ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT TYPE INEQUALITY WITH PARAMETERS BICHENG YANG. 1. Introduction. dxdy < x + y. sin(π/p) f p g q, (1)

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT TYPE INEQUALITY WITH PARAMETERS BICHENG YANG. 1. Introduction. dxdy < x + y. sin(π/p) f p g q, (1)"

Transcript

1 M atheatical I eualities & A licatios Volue 8, Nuber 2 (25, doi:.753/ia-8-32 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT TYPE INEQUALITY WITH PARAMETERS BICHENG YANG (Couicated by J. Pečarić Abstract. I this aer, by usig the way of weight coefficiets ad techiue of real aalysis ad colex aalysis, a ore accurate ultidiesioal discrete Hilbert-tye ieuality with a best ossible costat factor ad soe araeters is give. The euivalet for, the oerator exressio with the or are also cosidered.. Itroductio Assuig that >, + =, f (x, g(y, f L (R +, g L (R +, f = f (xdx} >, g >, we have the followig Hardy-Hilbert s itegral ieuality (cf. []: f (xg(y dxdy < x + y π si(π/ f g, ( π with the best ossible costat factor si(π/.ifa,b, a = a } = l, b = b } = l, a = = a } >, b >, the we have the followig Hardyπ Hilbert s ieuality with the sae best costat si(π/ (cf.[]: = = a b + < π si(π/ a b. (2 Ieualities ( ad(2 are iortat i aalysis ad its alicatios (cf. [], [2], [3], [4], [6], [7]. I 998, by itroducig a ideedet araeter λ (,], Yag[5] gavea extesio of ( for = = 2. Followig the results of [5], Yag [6] gave soe best extesios of (ad(2 as follows: Matheatics subject classificatio (2: 26D5, 47A7. Keywords ad hrases: Hilbert-tye ieuality, weight coefficiet, euivalet for, oerator, or. This work is suorted by the Natioal Natural Sciece Foudatio of Chia (No , ad 23 Kowledge Costructio Secial Foudatio Ite of Guagdog Istitutio of Higher Learig College ad Uiversity (No. 23KJCX4. c D l,zagreb Paer MIA

2 43 BICHENG YANG If λ,λ 2,λ R, λ + λ 2 = λ,k λ (x,y is a o-egative hoogeeous fuctio of degree λ, with k(λ = k λ (t,t λ dt R +, φ(x=x ( λ, ψ(x=x ( λ2, f (x, g(y, } f L,φ (R + = f ; f,φ := φ(x f (x dx} <, g L,ψ (R +, f,φ, g,ψ >, the k λ (x,y f (xg(ydxdy < k(λ f,φ g,ψ, (3 where the costat factor k(λ is the best ossible. Moreover, if k λ (x,y is fiite ad k λ (x,yx λ (k λ (x,yy λ2 is decreasig with resect to x > (y >, the for a, b, } a l,φ = a; a,φ := φ( a } <, = b = b } = l,ψ, a,φ, b,ψ >, it follows = = k λ (,a b < k(λ a,φ b,ψ, (4 where, the costat factor k(λ is still the best ossible. Clearly, for λ =, k (x,y= x+y, λ =, λ 2 =, ieuality (3 reduces to (, while (4 reduces to (2. Soe other results icludig the ultidiesioal Hilberttye itegral ieualities are rovided by [8] [22]. About half-discrete Hilbert-tye ieualities with the o-hoogeeous kerels, Hardy et al. rovided a few results i Theore 35 of []. But they did ot rove that the the costat factors are the best ossible. However, Yag [23] gave a result with the kerel ( < λ 2 by itroducig a variable ad roved that the costat factor (+x λ is the best ossible. I 2 Yag [24] gave the followig half-discrete Hardy-Hilbert s ieuality with the best ossible costat factor B(λ,λ 2 : f (x = a (x + λ dx < B(λ,λ 2 f,φ a,ψ, (5 where, λ >, < λ 2, λ + λ 2 = λ, B(u,v= ( + t u+vtu dt(u,v > is the beta fuctio. Zhog et al ([25] [7] ivestigated several half-discrete Hilberttye ieualities with articular kerels.

3 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 43 Usig the way of weight fuctios ad the techiues of discrete ad itegral Hilbert-tye ieualities with soe additioal coditios o the kerel, a half-discrete Hilbert-tye ieuality with a geeral hoogeeous kerel of degree λ R ad a best costat factor k (λ is obtaied as follows: f (x = k λ (x,a dx < k(λ f,φ a,ψ, (6 which is a extesio of (5 (see Yag ad Che [33]. At the sae tie, a half-discrete Hilbert-tye ieuality with a geeral o-hoogeeous kerel ad a best costat factor is give by Yag [34]. I this aer, by usig the way of weight coefficiets ad techiue of real aalysis, a ultidiesioal discrete Hilbert s ieuality with araeters ad a best ossible costat factor is give, which is a extesio of (4 fork λ (,= s k= (λ/s +c k λ/s. The euivalet for, the oerator exressio with the or are also cosidered. 2. Soe leas If i, j N(N is the set of ositive itegers,, >, we ut ( i x := x k (x =(x,,x i R i, (7 k= ( j y := y k (y =(y,,y j R j. (8 k= LEMMA. If s N, γ,m>, Ψ(u is a o-egative easurable fuctio i (,], ad D M := x R s s + ; x γ i M }, γ the we have (cf. [35] i= ( s ( xi γ Ψ D M dx dx s i= M = Ms Γ s ( γ γ s Γ( s γ Ψ(uu s γ du. (9 LEMMA 2. If s N,γ >, ε >, c =(c,,c s [, s, the we have Γ s ( c s ε γ γ = εs ε/γ γ s Γ( s γ + O((ε +. (

