Monotonicity for excited random walk in high dimensions
|
|
- Vanessa Stephens
- 5 years ago
- Views:
Transcription
1 Monotonicity for excite ranom walk in high imensions Remco van er Hofsta Mark Holmes March, 2009 Abstract We prove that the rift θ, β) for excite ranom walk in imension is monotone in the excitement parameter β [0, ], when is sufficiently large. We give an explicit criterion for monotonicity involving ranom walk Green s functions, an use rigorous numerical upper bouns provie by Hara [8] to verify the criterion for 9. Introuction In this paper we stuy excite ranom walk, where the ranom walker has a rift in the irection of the first component each time the walker visits a new site. It was shown that this process has ballistic behaviour when 2 in [4, 3, 4]. A nontrivial strong law of large numbers SLLN) can then be obtaine for 2 using renewal techniques see for example [5, 6]). For =, it is known that ERW is recurrent an iffusive [7] except in the trivial case β =. Aitional results on one-imensional multi)-excite ranom walks can be foun in [, 2, 3, 6]. In [0], a perturbative expansion was introuce an use to prove a weak law of large numbers an a central limit theorem for excite ranom walk in imensions > 5 an > 8 respectively, with sufficiently small excitement parameter. More recently, [5] explicitly prove a SLLN an establishe a functional central limit theorem in imensions 2. Inclue in [0] is an explicit representation of the rift in terms of the expansion coefficients. In this paper we use this representation, together with explicit simple ranom walk Green s function bouns [8, 9] to prove that in imensions 9, the rift for excite ranom walk is strictly) increasing in the excitement parameter β.. Main results The main result of this paper is the following theorem. Department of Mathematics an Computer Science, Einhoven University of Technology, P.O. Box 53, 5600 MB Einhoven, The Netherlans. rhofsta@win.tue.nl Department of Statistics, The University of Aucklan, Private Bag 9209, Aucklan 42, New Zealan. mholmes@stat.aucklan.ac.nz
2 Theorem. Monotonicity of the spee). For all 9, an β [0, ], the rift for excite ranom walk in imension with excitement parameter β is strictly increasing in β. We are also able to show that for 8, there exists β 0 ) such that the rift for ERW is strictly increasing in β [0, β 0 ]. Simulations [5] suggest that the limiting variance of the first coorinate is not monotone in the excitement parameter β in 2 imensions. We expect that using the approach introuce in this paper we can show that the variance is monotone ecreasing in β when the imension is taken sufficiently high. By [0], the variance of the first coorinate is equal to σβn 2 + o)) for some asymptotic variance σβ, 2 an base on our methos, we expect that σβ 2 = 2 β 2 + β 2 O 3 ), showing that, in sufficiently high, β σβ 2 is ecreasing. Although we only consier the once-excite ranom walk in this paper, the general multiexcite ranom walk can be hanle with very minor moifications, yieling a result at least as strong as Theorem.. A large part of the methoology in this paper can be applie more generally. See [2] for an application of the methos an results in this paper to the case where the rift on subsequent visits to a site is in the opposite irection to that inuce by the excitement parameter on the first visit. Given the present context, another natural example is a ranom walk in an environment that is ranom in the first few coorinates only, with the expecte rift inuce by the environment enote by β. Some progress has been mae in this irection [] making use of the fact that a SLLN has been prove for general versions of such ranom walks in ranom environment in [6]. We first introuce some notation. A nearest-neighbour ranom walk path η is a sequence {η i } for which η i Z an η i+ η i is a nearest-neighbour of the origin for all i 0. For a general nearest-neighbour path η with η 0 = 0, we write p η i x i, x i+ ) for the conitional probability that the walk steps from η i = x i to x i+, given the history of the path η i = η 0,..., η i ). We write, for β [0, ], p β x) = + βe x I { x =},.) 2 where e =, 0,..., 0), an x y is the inner-prouct between x an y. Thus, p β is the transition probability for a ranom walk having a rift β when stepping in the first coorinate. We write ω n for the n-step path of excite ranom walk ERW), an Q for the law of { ω n } n=0, i.e., for every n-step nearest-neighbour path η n, Q ω n = η n ) = n p η i η i, η i+ ),.2) where p 0 0, η ) = p β η ) is the probability to jump to η in the first step, an p η i η i, η i+ ) = p 0 η i+ η i )I {ηi η i } + p β η i+ η i )[ I {ηi η i }],.3) where I {ηi η i } enotes the inicator that η i = η j for some 0 j i. In wors, the ranom walker gets excite each time he/she visits a new site, an when the ranom walk is excite, it has a positive rift in the irection of the first coorinate. For a escription in terms of cookies, see [6]. 2
