - 10 Separation of Variabes, One dimensiona Wave Equation Initia Boundary Vaue Probem (IBVP) Reca: A physica probem governed by a PDE may contain both boundary
More informationPRIME TWISTS OF ELLIPTIC CURVES
PRIME TWISTS OF ELLIPTIC CURVES DANIEL KRIZ AND CHAO LI Abstract. For certain eiptic curves E/Q with E(Q)[2] = Z/2Z, we prove a criterion for prime twists of E to have anaytic rank 0 or 1, based on a mod
More informationOn colorings of the Boolean lattice avoiding a rainbow copy of a poset arxiv: v1 [math.co] 21 Dec 2018
On coorings of the Booean attice avoiding a rainbow copy of a poset arxiv:1812.09058v1 [math.co] 21 Dec 2018 Baázs Patkós Afréd Rényi Institute of Mathematics, Hungarian Academy of Scinces H-1053, Budapest,
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More informationIntegrating Factor Methods as Exponential Integrators
Integrating Factor Methods as Exponentia Integrators Borisav V. Minchev Department of Mathematica Science, NTNU, 7491 Trondheim, Norway Borko.Minchev@ii.uib.no Abstract. Recenty a ot of effort has been
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationRestricted weak type on maximal linear and multilinear integral maps.
Restricted weak type on maxima inear and mutiinear integra maps. Oscar Basco Abstract It is shown that mutiinear operators of the form T (f 1,..., f k )(x) = R K(x, y n 1,..., y k )f 1 (y 1 )...f k (y
More information(1 ) = 1 for some 2 (0; 1); (1 + ) = 0 for some > 0:
Answers, na. Economics 4 Fa, 2009. Christiano.. The typica househod can engage in two types of activities producing current output and studying at home. Athough time spent on studying at home sacrices
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationAn explicit Jordan Decomposition of Companion matrices
An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057
More informationTwo-Stage Least Squares as Minimum Distance
Two-Stage Least Squares as Minimum Distance Frank Windmeijer Discussion Paper 17 / 683 7 June 2017 Department of Economics University of Bristo Priory Road Compex Bristo BS8 1TU United Kingdom Two-Stage
More informationPRESENTING QUEER SCHUR SUPERALGEBRAS
PRESENTING QUEER SCHUR SUPERALGEBRAS JIE DU AND JINKUI WAN Abstract. Associated to the two types of finite dimensiona simpe superagebras, there are the genera inear Lie superagebra and the queer Lie superagebra.
More informationK a,k minors in graphs of bounded tree-width *
K a,k minors in graphs of bounded tree-width * Thomas Böhme Institut für Mathematik Technische Universität Imenau Imenau, Germany E-mai: tboehme@theoinf.tu-imenau.de and John Maharry Department of Mathematics
More informationarxiv: v1 [math.fa] 23 Aug 2018
An Exact Upper Bound on the L p Lebesgue Constant and The -Rényi Entropy Power Inequaity for Integer Vaued Random Variabes arxiv:808.0773v [math.fa] 3 Aug 08 Peng Xu, Mokshay Madiman, James Mebourne Abstract
More informationHomework #04 Answers and Hints (MATH4052 Partial Differential Equations)
Homework #4 Answers and Hints (MATH452 Partia Differentia Equations) Probem 1 (Page 89, Q2) Consider a meta rod ( < x < ), insuated aong its sides but not at its ends, which is initiay at temperature =
More informationThe Symmetric and Antipersymmetric Solutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B A l X l B l = C and Its Optimal Approximation
The Symmetric Antipersymmetric Soutions of the Matrix Equation A 1 X 1 B 1 + A 2 X 2 B 2 + + A X B C Its Optima Approximation Ying Zhang Member IAENG Abstract A matrix A (a ij) R n n is said to be symmetric
More informationProduct Cosines of Angles between Subspaces
Product Cosines of Anges between Subspaces Jianming Miao and Adi Ben-Israe Juy, 993 Dedicated to Professor C.R. Rao on his 75th birthday Let Abstract cos{l,m} : i cos θ i, denote the product of the cosines
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationMAT 167: Advanced Linear Algebra
< Proem 1 (15 pts) MAT 167: Advanced Linear Agera Fina Exam Soutions (a) (5 pts) State the definition of a unitary matrix and expain the difference etween an orthogona matrix and an unitary matrix. Soution:
More informationIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 2, FEBRUARY
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 2, FEBRUARY 206 857 Optima Energy and Data Routing in Networks With Energy Cooperation Berk Gurakan, Student Member, IEEE, OmurOze,Member, IEEE,
More informationApproximated MLC shape matrix decomposition with interleaf collision constraint
Approximated MLC shape matrix decomposition with intereaf coision constraint Thomas Kainowski Antje Kiese Abstract Shape matrix decomposition is a subprobem in radiation therapy panning. A given fuence
More informationUNIFORM CONVERGENCE OF MULTIPLIER CONVERGENT SERIES
royecciones Vo. 26, N o 1, pp. 27-35, May 2007. Universidad Catóica de Norte Antofagasta - Chie UNIFORM CONVERGENCE OF MULTILIER CONVERGENT SERIES CHARLES SWARTZ NEW MEXICO STATE UNIVERSITY Received :
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationThe Relationship Between Discrete and Continuous Entropy in EPR-Steering Inequalities. Abstract
The Reationship Between Discrete and Continuous Entropy in EPR-Steering Inequaities James Schneeoch 1 1 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 arxiv:1312.2604v1
More informationB. Brown, M. Griebel, F.Y. Kuo and I.H. Sloan
Wegeerstraße 6 53115 Bonn Germany phone +49 8 73-347 fax +49 8 73-757 www.ins.uni-bonn.de B. Brown, M. Griebe, F.Y. Kuo and I.H. Soan On the expected uniform error of geometric Brownian motion approximated
More informationConstruct non-graded bi-frobenius algebras via quivers
Science in China Series A: Mathematics Mar., 2007, Vo. 50, No. 3, 450 456 www.scichina.com www.springerink.com Construct non-graded bi-frobenius agebras via quivers Yan-hua WANG 1 &Xiao-wuCHEN 2 1 Department
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationMIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI
MIXING AUTOMORPHISMS OF COMPACT GROUPS AND A THEOREM OF SCHLICKEWEI KLAUS SCHMIDT AND TOM WARD Abstract. We prove that every mixing Z d -action by automorphisms of a compact, connected, abeian group is
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More informationAppendix of the Paper The Role of No-Arbitrage on Forecasting: Lessons from a Parametric Term Structure Model
Appendix of the Paper The Roe of No-Arbitrage on Forecasting: Lessons from a Parametric Term Structure Mode Caio Ameida cameida@fgv.br José Vicente jose.vaentim@bcb.gov.br June 008 1 Introduction In this
More informationFunctions of Several Variables
Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector
More informationMany-Help-One Problem for Gaussian Sources with a Tree Structure on their Correlation
Many-Hep-One Probem for Gaussian Sources with a Tree Structure on their Correation Yasutada Oohama arxiv:090988v csit 4 Jan 009 Abstract In this paper we consider the separate coding probem for + correated
More informationare left and right inverses of b, respectively, then: (b b 1 and b 1 = b 1 b 1 id T = b 1 b) b 1 so they are the same! r ) = (b 1 r = id S b 1 r = b 1
Lecture 1. The Category of Sets PCMI Summer 2015 Undergraduate Lectures on Fag Varieties Lecture 1. Some basic set theory, a moment of categorica zen, and some facts about the permutation groups on n etters.
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationIterative Decoding Performance Bounds for LDPC Codes on Noisy Channels
Iterative Decoding Performance Bounds for LDPC Codes on Noisy Channes arxiv:cs/060700v1 [cs.it] 6 Ju 006 Chun-Hao Hsu and Achieas Anastasopouos Eectrica Engineering and Computer Science Department University
More informationSmoothness equivalence properties of univariate subdivision schemes and their projection analogues
Numerische Mathematik manuscript No. (wi be inserted by the editor) Smoothness equivaence properties of univariate subdivision schemes and their projection anaogues Phiipp Grohs TU Graz Institute of Geometry
More information(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].
PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform
More informationCONGRUENCES. 1. History
CONGRUENCES HAO BILLY LEE Abstract. These are notes I created for a seminar tak, foowing the papers of On the -adic Representations and Congruences for Coefficients of Moduar Forms by Swinnerton-Dyer and
More informationarxiv: v1 [math.co] 17 Dec 2018
On the Extrema Maximum Agreement Subtree Probem arxiv:1812.06951v1 [math.o] 17 Dec 2018 Aexey Markin Department of omputer Science, Iowa State University, USA amarkin@iastate.edu Abstract Given two phyogenetic
More informationApproximated MLC shape matrix decomposition with interleaf collision constraint
Agorithmic Operations Research Vo.4 (29) 49 57 Approximated MLC shape matrix decomposition with intereaf coision constraint Antje Kiese and Thomas Kainowski Institut für Mathematik, Universität Rostock,
More informationarxiv:math/ v2 [math.pr] 6 Mar 2005
ASYMPTOTIC BEHAVIOR OF RANDOM HEAPS arxiv:math/0407286v2 [math.pr] 6 Mar 2005 J. BEN HOUGH Abstract. We consider a random wa W n on the ocay free group or equivaenty a signed random heap) with m generators
More informationSUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS
ISEE 1 SUPPLEMENTARY MATERIAL TO INNOVATED SCALABLE EFFICIENT ESTIMATION IN ULTRA-LARGE GAUSSIAN GRAPHICAL MODELS By Yingying Fan and Jinchi Lv University of Southern Caifornia This Suppementary Materia
More informationOnline Load Balancing on Related Machines
Onine Load Baancing on Reated Machines ABSTRACT Sungjin Im University of Caifornia at Merced Merced, CA, USA sim3@ucmerced.edu Debmaya Panigrahi Duke University Durham, NC, USA debmaya@cs.duke.edu We give
More informationChapter 5. Wave equation. 5.1 Physical derivation
Chapter 5 Wave equation In this chapter, we discuss the wave equation u tt a 2 u = f, (5.1) where a > is a constant. We wi discover that soutions of the wave equation behave in a different way comparing
More informationMonomial Hopf algebras over fields of positive characteristic
Monomia Hopf agebras over fieds of positive characteristic Gong-xiang Liu Department of Mathematics Zhejiang University Hangzhou, Zhejiang 310028, China Yu Ye Department of Mathematics University of Science
More informationReflection principles and kernels in R n _+ for the biharmonic and Stokes operators. Solutions in a large class of weighted Sobolev spaces
Refection principes and kernes in R n _+ for the biharmonic and Stokes operators. Soutions in a arge cass of weighted Soboev spaces Chérif Amrouche, Yves Raudin To cite this version: Chérif Amrouche, Yves
More informationMONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES
MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Abstract. For positive integers k and, a k-uniform hypergraph is caed a oose path of ength, and denoted by
More informationAge of Information: The Gamma Awakening
Age of Information: The Gamma Awakening Eie Najm and Rajai Nasser LTHI, EPFL, Lausanne, Switzerand Emai: {eie.najm, rajai.nasser}@epf.ch arxiv:604.086v [cs.it] 5 Apr 06 Abstract Status update systems is
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationConvergence Property of the Iri-Imai Algorithm for Some Smooth Convex Programming Problems
Convergence Property of the Iri-Imai Agorithm for Some Smooth Convex Programming Probems S. Zhang Communicated by Z.Q. Luo Assistant Professor, Department of Econometrics, University of Groningen, Groningen,
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationSupporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers
Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,
More informationCS 331: Artificial Intelligence Propositional Logic 2. Review of Last Time
CS 33 Artificia Inteigence Propositiona Logic 2 Review of Last Time = means ogicay foows - i means can be derived from If your inference agorithm derives ony things that foow ogicay from the KB, the inference
More informationResearch Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation
Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi:10.1155/007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi
More informationWeek 5 Lectures, Math 6451, Tanveer
Week 5 Lectures, Math 651, Tanveer 1 Separation of variabe method The method of separation of variabe is a suitabe technique for determining soutions to inear PDEs, usuay with constant coefficients, when
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationDo Schools Matter for High Math Achievement? Evidence from the American Mathematics Competitions Glenn Ellison and Ashley Swanson Online Appendix
VOL. NO. DO SCHOOLS MATTER FOR HIGH MATH ACHIEVEMENT? 43 Do Schoos Matter for High Math Achievement? Evidence from the American Mathematics Competitions Genn Eison and Ashey Swanson Onine Appendix Appendix
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 12. For problem (1), complete the calculation of the series in case j(t) = 0 and h(t) = e t.
Strauss PDEs e: Section 5.6 - Exercise Page 1 of 1 Exercise For probem (1, compete the cacuation of the series in case j(t = and h(t = e t. Soution With j(t = and h(t = e t, probem (1 on page 147 becomes
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationDiscrete Fourier Transform
Discrete Fourier Trasform ) Purpose The purpose is to represet a determiistic or stochastic siga u( t ) as a fiite Fourier sum, whe observatios of u() t ( ) are give o a reguar grid, each affected by a
More informationT.C. Banwell, S. Galli. {bct, Telcordia Technologies, Inc., 445 South Street, Morristown, NJ 07960, USA
ON THE SYMMETRY OF THE POWER INE CHANNE T.C. Banwe, S. Gai {bct, sgai}@research.tecordia.com Tecordia Technoogies, Inc., 445 South Street, Morristown, NJ 07960, USA Abstract The indoor power ine network
More information