4 432 BICHENG YANG Proof. For M > s /γ, we set, < u < s Ψ(u= M γ, (Mu /γ s ε s, M γ u. The by (9, it follows c s ε γ = x R s + ;x i +c i } u R s + ;u i } = li M x c s ε γ dx u s ε γ du D M Ψ ( s ( xi γ dx dx s i= M M s Γ s ( γ = li M γ s Γ( s γ (Mu/γ s ε u s γ du = s/m γ For s =, it follows < 2 = c ε γ < ; fors 2, Γ s ( γ εs ε/γ γ s Γ( s γ. < c s ε γ a + c (s (+ε γ N s ; i, i =,2} N s ; i 3} The we have Γ s ( γ a + ( + ε(s (+ε/γ γ s 2 Γ( s γ < (a R +. < c s ε γ N s ; i } = c s ε γ + c s ε γ N s ; i, i =,2} N s ; i 3} Õ(+ Γ s ( γ εs ε/γ γ s Γ( s γ (ε +. Hecewehave(. LEMMA 3. If C is the set of colex ubers ad C = C }, z k C\z Rez, Iz = } (k =,2,, are differet oits, the fuctio f (z is aalytic i C excet for z i (i =,2,,, ad z = is a zero oit of f (z whose order is ot less tha, the for R, we have f (xx dx = 2πi e 2πi k= Res[ f (zz,z k ], (

5 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 433 where, < Ilz = argz < 2π. I articular, if z k (k =,, are all oles of order, settig ϕ k (z=(z z k f (z(ϕ k (z k, the f (xx dx = π siπ k= ( z k ϕ k (z k. (2 Proof. By [36] (P. 8, we have (. We fid e 2πi = cos2π isi2π = 2isiπ(cosπ+ isiπ= 2ie iπ siπ. I articular, sice f (zz = z z k (ϕ k (zz, it is obvious that Res[ f (zz, a k ]=z k ϕ k (z k = e iπ ( z k ϕ k (z k. The by (, we obtai (2. EXAMPLE. For s N, we set k λ (x,y= s k= (x λ /s + c k y λ /s ( < c < < c s, < λ s. For < λ i,< λ 2 j, λ + λ 2 = λ, by (2, we fid k s (λ := u=t λ/s = s λ = s k= s πs t λ /s t λ dt + c k k= λ si( πsλ I articular, for s =, we obtai k (λ = λ u sλ u + c k s λ k= c λ du u (λ /λ u + c du = sλ λ s k j=( j k π λ si( πλ λ c R +. (3 c j c k λ λ. LEMMA 4. If ( i h (i (t > (t > ; i=,2, the for b >, <, we have ( i di dx i h((b + x > (x > ;i =,2. (4

6 434 BICHENG YANG Proof. We fid d dx h((b + x =h ((b + x (b + x x <, d 2 dx 2 h((b + x = d dx [h ((b + x (b + x x ] = h ((b + x (b + x 2 2 x 2 2 ( + h ((b + x (b + x 2 x 2 2 +( h ((b + x (b + x x 2 = h ((b + x (b + x 2 2 x 2 2 +b( h ((b + x (b + x 2 x 2 >. The lea is roved. DEFINITION. For s N, <,, < c < < c s,< λ s, < λ i, < λ 2 j, λ + λ 2 = λ,τ =(τ,,τ i (, 2 ]i, σ =(σ,,σ j (, 2 ] j, τ =( τ,, i τ i R i +, σ =( σ,, j σ j R j +, defie two weight coefficiets w λ (λ 2, ad W λ (λ, as follows: w λ (λ 2, := W λ (λ, := σ λ 2 τ λ i k= ( τ λ + c k σ λ /s (5, τ λ σ λ 2 j k= ( τ λ where, = i = = ad = j = =. where, + c k σ λ /s (6, LEMMA 5. Let the assutios as i Defiitio are fulfilled. The, we have (i w λ (λ 2, < K 2 ( N j, (7 W λ (λ, < K ( N i, (8 K := Γ j ( j Γ( j k s(λ, K 2 := Γi( i Γ( i k s(λ, (9 where, k s (λ is idicated by (3; (ii for >, < ε < iλ, λ 2 }, settig λ = λ ε, λ 2 = λ 2 + ε, we have < K 2 ( θ λ ( < w λ ( λ 2,, (2