3 2 An overview of the proof an the expansion In this section we recall some results an notation from [0]. If η an ω are two paths of length at least j an m respectively an such that η j = ω 0, then the concatenation η j ω m is efine by η j ω m ) i = { ηi when 0 i j, ω i j when j i m + j. 2.) Given η m such that Q ω m = η m ) > 0, we efine a probability measure Q ηm on paths starting from η m by specifying its value on particular cyliner sets in a consistent manner) as follows Q ηm ω n = µ n ) n p ηm µ i µ i, µ i+ ), 2.2) an extening the measure to all finite-imensional cyliner sets in the natural consistent) way. Then 2.2) is also Q ω m+n = η m µ n ω m = η m ). In [0], a perturbative expansion was erive for the two-point function c n x) = Qω n = x), giving rise to a recursion relation of the form c n+ x) = y Z p β y)c n x y) + n+ m=2 y Z π m y)c n+ m x y). 2.3) This expansion was use to prove a law of large numbers an central limit theorem for ERW. We next iscuss the coefficients π m y) an some results of this expansion. The expansion coefficients. Let N, an for i 0, let ω i) j i + be a path of length j i + Z +, where, by convention, j 0 = 0. Then efine N = p ωn ) + ωn) ω p N) ) ω N), ω N) +), 2.4) which epens on ω N ) + an ω N) + although this epenence is suppresse in the notation). The ifference 2.4) is ientically zero when the histories ω N ) + ω N) an ω N) give the same transition probabilities to go from ω N) when ω N) π N) has alreay been visite by ω N ) + but not by ω N) N = βe ω N) + ω N) [ ) 2 to ω N) +. For excite ranom walk, N is non-zero precisely, so that I N) {ω j / ω N ) N j ω N) N } I {ω N) ] / ω N) } β 2 I I β {ω N) + =ωn) j ±e } {ω N) N j ω N ) N j \ ω N) N } 2 I I. {ω N) + =ωn) j ±e } {ω N) N j ω N ) N j } N Define A m,n = {j,..., ) Z N + : N l= j l = m N }, A N = m A m,n an m x, y) = j A m,n ω 0) ω ) j + ω N) + I {ω N) =x,ω N) + =y}p βω 0) N ) n= j n n i n=0 2.5) p ωn ) j n + ωn) in ω n) i n, ω n) i n+). 2.6) 3
4 Then we efine π m x, y) = N= π N) m x, y), π N) x, y) = m π N) m x, y), an π m y) = π N) N= x Z m x, y). 2.7) Note that the quantities π m N) epen on β. Note further that are all zero when N + > m, an that all of the above quantities π m N) x, y) = 0, 2.8) y Z since ω N) N = 0 see also [0, 6.0)]). + The importance of these quantities is given by [0, Proposition 3.], which states that if lim nm=2 n x Z xπ mx) exists an n ω n converges in probability to θ, then θβ, ) = yp β y) + yπ m y) = βe y Z m=2 y Z + m=2 y Z yπ m y). 2.9) Strategy of the proof of Theorem.. We shall explicitly ifferentiate the right han sie of 2.9), an prove that this erivative is positive for all β [0, ], when 9. From 2.9) an using 2.7) 2.8), we have so that yπ m y) = y Z x,y Z y x)π m x, y), 2.0) θβ, ) = βe + m=2 N= x,y Z y x)π N) m x, y). 2.) Letting ϕ N) m x, y) = sums, we then have β πn) m x, y) an assuming that the limit can be taken through the infinite θ β β, ) = e + N= m=2 Since ϕ N) m x, y) 0 unless x y =, we have that x,y Z y x)ϕ N) m x, y). 2.2) θ β β, ) e ϕ N) m x, y). 2.3) N= m=2 x,y Z We conclue that θ θ β, ), which is the first coorinate of β, ), is positive for any β at which β β N= m=2 x,y Z ϕn) m x, y) <. This is what we shall prove in the remainer of this paper, which is organise as follows. In Section 3, we start by proving bouns on π m N). These bouns will be crucially use to prove bouns on ϕ N) m in Section 4. The results in Section 4 are use in Section 5 to prove Theorem.. 4
5 3 Boun on π Before proceeing to the proof of Theorem., we prove a new boun on x,y Z m π N) m x, y). The proof of this new boun makes use of Lemma 3. below. Let P enote the law of simple symmetric ranom walk in imensions, starting at the origin, an let D x) = I { x =} /2) be the simple ranom walk step istribution. We will make use of the convolution of functions, which is efine for absolutely summable functions f, g on Z by f g)x) = y Z fy)gx y). 3.) Let f k enote the k-fol convolution of f with itself, an let G x) = k=0 D k x) enote the Green s function for this ranom walk. We shall sometimes make use of the representation G i x) = k=0 m i :m + +m i =k D m + +m i ) x) = k=0 k + i )! P ω k = x), for i. 3.2) i )!k! Note that G i x) < if an only if > 2i. We shall often abbreviate G i let = G i 0). For i 0, ) i+g i+) E i ) = sup v) δ 0,v ). 3.3) v Z Lemma 3. Diagrammatic bouns for ERW). For excite ranom walk, uniformly in u Z an η m, for i 0, j=0 j= Q ηm ω j = u) i! ) i+g i+), 3.4) Q ηm ω j = u) i!e i ). 3.5) Proof. Define an increasing sequence of ranom variables N j, j 0, by letting j N j be the number of steps that the walk ω j takes in the first coorinate. Observe that inepenently of η, N j has a Binomialj, q ) istribution, where q = )/. If we consier ω j as the initial position an first j steps of an infinite walk ω, then the sequence {N j } j 0 is a ranom walk on Z + taking i.i.. steps that are either + or 0 with probability q an q respectively. The ranom time that such a walk spens at any level l has a Geometric istribution with parameter q. Thus, writing P for the law of {N j } j=0, we obtain that for every i 0, so that, for m l, PN j = l) = j=m l!j l)! ql q ) j l = q i PN j = l) = q i+) l + i)! PN j+i = l + i), l! l + i)!. 3.6) l! 5
