Some comments on rigorous quantum field path integrals in the analytical regularization scheme


 Emma Newton
 3 years ago
 Views:
Transcription
1 CBPFNF03/08 Some commets o rigorous quatum field path itegrals i the aalytical regularizatio scheme Luiz C.L. Botelho Departameto de Matemática Aplicada, Istituto de Matemática, Uiversidade Federal Flumiese, Rua Mario Satos Braga , Niterói, Rio de Jaeiro, Brazil Abstract Through the systematic use of the Milos theorem o the support of cylidrical measures o R, we produce several mathematically rigorous path itegrals i iteractig euclidea quatum fields with Gaussia free measures defied by geeralized powers of the Laplacea operator. 1 Itroductio Sice the result of R.P. Feyma o represetig the iitial value solutio of Schrodiger Equatio by meas of a aalytically time cotiued itegratio o a ifiite  dimesioal space of fuctios, the subject of Euclidea Fuctioal Itegrals represetatios for Quatum Systems has became the mathematical  operatioal framework to aalyze Quatum Pheomea ad stochastic systems as showed i the previous decades of research o Theoretical Physics ([1] [3]). Oe of the most importat ope problem i the mathematical theory of Euclidea Fuctioal Itegrals is that related to implemetatio of soud mathematical approximatios to these IfiiteDimesioal Itegrals by meas of FiiteDimesioal approximatios outside of the always used [computer orieted] SpaceTime Lattice approximatios (see [], [3]  chap. 9). As a first step to tackle upo the above cited problem it will be eeded to characterize mathematically the Fuctioal Domai where these Fuctioal Itegrals are defied. The purpose of this ote is to preset the formulatio of Euclidea Quatum Field theories as Fuctioal Fourier Trasforms by meas of the BocherMartiKolmogorov theorem for Topological Vector Spaces ([4], [5]  theorem 4.35) ad suitable to defie ad aalyze rigorously Fuctioal Itegrals by meas of the wellkow Milos theorem ([5]  theorem 4.31 ad [6]  part ) ad preseted i full i Appedix 1.
2 CBPFNF03/08 We thus preset ews results o the difficult problem of defiig rigorously ifiitedimesioal quatum field path itegrals i geeral space times Ω R ν (ν =, 4,... ) by meas of the aalytical regularizatio scheme. Some rigorous quatum field path itegral i the Aalytical regularizatio scheme Let us thus start our aalysis by cosiderig the Gaussia measure associated to the (ifrared regularized) αpower (α > 1) of the Laplacea actig o L (R ) as a operatioal quadratic form (the Stoe spectral theorem or the aalytical regularizig schem i QFT) ( ) α ε = (λ) α de(λ) (1a) ε IR λ α,ε IR [j] = exp 1 Z (0) = j, ( ) α ε α,εµ [ϕ] exp ( i j, ϕ j L (R ) L (R ) ) (1b) Here ε IR > 0 deotes the ifrared cut off. It is worth call the reader attetio that due to the ifrared regularizatio itroduced o eq (1a), the domai of the Gaussia measure ([4] [6]) is give by the space of square itegrable fuctios o R by the Milos theorem of Appedix 1, sice for α > 1, the operator ( ) α ε IR defies a classe trace operator o L (R ), amely T r H (( ) α 1 ε IR ) = d 1 k ( K α + ε IR ) < (1c) This is the oly poit of our aalysis where it is eeded to cosider the ifrared cut off cosidered o the spectral resolutio eq (1a). As a cosequece of the above remarks, oe ca aalize the ultraviolet reormalizatio program i the followig iteractig model proposed by us ad defied by a iteractio g bare V (ϕ(x)), with V (x) deotig a fuctio o R such, that it posseses a essetially bouded (compact support) Fourier trasform ad g bare deotig the positive bare couplig costat. Let us show that by defiig a reormalized couplig costat as (with g re < 1) g bare = g re (1 α) 1/ () oe ca show that the iteractio fuctio exp g bare (α) d x V (ϕ(x)) (3) is a itegrable fuctio o L 1 (L (R ), α,ε IR µ [ϕ]) ad leads to a welldefied ultraviolet path itegral i the limit of α 1.