7 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 435 where, θ λ ( = k s ( λ i λ/(s / σ λ/s v sλ λ dv s k= (v + c k = O σ λ, (2 K 2 = Γi ( k s( λ i Γ( i R +. (22 Proof. By Lea 4, Herite-Hadaard s ieuality (cf. [37], (9 ad(3, it follows w λ (λ 2, < = ( 2, i σ λ 2 x τ λ i k= ( x τ λ + c k σ λ /s dx u R i + ;u i > 2 τ i} σ Ri+ λ 2 u λ i σ λ 2 u λ i k= ( u λ + c k σ λ /s du k= ( u λ + c k σ λ /s du σ = li M DM λ 2 Mλ i [ j i= ( u i M ] (λ i / du s k= M s λ [ i i= ( u i M ] s λ s + c k σ λ } = li M = li M M i Γ i ( i Γ( i M λ Γ i ( i Γ( i t= σ M v s/λ = σ λ 2 Mλ i t (λ i / dt s k= (M λ s t λ s + c k σ λ s σ λ 2 t λ dt s k= (M λ s t λ s + c k σ λ s sγ i ( λ i Γ( i v sλ λ s k= (v + c k dv = Γi ( i Γ( i k s(λ =K 2. Hece, we have (7.Bythesaeway,wehave(8. By the decreasig roerty ad the sae way of obtaiig (, we have w λ ( λ 2, > x R i + ;x i +τ i } = σ λ 2 u R i + ;u i } σ λ 2 x τ λ i dx k= ( x τ λ + c k σ λ /s u λ i du k= ( u λ + c k σ λ /s

8 436 BICHENG YANG = sγi ( v sλ λ λ i Γ( i i λ/(s / σ λ/s s k= (v + c k dv = K 2 ( θ λ ( >, < θ s λ (= λ k s ( λ The lea is roved. s λ k s ( λ s k= c k = λ k s ( λ s k= c k i λ/(s / σ λ/s λ/(s i / σ λ/s i λ / σ λ. v sλ λ s k= (v + c k dv v s λ λ dv 3. Mai results ad oerator exressios Settig Φ( := τ (i λ i ( N i ad Ψ( := σ ( j λ 2 j ( N j, we have THEOREM. If s N, <,, < c < < c s, < λ s, < λ i, < λ 2 j, λ + λ 2 = λ, τ (, 2 ]i, σ (, 2 ] j, the for >, + =, a,b, < a,φ, b,ψ <, we have the followig ieuality a b I := k= ( τ λ + c k σ λ /s < K K 2 a,φ b,ψ, (23 where the costat factor K K 2 = [ Γ j ( j Γ( j is the best ossible (k s (λ is idicated by (3. ] [ Γ i ( i Γ( i Proof. By Hölder s ieuality (cf. [37], we have I = s k= /s ( τ λ + c k σ λ /s τ (i λ / σ ( j λ 2 / a ] k s (λ (24 σ ( j λ 2 / τ (i λ / b

9 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 437 } W λ (λ, τ (i λ i a w λ (λ 2, σ ( j λ 2 j b }. The by (7ad(8, we have (23. For < ε < iλ, λ 2 }, λ = λ ε, λ 2 = λ 2 + ε, we set ã = τ i +λ ε, b = σ j +λ 2 ε ( N i, N j. The by (ad(2, we obtai ã,φ b,ψ = τ (i λ i ã = τ i ε [ = Γ i ( ε Ĩ := } i ε/ i Γ( i + εo( } } σ ( j λ 2 j b } σ j ε ] Γ j ( j ε/ j Γ( j + εõ( ã k= ( τ λ + c k σ λ /s = w λ ( λ 2, σ j ε > K 2 O( σ λ σ j ε b, (25 Γ j ( = K 2 ε j ε/ + Õ( O(. (26 j Γ( j If there exists a costat K K K 2, such that (23 is valid as we relace K by K, the we have Γ j ( (K 2 + o( j ε/ j Γ( j [ Γ i ( < εk ã,ϕ b,ψ = K ] i ε/ i Γ( i + εo( + εõ( εo( < εĩ Γ j ( j ε/ j Γ( j + εõ( K 2.

10 438 BICHENG YANG For ε +, we fid Γ j ( j Γ( j ad the K K 2 K. Hece, K = K K 2 Γ i ( [ Γ i i Γ( i k ( s(λ K ] [ Γ j ( ] i Γ( i j Γ( j, is the best ossible costat factor of (23. THEOREM 2. With the assutios of Theore, for < a,φ <, we have the followig ieuality with the best costat factor K J := σ λ 2 j K 2 : a k= ( τ λ + c k σ λ /s < K K 2 a,φ, which is euivalet to (23. (27 Proof. We set b as follows: b := σ λ 2 j a k= ( τ λ + c k σ λ /s The it follows J = b,ψ. If J =, the (27 is trivially valid sice < a,φ < ; if J =, the it is a cotradictio sice the right had side of (27 isfiite. Suose that < J <. The by (23, we fid b,ψ = J = I < K K 2 a,φ b,ψ, K 2 aely, b,ψ = J < K K 2 a,φ, ad the (27 follows. O the other had, assuig that (27 is valid, by Hölder s ieuality, we have I = (Ψ( a [(Ψ( k= ( τ λ + c k σ λ /s b ] J b,ψ. (28 The by (27, we have (23. Hece (27 ad(23 are euivalet. By the euivalecy, the costat factor K i (27 is the best ossible. Otherwise, we would reach a cotradictio by (28 that the costat factor K ot the best ossible. K 2. i (23 is