6 Given u = u,..., u ) Z, we write u := u 2, u 3,..., u ) Z. To prove 3.4), note that j=0 j Q ηm ω j = u) = Q ηm ω j = u N j = l)pn j = l) j=0 P ω l = u ηm) PN j = l) j=l q i+) sup P ω l = v) v Z l + i)!. 3.7) l! By 3.2), 3.7) is equal to i!q i+) sup v Z G i+) v). By [9, Lemma B.3], the supremum occurs at v = 0. Since q = / ), this proves 3.4). The boun 3.5) is prove similarly. Inee, for i 0, we can write j= Q ηm ω j = u) sup v Z = sup v Z = sup v Z =i! sup v Z P ω l = v) j=l [ P ω l = v) j=l q i+) i+) q PN j = l) PN j = l) δ 0,l i!pn 0 = 0) ) l + i)! P ω l = v) i!δ 0,v l! G i+) v) δ 0,v ), 3.8) since PN 0 = 0) = an P ω l = v)δ 0,l = δ 0,v, an following the steps in 3.7) above. Define a = ) 2 G ) Proposition 3.2 Bouns on the expansion coefficients for ERW). For N =, x,y Z m π ) m x, y) β E 0 ), an, for N 2, x,y Z Given η m an z j+, efine m π N) m x, y) β N ) G E )a N ) z j+ ) = p ηm z j z j, z j+ ) p z j z j, z j+ ) ) I {z0 =η m}. 3.) We will use the following lemma to prove Proposition 3.2. ]) 6
7 Lemma 3.3 Ingreients for bouns on lace expansion coefficients). For any η s, j=0 z j+ z j+ ) j=0j + ) z j+ j= z j+ z j+ ) j=j + ) z j+ j p ηs z i z i, z i+ ) sβ G, 3.2) j z j+ ) p ηs z i z i, z i+ ) sβa, 3.3) j p ηs z i z i, z i+ ) sβ E 0), 3.4) j z j+ ) p ηs z i z i, z i+ ) sβ E ). 3.5) Proof. As in 2.5), the left han sie of 3.2) is boune above by j p ηs z i β z i, z i+ ) I {zj η s } I {zj+ =z j=0 z j 2 j ±e } 3.6) z j+ β j p ηs z i z i, z i+ ) I {zj η s }, j=0 z j since only two terms contribute to the rightmost sum in 3.6). From 2.2), this is equal to β Q ηs ω j = z j )I {zj η s } = β Q ηs ω j η s ) β s Q ηs ω j = η l ). 3.7) j=0 z j j=0 j=0 The inequality 3.2) then follows from 3.4) with i = 0. The inequality 3.3) is obtaine similarly, using 3.4) with i = at the last step, while 3.4) an 3.5) are obtaine using 3.5) with i = 0 an i = respectively at the last step. Proof of Proposition 3.2. It follows from 2.6) that x,y Z m π N) m x, y) is boune by p β ω 0) ) j = ω 0) ω ) j + j ) p ω0) ω) i ω ) i, ω ) i + i = =0 ω N) + N p ωn ) + ωn) i N ω N) i N, ω N) i N +), i N = 3.8) where the sums over j k, k 2 are all from 0 to. Note that can only be non-zero if j is o so in particular, non-zero). We procee by using Lemma 3.3 to successively boun the sums over j k of this expression, beginning with the sum over. If N = then we use 3.4) with s = to boun this sum by sβ E 0), an then p ω 0) β ω 0) ) = gives the result. If N > then we use 3.2) with s = + on the sum over, followe by repeate applications of 3.3) with s = j k + on the sums over j k with k = N,..., 2 respectively, then 3.5) with s = on the sum over j an again the result follows since p ω 0) β ω 0) ) =. Since the spee is known to exist [5], the following corollary is an easy consequence of [0, Propositions 3. an 6.] together with Proposition 3.2, an the fact that a 6 < since G 2 5 < 5 2 /6 [9]. 7
8 Corollary 3.4 Formula for the spee of ERW). For all 6 an β [0, ], θβ, ) = lim E[ω n+ ω n ] = βe n + m=2 x Z xπ m x). 3.9) In fact, the first equality in Corollary 3.4 hols for all 2 since the law µ n of the cookie environment as viewe by the ranom walker at time n is known to converge see e.g. [5]) an E[ω n+ ω n ] = E [ E[ω n+ ω n ω n ] ] [ ] βe = E I {ω n / ω n } = βe [ Pωn ω n ) ], 3.20) where the right han sie converges as n since Pω n ω n ) is the µ n -measure of the event that the cookie at the origin is absent. 4 The ifferentiation step To verify the exchange of limits in 2.2), it is sufficient to prove that x,y Zy x)πn) m x, y) is absolutely summable in m an N note that for every m an N the summations over x an y are finite) an that m=2 N= sup β [0,] x,y Zy x)ϕn) m x, y) <. By Proposition 3.2 an the fact that y x = for x, y nearest neighbours, the first conition hols provie that βa <. 4.) In fact we will see later on that this inequality for β = is sufficient to also establish the secon conition. We now ientify ϕ m N) x, y). Recall 2.6). Then we can write ϕ N) m x, y) = ϕ m N,) x, y) + ϕ N,2) m x, y) + ϕ N,3) m x, y), 4.2) where by Leibniz rule), ϕ m N,) x, y), ϕ m N,2) x, y) an ϕ N,3) m x, y) arise from ifferentiating p β ω 0) ), Nn= jn i n=0 p ωn ) j n + ωn) in ω n) i n, ω n) i n+) an Nn= n, respectively, with respect to β. Observe that if η m = x l then β p ηm β x l, x) = e x x l )I {xl / η m } 2 an hence, using I A I A C = I A C c β Clearly then β I { x xl =} = I {x l / η m } 2 we have I{x xl =e } I {x xl = e } ), 4.3) p η m β x l, x) p ωn ηm β x l, x) ) = 2 I ) {x l / η m,x l ω n } I{x xl =e } I {x xl = e }. 4.4) p η m β x l, x) p ωn ηm β x l, x) ) 2 I ) {x l ω n \ η m } I{x xl =e } + I {x xl = e }. 4.5) Let ρ N) be the quantity obtaine by replacing p β ω 0) ) in 2.6) with 2) I 0) {ω a boun =±e } on its erivative) an by bouning n by n for all n =,..., N. 8