3 CBPFNF03/08 3 The proof is based o the followig estimates. Sice almost everywhere we have the poitwise limit exp g bare (α) d x V (ϕ(x)) N ( 1) (g bare (α)) lim dk 1 dk Ṽ (k 1 ) N! Ṽ (k ) =0 R dx 1 dx e ik 1ϕ(x 1) ik ϕ(x) e R we have that the upperboud estimate below holds true with Zε α IR [g bare ] ( 1) (g bare (α)) =0 Zε α IR [g bare ] =! R dx 1 dx α,ε IR µ[ϕ] exp g bare (α) R dk 1 dk Ṽ (k 1 ) Ṽ (k ) NP (4) α,ε IR µ[ϕ](e i k l ϕ(x l ) l=1 ) (5a) d x V (ϕ(x)) (5b) we have, thus, the more suitable form after realizig the d k i ad α,ε IR µ[ϕ] itegrals respectivelly Zε α IR =0[g bare ] (g bare (α)) =0! dx 1 dx det 1 ( ) Ṽ L (R) [ ] G (N) α (x i, x j ) 1 i N 1 j N (6) Here [G (N) α (x i, x j )] 1 i N deotes the N N symmetric matrix with the (i, j) etry 1 j N give by the Greefuctio of the αlaplacea (without the ifrared cut off here! ad the eeded ormalizatio factors!). (1 α) Γ(1 α) G α (x i, x j ) = x i x j Γ(α) (7) At this poit, we call the reader attetio that we have the formulae o the asymptotic behavior for α 1. lim det 1 [G (N) α (x i, x j )] α 1 α>1 ( (1 α) N/ (N 1)( 1) N π N/ ) 1 After substitutig eq.(8) ito eq.(6) ad takig ito accout the hypothesis of the compact support of the oliearity Ṽ (k), oe obtais the fiite boud for ay value (8)
4 CBPFNF03/08 4 g rem > 0, ad producig a proof for the covergece of the perturbative expasio i terms of the reormalized couplig costat. lim Zε α ( Ṽ α 1 IR =0[g bare (α)] ( ) L (R)) g re (1 ) (1 α) /! (1 α) 1 =0 e πg re Ṽ L (R) < (9) Aother importat rigorously defied fuctioal itegral is to cosider the followig αpower Klei Gordo operator o Euclidea spacetime L = ( ) α + m (10) with m a positive mass parameters. Let us ote that L 1 is a operator of class trace o L (R ν ) if ad oly if the result below holds true T r L (R ν )(L 1 ) = d ν 1 k k α + m = C(ν) m ( ν π α ) α cosecνπ < (11) α amely if α > ν (1) I this case, let us cosider the double fuctioal itegral with fuctioal domai L (R ν ) Z[j, k] = exp G β[v(x)] ( ) α +v+m µ[ϕ] i d ν x (j(x) ϕ(x) + k(x) v(x)) where the Gaussia fuctioal itegral o the fields V (x) has a Gaussia geeratig fuctioal defied by a itegral operator with a positive defied kerel g( x y ), 1 amely Z (0) [k] = G i β[v(x)] exp d ν x k(x)v(x) = exp 1 d ν x d ν y (k(x) g( x y ) k(x)) (14) By a simple direct applicatio of the FubbiiToelli theorem o the exchage of the itegratio order o eq.(13), lead us to the effective λϕ 4  like welldefied fuctioal itegral represetatio Z eff [j] = (( ) α +m ) µ [ϕ(x)] exp 1 d ν xd ν y ϕ(x) g( x y ) ϕ(y) exp i d ν x j(x)ϕ(x) (15) (13)
5 CBPFNF03/08 5 Note that if oe itroduces from the begiig a bare mass parameters m bare depedig o the parameters α, but such that it always satisfies eq.(11) oe should obtais agai eq.(15) as a welldefied measure o L (R ν ). Of course that the usual pure Laplacea limit of α 1 o eq.(10), will eeded a reormalizatio of this mass parameters (lim m bare (α) = +!) as much as doe i the previous example. α 1 Let us cotiue our examples by showig agai the usefuless of the precise determiatio of the fuctioal  distributioal structure of the domai of the fuctioal itegrals i order to costruct rigorously these path itegrals without complicated limit procedures. Let us cosider a geeral R ν Gaussia measure defied by the Geeratig fuctioal o S(R ν ) defied by the αpower of the Laplacea operator actig o S(R ν ) with a of small ifrared regularizatio mass parameter µ Z (0) [j] = exp = 1 E alg (S(R ν )) j, (( ) α + µ 0) 1 j L (R ν ) α µ[ϕ] exp(i ϕ(j)) (16) A explicitly expressio i mometum space for the Gree fuctio of the αpower of ( ) α + µ 0 give by ( ) d (( ) +α + µ 0) 1 ν k 1 (x y) = (π) ν eik(x y) (17) k α + µ 0 Here C(ν) is a νdepedet (fiite for νvalues!) ormalizatio factor. Let us suppose that there is a rage of αpower values that ca be choose i such way that oe satisfies the costrait below α µ[ϕ]( ϕ L j (R ν )) j < (18) E alg (S(R ν )) with j = 1,,, N ad for a give fixed iteger N, the highest power of our poliomial field iteractio. Or equivaletly, after realizig the ϕgaussia fuctioal itegratio, with a spacetime cutt off volume Ω o the iteractio to be aalyzed o eq.(16) Ω ( d ν x[( ) α + µ 0] j d ν k (x, x) = vol(ω) k α + µ 0 ( = C ν (µ 0 ) ( ν π ) α ) α cosecνπ < (19) α For α > ν, oe ca see by the Milos theorem that the measure support of the Gaussia measure eq.(16) will be give by the itersectio Baach space of measurable Lebesgue fuctios o R ν istead of the previous oe E alg (S(R ν )) ([4] [6]). L N (R ν ) = ) j N (L j (R ν )) (0) j=1
6 CBPFNF03/08 6 I this case, oe obtais that the fiite  volume p(ϕ) iteractios N exp λ j (ϕ (x)) j dx 1 (1) j=1 Ω is mathematically welldefied as the usual poitwise product of measurable fuctios ad for positive couplig costat values λ j 0. As a cosequece, we have a measurable fuctioal o L 1 (L N (R ν ); α µ[ϕ]) (sice it is bouded by the fuctio 1). So, it makes sese to cosider mathematically the welldefied path  itegral o the full space R ν with those values of the power α satisfyig the cotrait eq.(17). N Z[j] = α µ[ϕ] exp λ j ϕ j (x)dx exp(i j(x)ϕ(x)) () L N (R ν ) Ω R ν j=1 Fially, let us cosider a iteractig field theory i a compact spacetime Ω R ν defied by a iteger eve power of the Laplacea operator with Dirichlet Boudary coditios as the free Gaussia kietic actio, amely Z (0) [j] = exp 1 j, ( ) j L (Ω) = W (Ω) () µ[ϕ] exp(i j, ϕ L (Ω)) (3) here ϕ W (Ω)  the Sobolev space of order which is the fuctioal domai of the cylidrical Fourier Trasform measure of the Geeratig fuctioal Z (0) [j], a cotiuous biliear positive form o W (Ω) (the topological dual of W (Ω)) ([4] [6]). By a straightforward applicatio of the wellkow Sobolev immersio theorem, we have that for the case of k > ν (4) icludig k a real umber the fuctioal Sobolev space W (Ω) is cotaied i the cotiuously fractioal differetiable space of fuctios C k (Ω). As a cosequece, the domai of the Bosoic fuctioal itegral ca be further reduced to C k (Ω) i the situatio of eq.(4) Z (0) [j] = () µ[ϕ] exp(i j, ϕ L (Ω)) (5) C k (Ω) That is our ew result geeralizig the Wieer theorem o Browia paths i the case of = 1, k = 1 ad ν = 1 Sice the bosoic fuctioal domai o eq.(5) is formed by real fuctios ad ot distributios, we ca see straightforwardly that ay iteractio of the form exp g F (ϕ(x))d ν x (6) with the oliearity F (x) deotig a lower bouded real fuctio (γ > 0) Ω F (x) γ (7)
7 CBPFNF03/08 7 is welldefied ad is itegrable fuctio o the fuctioal space (C k (Ω), () µ[ϕ]) by a direct applicatio of the Lebesque theorem exp g Ω F (ϕ(x)) d ν x exp+gγ (8) At this poit we make a subtle mathematical remark that the ifiite volume limit of eq.(5)  eq.(6) is very difficult, sice oe looses the Gardig  Poicaré iequalite at this limit for those elliptic operators ad, thus, the very importat Sobolev theorem. The probable correct procedure to cosider the thermodyamic limit i our Bosoic path itegrals is to cosider solely a volume cut off o the iteractio term Gaussia actio as i eq.