11 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 439 For >, we defie two real weight oral discrete saces l,ϕ ad l,ψ as follows: } l,ϕ := a = a }; a,φ = Φ(a } <, } l,ψ :=. b = b }; b,ψ = Ψ(b } < With the assutios of Theore, i view of J < K followig defiitio: K 2 a,φ, we have the DEFINITION 2. Defie a ultidiesioal Hilbert-tye oerator T : l,φ l,ψ as follows: For a l,φ, there exists a uiue reresetatio Ta l,ψ, satisfyig for N j, (Ta( := k= ( τ λ + c k σ λ /s (29. For b l,ψ, we defie the followig foral ier roduct of Ta ad b as follows: (Ta,b := a k= ( τ λ a b + c k σ λ /s (3. The by Theore ad Theore 2, for < a,ϕ, b,ψ <, we have the followig euivalet ieualities: (Ta,b < K K 2 a,φ b,ψ, (3 Ta,Ψ < K K 2 a,φ. (32 It follows that T is bouded sice Ta T := su,ψ a( θ l,φ a,φ K K 2. (33 Sice the costat factor K K 2 i (32 is the best ossible, we have COROLLARY. With the assutios of Theore 2, T is defied by Defiitio 2, it follows [ T = K K Γ j 2 = ( ] [ Γ i ( ] j Γ( j i Γ( i k s (λ. (34 REMARK. (i Settig Φ ( := (i λ i ( N i ad Ψ ( := ( j λ 2 j ( N j,

12 44 BICHENG YANG the uttig τ = σ = i(23 ad(27, we have the followig euivalet ieualities with the best costat factor K λ 2 j K 2 : a b k= ( λ + c k λ /s a < K k= ( λ + c k λ /s K 2 a,φ b,ψ, (35 Hece, (23ad(27 are ore accurate ieualities of (35ad(36. (ii Puttig i = j = i(32, we have ieuality = = Hece, (35 is a extesio of (4 for < K K 2 a,φ. (36 a b s k= (λ /s + c k λ /s < k s(λ a,φ b,ψ. (37 k λ (,= s k= (λ /s + c k λ /s. REFERENCES [] G. H. HARDY, J. E. LITTLEWOOG, G. PÓLYA, Ieualities, Cabridge Uiversity Press, Cabridge, 934. [2] D. S. MITRINOVIĆ, J. E. PEČARIĆ, A. M. FINK, Ieualities ivolvig fuctios ad their itegrals ad derivatives, Kluwer Acareic Publishers, Bosto, 99. [3] B. YANG, Hilbert-tye itegral ieualities, Betha Sciece Publishers Ltd., Dubai, 29. [4] B. YANG, Discrete Hilbert-tye ieualities, Betha Sciece Publishers Ltd., Dubai, 2. [5] B. YANG, O Hilbert s itegral ieuality, Joural of Matheatical Aalysis ad Alicatios, 22 (998, [6] B. YANG, The or of oerator ad Hilbert-tye ieualities, Sciece Press, Beiji, 29 (Chia. [7] B. YANG, Two tyes of ultile half-discrete Hilbert-tye ieualities, Labert Acadeic Publishig, Berli, 22. [8] B. YANG, I. BRNETIĆ, M. KRNIĆ, J. E. PEČARIĆ, Geeralizatio of Hilbert ad Hardy-Hilbert itegral ieualities, Math. Ie. ad Al., 8, 2 (25, [9] M. KRNIĆ, J. E. PEČARIĆ, Hilbert s ieualities ad their reverses, Publ. Math. Debrece, 67, 3 4 (25, [] B. YANG, TH. M. RASSIAS, O the way of weight coefficiet ad research for Hilbert-tye ieualities, Math. Ie. Al., 6, 4 (23, [] B. YANG, TH. M. RASSIAS, O a Hilbert-tye itegral ieuality i the subiterval ad its oerator exressio, Baach J. Math. Aal., 4, 2 (2,. [2] L. AZAR, O soe extesios of Hardy-Hilbert s ieuality ad Alicatios, Joural of Ieualities ad Alicatios, 29, o [3] B. ARPAD, O. HOONGHONG, Best costat for certai ulti liear itegral oerator, Joural of Ieualities ad Alicatios, 26, o [4] J. KUANG, L. DEBNATH, O Hilbert s tye ieualities o the weighted Orlicz saces, Pacific J. Al. Math.,, (27, [5] W. ZHONG, The Hilbert-tye itegral ieuality with a hoogeeous kerel of Labda-degree, Joural of Ieualities ad Alicatios, 28, o