9 For k =,..., N, let γ N) k be the quantity obtaine from 2.6) by bouning n by n for ω k) i k, ω k) i k +) with the following boun on its all n =,..., N an by replacing j k i k =0 p ωk ) j k + ωk) i k erivative j k I {ω k) l+ ωk) l =±e } 2 j k i k = 0 i k l p ωk ) j k + ωk) i k ω k) i k, ω k) i k +). 4.6) Similarly, let χ N) k be obtaine by replacing k in 2.6) by 2) I I a k) {ω j ω k ) k j } {ω k) k j k + ωk) j =±e } k boun on its erivative) an by bouning n for n k by n. Letting γ N) = N k= γ N) k an χ N) = N k= χ N) k, we obtain m x,y Z ϕ N,) m x, y) ρ N), m ϕ N,2) m x, y) γ N), an x,y Z m x,y Z ϕ N,3) We shall boun each of these terms separately, in Lemmas 4., 4.4 an 4.2 below. Lemma 4. Bouns on ρ N) ). For N =, ρ ) 2 βe 0 ), an, for N 2, m x, y) χ N). 4.7) ρ N) β N G E ) 2 ) an ) Proof. This is exactly the same as the proof of Proposition 3.2 except that at the very last step we use 2 I = {ω 0) =±e }. 4.9) ω 0) Lemma 4.2 Bouns on χ N) ). For N =, χ ) E 0 ), an, for N 2, χ N) Nβ N G E ) ) an ) Proof. Proceeing exactly as in the proof of Proposition 3.2, except that the boun on k is missing the β term, we obtain χ ) k E 0 ) an, for N 2, χ N) k β N ) G E )a N 2. 4.) The resulting boun on χ N), which is simply β N times 3.0)) is then easily obtaine by summing over k from to N. Before proceeing to the boun on γ N), we first nee a new lemma similar to Lemma 3.. Lemma 4.3. Let Q l, η s enote the law of a self-interacting ranom walk ω with history η s, where the transition probabilities are those of an ERW with history η s, except that Q l, η s ω l+ = ω l + e ω l ) = Q l, η s ω l+ = ω l e ω l ) = ) 9
10 Then, for all i 0, j= j Q l, η s ω j = u) i + )! ) i+2 G i+2). 4.3) Proof. Since one of the j steps is a simple ranom walk step in the first coorinate, the number of steps in the other coorinates has a Binomialj, ) istribution. Thus, j= j Q l, η s ω j = u) sup j j v Z j= l= k=0 = sup v Z k=0 = sup v Z PN j = k)p ω k = v) P ω k = v) j j=k+ P ω k = v) k=0 r=k PN j = k) r + i + )! PN r = k). 4.4) r! Now procee as in the proof of Lemma 3. to obtain the result. Define 2 ɛ) = ) G G E ) ) 2 G ) Lemma 4.4 Bouns on γ N) ). For N =, 2, γ ) N 3, β ) 2 G 2, γ 2) β 2 ɛ) an, for all γ N) ɛ)β 2 βa ) N 2 + N 2) 2β3 E ) ) 4 G G 3 βa ) N ) Proof. We procee as in the proof of Proposition 3.2 except that from the efinition of γ N) k, the prouct of transition probabilities insie the sum over j k in 3.8), is replace with 4.6). We use Lemma 4.3 instea of Lemma 3. to boun this sum. When N =, then also k = an γ ) is j p β ω 0) ) j = ω 0) β p 2 β ω 0) ω 0) ω ) j + I ) {ω l+ ω) =±e l } 2 j ) j = ω ) I ) {ω j j =ω 0) j i = 0 i l I ) {ω l+ ω) =±e l } 0 } 2 ) p ω0) ω) i ω ) i, ω ) i+ j i = 0 i l ) p ω0) ω) i ω ) i, ω ) i+ 4.7) where we have use the usual boun 2.5) an the fact that I ω ) ) j + {ω j + =ω) j ±e } observe that = 2. Now I {ω ) l+ ω) l =±e } 2 j i = 0 i l ) p ω0) ω) i ω ) i, ω ) i+ = Q l, ω 0) ωj = ω ) j ), 4.8) 0
11 so that 4.7) is equal to β 2 ω 0) p β ω 0) j ) I ) {ω j = ω ) j =ω 0) j = β p 2 β ω 0) ω 0) j ) j = 0) l, ω 0 }Q ωj = ω ) Q l, ω 0) ωj = ω 0) 0 ). Now use 4.3) with i = 0 to get the require boun. For the remaining cases we begin by ajusting the sum over j k in 3.8) as in 4.7) an 4.8). When N > an k = we use the same bouns as in the proof of Proposition 3.2 except that we use 4.3) with i = on the sum over j. This gives us a boun on γ N) when N > ) of 2β 2 ) 3 G 3 β G N i=2 j ) βa. 4.9) When N > an k = N, we use the same bouns as in the proof of Proposition 3.2 except that we use 4.3) with i = 0 on the sum over in 3.8). This gives us a boun on γ N) N when N > ) of β ) 2 G 2 β E ) N i=2 βa. 4.20) Similarly when N > an k N so N > 2) we use 4.3) on the sum over j k get a boun on γ N) k of the form in 3.8) to β G β E ) 2β 2 ) 3 N G 3 i=2 i k βa. 4.2) Simplifying these expressions an summing over k completes the proof of the lemma. Corollary 4.5 Summary of bouns). For all β [0, ], an such that a < N= N= N= ρ N) E 0) + G E ) ) a ), 4.22) χ N) E 0 ) + G E )2 a ) ) a ) 2, 4.23) γ N) G 2 ) 2 + ɛ) a + 2E )G G 3 ) 4 a ) ) Proof. Firstly note that the conition on a ensures that ρ N), χ N) an γ N) are all summable over N, an in all cases the supremum over β occurs at β = see Lemmas 4., 4.2 an 4.4). The results are then easily obtaine by summing each of the bouns in Lemmas 4., 4.2 an 4.4 over N.
12 5 Proof of Theorem. For such that a <, the bouns of Corollary 4.5 hol. From 4.7) we have the require absolute summability conitions in the iscussion after 2.2), an in particular 2.2) hols for all β [0, ]. To complete the proof of the theorem, it remains to show that the right han sie of 2.3) is no more than. By 4.7) an Corollary 4.5, we have boune times the right han sie of 2.3) by the sum of the right han sies of the bouns in Corollary 4.5. Since these terms all involve simple ranom walk Green s functions quantities, we will nee to use estimates of these quantities. In orer to boun E i ), we shall first prove that, for all i 0, E i ) = ) i+g i+). 5.) In orer to prove 5.), we first make use of [9, Lemma B.3], which states that G n x) is nonincreasing in x i for every i =,...,, so that the supremum in 3.3) can be restricte to v = 0 an v = e for any neighbour e of the origin. In orer to boun G n e), we make use of the fact that for any function x fx) for which fe) is constant for all e Z with e =, we have fe) = D f)0), so that { ) i+g i+) E i ) = max 0), ) } i+d G i+) )0). 5.2) Note that since G x) = δ 0,x + D G )x) δ 0,x, we have that G i 0) an G i+) 0) = G i 0) + D G i+) )0). Therefore, ) i+g i+) 0) = which is strictly larger than By [9, Lemma C.], G n ) i+d G i+) )0) + ) i+g i 0), 5.3) ) i+d G i+) )0) an thus proves 5.). is monotone ecreasing in for each n, so that it suffices to show that the sum of terms on the right han sies of 4.22), 4.23) an 4.24) is boune by for = 9. For this we use the following rigorous Green s functions estimates [8, 9] for = 8: G.07865, G 2.289, G ) Putting in these values for = 8 we get that the sum of the right han sies of the bouns in Corollary 4.5 is at most 0.97, whence the result follows for 9. To prove monotonicity for β [0, β 0 ] for some β 0 ) for each 8, it is sufficient to prove that χ ) < when 8 an that the other terms are boune), since this is the only term that oes not contain a factor β that can be mae arbitrarily small by choosing β 0 small. Since χ ) E 0 ), it is enough to show that E 0 ) < for = 8, since the right han sies of 4.22), 4.23) an 4.24) are boune for 8. From [9] we have 6G 5 5 < 6.57) <, an since 5 E 0 ) is ecreasing in, this completes the result. Acknowlegements. The work of RvH an MH was supporte in part by Netherlans Organisation for Scientific Research NWO). The work of MH was also supporte by a FRDF grant from the University of Aucklan. The authors thank Itai Benjamini for suggesting this problem to us, Takashi Hara for proviing the Green s functions upper bouns in 5.4), an an anonymous referee for many helpful suggestions that significantly improve the presentation of this paper. 2
13 References [] T. Antal an S. Rener. The excite ranom walk in one imension. J. Phys. A: Math. Gen., 38: , [2] A.-L. Basevant an A. Singh. On the spee of a cookie ranom walk. Probab. Theory Relat. Fiels., 43-4): , [3] A.-L. Basevant an A. Singh. Rate of growth of a transient cookie ranom walk. Electr. Journ. Probab., 3:8 85, [4] I. Benjamini an D. B. Wilson. Excite ranom walk. Electron. Comm. Probab., 8:86 92 electronic), [5] J. Bérar an A. Ramírez. Central limit theorem for excite ranom walk in imension 2. Electr. Comm. Probab., 2:300 34, [6] E. Bolthausen, A.-S. Sznitman, an O. Zeitouni. Cut points an iffusive ranom walks in ranom environment. Ann. Inst. H. Poincaré Probab. Statist., 393): , [7] B. Davis. Brownian motion an ranom walk perturbe at extrema. Probab. Theory Relat. Fiels., 3:50 58, 999. [8] T. Hara. Private communication, [9] T. Hara an G. Slae. The lace expansion for self-avoiing walk in five or more imensions. Reviews in Math. Phys., 4: , 992. [0] R. van er Hofsta an M. Holmes. An expansion for self-interacting ranom walks. arxiv: v2 [math.pr], [] M. Holmes. A monotonicity property for a ranom walk in a partially ranom environment. Preprint, [2] M. Holmes. Excite against the tie: A ranom walk with competing rifts. Preprint, [3] G. Kozma. Excite ranom walk in three imensions has positive spee. unpublishe, [4] G. Kozma. Excite ranom walk in two imensions has linear spee. arxiv:math/052535v [math.pr], [5] A.-S. Sznitman an M. Zerner. A law of large numbers for ranom walks in ranom environment. Ann. Probab., 27:85 869, 999. [6] M. Zerner. Multi-excite ranom walks on integers. Probab. Theory Relat. Fiels., 33:98 22,
Excited against the tide: A random walk with competing drifts
Excite against the tie: A rano walk with copeting rifts arxiv:0901.4393v1 [ath.pr] 28 Jan 2009 Mark Holes January 28, 2009 Abstract We stuy a rano walk that has a rift β to the right when locate at a previously
More informationLeast-Squares Regression on Sparse Spaces
Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction
More informationSharp Thresholds. Zachary Hamaker. March 15, 2010
Sharp Threshols Zachary Hamaker March 15, 2010 Abstract The Kolmogorov Zero-One law states that for tail events on infinite-imensional probability spaces, the probability must be either zero or one. Behavior
More informationarxiv: v2 [math.pr] 27 Nov 2018
Range an spee of rotor wals on trees arxiv:15.57v [math.pr] 7 Nov 1 Wilfrie Huss an Ecaterina Sava-Huss November, 1 Abstract We prove a law of large numbers for the range of rotor wals with ranom initial
More informationConvergence of Random Walks
Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of
More informationLower bounds on Locality Sensitive Hashing
Lower bouns on Locality Sensitive Hashing Rajeev Motwani Assaf Naor Rina Panigrahy Abstract Given a metric space (X, X ), c 1, r > 0, an p, q [0, 1], a istribution over mappings H : X N is calle a (r,
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS
ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationLower Bounds for the Smoothed Number of Pareto optimal Solutions
Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationRange and speed of rotor walks on trees
Range an spee of rotor wals on trees Wilfrie Huss an Ecaterina Sava-Huss May 15, 1 Abstract We prove a law of large numbers for the range of rotor wals with ranom initial configuration on regular trees
More informationTractability results for weighted Banach spaces of smooth functions
Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena Ernst-Abbe-Platz 2, 07740 Jena, Germany email: markus.weimar@uni-jena.e March
More informationarxiv: v2 [math.pr] 4 Sep 2017
arxiv:1708.08576v2 [math.pr] 4 Sep 2017 On the Speed of an Excited Asymmetric Random Walk Mike Cinkoske, Joe Jackson, Claire Plunkett September 5, 2017 Abstract An excited random walk is a non-markovian
More informationLecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012
CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationHITTING TIMES FOR RANDOM WALKS WITH RESTARTS
HITTING TIMES FOR RANDOM WALKS WITH RESTARTS SVANTE JANSON AND YUVAL PERES Abstract. The time it takes a ranom walker in a lattice to reach the origin from another vertex x, has infinite mean. If the walker
More informationLecture 5. Symmetric Shearer s Lemma
Stanfor University Spring 208 Math 233: Non-constructive methos in combinatorics Instructor: Jan Vonrák Lecture ate: January 23, 208 Original scribe: Erik Bates Lecture 5 Symmetric Shearer s Lemma Here
More informationA Sketch of Menshikov s Theorem
A Sketch of Menshikov s Theorem Thomas Bao March 14, 2010 Abstract Let Λ be an infinite, locally finite oriente multi-graph with C Λ finite an strongly connecte, an let p
More informationFLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction
FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationConvergence of random variables, and the Borel-Cantelli lemmas
Stat 205A Setember, 12, 2002 Convergence of ranom variables, an the Borel-Cantelli lemmas Lecturer: James W. Pitman Scribes: Jin Kim (jin@eecs) 1 Convergence of ranom variables Recall that, given a sequence
More informationOn the number of isolated eigenvalues of a pair of particles in a quantum wire
On the number of isolate eigenvalues of a pair of particles in a quantum wire arxiv:1812.11804v1 [math-ph] 31 Dec 2018 Joachim Kerner 1 Department of Mathematics an Computer Science FernUniversität in
More informationOn a limit theorem for non-stationary branching processes.
On a limit theorem for non-stationary branching processes. TETSUYA HATTORI an HIROSHI WATANABE 0. Introuction. The purpose of this paper is to give a limit theorem for a certain class of iscrete-time multi-type
More informationON TAUBERIAN CONDITIONS FOR (C, 1) SUMMABILITY OF INTEGRALS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 54, No. 2, 213, Pages 59 65 Publishe online: December 8, 213 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS Abstract. We investigate some Tauberian
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationIPA Derivatives for Make-to-Stock Production-Inventory Systems With Backorders Under the (R,r) Policy
IPA Derivatives for Make-to-Stock Prouction-Inventory Systems With Backorers Uner the (Rr) Policy Yihong Fan a Benamin Melame b Yao Zhao c Yorai Wari Abstract This paper aresses Infinitesimal Perturbation
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationGeneralized Tractability for Multivariate Problems
Generalize Tractability for Multivariate Problems Part II: Linear Tensor Prouct Problems, Linear Information, an Unrestricte Tractability Michael Gnewuch Department of Computer Science, University of Kiel,
More informationSome Examples. Uniform motion. Poisson processes on the real line
Some Examples Our immeiate goal is to see some examples of Lévy processes, an/or infinitely-ivisible laws on. Uniform motion Choose an fix a nonranom an efine X := for all (1) Then, {X } is a [nonranom]
More informationUnit vectors with non-negative inner products
Unit vectors with non-negative inner proucts Bos, A.; Seiel, J.J. Publishe: 01/01/1980 Document Version Publisher s PDF, also known as Version of Recor (inclues final page, issue an volume numbers) Please
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationDIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10
DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10 5. Levi-Civita connection From now on we are intereste in connections on the tangent bunle T X of a Riemanninam manifol (X, g). Out main result will be a construction
More informationImplicit Differentiation
Implicit Differentiation Implicit Differentiation Using the Chain Rule In the previous section we focuse on the erivatives of composites an saw that THEOREM 20 (Chain Rule) Suppose that u = g(x) is ifferentiable
More informationREAL ANALYSIS I HOMEWORK 5
REAL ANALYSIS I HOMEWORK 5 CİHAN BAHRAN The questions are from Stein an Shakarchi s text, Chapter 3. 1. Suppose ϕ is an integrable function on R with R ϕ(x)x = 1. Let K δ(x) = δ ϕ(x/δ), δ > 0. (a) Prove
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationLower Bounds for Local Monotonicity Reconstruction from Transitive-Closure Spanners
Lower Bouns for Local Monotonicity Reconstruction from Transitive-Closure Spanners Arnab Bhattacharyya Elena Grigorescu Mahav Jha Kyomin Jung Sofya Raskhonikova Davi P. Wooruff Abstract Given a irecte
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationSolutions to Math 41 Second Exam November 4, 2010
Solutions to Math 41 Secon Exam November 4, 2010 1. (13 points) Differentiate, using the metho of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (4 points) Using the rule for the erivative of
More informationII. First variation of functionals
II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent
More informationA note on asymptotic formulae for one-dimensional network flow problems Carlos F. Daganzo and Karen R. Smilowitz
A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationDiscrete Mathematics
Discrete Mathematics 309 (009) 86 869 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/isc Profile vectors in the lattice of subspaces Dániel Gerbner
More informationarxiv: v4 [math.pr] 27 Jul 2016
The Asymptotic Distribution of the Determinant of a Ranom Correlation Matrix arxiv:309768v4 mathpr] 7 Jul 06 AM Hanea a, & GF Nane b a Centre of xcellence for Biosecurity Risk Analysis, University of Melbourne,
More informationOn the enumeration of partitions with summands in arithmetic progression
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 8 (003), Pages 149 159 On the enumeration of partitions with summans in arithmetic progression M. A. Nyblom C. Evans Department of Mathematics an Statistics
More informationSINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
Communications on Stochastic Analysis Vol. 2, No. 2 (28) 289-36 Serials Publications www.serialspublications.com SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
More informationAn Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback
Journal of Machine Learning Research 8 07) - Submitte /6; Publishe 5/7 An Optimal Algorithm for Banit an Zero-Orer Convex Optimization with wo-point Feeback Oha Shamir Department of Computer Science an
More informationAcute sets in Euclidean spaces
Acute sets in Eucliean spaces Viktor Harangi April, 011 Abstract A finite set H in R is calle an acute set if any angle etermine by three points of H is acute. We examine the maximal carinality α() of
More informationFurther Differentiation and Applications
Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle
More informationPDE Notes, Lecture #11
PDE Notes, Lecture # from Professor Jalal Shatah s Lectures Febuary 9th, 2009 Sobolev Spaces Recall that for u L loc we can efine the weak erivative Du by Du, φ := udφ φ C0 If v L loc such that Du, φ =
More informationCHAPTER 1 : DIFFERENTIABLE MANIFOLDS. 1.1 The definition of a differentiable manifold
CHAPTER 1 : DIFFERENTIABLE MANIFOLDS 1.1 The efinition of a ifferentiable manifol Let M be a topological space. This means that we have a family Ω of open sets efine on M. These satisfy (1), M Ω (2) the
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationRobustness and Perturbations of Minimal Bases
Robustness an Perturbations of Minimal Bases Paul Van Dooren an Froilán M Dopico December 9, 2016 Abstract Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important
More informationSurvey Sampling. 1 Design-based Inference. Kosuke Imai Department of Politics, Princeton University. February 19, 2013
Survey Sampling Kosuke Imai Department of Politics, Princeton University February 19, 2013 Survey sampling is one of the most commonly use ata collection methos for social scientists. We begin by escribing
More informationLATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More information7.1 Support Vector Machine
67577 Intro. to Machine Learning Fall semester, 006/7 Lecture 7: Support Vector Machines an Kernel Functions II Lecturer: Amnon Shashua Scribe: Amnon Shashua 7. Support Vector Machine We return now to
More informationJointly continuous distributions and the multivariate Normal
Jointly continuous istributions an the multivariate Normal Márton alázs an álint Tóth October 3, 04 This little write-up is part of important founations of probability that were left out of the unit Probability
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS
ANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS MICHAEL HOLST, EVELYN LUNASIN, AND GANTUMUR TSOGTGEREL ABSTRACT. We consier a general family of regularize Navier-Stokes an Magnetohyroynamics
More informationEquilibrium in Queues Under Unknown Service Times and Service Value
University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 1-2014 Equilibrium in Queues Uner Unknown Service Times an Service Value Laurens Debo Senthil K. Veeraraghavan University
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationSHARP BOUNDS FOR EXPECTATIONS OF SPACINGS FROM DECREASING DENSITY AND FAILURE RATE FAMILIES
APPLICATIONES MATHEMATICAE 3,4 (24), pp. 369 395 Katarzyna Danielak (Warszawa) Tomasz Rychlik (Toruń) SHARP BOUNDS FOR EXPECTATIONS OF SPACINGS FROM DECREASING DENSITY AND FAILURE RATE FAMILIES Abstract.
More informationSecond order differentiation formula on RCD(K, N) spaces
Secon orer ifferentiation formula on RCD(K, N) spaces Nicola Gigli Luca Tamanini February 8, 018 Abstract We prove the secon orer ifferentiation formula along geoesics in finite-imensional RCD(K, N) spaces.
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationTopic 7: Convergence of Random Variables
Topic 7: Convergence of Ranom Variables Course 003, 2016 Page 0 The Inference Problem So far, our starting point has been a given probability space (S, F, P). We now look at how to generate information
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More information1 Math 285 Homework Problem List for S2016
1 Math 85 Homework Problem List for S016 Note: solutions to Lawler Problems will appear after all of the Lecture Note Solutions. 1.1 Homework 1. Due Friay, April 8, 016 Look at from lecture note exercises:
More information1. Aufgabenblatt zur Vorlesung Probability Theory
24.10.17 1. Aufgabenblatt zur Vorlesung By (Ω, A, P ) we always enote the unerlying probability space, unless state otherwise. 1. Let r > 0, an efine f(x) = 1 [0, [ (x) exp( r x), x R. a) Show that p f
More informationThe Generalized Incompressible Navier-Stokes Equations in Besov Spaces
Dynamics of PDE, Vol1, No4, 381-400, 2004 The Generalize Incompressible Navier-Stokes Equations in Besov Spaces Jiahong Wu Communicate by Charles Li, receive July 21, 2004 Abstract This paper is concerne
More informationLeast Distortion of Fixed-Rate Vector Quantizers. High-Resolution Analysis of. Best Inertial Profile. Zador's Formula Z-1 Z-2
High-Resolution Analysis of Least Distortion of Fixe-Rate Vector Quantizers Begin with Bennett's Integral D 1 M 2/k Fin best inertial profile Zaor's Formula m(x) λ 2/k (x) f X(x) x Fin best point ensity
More informationWUCHEN LI AND STANLEY OSHER
CONSTRAINED DYNAMICAL OPTIMAL TRANSPORT AND ITS LAGRANGIAN FORMULATION WUCHEN LI AND STANLEY OSHER Abstract. We propose ynamical optimal transport (OT) problems constraine in a parameterize probability
More informationIterated Point-Line Configurations Grow Doubly-Exponentially
Iterate Point-Line Configurations Grow Doubly-Exponentially Joshua Cooper an Mark Walters July 9, 008 Abstract Begin with a set of four points in the real plane in general position. A to this collection
More informationPolynomial Inclusion Functions
Polynomial Inclusion Functions E. e Weert, E. van Kampen, Q. P. Chu, an J. A. Muler Delft University of Technology, Faculty of Aerospace Engineering, Control an Simulation Division E.eWeert@TUDelft.nl
More informationInterconnected Systems of Fliess Operators
Interconnecte Systems of Fliess Operators W. Steven Gray Yaqin Li Department of Electrical an Computer Engineering Ol Dominion University Norfolk, Virginia 23529 USA Abstract Given two analytic nonlinear
More informationTHE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE
MOSCOW MATHEMATICAL JOURNAL Volume 1, Number 1, January March 2001, Pages 27 47 THE DUCK AND THE DEVIL: CANARDS ON THE STAIRCASE J. GUCKENHEIMER AND YU. ILYASHENKO Abstract. Slow-fast systems on the two-torus
More informationHigh-Dimensional p-norms
High-Dimensional p-norms Gérar Biau an Davi M. Mason Abstract Let X = X 1,...,X be a R -value ranom vector with i.i.. components, an let X p = j=1 X j p 1/p be its p-norm, for p > 0. The impact of letting
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationTensors, Fields Pt. 1 and the Lie Bracket Pt. 1
Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 PHYS 500 - Southern Illinois University September 8, 2016 PHYS 500 - Southern Illinois University Tensors, Fiels Pt. 1 an the Lie Bracket Pt. 1 September 8,
More information3 The variational formulation of elliptic PDEs
Chapter 3 The variational formulation of elliptic PDEs We now begin the theoretical stuy of elliptic partial ifferential equations an bounary value problems. We will focus on one approach, which is calle
More informationSpectral properties of a near-periodic row-stochastic Leslie matrix
Linear Algebra an its Applications 409 2005) 66 86 wwwelseviercom/locate/laa Spectral properties of a near-perioic row-stochastic Leslie matrix Mei-Qin Chen a Xiezhang Li b a Department of Mathematics
More informationNonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity
Physica D 212 (2005) 195 204 www.elsevier.com/locate/phys Nonlinear Schröinger equation with a white-noise potential: Phase-space approach to sprea an singularity Albert C. Fannjiang Department of Mathematics,
More informationAn extension of Alexandrov s theorem on second derivatives of convex functions
Avances in Mathematics 228 (211 2258 2267 www.elsevier.com/locate/aim An extension of Alexanrov s theorem on secon erivatives of convex functions Joseph H.G. Fu 1 Department of Mathematics, University
More informationAsymptotic determination of edge-bandwidth of multidimensional grids and Hamming graphs
Asymptotic etermination of ege-banwith of multiimensional gris an Hamming graphs Reza Akhtar Tao Jiang Zevi Miller. Revise on May 7, 007 Abstract The ege-banwith B (G) of a graph G is the banwith of the
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More informationA FURTHER REFINEMENT OF MORDELL S BOUND ON EXPONENTIAL SUMS
A FURTHER REFINEMENT OF MORDELL S BOUND ON EXPONENTIAL SUMS TODD COCHRANE, JEREMY COFFELT, AND CHRISTOPHER PINNER 1. Introuction For a prime p, integer Laurent polynomial (1.1) f(x) = a 1 x k 1 + + a r
More informationarxiv:math/ v1 [math.pr] 19 Apr 2001
Conitional Expectation as Quantile Derivative arxiv:math/00490v math.pr 9 Apr 200 Dirk Tasche November 3, 2000 Abstract For a linear combination u j X j of ranom variables, we are intereste in the partial
More informationLEGENDRE TYPE FORMULA FOR PRIMES GENERATED BY QUADRATIC POLYNOMIALS
Ann. Sci. Math. Québec 33 (2009), no 2, 115 123 LEGENDRE TYPE FORMULA FOR PRIMES GENERATED BY QUADRATIC POLYNOMIALS TAKASHI AGOH Deicate to Paulo Ribenboim on the occasion of his 80th birthay. RÉSUMÉ.
More informationON ISENTROPIC APPROXIMATIONS FOR COMPRESSIBLE EULER EQUATIONS
ON ISENTROPIC APPROXIMATIONS FOR COMPRESSILE EULER EQUATIONS JUNXIONG JIA AND RONGHUA PAN Abstract. In this paper, we first generalize the classical results on Cauchy problem for positive symmetric quasilinear
More informationResistant Polynomials and Stronger Lower Bounds for Depth-Three Arithmetical Formulas
Resistant Polynomials an Stronger Lower Bouns for Depth-Three Arithmetical Formulas Maurice J. Jansen an Kenneth W.Regan University at Buffalo (SUNY) Abstract. We erive quaratic lower bouns on the -complexity
More informationMonotonicity of facet numbers of random convex hulls
Monotonicity of facet numbers of ranom convex hulls Gilles Bonnet, Julian Grote, Daniel Temesvari, Christoph Thäle, Nicola Turchi an Florian Wespi arxiv:173.31v1 [math.mg] 7 Mar 17 Abstract Let X 1,...,
More information