() ad there search for vol(ω) ([7] [10]). As a last remark related to eq.(3) oe ca see that a kid of fishet expoetial geeratig fuctioal Z (0) [j] = exp 1 j, exp α j (9) L (Ω) has a Fourier trasformed fuctioal itegral represetatio defied o the space of the ifiitelly differetiable fuctios C (Ω), which physically meas that all field cofiguratios makig the domai of such path itegral has a strog behavior like purely ice smooth classical field cofiguratios. As a last importat poit of this ote, we preset a importat result o the geometrical characterizatio of massive free field o a Euclidea SpaceTime ([10]). Firstly we aoucig a slightly improved versio of the usual Milos Theorem ([4]). Theorem 3. Let E be a uclear space of tests fuctios ad dµ a give σmeasure o its topologic dual with the strog topology. Let, 0 be a ier product i E, iducig a Hilbertia structure o H 0 = (E,, 0 ), after its topological completatio. We suppose the followig: a) There is a cotiuous positive defiite fuctioal i H 0, Z(j), with a associated cylidrical measure dµ. b) There is a HilbertSchmidt operator T : H 0 H 0 ; ivertible, such that E Rage (T ), T 1 (E) is dese i H 0 ad T 1 : H 0 H 0 is cotiuous. We have thus, that the support of the measure satisfies the relatioship support dµ (T 1 ) (H 0 ) E (30) At this poit we give a otrivial applicatio of ours of the above cited Theorem 3. Let us cosider a differetial iversible operator L: S (R N ) S(R), together with a positive iversible selfadjoit elliptic operator P : D(P ) L (R N ) L (R N ). Let H α be the followig Hilbert space H α = S(R N ), P α ϕ, P α ϕ L (R N ) =, α, for α a real umber. (31) We ca see that for α > 0, the operators below P α : L (R N ) H +α ϕ (P α ϕ) (3)
8 CBPFNF03/08 8 P α : H +α L (R N ) are isometries amog the followig subspaces ϕ (P α ϕ) (33) sice ad D(P α ),, L ) ad H +α P α ϕ, P α ϕ H+α = P α P α ϕ, P α P α ϕ L (R N ) = ϕ, ϕ L (R N ) (34) P α f, P α f L (R N ) = f, f H +α (35) If oe cosiders T a give HilbertSchmidt operator o H α, the composite operator T 0 = P α T P α is a operator with domai beig D(P α ) ad its image beig the Rage (P α ). T 0 is clearly a ivertible operator ad S(R N ) Rage (T ) meas that the equatio (T P α )(ϕ) = f has always a ozero solutio i D(P α ) for ay give f S(R N ). Note that the coditio that T 1 (f) be a dese subset o Rage (P α ) meas that T 1 f, P α ϕ L (R N ) = 0 (36) has as uique solutio the trivial solutio f 0. Let us suppose too that T 1 : S(R N ) H α be a cotiuous applicatio ad the biliear term (L 1 (j))(j) be a cotiuous applicatio i the Hilbert spaces H +α S(R N ), L amely: if j j, the L 1 : P α L j L 1 P α j, for j Z ad j S(R N ). By a direct applicatio of the Milos Theorem, we have the result Z(j) = exp 1 [L 1 (j)(j)] = dµ(t ) exp(it (j)) (37) (T 1 ) H α Here the topological space support is give by [ (P (T 1 ) H α = α T 0 P α) ( ) 1] (P α (S(R N ))) = [ (P α ) (T 1 0 ) (P α) ) ] P α (S(R N )) = P α T 1 0 (L (R N )) (38) I the importat case of L = ( + m ): S (R N S(R N ) ad T 0 T0 = ( + m ) β ( ) N 1 (L (R N )) sice T r(t 0 T0 1 m Γ( N ) = )Γ(β N ) < for β > (m ) β 1 Γ(β) N 4 with the choice P = ( + m ), we ca see that the support of the measure i the pathitegral represetatio of the Euclidea measure field i R N may be take as the measurable subset below supp d ( + m ) u(ϕ) = ( + m ) α ( + m ) +β (L (R N )) (39) sice L 1 P α = ( + m ) 1 α is always a bouded operator i L (R N ) for α > 1.