13 ON A MORE ACCURATE MULTIDIMENSIONAL HILBERT-TYPE INEQUALITY 44 [6] Y. HONG, O Hardy-Hilbert itegral ieualities with soe araeters, J. Ie. i Pure & Alied Math., 6, 4 (25 Art. 92,. [7] W. ZHONG, B. YANG, O ultile Hardy-Hilbert s itegral ieuality with kerel, Joural of Ieualities ad Alicatios, Vol. 27, Art. ID 27962, 7 ages, doi:.55/ 27/27. [8] B. YANG, M. KRNIĆ, O the Nor of a Mult-diesioal Hilbert-tye Oerator, Sarajevo Joural of Matheatics, 7, 2 (2, [9] M. KRNIĆ, J. E. PEČARIĆ, P. VUKOVIĆ, O soe higher-diesioal Hilbert s ad Hardy-Hilbert s tye itegral ieualities with araeters, Math. Ieual. Al., (28, [2] M. KRNIĆ, P. VUKOVIĆ, O a ultidiesioal versio of the Hilbert-tye ieuality, Aalysis Matheatica, 38 (22, [2] M. TH. RASSIAS, B. YANG, A ultidiesioal half-discrete Hilbert-tye ieuality ad the Riea zeta fuctio, Alied Matheatics ad Coutatio, 225 (23, [22] Y. LI, B. HE, O ieualities of Hilbert s tye, Bulleti of the Australia Matheatical Society, 76, (27, 3. [23] B. YANG, A ixed Hilbert-tye ieuality with a best costat factor, Iteratioal Joural of Pure ad Alied Matheatics, 2, 3 (25, [24] B. YANG, A half-discrete Hilbert-tye ieuality, Joural of Guagdog Uiversity of Educatio, 3, 3 (2, 7. [25] W. ZHONG, A ixed Hilbert-tye ieuality ad its euivalet fors, Joural of Guagdog Uiversity of Educatio, 3, 5 (2, [26] W. ZHONG, A half discrete Hilbert-tye ieuality ad its euivalet fors, Joural of Guagdog Uiversity of Educatio, 32, 5 (22, 8 2. [27] J. ZHONG, B. YANG, O a extesio of a ore accurate Hilbert-tye ieuality, Joural of Zhejiag Uiversity (Sciece Editio, 35, 2 (28, [28] J. ZHONG, Two classes of half-discrete reverse Hilbert-tye ieualities with a o-hoogeeous kerel, Joural of Guagdog Uiversity of Educatio, 32, 5 (22, 2. [29] W. ZHONG, B. YANG, A best extesio of Hilbert ieuality ivolvig several araeters, Joural of Jia Uiversity (Natural Sciece, 28, (27, [3] W. ZHONG, B. YANG, A reverse Hilbert s tye itegral ieuality with soe araeters ad the euivalet fors, Pure ad Alied Matheatics, 24, 2 (28, [3] M. TH. RASSIAS,B. YANG, O half-discrete Hilbert s ieuality, Alied Matheatics ad Coutatio, 22 (23, [32] W. ZHONG, B. YANG, O ultile Hardy-Hilbert s itegral ieuality with kerel, Joural of Ieualities ad Alicatios, Vol. 27, Art. ID 27962, 7 ages, doi:.55/27/27. [33] B. YANG, Q. CHEN, A half-discrete Hilbert-tye ieuality with a hoogeeous kerel ad a extesio, Joural of Ieualities ad Alicatios, 24 (2, doi:.86/29-242x [34] B. YANG, A half-discrete Hilbert-tye ieuality with a o-hoogeeous kerel ad two variables, Mediterraea Joural of Metheatics, (23, [35] B. YANG, Hilbert-tye itegral oerators: ors ad ieualities (I Chater 42 of Noliear Aalysis, stability, aroxiatio, ad ieualities (P. M. Paralos et al., Sriger, New York, , 22. [36] Y. PAN, H. WANG, F. WANG, O colex fuctios, Sciece Press, Beijig, 26 (Chia. [37] J. KUANG, Alied ieualities, Shagdog Sciece Techic Press, Jia, 24 (Chia. (Received Deceber 2, 23 Bicheg Yag Deartet of Matheatics Guagdog Uiversity of Educatio Guagzhou, Guagdog 533, P.R. Chia e-ail: bcyag@gdei.edu.c, bcyag88@63.co Matheatical Ieualities & Alicatios ia@ele-ath.co

An extension of a multidimensional Hilbert-type inequality

An extension of a multidimensional Hilbert-type inequality Zhog ad Yag Joural of Ieualities ad Alicatios 27 27:78 DOI.86/s366-7-355-6 R E S E A R C H Oe Access A extesio of a ultidiesioal Hilbert-tye ieuality Jiahua Zhog ad Bicheg Yag * * Corresodece: bcyag@gdei.edu.c

Διαβάστε περισσότερα

On Generating Relations of Some Triple. Hypergeometric Functions

On Generating Relations of Some Triple. Hypergeometric Functions It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade

Διαβάστε περισσότερα

A study on generalized absolute summability factors for a triangular matrix

A study on generalized absolute summability factors for a triangular matrix Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.ee/proceedigs A study o geeralized absolute suability factors for a triagular atrix Ere Savaş

Διαβάστε περισσότερα

A New Class of Analytic p-valent Functions with Negative Coefficients and Fractional Calculus Operators

A New Class of Analytic p-valent Functions with Negative Coefficients and Fractional Calculus Operators Tamsui Oxford Joural of Mathematical Scieces 20(2) (2004) 175-186 Aletheia Uiversity A New Class of Aalytic -Valet Fuctios with Negative Coefficiets ad Fractioal Calculus Oerators S. P. Goyal Deartmet

Διαβάστε περισσότερα

1. For each of the following power series, find the interval of convergence and the radius of convergence:

1. For each of the following power series, find the interval of convergence and the radius of convergence: Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.

Διαβάστε περισσότερα

On Certain Subclass of λ-bazilevič Functions of Type α + iµ

On Certain Subclass of λ-bazilevič Functions of Type α + iµ Tamsui Oxford Joural of Mathematical Scieces 23(2 (27 141-153 Aletheia Uiversity O Certai Subclass of λ-bailevič Fuctios of Type α + iµ Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua College of Mathematics ad

Διαβάστε περισσότερα

ESTIMATES FOR WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS

ESTIMATES FOR WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS ESTIMATES FO WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS V F Babeo a S A Sector Let ψ D be orthogoal Daubechies wavelets that have zero oets a let W { } = f L ( ): ( i ) f ˆ( ) N We rove that li

Διαβάστε περισσότερα

Presentation of complex number in Cartesian and polar coordinate system

Presentation of complex number in Cartesian and polar coordinate system 1 a + bi, aεr, bεr i = 1 z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real:

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

L.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:

Διαβάστε περισσότερα

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1) 8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r

Διαβάστε περισσότερα

ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY. Gabriel STAN 1

ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY. Gabriel STAN 1 Bulleti of the Trasilvaia Uiversity of Braşov Vol 5) - 00 Series III: Mathematics, Iformatics, Physics, -4 ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY Gabriel STAN Abstract A extesio ad a refiemet

Διαβάστε περισσότερα

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

On Inclusion Relation of Absolute Summability

On Inclusion Relation of Absolute Summability It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, 2641-2646 O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia-243006 aditya jauhari@rediffmail.com

Διαβάστε περισσότερα

Solve the difference equation

Solve the difference equation Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y

Διαβάστε περισσότερα

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6 SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si

Διαβάστε περισσότερα

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University)

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University) Itroductio of Numerical Aalysis #03 TAGAMI, Daisuke (IMI, Kyushu Uiversity) web page of the lecture: http://www2.imi.kyushu-u.ac.jp/~tagami/lec/ Strategy of Numerical Simulatios Pheomea Error modelize

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

J. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1.

J. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1. Vol. 35 ( 205 ) No. 4 J. of Math. (PRC), (, 243002) : a.s. Marov Borel-Catelli. : Marov ; Borel-Catelli ; ; ; MR(200) : 60F5 : O2.4; O236 : A : 0255-7797(205)04-0969-08 Shao-McMilla,. Shao 948 [],, McMilla

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

LAD Estimation for Time Series Models With Finite and Infinite Variance

LAD Estimation for Time Series Models With Finite and Infinite Variance LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

Certain Sequences Involving Product of k-bessel Function

Certain Sequences Involving Product of k-bessel Function It. J. Appl. Coput. Math 018 4:101 https://doi.org/10.1007/s40819-018-053-8 ORIGINAL PAPER Certai Sequeces Ivolvig Product of k-bessel Fuctio M. Chad 1 P. Agarwal Z. Haouch 3 Spriger Idia Private Ltd.

Διαβάστε περισσότερα

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Dr Neetu Vishwakarma a Dr M S Chauha Sagar Istitute of

Διαβάστε περισσότερα

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ.

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ. Chapter 3. Biorthogoal Wavelets ad Filter Baks via PFFS 3.0 PFFS applied to shift-ivariat subspaces Defiitio: X is a shift-ivariat subspace if h X h( ) τ h X. Ex: Multiresolutio Aalysis (MRA) subspaces

Διαβάστε περισσότερα

Homework for 1/27 Due 2/5

Homework for 1/27 Due 2/5 Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where

Διαβάστε περισσότερα

Apr Vol.26 No.2. Pure and Applied Mathematics O157.5 A (2010) (d(u)d(v)) α, 1, (1969-),,.

