An upper bound on the number of high-dimensional permutations
|
|
- Brianna Freeman
- 5 years ago
- Views:
Transcription
1 An uppe bound on the numbe of high-dimensional pemutations Nathan Linial Zu Luia Abstact What is the highe-dimensional analog of a pemutation? If we think of a pemutation as given by a pemutation matix, then the following definition suggests itself: A d-dimensional pemutation of ode n is an n n n = [n] d+1 aay of zeos and ones in which evey line contains a unique 1 enty A line hee is a set of enties of the fom {(x 1,, x i 1, y, x i+1,, x d+1 ) n y 1} fo some index d + 1 i 1 and some choice of x j [n] fo all j i It is easy to obseve that a one-dimensional pemutation is simply a pemutation matix and that a two-dimensional pemutation is synonymous with an ode-n Latin squae We seek an estimate fo the numbe of d- dimensional pemutations Ou main esult is the following uppe bound on thei numbe ((1 + o(1)) n ) n d e d We tend to believe that this is actually the coect numbe, but the poblem of poving the complementay lowe bound emains open Ou main tool is an adaptation of Bègman s [1] poof of the Minc conjectue on pemanents Moe concetely, ou appoach is vey close in spiit to Schijve s [11] and Radhakishnan s [10] poofs of Bègman s theoem Depatment of Compute Science, Hebew Univesity, Jeusalem 91904, Isael nati@cshujiacil Suppoted by ISF and BSF gants Depatment of Compute Science, Hebew Univesity, Jeusalem 91904, Isael zluia@cshujiacil 1
2 1 Intoduction The pemanent of an n n matix A = (a ij ) is defined by P e(a) = σ S n n a i,σi Pemanents have attacted a lot of attention [9] They play an impotant ole in combinatoics Thus if A is a 0-1 matix, then P e(a) counts pefect matchings in the bipatite gaph whose adjacency matix is A They ae also of geat inteest fom the computational pespective It is #P -had to calculate the pemanent of a given 0-1 matix [12], and following a long line of eseach, an appoximation scheme was found [6] fo the pemanents of nonnegative matices Bounds on pemanents have also been studied at geat depth Van de Waeden conjectued that P e(a) n! fo evey n n n n doubly stochastic matix A, and this was established moe than fifty yeas late by Falikman and by Egoychev [4, 3] Moe ecently, Guvitz [5] discoveed a new conceptual poof fo this conjectue (see [8] fo a vey eadable pesentation) What is moe elevant fo us hee ae uppe bounds on pemanents These ae the subject of Minc s conjectue which was poved by Bègman Theoem 11 If A is an n n 0-1 matix with i ones in the i-th ow, then P e(a) n ( i!) 1/ i In the next section we eview Radhakishnan s poof, which uses the entopy method Ou plan is to imitate this poof fo a d-dimensional analogue of the pemanent To this end we need the notion of d-dimensional pemutations Definition 12 fom 1 Let A be an [n] d aay A line of A is vecto of the (A(i 1,, i j 1, t, i j+1,, i d )) n t=1, whee 1 j d and i 1,, i j 1, i j+1,, i d [n] 2
3 2 A d-dimensional pemutation (o d-pemutation) of ode n is an [n] d+1 aay P of zeos and ones such that evey line of P contains a single one and n 1 zeos Denote the set of all d-dimensional pemutations of ode n by S d,n Fo example, a two dimensional aay is a matix It has two kinds of lines, usually called ows and columns Thus a 1-pemutation is an n n 0-1 matix with a single one in each ow and a single one in each column, namely a pemutation matix A 2-pemutation is identical to a Latin squae and S 2,n is the same as the set L n, of ode-n Latin squaes We now explain the coespondence between the two sets If X is a 2-pemutation of ode n, then we associate with it a Latin squae L, whee L(i, j) as the (unique) index of a 1 enty in the line A(i, j, ) Fo moe on the subject of Latin squaes, see [13] The same definition yields a one-to-one coespondence between 3-dimensional pemutations and Latin cubes In geneal, d-dimensional pemutations ae synonymous with d-dimensional Latin hypecubes Fo moe on d-dimensional Latin hypecubes, see [14] To summaize, the following is an equivalent definition of a d-pemutation It is an [n] d aay with enties fom [n] in which evey line contains each i [n] exactly once We intechange feely between these two definitions accoding to context Ou main concen hee is to estimate S d,n, the numbe of d-pemutations of ode n By Stiling s fomula ( S 1,n = n! = (1 + o(1)) n ) n e As we saw, S 2,n is the numbe of ode n Latin squaes The best known estimate [13] is S 2,n = L n = ((1 + o(1)) n ) n 2 e 2 This elation is poved using bounds on pemanents Bégman s theoem fo the uppe bound, and the Falikman-Egoychev theoem fo the lowe bound This suggests Conjectue 13 S d,n = ((1 + o(1)) n ) n d e d In this pape we pove the uppe bound 3