9 CBPFNF03/08 9 As a cosequece each field cofiguratio ca be cosidered as a kid of fractioal distributioal derivative of a square itegrable fuctio as writte below [ ( ϕ(x) = ) + m N 4 f] +ε 1 (x) (40) with a fuctio f(x) L (R N ) ad ay give ε > 0, eve if origially all fields cofiguratios eterig ito the pathitegral were elemets of the Schwartz Tempered Distributio Spaces S (R N ) certaily very rough mathematical objects to characterize from a rigorous geometrical poit of view. We have, thus, make a further reductio of the fuctioal domai of the free massive Euclidea scalar field of S (R N ) to the measurable subset as give by eq.(130) deoted by W (R N ) exp [ 1 ( + m ) 1 j ] (j) = d ( +m )µ(ϕ) e i ϕ(j) S (R N ) = d ( +m ) µ(f) e i f,( +m ) N 4 +ε 1 f L (R N ) W (R N ) S (R N ) (41)
10 CBPFNF03/08 10 Refereces [1] B. Simo, Fuctioal Itegratio ad Quatum Physics  Academic Press, (1979). [] B. Simo, The P (φ) Euclidea (Quatum) Field Theory  Priceto Series i Physics, (1974). [3] J. Glimm ad A. Jaffe, Quatum Physics  A Fuctioal Itegral Poit of View Spriger Verlag, (1988). [4] Y. Yamasaki, Measure o Ifiite Dimesioal Spaces  World Scietific Vol. 5, (1985). [5] Xia Dao Xig, Measure ad Itegratio Theory o Ifiite Dimesioal Space Academic Press, (197). [6] L. Schwartz, Radom Measure o Arbitrary Topological Space ad Cylidrical Measures  Tata Istitute  Oxford Uiversity Press, (1973). [7] K. Symazik, J. Math., Phys., 7, 510 (1966). [8] B. Simo, Joural of Math. Physics, vol. 1, 140 (1971). [9] Luiz C.L. Botelho, Phys., Rev. 33D, 1195 (1986). [10] E Nelso, Regular probability measures o fuctio space; Aals of Mathematics (), 69, , (1959).
11 CBPFNF03/08 11 APPENDIX Some Commets o the Support of Fuctioal Measures i Hilbert Space Let us commet further o the applicatio of the Milos Theorem i Hilbert Spaces. I this case oe has a very simple proof which holds true i geeral Baach Spaces (E, ). Let us thus, give a cylidrical measures d µ(x) i the algebraic dual E alg of a give Baach Space E ([4] [6]). Let us suppose either that the fuctio x belogs to L 1 (E alg, d µ(x)). The the support of this cylidrical measures will be the Baach Space E. The proof is the followig: Let A be a subset of the vectorial space E alg (with the topology of potual covergece), such that A E c (so x = + ). Let be the sets A µ = x E alg x. The we have the set iclusio A =0 A, so its measure satisfies the estimates below: µ(a) lim if = lim if lim if = lim if µ(a ) µx E alg x 1 x d µ(x) E alg x L 1 (E alg,d µ ) = 0. (1) Leadig us to the Milos theorem that the support of the cylidrical measure i E alg is reduced to the ow Baach Space E. Note that by the Mikowisky iequality for geeral itegrals, we have that x L 1 (E alg, d µ(x)). Now it is elemetary evaluatio to see that if A 1 (M), whe 1 E = M, a give Hilbert Space, we have that d A µ(x) x = T r M (A 1 ) <. () M alg This result produces aother criterium for supp d A µ = M (the Milos Theorem), whe E = M is a Hilbert Space. It is easy too to see that if x d µ(x) < (3) the the FourierTrasformed fuctioal Z(j) = is cotiuous i the orm topology of M. M M e i(j,x) M d µ(x) (4)
12 CBPFNF03/08 1 Otherwise, if Z(j) is ot cotiuous i the origi 0 M (without loss of geerality), the there is a sequece j M ad δ > 0, such that j 0 with δ Z(j ) 1 e i(j,x) M 1 d µ(x) M (j, x) d µ(x) M ( ) j x d µ(x) 0, (5) a cotradictio with δ > 0. Fially, let us cosider a elliptic operator B (with iverse) from the Sobelev space M m (Ω) to M m (Ω). The by the criterium give by eq.