Apr Vol.26 No.2. Pure and Applied Mathematics O157.5 A (2010) (d(u)d(v)) α, 1, (1969-),,. 2010 4 26 2 Pure and Applied Matheatics Apr. 2010 Vol.26 No.2 Randić 1, 2 (1., 352100; 2., 361005) G Randić 0 R α (G) = v V (G) d(v)α, d(v) G v,α. R α,, R α. ; Randić ; O157.5 A 1008-5513(2010)02-0339-06

Διαβάστε περισσότερα

IIT JEE (2013) (Trigonomtery 1) Solutions

IIT JEE (2013) (Trigonomtery 1) Solutions L.K. Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

α β

α β 6. Eerg, Mometum coefficiets for differet velocit distributios Rehbock obtaied ) For Liear Velocit Distributio α + ε Vmax { } Vmax ε β +, i which ε v V o Give: α + ε > ε ( α ) Liear velocit distributio

Διαβάστε περισσότερα

Outline. M/M/1 Queue (infinite buffer) M/M/1/N (finite buffer) Networks of M/M/1 Queues M/G/1 Priority Queue

Outline. M/M/1 Queue (infinite buffer) M/M/1/N (finite buffer) Networks of M/M/1 Queues M/G/1 Priority Queue Queueig Aalysis Outlie M/M/ Queue (ifiite buffer M/M//N (fiite buffer M/M// (Erlag s B forula M/M/ (Erlag s C forula Networks of M/M/ Queues M/G/ Priority Queue M/M/ M: Markovia/Meoryless Arrival process

Διαβάστε περισσότερα

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period.

Διαβάστε περισσότερα

Degenerate Perturbation Theory

Degenerate Perturbation Theory R.G. Griffi BioNMR School page 1 Degeerate Perturbatio Theory 1.1 Geeral Whe cosiderig the CROSS EFFECT it is ecessary to deal with degeerate eergy levels ad therefore degeerate perturbatio theory. The

Διαβάστε περισσότερα

The Neutrix Product of the Distributions r. x λ

The Neutrix Product of the Distributions r. x λ ULLETIN u. Maaysia Math. Soc. Secod Seies 22 999 - of the MALAYSIAN MATHEMATICAL SOCIETY The Neuti Poduct of the Distibutios ad RIAN FISHER AND 2 FATMA AL-SIREHY Depatet of Matheatics ad Copute Sciece

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μαθηματικά Ικανές και αναγκαίες συνθήκες Έστω δυο προτάσεις Α και Β «Α είναι αναγκαία συνθήκη για την Β» «Α είναι ικανή συνθήκη για την Β» Α is ecessary for

Διαβάστε περισσότερα

Uniform Estimates for Distributions of the Sum of i.i.d. Random Variables with Fat Tail in the Threshold Case

Uniform Estimates for Distributions of the Sum of i.i.d. Random Variables with Fat Tail in the Threshold Case J. Math. Sci. Uiv. Tokyo 8 (2, 397 427. Uiform Estimates for Distributios of the Sum of i.i.d. om Variables with Fat Tail i the Threshold Case By Keji Nakahara Abstract. We show uiform estimates for distributios

Διαβάστε περισσότερα

Ψηφιακή Επεξεργασία Εικόνας

Ψηφιακή Επεξεργασία Εικόνας ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Φιλτράρισμα στο πεδίο των συχνοτήτων Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing. Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis

Διαβάστε περισσότερα

MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra

MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log

Διαβάστε περισσότερα

INTEGRATION OF THE NORMAL DISTRIBUTION CURVE

INTEGRATION OF THE NORMAL DISTRIBUTION CURVE INTEGRATION OF THE NORMAL DISTRIBUTION CURVE By Tom Irvie Email: tomirvie@aol.com March 3, 999 Itroductio May processes have a ormal probability distributio. Broadbad radom vibratio is a example. The purpose

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

Homomorphism of Intuitionistic Fuzzy Groups

Homomorphism of Intuitionistic Fuzzy Groups International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that

Διαβάστε περισσότερα

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X. Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

Notations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation. Generalized hypergeometric function

Notations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation. Generalized hypergeometric function HyergeometricPFQ Notatios Traditioal ame Geeralied hyergeometric fuctio Traditioal otatio F a 1,, a ; b 1,, b ; Mathematica StadardForm otatio HyergeometricPFQa 1,, a, b 1,, b, Primary defiitio 07.31.0.0001.01

Διαβάστε περισσότερα

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

Bessel function for complex variable

Bessel function for complex variable Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by {

Διαβάστε περισσότερα

Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

Homomorphism in Intuitionistic Fuzzy Automata

Homomorphism in Intuitionistic Fuzzy Automata International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic

Διαβάστε περισσότερα

6.3 Forecasting ARMA processes

6.3 Forecasting ARMA processes 122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

Abstract Storage Devices

Abstract Storage Devices Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD

Διαβάστε περισσότερα

Binet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods

Binet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods DOI: 545/mjis764 Biet Type Formula For The Sequece of Tetraacci Numbers by Alterate Methods GAUTAMS HATHIWALA AND DEVBHADRA V SHAH CK Pithawala College of Eigeerig & Techology, Surat Departmet of Mathematics,

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω 0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

Data Dependence of New Iterative Schemes

Data Dependence of New Iterative Schemes Mathematics Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X Data Depedece of New Iterative Schemes KEYWORDS CR Iteratio Data Depedece New Multistep Iteratio Quasi Cotractive * Aarti Kadia Assistat Professor