4 Theoem 14 S d,n ((1 + o(1)) n ) n d e d As mentioned, ou method of poof is an adaptation of [10] We fist need Definition 15 M 2 if 1 An [n] d aay M 1 is said to suppot an aay M 2 (i 1,, i d+1 ) = 1 M 1 (i 1,, i d+1 ) = 1 2 The d-pemanent of a [n] d aay A is P e d (A) = The numbe of d-pemutations suppoted by A Note that in the one-dimensional case, this is indeed the usual definition of P e(a) It is not had to see that fo d = 1 the following theoem coincides with Bègman s theoem Theoem 16 Define the function f : N 0 N R ecusively by: f(0, ) = log(), whee the logaithm is in base e f(d, ) = 1 f(d 1, k) Let A be an [n] d aay with i1,i d ones in the line A(i 1,, i d, ) Then P e d (A) ) i 1,,i d e f(d,i1,id We will deive below faily tight bounds on the function f that appeas in theoem 16 It is then an easy matte to pove theoem 14 by applying theoem 16 to the all-ones aay What about poving a matching lowe bound on S d,n (and thus poving conjectue 13)? In ode to follow the footsteps of [13], we would need a lowe bound on P e d (A), namely, a highe-dimensional analog of the van de Waeden conjectue The enties of a multi-stochastic aay ae nonnegative eals and the sum of enties along evey line is 1 This is the highe-dimensional countepat of a doubly-stochastic matix It should be clea how to extend the notion of P e d (A) to eal-valued aays In this appoach we would need a lowe bound on P e d (A) that holds fo evey multi-stochastic aay A 4
5 Howeve, this attempt (o at least its most simplistic vesion) is bound to fail An easy consequence of Hall s theoem says that a 0-1 matix in which evey line o column contains the same (positive) numbe of 1-enties, has a positive pemanent (We still do not know exactly how small such a pemanent can be, see [8] fo moe on this) Howeve, the highe dimensional analog of this is simply incoect Thee exist multi-stochastic aays whose d-pemanent vanishes, as can easily be deduced eg, fom [7] We can, howeve, deive a lowe bound of S d,n exp(ω(n d )) fo even n Conside the following constuction: Let n be an even intege, and let P be a d-pemutation of ode [ ] n d 2 It is easy to see that such a P exists Simply set P (i 1,, i d ) = (i i d ) mod n 2 Now we constuct a d-pemutation Q of ode [n] d by eplacing each element of P with a [2] d block If P (i 1,, i d ) = j, then the coesponding block contains the values j and j + n It is easy to see that thee ae exactly 2 two ways to aange these values in each block, and that Q is indeed a d- pemutation of ode [n] d Thee ae ( ) n d 2 blocks, and so the numbe of possible Q s is 2 ( n 2 ) d Fo a constant d this is exp(ω(n d )) In section 2 we pesent Radhakishnan s poof of the Bègman bound In section 3 we pove theoem 16 In section 4 we use this bound to pove theoem 14 2 Radhakishnan s poof of Bègman s theoem 21 Entopy - Some basics We eview the basic mateial concening entopy that is used hee and efe the eade to [2] fo futhe infomation on the topic Definition 21 The entopy of a discete andom vaiable X is given by H(X) = ( ) 1 P(X = x) log P(X = x) x 5
6 Fo andom vaiables X and Y, the conditional entopy of X given Y is H(X Y ) = E[H(X Y = y)] = y P(Y = y)h(x Y = y) In this pape we will always conside the base e entopy of X which simply means that the logaithm is in base e Theoem 22 1 If X is a discete andom vaiable, then H(X) log ange(x), with equality iff X has a unifom distibution 2 If X 1,, X n is a sequence of andom vaiables, then H(X 1,, X n ) = H(X i X 1,, X i 1 ) 3 The inequality H(X Y ) H(X f(y )) holds fo evey two discete andom vaiables X and Y and evey eal function f( ) The following is a geneal appoach using entopy that is useful fo a vaiety of appoximate counting poblems Suppose that we need to estimate the cadinality of some set S If X is a andom vaiable which takes values in S unde the unifom distibution on S, then H(X) = log( S ) So, a good estimate on H(X) yields bounds on S This appoach is the main idea of both Radhakishnan s poof and ou wok 22 Radhakishnan s poof Let A be an n n 0-1 matix with i ones in the i-th ow Ou aim is to pove the uppe bound n P e(a) ( i!) 1 i 6
7 Let M be the set of pemutation matices suppoted by A, and let X be a unifomly sampled andom element of M Ou plan is to evaluate H(X) using the chain ule and estimate M using the fact (theoem 22) that H(X) = log( M ) Let X i be the unique index j such that X(i, j) = 1 We conside a pocess whee we scan the ows of X in sequence and estimate H(X) = H(X 1,, X n ) using the chain ule in the coesponding ode To cay out this plan, we need to bound the contibution of the tem involving X i conditioned on the peviously obseved ows That is, we wite H(X) = H(X i X 1,, X i 1 ) Let Let R i be the set of indices of the 1-enties in A s i-th ow That is, R i = {j : A(i, j) = 1} Z i = {j R i : X i = j fo some i < i} Note that X i R i, because X is suppoted by A In addition, given that we have aleady exposed the values X i fo i < i, it is impossible fo X i to take any value j Z i, o else the column X(, j) contains moe than a single 1-enty Theefoe, given the vaiables that pecede it, X i must take a value in R i Z i The cadinality N i = R i Z i is a function of X 1,, X i 1 and so by theoem 22, = H(X) = H(X i X 1,, X i 1 ) P(X 1 = x 1,, X i 1 = x i 1 )H(X i X 1 = x 1,, X i 1 = x i 1 ) x 1,,x i 1 P(X 1 = x 1,, X i 1 = x i 1 ) log(n i ) x 1,,x i 1 = E X1,,X i 1 [log(n i )] = E X [log(n i )] 7
8 It is not clea how we should poceed fom hee, fo how can we bound log(n i ) fo a geneal matix? Moeove, diffeent odeings of the ows will give diffeent bounds We use this fact to ou advantage and conside the expectation of this bound ove all possible odeings Associated with a pemutation σ S n is an odeing of the ows whee X j is evealed befoe X i if σ(j) < σ(i) We edefine Z i and N i to take the odeing σ into account Let Z i (σ) = {j R i : X i = j fo some σ(i ) < σ(i)} N i (σ) = R i Z i (σ) Then N i (σ) is the numbe of available values fo X i, given all the vaiables X j fo j such that σ(j) < σ(i) As befoe, using the chain ule we obtain the inequality H(X) = H(X i X j : σ(j) < σ(i)) E X [log(n i (σ))] The inequality emains tue if we take the expected value of both sides when σ is a andom pemutation sampled fom the unifom distibution on S n H(X) E σ [E X [log(n i (σ))]] = E X [E σ [log(n i (σ))]] Thus, the bound we get on H(X) depends on the distibution of the andom vaiable N i (σ) The final obsevation that we need is that the distibution of N i (σ) is vey simple and that it does not depend on X Consequently we can eliminate the step of taking expectation with espect to the choice of X Let us fix a specific X Let W i denote the set of i 1 ow indices j i fo which X j R i Note that N i is equal to i minus the numbe of indices in W i that pecede i in the andom odeing σ Since σ was chosen unifomly, this numbe is distibuted unifomly in {0,, i 1} Thus, N i is unifom on the set {1,, i } Theefoe Hence E σ [log(n i (σ))] = H(X) i [ ] 1 E X log( i!) = i 1 i log(k) = 1 i log( i!) 8 1 i log( i!)
9 which implies the Bègman bound 3 The d-dimensional case 31 An infomal discussion The coe of the above-descibed poof of the Bègman bound can be viewed as follows Let us pick fist a 1-pemutation X that is contained in the matix A and conside the set R i of the i 1-enties in A s i-th ow Thee ae exactly i indices j fo which X j R i The andom odeing of the ows detemines which of these will pecede the i-th ow (o will cast its shadow on the i-th ow) The andom numbe u i of ows that cast a shadow on the i-th ow is unifomly distibuted in the ange {0,, i 1} The contibution of this ow to the uppe bound on H(X) is E σ [log N i ], whee N i = i u i is the numbe of 1-enties in the i-th ow that ae still unshaded The expectation of log N i is exactly 1 i i j=1 log j = 1 i log( i!) How should we modify this agument to deal with d-dimensional pemutations? We fix a d-pemutation X that is contained in A and conside a andom odeing of all lines of the fom A(i 1,, i d, ) Given such an odeing, we use the chain ule to deive an uppe bound on H(X) Each odeing yields a diffeent bound Howeve, as in the one dimensional case, the key insight is that aveaging ove all possible odeings (in a class that we late define) gives us a simple bound on H(X) The oveall stuctue of the agument emains the same We conside a concete line A(i 1,, i d, ) Its contibution to the estimate of the entopy is log N whee N is the numbe of 1-enties that emain unshaded at the time (accoding to the chosen odeing) at which we compute the coesponding tem in the chain ule fo the entopy Howeve, now shade can fall fom d diffeent diections The contibution of the line to the entopy will be the expected logaithm of the numbe of ones that emain unshaded afte each of the d dimensions has cast its shade on it The lines ae odeed by a andom lexicogaphic odeing At the coasest level lines ae odeed accoding to thei fist coodinate i 1 This odeing is chosen unifomly fom S n To undestand how many 1 s emain unshaded in a given line, we fist conside the shade along the fist coodinate If it initially has 1-enties, then the numbe of unshaded 1-enties afte this stage is unifomly distibuted on [] We then ecuse with the emaining 9
10 1-enties and poceed on the subcube of codimension 1 that is defined by the value of the fist coodinate It is not had to see how the ecusive expession fo f(d, ) eflects this calculation 32 In detail Let A be a [n] d+1 -dimensional aay of zeos and ones, and X is a andom d-pemutation sampled unifomly fom the set of d-pemutations contained in A Then H(X) = log(p e d (A)) by theoem 22 and again we seek an uppe bound on H(X) We think of X as an [n] d aay each line of which contains each membe of [n] exactly once The poof does its accounting using lines of the fom A(i 1,, i d, ), ie, lines in which the (d + 1)-st coodinate vaies Such a line is specified by i = (i 1,, i d ) The andom vaiable X i is defined to be the value of X(i 1,, i d ) We think of the vaiables X i as being evealed to us one by one Thus, X i1,,i d must belong to R i = R i1,,i d = {j : A(i 1,, i d, j) = 1} the set of 1-enties in this line In the poof we scan these lines in a paticula andomly chosen ode Let us ignoe this issue fo a moment and conside some fixed odeing of these lines Initially, the numbe of 1-enties in this line is i As we poceed, some of these 1 s become unavailable to X i, since choosing them would esult in a conflict with the choice made in some peviously evealed line We say that these 1 s ae in the shade of peviously consideed lines This shade can come fom any of the d possible diections Thus we denote by Z i R i the set of the indices of the 1-enties in R i that ae unavailable to X i given the values of the peceding vaiables We can expess Z i = d Zk i whee enties in Zi k ae shaded fom diection k Namely, a membe j of R i belongs to Zi k if thee is an aleady scanned line indexed by i with X i = j and whee i and i coincide on all coodinates except the k-th Thus, given the values of the peviously consideed vaiables, thee ae at most N i = R i Z i values that ae available to X i We next tun to the andom odeing of the lines Now, howeve, we do not select a completely andom odeing, but opt fo a andom lexicogaphic 10
11 odeing Namely, we select d andom pemutations σ 1,, σ d S n The line A(i 1,, i d, ) pecedes A(i 1,, i d, ) if thee is a k [n] such that σ k(i k ) < σ k (i k ) and i j = i j fo all j < k Thus a choice of the odeings σ k induces a total ode on the lines A(i 1,, i d, ) Denote this ode by That is, we wite i j if i comes befoe j We wite i k j if i j and i and j diffe only in the k-th coodinate We think of X i as being evealed to us accoding to this ode We tun to the definition of R i, Zi k and N i Thei definitions ae affected by the chosen odeing of the lines In addition, fo easons to be made clea late, we genealize the definition of N i It is defined as the numbe of values available to X i (given the peceding lines) fom a given index set W R i In the discussion below, we fix X, a d-pemutation that is contained in A Definition 31 The index set of the 1-enties in the line A(i 1,, i d, ) is denoted by R i = R i1,,i d = {j : A(i 1,, i d, j) = 1}, and its cadinality is i = R i Let W R i with i = (i 1,, i d ), and suppose that X i W Fo a given odeing, let Z k i (X, ) = {j R i : X i = j fo some i k i} N i (W, X, ) = W d Z k i (X, ) Thus, N i is a function of W R i, X and the odeing Each vaiable X i specifies a 1 enty of the line A(i 1,, i d, ) The enty thus specified must confom to the values taken by the peceding vaiables Namely, no line of X can contain moe than a single 1 enty We conside the numbe of values that the vaiable X i can take, given the values that pecede it Fix an index tuple i = (i 1,, i d ) The vaiable X i must specify an index i d+1 with A(i 1,, i d+1 ) = 1, ie, an element of R i Conside some element j R i If X i = j, fo some i k i and k d then clealy X i j, o else the line X(i 1,, i k 1,, i k+1,, i d ) contains moe than a single j-enty In othe wods, X i cannot specify an element of Zi k (X, ) and is esticted to the set R i d Zk i (X, ) Theefoe, thee ae at most N i(r i, X, ) possible values that X i can take given the vaiables that pecede it in the ode 11
12 Fo a given ode, we can use the chain ule to deive H(X) = i H(X i X j : j i) By theoem 22, H(X i X j : j i) = E Xj :j i [H(X i X j = x j : j i)] E Xj :j i [log(n i (R i, X, ))] = E X [log(n i (R i, X, ))] The last equality holds because N i depends only on the lines of X that pecede X i, and so taking the expectation ove the est of X doesn t change anything As in the one dimensional case, the next step is to take the expectation of both sides of the above inequality ove H(X) i E [E X [log(n i (R i, X, ))]] = i E X [E [log(n i (R i, X, ))]] The key to unaveling this expession is the insight that the andom vaiable N i has a simple distibution (as a function of ), and moeove, that this distibution does not depend on X Recall that in the one dimensional case, we obtained the distibution of N i as follows Initially, the numbe of ones in the i-th ow was i Then the ows peceding the i-th ow wee evealed, and some of the ones in the i-th ow became unavailable to X, because some othe ow had placed a one in thei column We defined N i = R i Z i (σ) The size of Z i (σ) was shown to be unifomly distibuted ove {0,, i 1}, and thus the distibution of N i was shown to be unifom ove {1,, i } A simila agument woks in the d dimensional case, but the distibution of N i is no longe unifom Recall that the function f is defined ecusively by f(0, ) = log() f(d, ) = 1 f(d 1, k) 12
13 Claim 32 Let X be a d-pemutation, i = (i 1,, i d ) and let W R i be an index set such that X i W Then E [log(n i (W, X, ))] depends only on d and = W, and E [log(n i (W, X, ))] = f(d, ) Poof The poof poceeds by induction on d Fist, note that if W = and d = 0, then N i (W, X, ) = W = by definition, and theefoe E [log(n i (W, X, ))] = log() = f(0, ) In ode to poceed with the induction step, we must descibe N i (W, X, ) in tems of paametes of dimension d 1 instead of d To this end we need the following definitions: X = X(i 1,,, ) Note that X is a (d 1)-dimensional pemutation W = W Z 1 i (X, ) Note that W actually depends only on σ 1, the odeing of the fist coodinate Let i = (i 1,, i d 1 ) = (i 2,, i d ) Given an odeing, let be the odeing on the index tuples (i 1,, i d 1 ) defined by the odeings σ 2, σ 3,, σ d Note that fo evey X,W, i and we have N i (W, X, ) = N i (W, X, ) This equality follows diectly fom the definition of N Now, E [log(n i (W, X, ))] = E σ1 [E [log(n i (W, X, ))]] = E σ1 [E [log(n i (W, X, ))]] = E σ1 [f(d 1, W )] The last step follows fom the induction hypothesis Consequently, E [log(n i (W, X, ))] = k P( W = k)f(d 1, k) The only emaining question is to detemine the distibution of W as a function of σ 1 Note, howeve, that we have aleady answeed this question in the one dimensional poof, namely, W is unifomly distibuted on {1,, } Indeed, W = W Zi 1(X, ), and Z1 i (X, ) is the set of indices s such that: 13
14 Fo some j W, X(s, i 2,, i d ) = j (thee ae 1 such indices, one fo each j W ) The andom odeing σ 1 places s befoe i 1 In a andom odeing, the position of i 1 is unifomly distibuted Theefoe Zi 1(X, ) is unifomly distibuted on {0,, 1}, and P( W = k) = 1 fo evey 1 k Putting this togethe, we have shown that E [log(n i (W, X, ))] = 1 f(d 1, k) = f(d, ) In conclusion, we have shown that H(X) i E X [E [log(n i (R i, X, ))]] = i E X [f(d, i )] = i f(d, i ), is the numbe of ones in the vecto A(i 1,, i d, ) Thee- whee i = i1,,i d foe, P e d (A) i e f(d, i) 4 The numbe of d-pemutations An uppe bound As mentioned, the uppe bound on the numbe of d-dimensional pemutations is deived by applying theoem 16 to the all-ones aay J The main technical step is a deivation of an uppe bound on the function f(d, ) Theoem 41 Fo evey d thee exist constants c d and d such that fo all d, log d () f(d, ) log() d + c d One possible choice that we adopt hee is d = e d fo evey d, c 1 = 5, c 2 = 8, and c d = d3 (11) d fo d 3 d! 14
15 Poof A staightfowad induction on d yields the weake bound f(d, ) log() fo all d, Fo d = 0 thee is equality and the geneal case follows since f(d, ) = 1 f(d 1, k) 1 log(k) log() This simple bound seves us to deal with the ange of small s (below d 1 ) We tun to the main pat of the poof f(d, ) = 1 f(d 1, k) = 1 d 1 f(d 1, k) + k= d 1 +1 f(d 1, k) 1 log d 1 (k) d 1 log( d 1 ) + log(k) (d 1) + c d 1 k k= d 1 [ ] 1 log d 1 (k) d 1 log( d 1 ) + d 1 (d 1) + log(k) (d 1) + c d 1 k ξ + 1 log(!) (d 1) + c d 1 log d 1 (k) k whee ξ = d 1 log( d 1 )+ d 1 (d 1) = 2(d 1)e d 1 It is easily veified that fo d 3 thee holds log(!) log() + 2 log() We can poceed with ξ + log() + 2 log() d + c d 1 log d 1 (k) k log d 1 (x)dx x = We now bound the sum log d 1 (k) by means of the integal k 1 Note that the integand is unimodal and its maximal value is γ = d log d () ( d 1 e ) d 1 Thus, c d 1 log d 1 (k) k c d 1 ( log d () d ) + γ Putting this togethe, we have the inequality ( ) 2 log() + ξ + c d 1 γ + logd () d f(d, ) log() d + 15
16 Theefoe it is sufficient to choose c d such that fo evey e d ( ) 2 log() + ξ + c d 1 γ + logd () c d log d () d ie, ( 2 log d 1 () + ξ γ log d () + c d 1 log d () + 1 ) c d d The left hand side of the above inequality is clealy a deceasing function of Theefoe it is sufficient to veify the inequality fo = e d Plugging this and the values of the constants ξ and γ into the left hand side of the above inequality, we get Thus, we may take 2 2(d 1)ed 1 + dd 1 d d ( e d 1 d c d = ) cd 1 ( (d 1) d 1 + c d ) e d 1 d d d ( 2 ( e ) ) d + d d + d d ( ) ( cd 1 2 ( e ) ) d e d 1 d + d d + d d Calculating c d using this ecusion and the fact that c 0 = 0, we get that c 1 = 2 + e 5, c 2 8, and c d d3 (11) d fo 3 d 10 Poceeding by d! induction, ( c d = ) ( (d 1) 3 (11) d 1 2 ( e ) ) d + d e d 1 d! d + d d (11)d (d 1) 3 ( e d (11) + 2d d! d) d (d 1) 3 + 2d 2 (11)d d 3 d! d! In the inequality befoe the last one, we used the fact that fo d 10, ( e ) d d d d! Fo the [n] d+1 all ones aay J, i1,,i d = n fo evey tuple (i 1,, i d ), and so fo lage enough n we have the bound P e d (J) i 1,,i d e f(d,n) = ( e f(d,n) ) nd 16 ( [ log d ]) n d (n) exp log(n) d + c d n
17 log Fo a constant d, letting n go to infinity, c d (n) d = o(1) and theefoe the n numbe of d-pemutations is at most ((1 + o(1)) n ) n d e d Refeences [1] L M Bègman, Cetain popeties of nonnegative matices and thei pemanents, Dokl Akad Nauk SSSR 211 (1973), MR MR (48 #6130) [2] T M Cove and J A Thomas, Elements of Infomation Theoy, Wiley, New Yok, 1991 [3] GP Egoichev, Poof of the van de Waeden conjectue fo pemanents, Sibeian Math J 22 (1981), [4] DI Falikman, A poof of the van de Waeden conjectue egading the pemanent of a doubly stochastic matix, Math Notes Acad Sci USSR 29 (1981), [5] L Guvits, Van de Waeden/SchijveValiant like conjectues and stable (aka hypebolic) homogeneous polynomials: one theoem fo all With a coigendum, Electon J Combin 15 (2008), R66 (26 pp) [6] M Jeum, A Sinclai, and E Vigoda, A polynomial-time appoximation algoithm fo the pemanent of a matix with nonnegative enties, J ACM, [7] M Kochol, Relatively naow Latin paallelepipeds that cannot be extended to a Latin cube, As Combin 40 (1995), [8] Lauent, M and Schijve, A, On Leonid Guvitss poof fo pemanents, The Ameican Mathematical Monthly 10 (2010), [9] H Minc, Pemanents, Encyclopedia of Mathematics and Its Applications Vol 6, Addison-Wesley, Reading, Mass, 1978 [10] Jaikuma Radhakishnan, An entopy poof of Begman s theoem, J Combinatoial Theoy Se A 77 (1997), no 1, MR MR (97m:15006) 17
18 [11] Schijve, A, A shot poof of Minc s conjectue, J Comb Theoy Se A 25 (1978), [12] L G Valiant, The complexity of computing the pemanent, Theoet Comput Sci 8 (1979), [13] J H van Lint and R M Wilson, A Couse in Combinatoics, Cambidge UP, 1992 [14] BD McKay and IM Wanless, A census of small latin hypecubes, SIAM J Discete Math 22 (2008), pp
arxiv: v1 [math.co] 6 Mar 2008
An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More informationA Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
More informationNew problems in universal algebraic geometry illustrated by boolean equations
New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationStanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012
Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,
More informationOn a quantity that is analogous to potential and a theorem that relates to it
Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich
More informationarxiv: v1 [math.co] 1 Apr 2011
Weight enumeation of codes fom finite spaces Relinde Juius Octobe 23, 2018 axiv:1104.0172v1 [math.co] 1 Ap 2011 Abstact We study the genealized and extended weight enumeato of the - ay Simplex code and
More informationarxiv: v1 [math.co] 4 May 2017
On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has
More informationKOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS
Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Septembe 5, 011 Abstact To study how balanced o unbalanced a maximal intesecting
More informationFall 2014 Randomized Algorithms Oct 8, Lecture 3
Fall 204 Randomized Algoithms Oct 8, 204 Lectue 3 Pof. Fiedich Eisenband Scibes: Floian Tamè In this lectue we will be concened with linea pogamming, in paticula Clakson s Las Vegas algoithm []. The main
More informationMATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form
MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and
More informationThe Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr
The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi
Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated
More informationQuasi-Randomness and the Distribution of Copies of a Fixed Graph
Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationInformation Retrieval Advanced IR models. Luca Bondi
Advanced IR models Luca Bondi Advanced IR models 2 (LSI) Pobabilistic Latent Semantic Analysis (plsa) Vecto Space Model 3 Stating point: Vecto Space Model Documents and queies epesented as vectos in the
More informationJournal of Inequalities in Pure and Applied Mathematics
Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics
More informationOn the integration of the equations of hydrodynamics
Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious
More informationA Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction
A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić
More informationMultiple Criteria Secretary Problem: A New Approach
J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and
More informationResearch Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function
Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming
More informationOn the ratio of maximum and minimum degree in maximal intersecting families
On the atio of maximum and minimum degee in maximal intesecting families Zoltán Lóánt Nagy Lale Özkahya Balázs Patkós Máté Vize Mach 6, 013 Abstact To study how balanced o unbalanced a maximal intesecting
More informationRelating Branching Program Size and. Formula Size over the Full Binary Basis. FB Informatik, LS II, Univ. Dortmund, Dortmund, Germany
Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de
More informationDuality between Statical and Kinematical Engineering Systems
Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland.
More informationOn the Quasi-inverse of a Non-square Matrix: An Infinite Solution
Applied Mathematical Sciences, Vol 11, 2017, no 27, 1337-1351 HIKARI Ltd, wwwm-hikaicom https://doiog/1012988/ams20177273 On the Quasi-invese of a Non-squae Matix: An Infinite Solution Ruben D Codeo J
More informationA Multivariate Normal Law for Turing s Formulae
A Multivaiate Nomal Law fo Tuing s Fomulae Zhiyi Zhang Depatment of Mathematics and Statistics Univesity of Noth Caolina at Chalotte Chalotte, NC 28223 Abstact This pape establishes a sufficient condition
More informationLecture 28: Convergence of Random Variables and Related Theorems
EE50: Pobability Foundations fo Electical Enginees July-Novembe 205 Lectue 28: Convegence of Random Vaiables and Related Theoems Lectue:. Kishna Jagannathan Scibe: Gopal, Sudhasan, Ajay, Swamy, Kolla An
More informationSurveillance Points in High Dimensional Spaces
Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationFractional Zero Forcing via Three-color Forcing Games
Factional Zeo Focing via Thee-colo Focing Games Leslie Hogben Kevin F. Palmowski David E. Robeson Michael Young May 13, 2015 Abstact An -fold analogue of the positive semidefinite zeo focing pocess that
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More information6 Matrix Concentration Bounds
6 Matix Concentation Bounds Concentation bounds ae inequalities that bound pobabilities of deviations by a andom vaiable fom some value, often its mean. Infomally, they show the pobability that a andom
More informationA solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane
A solution to a poblem of Günbaum and Motzkin and of Edős and Pudy about bichomatic configuations of points in the plane Rom Pinchasi July 29, 2012 Abstact Let P be a set of n blue points in the plane,
More informationPearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms
Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two
More informationOn decompositions of complete multipartite graphs into the union of two even cycles
On decompositions of complete multipatite gaphs into the union of two even cycles A. Su, J. Buchanan, R. C. Bunge, S. I. El-Zanati, E. Pelttai, G. Rasmuson, E. Spaks, S. Tagais Depatment of Mathematics
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationCentral Coverage Bayes Prediction Intervals for the Generalized Pareto Distribution
Statistics Reseach Lettes Vol. Iss., Novembe Cental Coveage Bayes Pediction Intevals fo the Genealized Paeto Distibution Gyan Pakash Depatment of Community Medicine S. N. Medical College, Aga, U. P., India
More informationSemicanonical basis generators of the cluster algebra of type A (1)
Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:
More informationTHE NUMBER OF TWO CONSECUTIVE SUCCESSES IN A HOPPE-PÓLYA URN
TH NUMBR OF TWO CONSCUTIV SUCCSSS IN A HOPP-PÓLYA URN LARS HOLST Depatment of Mathematics, Royal Institute of Technology S 100 44 Stocholm, Sweden -mail: lholst@math.th.se Novembe 27, 2007 Abstact In a
More informationFailure Probability of 2-within-Consecutive-(2, 2)-out-of-(n, m): F System for Special Values of m
Jounal of Mathematics and Statistics 5 (): 0-4, 009 ISSN 549-3644 009 Science Publications Failue Pobability of -within-consecutive-(, )-out-of-(n, m): F System fo Special Values of m E.M.E.. Sayed Depatment
More informationCOLLAPSING WALLS THEOREM
COLLAPSING WALLS THEOREM IGOR PAK AND ROM PINCHASI Abstact. Let P R 3 be a pyamid with the base a convex polygon Q. We show that when othe faces ae collapsed (otated aound the edges onto the plane spanned
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More informationThe Chromatic Villainy of Complete Multipartite Graphs
Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at:
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More informationA generalization of the Bernstein polynomials
A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This
More informationNOTE. Some New Bounds for Cover-Free Families
Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment
More informationEQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS
EQUI-PARTITIONING OF HIGHER-DIMENSIONAL HYPER-RECTANGULAR GRID GRAPHS ATHULA GUNAWARDENA AND ROBERT R MEYER Abstact A d-dimensional gid gaph G is the gaph on a finite subset in the intege lattice Z d in
More informationA STUDY OF HAMMING CODES AS ERROR CORRECTING CODES
AGU Intenational Jounal of Science and Technology A STUDY OF HAMMING CODES AS ERROR CORRECTING CODES Ritu Ahuja Depatment of Mathematics Khalsa College fo Women, Civil Lines, Ludhiana-141001, Punjab, (India)
More informationEnumerating permutation polynomials
Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem
More informationUsing Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu
Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of
More informationCompactly Supported Radial Basis Functions
Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically
More informationMEASURING CHINESE RISK AVERSION
MEASURING CHINESE RISK AVERSION --Based on Insuance Data Li Diao (Cental Univesity of Finance and Economics) Hua Chen (Cental Univesity of Finance and Economics) Jingzhen Liu (Cental Univesity of Finance
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationConstruction and Analysis of Boolean Functions of 2t + 1 Variables with Maximum Algebraic Immunity
Constuction and Analysis of Boolean Functions of 2t + 1 Vaiables with Maximum Algebaic Immunity Na Li and Wen-Feng Qi Depatment of Applied Mathematics, Zhengzhou Infomation Engineeing Univesity, Zhengzhou,
More informationConservative Averaging Method and its Application for One Heat Conduction Problem
Poceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER THERMAL ENGINEERING and ENVIRONMENT Elounda Geece August - 6 (pp6-) Consevative Aveaging Method and its Application fo One Heat Conduction Poblem
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationq i i=1 p i ln p i Another measure, which proves a useful benchmark in our analysis, is the chi squared divergence of p, q, which is defined by
CSISZÁR f DIVERGENCE, OSTROWSKI S INEQUALITY AND MUTUAL INFORMATION S. S. DRAGOMIR, V. GLUŠČEVIĆ, AND C. E. M. PEARCE Abstact. The Ostowski integal inequality fo an absolutely continuous function is used
More informationFunctions Defined on Fuzzy Real Numbers According to Zadeh s Extension
Intenational Mathematical Foum, 3, 2008, no. 16, 763-776 Functions Defined on Fuzzy Real Numbes Accoding to Zadeh s Extension Oma A. AbuAaqob, Nabil T. Shawagfeh and Oma A. AbuGhneim 1 Mathematics Depatment,
More informationONE-POINT CODES USING PLACES OF HIGHER DEGREE
ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU
More informationDeterministic vs Non-deterministic Graph Property Testing
Deteministic vs Non-deteministic Gaph Popety Testing Lio Gishboline Asaf Shapia Abstact A gaph popety P is said to be testable if one can check whethe a gaph is close o fa fom satisfying P using few andom
More informationUnobserved Correlation in Ascending Auctions: Example And Extensions
Unobseved Coelation in Ascending Auctions: Example And Extensions Daniel Quint Univesity of Wisconsin Novembe 2009 Intoduction In pivate-value ascending auctions, the winning bidde s willingness to pay
More informationSyntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)
Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationMULTINOMIAL PROBABILITIES, PERMANENTS AND A CONJECTURE OF KARLIN AND RINOTT R. B. BAPAT
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Numbe 3, Mach 1988 MULTINOMIAL PROBABILITIES, PERMANENTS AND A CONJECTURE OF KARLIN AND RINOTT (Communicated R. B. BAPAT by Bet E. Fistedt)
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationOn the Number of Rim Hook Tableaux. Sergey Fomin* and. Nathan Lulov. Department of Mathematics. Harvard University
Zapiski Nauchn. Seminaov POMI, to appea On the Numbe of Rim Hook Tableaux Segey Fomin* Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239 Theoy of Algoithms Laboatoy SPIIRAN,
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationSolving Some Definite Integrals Using Parseval s Theorem
Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s
More information5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS
5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationLikelihood vs. Information in Aligning Biopolymer Sequences. UCSD Technical Report CS Timothy L. Bailey
Likelihood vs. Infomation in Aligning Biopolyme Sequences UCSD Technical Repot CS93-318 Timothy L. Bailey Depatment of Compute Science and Engineeing Univesity of Califonia, San Diego 1 Febuay, 1993 ABSTRACT:
More informationA NEW VARIABLE STIFFNESS SPRING USING A PRESTRESSED MECHANISM
Poceedings of the ASME 2010 Intenational Design Engineeing Technical Confeences & Computes and Infomation in Engineeing Confeence IDETC/CIE 2010 August 15-18, 2010, Monteal, Quebec, Canada DETC2010-28496
More informationLecture 18: Graph Isomorphisms
INFR11102: Computational Complexity 22/11/2018 Lectue: Heng Guo Lectue 18: Gaph Isomophisms 1 An Athu-Melin potocol fo GNI Last time we gave a simple inteactive potocol fo GNI with pivate coins. We will
More informationOn the Poisson Approximation to the Negative Hypergeometric Distribution
BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution
More informationChapter 3: Theory of Modular Arithmetic 38
Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences
More informationParameter identification in Markov chain choice models
Poceedings of Machine Leaning Reseach 76:1 11, 2017 Algoithmic Leaning Theoy 2017 Paamete identification in Makov chain choice models Aushi Gupta Daniel Hsu Compute Science Depatment Columbia Univesity
More informationNumerical approximation to ζ(2n+1)
Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationExceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs
(May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett@math.umn.edu http://www.math.umn.edu/ gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving
More informationOSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
More informationTHE INFLUENCE OF THE MAGNETIC NON-LINEARITY ON THE MAGNETOSTATIC SHIELDS DESIGN
THE INFLUENCE OF THE MAGNETIC NON-LINEARITY ON THE MAGNETOSTATIC SHIELDS DESIGN LIVIU NEAMŢ 1, ALINA NEAMŢ, MIRCEA HORGOŞ 1 Key wods: Magnetostatic shields, Magnetic non-lineaity, Finite element method.
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationand the initial value R 0 = 0, 0 = fall equivalence classes ae singletons fig; i = 1; : : : ; ng: (3) Since the tansition pobability p := P (R = j R?1
A CLASSIFICATION OF COALESCENT PROCESSES FOR HAPLOID ECHANGE- ABLE POPULATION MODELS Matin Mohle, Johannes Gutenbeg-Univesitat, Mainz and Seik Sagitov 1, Chalmes and Gotebogs Univesities, Gotebog Abstact
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationApplication of Parseval s Theorem on Evaluating Some Definite Integrals
Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at http://pubs.sciepub.com/tjant/// Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating
More informationAn intersection theorem for four sets
An intesection theoem fo fou sets Dhuv Mubayi Novembe 22, 2006 Abstact Fix integes n, 4 and let F denote a family of -sets of an n-element set Suppose that fo evey fou distinct A, B, C, D F with A B C
More informationGROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS
Annales Academiæ Scientiaum Fennicæ Mathematica Volumen 32, 2007, 595 599 GROWTH ESTIMATES THROUGH SCALING FOR QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS Teo Kilpeläinen, Henik Shahgholian and Xiao Zhong
More informationAppendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More informationA THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM
A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More information