() if M T r L (Ω)[(I + ) + m B 1 (I + ) + m ] <, (6) we will have that the path itegral below writte is welldefied for x M +m (Ω) ad j M m (Ω). Namely exp( 1 (j, B 1 j) L (Ω)) = d B µ(x) exp(i(j, x) L (Ω)). (7) M +m (Ω) By the Sobolev theorem which meas that the embeeded below is cotiuous (with Ω R ν deotig a smooth domai), oe ca further reduce the measure support to the Hölder α cotiuous fuctio i Ω if m ν > α. Namely, we have a easy proof of the famous Wieer Theorem o sample cotiuity of certai path itegrals i Sobolev Spaces M m (Ω) C α (Ω) (8a) The above Wieer Theorem is fudametal i order to costruct otrivial examples of mathematically rigorous euclideas path itegrals i spaces R ν of higher dimesioality, sice it is a trivial cosequece of the Lebesgue theorem that positive cotiuous fuctios V (x) geerate fuctioals itegrable i M m (Ω), d B µ(ϕ) of the form below exp V (ϕ(x))dx L 1 (M m (Ω), d B µ(ϕ)). (8b) Ω As a last importat remark o Cylidrical Measures i Separable Hilbert Spaces, let us poit at to our reader that the support of such above measures is always a σ compact set i the orm topology of M. I order to see such result let us cosider a give dese set of M, amely x k k I +. Let δ k k I + be a give sequece of positive real umbers with δ k 0. Let ε aother sequece of positive real umbers such that =1 ε < ε. Now it is straightforward to see that M h=1 B(x k, δ k ) M ad thus lim sup µ ( k=1 B(x k, δ k ) = µ(m) = 1. As a cosequece, for each, there is a k, such k ) that µ k=1 B(x k, δ k ) 1 ε. Now the sets K µ = [ k ] =1 k=1 B(x k, δ k ) are closed ad totally bouded, so they are compact sets i M with µ(m) 1 ε. Let is ow choose ε = 1 ad the associated
13 CBPFNF03/08 13 compact sets K,µ. Let us further cosider the compact sets ˆK,µ = l=1 K l,µ. We have that ˆK,µ ˆK +1,µ, for ay ad lim sup µ( ˆK,µ ) = 1. So, Supp dµ = ˆK =1,µ, a σcompact set of M. We cosider ow a eumerable family of cylidrical measures dµ i M satisfyig the chai iclusio relatioship for ay I + Supp dµ Supp dµ +1. () Now it is straightforward to see that the compact sets ˆK, where Supp dµ m = =1 is such that Suppdµ m () =1 ˆK, for ay m I +. Let us cosider the family of fuctioals iduced by the restrictio of this sequece of measures i ay compact ˆK (). Namely µ L () (f) = ˆK () f(x) dµ p (x). ˆK (m), (8c) () Here f C b ( ˆK ). Note that all the above fuctioals i =1 C () b( ˆK ) are bouded by 1. ( By the AlaogluBourbaki theorem they form a compact set i the weak star topology of =1 C (), b( ˆK )) so there is a subsequece (or better the whole sequece) covergig to a uique cylidrical measure µ(x). Namely f(x)dµ (x) = f(x)d µ(x) (8d) for ay f =1 C () b( ˆK ). lim M M
Convexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationAdvanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σalgebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationMeasurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationMetric, Normed, and Topological Spaces
Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolutevalue metric
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationA note on the boundary behavior for a modiﬁed Green function in the upperhalf space
Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s136610150363z RESEARCH Ope Access A ote o the boudary behavior for a modiﬁed Gree fuctio i the upperhalf space Yulia Zhag1 ad Valery
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More information2.3. GEOMETRIC SERIES
6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More informationA Study for the (μ,s) n Relation for Tent Map
Applied Mathematical Scieces, Vol. 8, 04, o. 60, 3009305 HIKARI Ltd, www.mhikari.com http://dx.doi.org/0.988/ams.04.4437 A Study for the (μ,s) Relatio for Tet Map Saba Noori Majeed Departmet of Mathematics
More information13 Fast Fourier Transform (FFT)
13 Fast Fourier Trasform FFT) The fast Fourier trasform FFT) is a algorithm for the efficiet implemetatio of the discrete Fourier trasform. We begi our discussio oce more with the cotiuous Fourier trasform.
More informationMeasure Theory, MA 359 Handout 1
Measure Theory, M 359 Hadout 1 Valeriy Slastikov utum, 2005 1 Measure theory 1.1 Geeral costructio of Lebesgue measure I this sectio we will do the geeral costructio of σadditive complete measure by extedig
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationNumerical Solution of Equations
School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationπ d i (b i z) (n 1)π )... sin(θ + )
SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationA sharp TrudingerMoser type inequality for unbounded domains in R n
A sharp TrudigerMoser type iequality for ubouded domais i R Yuxiag Li ad Berhard Ruf Abstract The TrudigerMoser iequality states that for fuctios u H, 0 (Ω) (Ω R a bouded domai) with Ω u dx oe has Ω
More informationOn the L p conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/0302376, ublished olie 2007080 DOI 0.007/s0003007993x Archiv der Mathematik O the L cojecture for locally comact
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationSection 7: Free electron model
Physics 97 Sectio 7: ree electro model A free electro model is the simplest way to represet the electroic structure of metals. Although the free electro model is a great oversimplificatio of the reality,
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423432 HIKARI Ltd, www.mhikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More information1 n. n > dt. t < n 1 + n=1
Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more
More informationReview of Fourier Series and Its Applications in Mechanical Engineering Analysis
ME 3 Applied Egieerig Aalysis Chapter 6 Review of Fourier Series ad Its Applicatios i Mechaical Egieerig Aalysis TaiRa Hsu, Professor Departmet of Mechaical ad Aerospace Egieerig Sa Jose State Uiversity
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More informationSyllabus S.Y.B.Sc. (C.S.) Mathematics Paper II Linear Algebra
Syllabus S.Y.B.Sc. (C.S.) Mathematics Paper II Liear Algebra Uit : Systems of liear equatios ad matrices (a) Systems of homogeeous ad ohomogeeous liear equatios (i) The solutios of systems of m homogeeous
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationMatrix Transforms of Astatistically Convergent Sequences with Speed
Filomat 27:8 2013, 1385 1392 DOI 10.2298/FIL1308385 Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia vailable at: http://www.pmf.i.ac.rs/filomat Matrix Trasforms of statistically
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationAlternate treatments of jacobian singularities in polar coordinates within finitedifference schemes
ISSN 1 746733, Eglad, UK World Joural of Modellig ad Simulatio Vol. 8 (1) No. 3, pp. 163171 Alterate treatmets of jacobia sigularities i polar coordiates withi fiitedifferece schemes Alexys BruoAlfoso
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationFunctional Analysis Basics c 2007 Ulrich G. Schuster SVN Revision: 2109, December 11, 2007 Vector Space Concepts
Fuctioal Aalysis Basics c 2007 Ulrich G. Schuster SVN Revisio: 2109, December 11, 2007 Vector Space Cocepts icomplete liear spaces ormed spaces ier product spaces metric spaces Basis ad Dimesio [2, 1].
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information12. FRAMES IN HILBERT SPACES
1. FRAMES IN HILBERT SPACES 81 1. FRAMES IN HILBERT SPACES Frames were itroduced by Duffi ad Schaeffer i the cotext of oharmoic Fourier series [DS5]. They were iteded as a alterative to orthoormal or Riesz
More informationAn Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function
A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationAlternatives To Pearson s and Spearman s Correlation Coefficients
Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives
More informationwhere w^o and q (0<q<l) are real constants. I prove that, for large z,
JACOBIAN FUNCTIONS IN POWERS OF THE MODULI. 115 of fidig a place i the stadard treatises have had to be rediscovered is perhaps due partly to the fact that the joural to which they were commuicated was
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More informationIntroductory Explorations of the Fourier Series by
page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ 7764898 tzielis@momouth.edu Copyright
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More information0,1 is an accumulation
Sectio 5.4 1 Accumulatio Poits Sectio 5.4 BolzaoWeierstrass ad HeieBorel Theorems Purpose of Sectio: To itroduce the cocept of a accumulatio poit of a set, ad state ad prove two major theorems of real
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More information