Διαβάστε περισσότερα

Gauss Radau formulae for Jacobi and Laguerre weight functions

Gauss Radau formulae for Jacobi and Laguerre weight functions Mathematics ad Computers i Simulatio 54 () 43 41 Gauss Radau formulae for Jacobi ad Laguerre weight fuctios Walter Gautschi Departmet of Computer Scieces, Purdue Uiversity, West Lafayette, IN 4797-1398,

Διαβάστε περισσότερα

MATHEMATICS. Received February 26, 2016

MATHEMATICS. Received February 26, 2016 IN 064-5624 Doklady Mathematics 206 Vol. 94 No.. 387 392. Pleiades Publishig Ltd. 206. Published i Russia i Doklady Akademii Nauk 206 Vol. 469 No. 2. 48 53. MATHEMATIC Homogeizatio of the -Lalacia with

Διαβάστε περισσότερα

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig?

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

F19MC2 Solutions 9 Complex Analysis

F19MC2 Solutions 9 Complex Analysis F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

Solution Series 9. i=1 x i and i=1 x i.

Solution Series 9. i=1 x i and i=1 x i. Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =.

Διαβάστε περισσότερα

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5 Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2

Διαβάστε περισσότερα

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,, Y satisfy Y i = βx i + ε i : i,, where x,, x R are fixed values ad ε,, ε Normal0, σ ) with σ R + kow Fid ˆβ = MLEβ) IND Solutio: Observe that Y

Διαβάστε περισσότερα

Lecture 34 Bootstrap confidence intervals

Lecture 34 Bootstrap confidence intervals Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Outline. Detection Theory. Background. Background (Cont.)

Outline. Detection Theory. Background. Background (Cont.) Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear

Διαβάστε περισσότερα

Discrete-Time Markov Chains

Discrete-Time Markov Chains Markov Processes ad Alicatios Discrete-Time Markov Chais Cotiuous-Time Markov Chais Alicatios Queuig theory Performace aalysis ΠΜΣ54: Μοντελοποίηση και Ανάλυση Απόδοσης Δικτύων (Ι. Σταυρακάκης / Α. Παναγάκης

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p) Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities

Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2008, Article ID 598632, 13 pages doi:10.1155/2008/598632 Research Article Fiite-Step Relaxed Hybrid Steepest-Descet Methods for

Διαβάστε περισσότερα

The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality

The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

B.A. (PROGRAMME) 1 YEAR

B.A. (PROGRAMME) 1 YEAR Graduate Course B.A. (PROGRAMME) YEAR ALGEBRA AND CALCULUS (PART-A : ALGEBRA) CONTENTS Lesso Lesso Lesso Lesso Lesso Lesso : Complex Numbers : De Moivre s Theorem : Applicatios of De Moivre s Theorem 4

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

A Laplace Type Problem for a Lattice with Cell Composed by Three Triangles with Obstacles

A Laplace Type Problem for a Lattice with Cell Composed by Three Triangles with Obstacles Applied Matheatical Sciences Vol. 11 017 no. 6 65-7 HIKARI Ltd www.-hikari.co https://doi.org/10.1988/as.017.6195 A Laplace Type Proble for a Lattice with Cell Coposed by Three Triangles with Obstacles

Διαβάστε περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8  questions or comments to Dan Fetter 1 Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

3.4 Αζηίεξ ημζκςκζηήξ ακζζυηδηαξ ζημ ζπμθείμ... 64 3.4.1 Πανάβμκηεξ πνμέθεοζδξ ηδξ ημζκςκζηήξ ακζζυηδηαξ... 64 3.5 οιαμθή ηςκ εηπαζδεοηζηχκ ζηδκ

3.4 Αζηίεξ ημζκςκζηήξ ακζζυηδηαξ ζημ ζπμθείμ... 64 3.4.1 Πανάβμκηεξ πνμέθεοζδξ ηδξ ημζκςκζηήξ ακζζυηδηαξ... 64 3.5 οιαμθή ηςκ εηπαζδεοηζηχκ ζηδκ 2 Πεξηερόκελα Δονεηήνζμ πζκάηςκ... 4 Δονεηήνζμ δζαβναιιάηςκ... 5 Abstract... 6 Πενίθδρδ... 7 Δζζαβςβή... 8 ΘΔΩΡΗΣΙΚΟ ΜΔΡΟ... 12 Κεθάθαζμ 1: Θεςνδηζηέξ πνμζεββίζεζξ βζα ηδκ ακζζυηδηα ζηδκ εηπαίδεοζδ...

Διαβάστε περισσότερα

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits. EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.

Διαβάστε περισσότερα

Affine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik

Affine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015 Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero

Διαβάστε περισσότερα

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators ECE 830 Fall 2011 Statistical Sigal Processig istructor: R. Nowak, scribe: Iseok Heo Lecture 17: Miimum Variace Ubiased (MVUB Estimators Ultimately, we would like to be able to argue that a give estimator

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα