Nodal domains on graphs - How to count them and why?

Size: px
Start display at page:

Download "Nodal domains on graphs - How to count them and why?"

Transcription

1 Proeedings of Symposia in Pure Mathematis Nodal domains on graphs - How to ount them and why? Ram Band, Idan Oren and Uzy Smilansky, Abstrat. The purpose of the present manusript is to ollet known results and present some new ones relating to nodal domains on graphs, with speial emphasize on nodal ounts. Several methods for ounting nodal domains will be presented, and their relevane as a tool in spetral analysis will be disussed. 1. Introdution Spetral graph theory deals with the spetrum and eigenfuntions of the Laplae operator defined on graphs. The study of the eigenfuntions, and, in partiular, their nodal domains is an exiting and rapidly developing researh diretion. It is an extension to graphs of the investigations of nodal domains on manifolds, whih started already in the 19th entury by the pioneering work of Chladni on the nodal strutures of vibrating plates. Counting nodal domains started with Sturm s osillations theorem whih states that a vibrating string is divided into exatly n nodal intervals by the zeros of its n th vibrational mode. In an attempt to generalize Sturm s theorem to manifolds in more than one dimension, Courant formulated his nodal domains theorem for vibrating membranes [1], whih bounds the number of nodal domains of the n th eigenfuntion by n. Pleijel [] have shown later that Courant s bound an be realized only for a rare subsequene of eigenfuntions. The study of nodal domains ounts was revived after Blum et al have shown that nodal ount statistis an be used as a riterion for quantum haos [3]. A subsequent paper by Bogomolny and Shmit [4] illuminated another fasinating onnetion between nodal statistis and perolation theory. A reent paper by Nazarov and Sodin addresses the ounting of nodal domains of eigenfuntions of the Laplaian on S [5]. They prove that on average, the number of nodal domains inreases linearly with n, and the variane about the mean is bounded. At the same time, it was shown that the nodal sequene - the sequene of numbers of nodal domains ordered by the orresponding spetral parameters - stores geometrial information about the domain [6]. Moreover, there is a growing body of numerial and theoretial evidene whih shows that the nodal sequene an be used to distinguish between isospetral manifolds. [7, 8]. In the present paper we shall fous on the study of nodal domains on graphs, and show to what extent it goes hand in hand or omplements the orresponding results obtained for Laplaians on manifolds (opyright holder)

2 RAM BAND, IDAN OREN AND UZY SMILANSKY, The paper is designed as follows: The next hapter summarizes some elementary definitions and bakground material neessary to keep this paper self ontained. Next, we survey the known results regarding ounting nodal domains on graphs. After these preliminaries, we present a few ounting methods of nodal domains on graphs. Finally, the intimate onnetion between nodal sequenes and isospetrality on graphs will be reviewed, and some open problems will be formulated.. Definitions, notations and bakground A graph G = (V, B) is a set of verties V = {1,, V } of size V V and a set of undireted bonds (edges) B of size B B, suh that (i,j) B if the verties i and j are onneted by a bond. In this ase we say that verties i and j are adjaent and denote this by i j. The degree (valeny) of a vertex is the number of bonds whih are onneted to it. A graph is alled v-regular if all its verties are of degree v. Throughout the artile, and unless otherwise stated, we deal with onneted graphs with no multiple bonds or loops (a bond whih onnets a vertex to itself). A well known fat in graph theory is that the number of independent yles in a graph, denoted by r is equal to: (1) r = B V + Co where Co is the number of onneted omponents in G. We note that r is also the rank of the fundamental group of the graph. A tree is a graph for whih r = 0. Let g be a subgraph of G. We define the interior of g as the set of verties whose neighbors are also in g. The boundary of g is the set of verties in g whih are not in its interior. A graph G is said to be properly olored if eah vertex is olored so that adjaent verties have different olors. G is k-olorable if it an be properly olored by k olors. The hromati number χ(g) is k if G is k-olorable and not (k-1)-olorable. A very simple observation, whih we will use later, is that χ(g) V. G is alled bipartite if its hromati number is 1 or. However, sine a hromati number 1 orresponds to a graph with no bonds, and we are dealing only with onneted graphs, we an exlude this trivial ase and say that for a bipartite graph, χ =. The vertex set of a bipartite graph G an be partitioned into two disjoint sets, say V 1 and V, in suh a way that every bond of G onnets a vertex from V 1 with a vertex from V. We then have the following notation: G = (V 1 V, B) [0]. The adjaeny matrix (onnetivity) of G is the symmetri V V matrix C = C(G) whose entries are given by: { 1, if i and j are adjaent C ij = 0, otherwise Laplaians on graphs an be defined in various ways. The most elementary way relies only on the topology (onnetivity) of the graph, and the resulting Laplaian is an operator on a disrete and finite-dimensional Hilbert spae. These operators or their generalizations to be introdued below will be referred to as Disrete or ombinatorial Laplaians. One an onstrut the Laplaian operator as a differential operator if the bonds are endowed with a metri, and appropriate boundary onditions are required at the verties. The resulting operator should be referred

3 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 3 to as the Metri Laplaian. However, beause the metri Laplaian is idential with the Shrödinger operator on the graph, one often refers to this system as a Quantum Graph - a misnomer whih is now hard to eradiate. In the sequel we shall properly define and disuss the relevant versions of Laplaians on graphs. The disrete Laplaian, of G, is the matrix () L(G) = D C, where D is the diagonal matrix whose i th diagonal entry is the degree of the vertex v i, and C is the adjaeny matrix of G. A generalized Laplaian, L is a symmetri V V matrix with off-diagonal elements defined by: L ij < 0 if verties i and j are adjaent, and L ij = 0 otherwise. There are no onstraints on the diagonal elements of L. The eigenvalues of L(G) together with their multipliities, are known as the spetrum of G. To eah eigenvalue orresponds (at least one) eigenvetor whose entries are labeled by the verties indexes. It is well known that the eigenvalues of the ombinatorial Laplaian are non-negative. Zero is always an eigenvalue and its multipliity is equal to the number of onneted omponents of G. An important property regarding spetra of large v-regular graphs is that the limiting spetral distribution is symmetri about λ = v, and is supported in the interval [v v 1,v + v 1] [18]. An extensive survey of the spetral theory of disrete Laplaians an be found in [1,, 3]. To define quantum graphs a metri is assoiated to G. That is, eah bond is assigned a positive length: L b (0, ). The oordinate along the bond b is denoted by x b. The total length of the graph will be denoted by L = b B L b. This enables to define the metri Laplaian (or Shrödinger) operator on the graph as the Laplaian in 1-d d dx on eah bond. The domain of the Shrödinger operator on the graph is the spae of funtions whih onnet ontinuously aross verties and whih belong to the Sobolev spae H (b) on eah bond b. Moreover, vertex boundary onditions are imposed to render the operator self adjoint. We shall onsider in this paper the Neumann and Dirihlet boundary onditions: (3) Neumann ondition on the vertex i : ψ b (x b ) = 0, dx b (4) b S (i) d xb =0 Dirihlet ondition on the vertex i : ψ b (x b ) xb =0 = 0, where S (i) denotes the group of bonds whih emerge from the vertex i and the derivatives in (3) are direted out of the vertex i. The eigenfuntions are the solutions of the bond Shrödinger equations: (5) b B d dx ψ b = k ψ b, whih satisfy at eah vertex boundary onditions of the type (3) or (4). The spetrum {kn} n=1 is disrete, non-negative and unbounded. One an generalize the Shrödinger operator by inluding potential and magneti flux that are defined on the bonds. Other forms of boundary onditions an also be used. However, these generalizations will not be addressed here, and the interested reader is referred to two reent reviews [13, 14].

4 4 RAM BAND, IDAN OREN AND UZY SMILANSKY, Finally, two graphs, G and H, are said to be isospetral if they posses the same spetrum (same eigenvalues with the same multipliities). This definition holds both for disrete and quantum graphs. 3. Nodal domains on graphs Nodal domains on graphs are defined differently for disrete and metri graphs. Disrete graphs: Let G = (V, B) be a graph and let f=(f 1,f,...,f V ) be a real vetor. We assoiate the real numbers f i to the verties of G with i = 1,,...,V. A nodal domain is a maximally onneted subgraph of G suh that all verties have the same sign with respet to f. The number of nodal domains with respet to a vetor f is alled a nodal domains ount, and will be denoted by ν(f). The maximal number of nodal domains whih an be ahieved by a graph G will be denoted by ν G. The nodal sequene of a graph is the number of nodal domains of eigenvetors of the Laplaian, arranged by inreasing eigenvalues. This sequene will be denoted by {ν n } V n=1. The definition of nodal domains should be sharpened if we allow zero entries in f. Two definitions are then natural: A strong positive (negative) nodal domain is a maximally onneted subgraph H of G suh that f i > 0 (f i < 0) for all i H. A weak positive (negative) nodal domain is a maximally onneted subgraph H of G suh that f i 0 (f i 0) for all i H. In both ases, a positive (negative) nodal domain must onsist of at least one positive (negative) vertex. Aording to these definitions, it is lear that the weak nodal domains ount is always smaller or equal to the strong one. Metri graphs: Nodal domains are onneted domains of the metri graph where the eigenfuntion has a onstant sign. The nodal domains of the eigenfuntions are of two types. The ones that are onfined to a single bond are rather trivial. Their length is exatly half a wavelength and their number is on average kl π. The nodal domains whih extend over several bonds emanating from a single vertex vary in length and their existene is the reason why ounting nodal domains on graphs is not a trivial task. The number of nodal domains of a ertain eigenfuntion on a general graph an be written as (6) ν n = 1 i b S (i) { k nl b π + 1 ( 1 ( 1) knl b π sign[φ i ]sign[φ j ]) } B + V where x stands for the largest integer whih is smaller than x, and φ i,φ j are the values of the eigenfuntion at the verties onneted by the bond b = {i,j} [17]. (6) holds for the ase of an eigenfuntion whih does not vanish on any vertex: i φ i 0, and there is no yle of the graph on whih the eigenfuntion has a onstant sign. The last requirement is true for high enough eigenvalues where half the wavelength is smaller than the length of the shortest bond. This restrition, whih is important for low eigenvalues, was not stated in [17]. Nodal domains on quantum graphs an be also defined and ounted in an alternative way. The values of the eigenfuntions on the verties assoiate the vetor φ = (φ 1,...,φ V ) to the vertex set, and its number of nodal domains is ounted as was explained above. The reasoning behind this way of ounting is that the values of the eigenfuntion on the verties {φ i } together with the eigenvalue k

5 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 5 store the omplete information about the values of the eigenfuntion everywhere on the graph. We thus have two independent ways to define and ount nodal domains. To distinguish between them we shall refer to the first as metri nodal domains, and the number of metri domains in the n th eigenfuntion will be denoted by µ n. The domains defined in terms of the values of the eigenfuntion on the verties will be referred to as the disrete nodal domains. The number of disrete nodal domains of the n th eigenfuntion will be denoted by ν n, similar to the notation of this ount on the disrete graphs. As far as ounting nodal domains is onerned, trees behave as one dimensional manifolds, and the analogue of Sturm s osillation theory applies for the eigenfuntions of the disrete [9] and the metri Laplaians [10, 11, 1, 15], as long as the eigenvetor (or the eigenfuntion) does not vanish at any vertex. Thus we have ν n = n for disrete tree graphs and µ n = n for metri ones. Similarly, Courant s theorem applies for the eigenfuntions of both the disrete and the metri versions of the Laplaian on any graph, [16, 17]. However, sharper lower and upper bounds for the number of nodal domains were disovered reently. Berkolaiko [7] provided a lower bound for the nodal domains ount for both the disrete and the metri ases. He showed that the nodal domains ount of the n th eigenfuntion of a Laplaian (either disrete or metri) has no less than n r nodal domains. Again, this is valid if the eigenfuntion has no zero entries and it belongs to a simple eigenvalue. When n r < 0, this result is trivial sine a nodal domains ount is positive by definition. We note that for metri graphs this theorem holds only when the metri ount is used. A global upper bound for the nodal domains ount of a graph G was derived in [8]: The maximal number of nodal domains on G was proven to be smaller or equal to V χ +, where χ is the hromati number of G. This bound is valid for any vetor, not only for Laplaian eigenvetors. To end this setion we shall formulate and prove a few results whih show that not all possible subgraphs an be nodal domains of eigenvetors of the disrete Laplaians. The topology and onnetivity of nodal domains are restrited, and the restritions depend on whether the eigenvalue is larger or smaller than the spetral mid-point v. Theorem 3.1. Let G be a v-regular graph, and let f be an eigenvetor of the disrete Laplaian, orresponding to an eigenvalue λ. Let g be a nodal domain of f. Then the following statements hold: i. For all eigenvetors with eigenvalue λ > v the nodal domains do not have interior verties. ii. For all eigenvetors with eigenvalue λ < v, all the nodal domains onsist of at least two verties. iii. For all eigenvetors with eigenvalue λ < v k (and k < v), in every nodal domain there exists at least one vertex with a degree (valeny) whih is larger than k.

6 6 RAM BAND, IDAN OREN AND UZY SMILANSKY, Proof. i. Assume that i is an interior vertex in g. Hene, the signs of f j for all j i are the same as the sign of f i. This is not ompatible with (7) f j = (λ v)f i j i for λ > v. Hene g annot have any interior verties. ii. Assume that the subgraph g onsists of a single vertex i. Thus on all its neighbors j i, the sign of f j is different from the sign of f i. This is not ompatible with (8) f j = (v λ)f i j i for λ < v. Hene g annot onsist of a single vertex. iii. Denote the omplement of the nodal domain (subgraph) g in V by g. For all the verties i in g (9) (Lf) i = vf i C j,i f j C l,i f l = λf i. j g l g Summing over i g we get: (10) (v λ) f i = C j,i f j + C l,i f l i g i g j g l g Assuming for onveniene that f i are positive for i g, the rightmost sum in the equation above is non positive, and therefore (11) (v λ) f i C j,i f j = v i f i ˆv f i. i g i g j g i g i g Here, v i = j g C j,i is the valeny (degree) of the i th vertex in g, and ˆv denotes the largest valeny in the subgraph. Sine it is assumed that λ < v k we get (1) k < ˆv, whih ompletes the proof. The ase λ = v deserves speial attention. As long as the nodal domain under study has no vanishing entries, it annot onsist of a single vertex nor an it have interior verties. Namely, for λ = v, statements i and ii of Theorem 3.1. are valid simultaneously. Otherwise, one should treat separately the strong and the weak ounts. For the strong ount, and λ = v a nodal domain annot have an interior vertex. However, using the weak ount for λ = v one finds that no single vertex domains an exists, as in Theorem 3.1.ii. Item iii of Theorem 3.1 an be used to provide a λ dependent bound on the number of nodal domains of eigenvetors orresponding to eigenvalues λ < v. Define the integer k as k = v λ. Theorem 3.1.iii implies that every nodal domain oupies at least k + verties. Thus, their number is bounded by V k+. Courant theorem guarantees that the number of nodal domains is bounded by the spetral

7 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 7 ount N(λ). This information together with the known expression for the expetation value of N(λ) over the ensemble of random graphs, enable us to show that for large v and V, the bound V k+ is more restritive than the Courant bound. Unlike Pleijel s result, this bound is not uniform for the entire spetrum, and it applies only to the lower half of the eigenvalues with λ < v. Theorem 3.1 an be easily extended to the nodal properties of the eigenvetors of the generalized Laplaian, provided that the weights at eah vertex sum up to a onstant v whih is the same for the entire graph. 4. How to ount nodal domains on graphs? When disussing nodal domains ounting, we must make a lear distintion between algorithmi and analyti methods. In the first lass, we inlude omputer algorithms. They vary in effiieny and reliability, but they have one feature in ommon, namely, that the number of nodal domains is provided not as a result of a omputation, but rather, it follows from a systemati ounting proess. The most widely used method is the Hoshen-Kopelman algorithm (HK) for ounting nodal domains on -dimensional domains [30]. Analyti methods provide the number of nodal domains as a funtional of the funtion and the domain under study. The funtional might be quite ompliated, and not effiient when implemented numerially. An example of an analytial method for nodal domains ounting in one dimension, is given by (13) ν = b a δ (f(x)) df(x) dx dx + 1, where the nodal domains of f(x), in the interval [a,b] are provided (assuming that f(a)f(b) 0). While ounting in 1-d is simple, there is no analyti ounting method for omputing the number of nodal domains in higher dimensions: the ompliated onnetivity allowed in high dimensions renders the ounting operation too non loal. Graphs, whih are in some sense intermediate between one and two dimensions still allow several analyti ounting methods whih we disuss here. An example of an analytial ount is given by (6). The HK algorithm is well suited for graphs whih are grids. However, it is not as effiient when the graph under study is highly irregular. Although the HK algorithm fails for very omplex graphs, other algorithms, alled labeling algorithms, display linear effiieny ([46],[47]). Method III. in the following list, in addition of providing an analytial expression for the nodal domain ount, an also be implemented as a omputer algorithm. We show that it performs as effiiently as the labeling algorithm. The ounting methods that we present here are aimed for the disrete ounting of both disrete and metri graphs. In what follows, we assume that a vetor f is assoiated to the vertex set with entries f i. The nodal domains are defined with respet to f Method I. - Counting nodal domains in terms of flips. We define a flip as a bond on the graph whih onnets verties of opposite signs with respet to a vetor f. The sign vetor of f, denoted by f, is defined by f i Sign(f i ). For the time being, it is assumed that f has no zero entries. The general situation will

8 8 RAM BAND, IDAN OREN AND UZY SMILANSKY, be disussed later. We denote the set of flips on the graph by F(f): (14) F(f) = {(u,v) B f v f u < 0}. The ardinality of F(f) will be denoted by F(f). Lemma 4.1. The number of flips of a sign vetor f, an be expressed as (15) F(f) = F( f) = 1 4 ( f,l f). Proof. Using: (16) (17) ( f,l f) = 1 ( f v f u ) v u ( f v f u ) 4, if f v and f u have opposite signs = 0, if f v and f u have the same sign Using the number of flips, one an get an expression for the number of nodal domains: Theorem 4.. Given a onneted graph G on V verties, B bonds (and r yles) and a vetor f, then: (18) ν(f) = 1 4 ( f,l f) + V B + l(f) = 1 4 ( f,l f) (r l(f)) + 1 where l(f) is the number of independent yles in G of onstant sign (with respet to f). The seond equality above is based on equation (1). Proof. Let us remove all the flips from the graph. We are now left with a possibly disonneted graph G. There is a bijetive mapping between omponents of G and nodal domains of G. Hene, the number of omponents in G is equal to the nodal domains ount of G with respet to f. Let the number of nodal domains in G be denoted by ν(f). Using (1), it is lear that for the i th omponent (where i = 1,,...,ν(f)): r i = B i V i + 1 where r i, B i and V i are the number of yles, bonds and verties of the i th omponent, respetively. It is also lear that all the yles in G are of onstant sign, sine there are no flips in G. Thus, by our notation r i = l i. Let us sum over the omponents: ν ν (19) l(f) = l i = (B i V i + 1) = (B F(f)) V + ν(f) Combining (15) with (19), we get (18).

9 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 9 (18) is valid only for vetors f with no zero entries. In order to be able to handle a zero entry in f, we must perform a transformation on the graph. For a strong nodal ount, we simply delete all the zero verties along with the bonds onneted to them from the graph, and then apply (18) on the new graph (with the new Laplaian). For a weak nodal ount we replae eah zero vertex by two verties, one positive and one negative (not onneted to eah other), and onnet them to all verties whih were onneted to the original one. Now we an apply (18), and get the desired result. Notie that this orretion fails in the ase of a zero vertex whose neighbors are of the same sign (in this ase an artifiial nodal domain is added). However, the situation above an not our for an eigenvetor of a disrete Laplaian. Using (18), we an write some immediate onsequenes: (0) (1) F(f n ) + 1 r ν n F(f n ) + 1 n r 1 F(f n ) n + r 1 (0) results from the obvious fat that 0 l r, while (1) is a onsequene of Courant s nodal domains theorem and Berkolaiko s theorem whih states that n r ν n. In order to make use of (18), one must ompute l(f) whih is not given expliitly in terms of f. Thus, it annot be onsidered as an analyti ounting method, nor does it offer omputational advantage (There is no known effiient algorithm whih ounts all the yles of onstant sign with respet to f). However, it offers a useful analytial tool for deriving other results, and it makes a useful onnetion between various quantities defined on the graph. 4.. Method II. Partition funtion approah. Foltin [3] derived a partition funtion approah to ounting nodal domains of real funtions in two dimensions. It an be adapted for graphs in the following way: Eah vertex, i, is assigned an auxiliary spin variable s i where s i = ±1 (a so alled Ising-spin). Thus, given a ertain funtion f on the graph, eah vertex is assigned with two spins s i and f i. Let s denote the auxiliary spin vetor: s = (s 1,s,...,s V ). Foltin introdued a weight to eah onfiguration of the spins model. It assigns the value 1 to onfigurations in whih all spins s i belonging to the same nodal domain (with respet to f) are parallel, while spins of different domains might have different values. The weight reads: () w(f,s) = i,j C i,j [1 1 + f i fj 1 s is j ]. It an be easily heked that this form satisfies the requirements stated above: The weight an take the values one or zero. It is one if and only if eah fator in the produt is equal to one. A ertain fator is one, in either one of the two ases: if f i f j (i,j belong to different domains) - this allows the Ising-spins in different domains to be independent of eah other. The seond ase is if i,j are in the same domain, fi = f j and the orresponding Ising-spins are equal, s i = s j. Let us now sum over all possible spin onfigurations {s i } to get the partition funtion (3) Z(f) {s} w(f,s).

10 10 RAM BAND, IDAN OREN AND UZY SMILANSKY, For the onfigurations whose weight has the value one, the Ising-spins an take both signs, as long as they are all equal on a domain. Different domains are independent of eah other. Hene, the total number of suh onfigurations is: (4) Z(f) = ν(f), where ν(f) is the number of nodal domains of the vetor f. The nodal domains ount is: (5) ν(f) = 1 lnz(f) 1.44ln Z(f). ln The partition funtion approah provides an expliit formula for the number of nodal domains, and therefore it belongs to the analyti and not to the algorithmi ounting methods. As a matter of fat, it is highly ineffiient for pratial omputations. It involves running over all possible spin onfigurations {s i }, where s i = ±1. There are V different onfigurations, and as V inreases the effiieny deteriorates rapidly. The partition funtion approah an be used as a basis for the derivation of some identities involving the graph and a vetor f defined on it. It is onvenient to introdue the following notations: b (i,j) ϕ b 1 + f i fj σ b 1 s is j Where f and s are as before, and b is a direted bond (from i to j). We generalize the partition funtion by introduing a new parameter x into the definitions (6) (7) (8) w(f,s;x) = b B[1 ϕ b σ b x] Z(f;x) {s} Z(f;1) = ν(f) w(f,s;x) = {s} (1 ϕ b σ b x) where x an assume any real or omplex value. At x = 1, the generalized partition funtion is idential to (3). Let us now perform the summation over all the vetors s, and ompute the oeffiient of x k. To get all the ontributing terms we have to sum over all hoies of k brakets from (7), in whih x appears. Non vanishing ontributions our whenever both ϕ b and σ b are equal to one. Sine we are only summing over s, we only need to hek when σ b = 1. This happens if and only if the s vetor has a flip on the bond b. Sine we hoose k brakets (whih is equivalent to hoosing k bonds), we need to ount how many s vetors have flips on all these k bonds. The signs of those s vetors on bonds whih are not ontained in this hoie of k bonds are irrelevant. If we observe the hoie of k bonds (b 1,...,b k ), we notie that eah onneted omponent, within this hoie, ontributes a fator of, sine the symmetry of turning eah plus to minus and vie versa, does not hange the flip properties. Using (1) we see that the number of onneted omponents with respet to the hoie of k bonds is: Co(b 1,...,b k ) = V k + r(b 1,...,b k ), where r(b 1,...,b k ) is the number of independent yles that are ontained in this hoie. b B

11 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 11 Finally we notie that a yle of odd length annot have a flip on all its bonds, so we will not sum over hoies of b 1,...,b k whih ontain a yle of odd length. Thus, the sum over s yields: B k Z(f; x) = (ϕ bi ) V k+r(b1,...,bk) ( 1) k x k (9) k=0 b 1,...,b k B = B V ( 1) k k=0 k b 1,...,b k B k (ϕ bi ) r(b1,...,b k) x k Where stands for summation on all the possibilities to hoose k bonds b 1,...,b k B b 1,...,b k B suh that the subgraph they form do not ontain an odd yle. We an now derive some immediate properties of the generalized partition funtion. To start, we ompute the leading four derivatives at x = 0 to demonstrate the ounting tehniques involved. Some more effort is required to ompute the higher derivatives. (30) (31) (3) (33) (34) Z(f;0) = V Z (1) (f;0) = V 1 ϕ b = V 1 (B F(f)) b B Z () (f;0) =! V b 1,b B Z (3) (f;0) = 3! V 3 Z (4) (f;0) = 4! V 4 ( ) B F(f) (ϕ bi ) = V 1 b 1,...,b 3 B b 1,...,b 4 B 3 [( ) ] B F(f) (ϕ bi ) = 3! V 3 C (ϕ bi ) [( ) ]} B F(f) = 4! {C V C 4 4 C 3 [( ) ] B F(f) = 4! V 4 + C 4 4 C 3 Where C 3 is the number of triangles of onstant sign, C 4 is the number of yles of length 4 of onstant sign, and C 3 = C 3 (B F(f) 3) is the number of hoies of 4 non-flips bonds whih ontain a triangle. In the evaluation of the funtion and its first three derivatives we have used the identity n 3 r(b 1,...,b n ) = 0 when b 1,...,b n ontain no odd yles. The partition funtion is a polynomial of degree less than or equal to the number of bonds. For a graph G with B bonds and P onneted omponents, the polynomial is of degree B if and only if G is bipartite and the funtion f is of onstant sign on eah onneted omponent. This follows from two observations: first, a well known theorem in graph theory states that a graph G is bipartite if and only if it ontains no yles of odd length. Thus we an hoose all the bonds in G without having an odd yle ontained in this hoie. Seond, unless f is of onstant sign on eah onneted omponent, we will enounter a flip and hene the multipliation over all ϕ bi will vanish. If G is bipartite, the oeffiient of x B is P.

12 1 RAM BAND, IDAN OREN AND UZY SMILANSKY, While the value of the polynomial at x = 1 has an immediate appliation through (8), we an evaluate it for other values. Let us hoose f to be a vetor of onstant sign. This way ϕ bi = 1 for all i = 1,,...,B. Hene (9) redues to: (35) B V ( 1) k x k. k=0 k b 1,...,b k B r(b1,...,b k) On the other hand, (7) equals: (36) (1 σ b x). {s} b B Choosing x = n + 1 where n Z and using the fat that (35) and (36) are equal, we get: (37) 1 B V n F(s) ( 1) k = k r(b1,...,b k) ( n + 1) k. {s} k=0 b 1,...,b k B For random vetors s, uniformly distributed, the left hand side an be interpreted as the average over this ensemble of the quantity: n F(s). Let us hoose two speial values for n. If we hoose n = 1, the left hand side is just the differene between the probability that a random vetor s on the graph has an even number of flips, and an odd number of flips: prob(f(s) =even)-prob(f(s) =odd). The right hand side is: (38) B ( 1) k r(b1,...,bk). k=0 b 1,...,b k B For n =, we get: (39) 1 V F(s) = {s} B k=0 1 k r(b1,...,bk). b 1,...,b k B An example of the use of the polynomial ould be in proving that on a tree, the probability of an even number of flips is equal to the probability of an odd number of flips. We use Eq. (38) and see that this differene of probabilities is equal to: B k=0 ( 1)k( B k) = 0. Other possible identities we an derive are for the omplete graph, K V. A funtion f of onstant sign indues one nodal domain on K V, while any other funtion indues two nodal domains. Therefore, for x = 1: (40) (41) V V ± 1 f {±1} V V B k=0 B ( 1) k k=0 k b 1,...,b k B ( 1) k k b 1,...,b k B ( k ) ϕ bi (f) r(b1,...,b k) = r(b1,...,b k) = 4

13 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 13 Where ϕ bi (f) = 1+fufv for b i = (u,v). Equivalently, running over all the funtions f (inluding those of onstant sign) we get: B V ( 1) k ( k ) (4) ϕ bi (f) r(b1,...,b k) = 4( V 1) f {±1} V k=0 k b 1,...,b k B 4.3. Method III. Breaking up the graph. We begin again by deleting all the flips from the graph G. This way we are left with a (possibly) disonneted graph, G in whih eah onneted omponent orresponds uniquely to a nodal domain in the original graph. The onnetivity matrix C and the disrete Laplaian L of G are given by (43) (44) C ij = C ij 1 + f i fj L ij = C V ij + δ ij Where f is the sign vetor, and it is assumed for the moment that none of the entries of f vanish. We now make use of the theorem whih states that the lowest eigenvalue of the Laplaian is 0 with multipliity whih equals the number of onneted omponents in the graph. Therefore, finding the nodal domains ount redues to finding the multipliity of zero as an eigenvalue of L. An analyti ounting formula an be derived by onstruting the harateristi polynomial of L: (45) det(λi V L) The multipliity of its 0 eigenvalue provides the nodal domains ount: (46) ν(f) = lim λ 0 λ d dλ lndet(λi V L). The appliation of the method an be extended to ases where some entries of f vanish. Then we would like to ount both weak and strong nodal domains. For the strong nodal ount, we an redue the dimension of our problem. We simply remove the zero entries from f, along with the orresponding rows and olumns in L. Now we onstrut L, and ontinue as before, with the new matrix L of dimension (V z) (V z), z being the number of zero entries in f. The weak nodal domains ount is obtained in the opposite manner: sine a zero entry partiipates both in positive and negative nodal domains, we replae eah zero vertex by two verties, one positive and one negative, both onneted to all verties adjaent to the original one. Thus, for the weak nodal ount, the dimension of the new problem is (V + z) (V + z), z is defined as before. Noting again that this fails in the ase of a zero vertex whose neighbors are of the same sign (see orresponding remark in subsetion 4.1). This method of ounting, whih provides the analytial expression (46) for the nodal ount, is also the basis for a omputational algorithm whih turns out to be very effiient. It relies on the effiieny of state of the art algorithms to ompute the spetrum (inluding multipliity) of sparse, real and symmetri matries. To k=1 C ik

14 14 RAM BAND, IDAN OREN AND UZY SMILANSKY, estimate the dependene of the effiieny on the dimension V of the graph, we have to onsider the osts of the various steps in the omputation. The onstrution of the matrix L, takes O(V ) operations, and storing the information requires O(V ) memory ells as well. It takes O(V α ) operations to find all its eigenvalues where α.3 (and at worst ase α 3) [9]. In figure 1, this polynomial dependene is shown for graphs of two different onnetivity densities, with r V equals 0.5 and 5. In this figure the logarithm of the time needed to find all eigenvalues of L (defined for a random vetor), is plotted against the logarithm of the number of verties. The slope whih is the exponent of the polynomial dependene is smaller than 3. The eigenvalues in these two examples were attained using the Matlab ommand eig. As will be shown below, there are more effiient ways of finding the spetrum of sparse, real and symmetri matries. Thus, the effiieny stated above an be improved for graphs with sparse Laplaians. (a) (b) Figure 1. The time it takes to ompute the spetrum of L as a funtion of the number of verties, for two different onnetivity densities: (a) r V = 0.5 (b) r V = 5. Finally, we would like to show that the present method an be applied for ounting nodal domains of funtions defined on two dimensional grids and that its effiieny is omparable to that of the ommonly used HK algorithm. Given a funtion f on a two dimensional domain, we have to ompute its values on a retangular grid with V V points. The HK algorithm ounts the nodal domains in O(V ) operations [30]. Using our method, we onsider the retangular grid as a graph with V verties. Assuming for simpliity periodi boundary onditions, the valeny of all the verties is 4. The orresponding L matrix is a V V matrix whih is sparse (as long as V 4), and due to the periodi boundary onditions it takes the expliit form: (47) L i,j = 4δ i,j δ i,j 1 δ i,j+1 δ i V,j δ i+v,j. Thus, storing L takes only O(V ) memory ells, and onstruting L takes O(V ) operations. We mentioned above that for a general real symmetri matrix, the number of operations needed is O(V α ), where α.3 and at worst ase α 3. However, the sparse nature of L signifiantly simplifies the problem. The most

15 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 15 well-known eigenvalue method for sparse-real-symmetri matries is the Lanzos method. In addition, in reent years, new effiient algorithms were disovered for the same problem. In [31], it is proven that finding the eigenvalues of a sparse symmetri matrix takes only O(V ) operations. Combining the osts, we find that it takes our algorithm O(V ) operations in order to ompute the nodal domains ount, and therefore it is omparable in effiieny to the HK algorithm. As mentioned earlier, the labeling algorithms also display linear effiieny ([46],[47]). The labeling algorithms have the advantage that they are simpler in a sense, and that they are implemented quite easily as omputer programs. In addition, the labeling algorithms maintain their linear effiieny even for graphs with dense Laplaians. It is worth mentioning, however, that our algorithm has the advantage that it provides an analyti expression of the nodal domains ount (46) Method IV. A geometri point of view. The ounting method proposed here uses a geometri point of view whih starts by onsidering the V dimensional Eulidean spae, and dividing it into V setors using the following onstrution. Consider the V vetors e (α) = (e (α) 1,e (α),...,e (α) V ) where e(α) i = {1, 1}, α = 1,,..., V. A vetor x R n is in the setor α if x e (α) = n x i. In two dimensions, the setors are the standard quadrants. We shall refer to the vetors e (α) as the indiators. Given a graph G with V verties, we partition the V indiators into disjoint sets: γ n = {e (α) : ν(e (α) ) = n} where ν(e (α) ) denotes the nodal domains ount of the indiator e (α) with respet to G. As shown before {γ n } V χ +, where χ is the hromati number of G, and also some of the γ is might be empty. Let f be a vetor with non-zero entries defined on the vertex set of G. Then, the main observation is that ν(f) = n if and only if: (48) V ˆδ(< e α f > f i ) = 1, e α γ n where, ǫ (49) ˆδ(x) = lim ǫ 0 x sin x { 1, if x = 0 ǫ = 0, if x 0. In other words, by finding the setor to whih f belongs and knowing from a preliminary omputation the number of nodal domains in eah setor, one obtains the desired nodal ount. Thus, the present method requires a preliminary omputation in whih the setors are partitioned into equi-nodal sets γ n. This should be arried out one for any graph. Therefore the method is useful when the nodal ounts of many vetors is required. In several appliations, one is given a vetor field (of unit norm for simpliity) f S V 1 whih is distributed on the V 1-sphere with a given probability distribution p(f), and one is asked to ompute the distribution of nodal ounts, (50) P(n) = p(f)ˆδ(ν(f) n)d V 1 f. S V 1

16 16 RAM BAND, IDAN OREN AND UZY SMILANSKY, In suh ases, the preliminary task of omputing the equi-nodal pays off, and one obtains the following analyti expression for the distribution of the nodal ounts. (51) P(n) = p(f)d V 1 f V ˆδ(< e α f > f i ) = S V 1 e α γ n p(f)δ(1 f )d V f V ˆδ(< e α f > f i ) R V e α γ n where: p(f) = p( f f ) f. (51) an also be formulated as: (5) V P(n) = p(f β )δ(1 f )d V f V ˆδ(< e α f β > f i ) = β=1 f i 0 e α γ n + p(f β )δ(1 f )d V f V ˆδ(< e α f β > f i ) e α γ n e β γ n Where + = f i 0 means integration on the first setor (the vetors with all entries positive) And f µ is the dot produt of f and e µ : f µ = (f 1 e µ 1,f e µ,...,f V e µ V ) Equations (51) and (5) are the general equations governing the nodal domains ount distribution. In order to make further progress, we need to speify the distribution from whih f is taken. This means that we need to speify p(f) in (51) for example. Let us disuss two examples: A uniform distribution over the V 1 dimensional sphere: In this ase, we an solve Equation (51) and get that P(n) = γn. Note that for a tree, we an solve V this problem by other means. Using (18), we see that for a tree, the number of nodal domains is equal to the number of flips plus one. Sine f is taken from the uniform distribution, then the probability of a flip is half. The number of flips in a vetor f is thus a binomial variable: F(f) Binomial(N,p) with N = V 1 is the number of bonds, and p = 1. For large enough V this approahes the Gaussian distribution: F(f) Gaussian(µ,σ ) with µ = V 1 and σ = V 1 4. From this result we an infer that: (53) (54) V +1 (n P(n) exp ( ) ) π(v 1) V 1 V +1 V +1 (n γ n exp ( ) ) π(v 1) V 1 For the other extreme, the omplete graph, K V, the only possible nodal domains ounts are one and two [8]. The vetors whih yield a nodal domains ount of one are vetors of onstant sign. All other vetors yield a nodal domains ount of two. Indeed, using (51) or (5) it is easy to be onvined that for the omplete graph, γ 1 = while γ = V. All other γ s are empty. Miro-anonial ensemble: In this ase the vetors f are uniformly distributed on

17 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 17 the energy shell, where we an also define a measurement tolerane fator, : (55) p E (f) = δ(e < f L f > )δ(1 f ) d S V 1 fδ(e < f L f > ) V 1 In order to make use of this ensemble, further work must be done, for example, a natural way to order the funtions of the ensemble. 5. The resolution of isospetrality There are several known methods to onstrut isospetral yet different graphs. A review of this problem an be found in ([33]). The onditions under whih the spetral inversion of quantum graphs is unique were studied previously. In [38, 39] it was shown that in general, the spetrum does not determine uniquely the length of the bonds and their onnetivity. However, it was shown in [35] that quantum graphs whose bond lengths are rationally independent an be heard - that is - their spetra determine uniquely their onnetivity matries and their bond lengths. This fat follows from the existene of an exat trae formula for quantum graphs [40, 41]. Thus, isospetral pairs of non ongruent graphs, must have rationally dependent bond lengths. An example of a pair of metrially distint graphs whih share the same spetrum was already disussed in [35]. The main method of onstrution of isospetral pairs is due to Sunada [34]. This method enabled the onstrution of the first pair of planar isospetral domains in R [36] whih gave a negative answer to Ka s original question: Can one hear the shape of a drum? [37]. Later, it was shown that all the known isospetral domains in R [4, 43] whih were also onstruted using the Sunada method have orresponding isospetral pairs of quantum graphs [44]. An example of this orrespondene is shown in figure. As mentioned in the introdution, it is onjetured [7, 19] that nodal domains sequenes resolve between isospetral domains. For flat tori in 4-d, this was proven [8]. We present here three additional known results for the validity of the onjeture for graphs. The first result is for the quantum graphs shown in figure (). Both graphs of this isospetral pair are tree graphs and therefore have the same metri nodal ount µ n = n ([15]). This demonstrates the need to use the disrete nodal ount in order to resolve isospetrality in this ase. Indeed numerial examination of this ase shows that for the first 6600 eigenfuntions there is a different disrete nodal ount for 19 % of the eigenfuntions. Similar numerial results exist for two other pairs of isospetral graphs that are onstruted from the isospetral domains in [4, 43]. The exat results are desribed in [19]. Another result is also in the field of quantum graphs [19]. The graphs in figure 3 are the simplest isospetral pair of quantum graphs known so far. The simpliity of these graphs enables the omparison between the nodal ounts of both graphs. It was proved that the nodal ount is different between these graphs for half of the spetrum. This result was proved separately for the disrete ount and for the metri ount. The proof does not ontain an expliit formula for the nodal ount but rather deals with the differene of the nodal ount between the graphs averaged over the whole spetrum.

18 18 RAM BAND, IDAN OREN AND UZY SMILANSKY, I II (a) a b (b) a b b a a b a b b a a b a b II b a b a a b b a b a b a a b I () Figure. (a) Planar isospetral domains of the 7 3 type. (b) Reduing the building blok to a 3-star. () The resulting isospetral quantum graphs. I II Figure 3. The isospetral pair with boundary onditions. D stands for Dirihlet and N for Neumann. The bonds lengths are determined by the parameters a,b, Examining the nodal sequenes for the graph II for various values of the length parameters a,b,, we observed that the formula (56) µ II n = n 1 1 ( 1) b+ a+b+ n reprodues the entire data set without any flaw 1. Assuming it is orret (whih is not yet proved rigorously) we first see that it provides an easy explanation for the previously disussed result that for rationally independent values for the parameters a,b, one gets that µ II n n for half of the spetrum. This, together with µ I n = n (sine graph I is a tree) we see again that for half of the spetrum the nodal 1 This result was obtained with A. Aronovith.

19 NODAL DOMAINS ON GRAPHS - HOW TO COUNT THEM AND WHY? 19 domain sequenes are different. Expression (56) is a periodi funtion of n with period proportional to the length of the only loop orbit on the graph (the length is measured in units of the its total length). It an be expanded and brought to a form whih is similar in struture to a trae formula where the length of this orbit and its repetitions are the osillation frequenies. A similar trae formula for the nodal ounts of the Laplaian eigenfuntions on surfaes of revolution was reently derived [6]. Finally, we diret our attention to disrete Laplaians. It was reently shown [8] that if G and H are two isospetral graphs where one of them is bipartite and the other one is not (without loss of generality, G is bipartite and H is not). Then for the eigenvetor orresponding to the largest eigenvalue λ V, their nodal domains ount will differ (ν(g) = V, ν(h) < V ). The proof of this theorem is based on another interesting result derived in [8]: Denote by f V the eigenvetor orresponding to the largest eigenvalue of the Laplaian of a onneted graph G. Then ν(f V ) = ν G = V, if and only if G is bipartite. Figure 4. illustrates this result. Figure 4. The upper figure presents a pair of isospetral graphs taken from [6]. Graph G, on the right is bipartite, whereas graph H, on the left, is not. The lower figure presents the nodal domains ount, ν(f n ) vs. the index n.

20 0 RAM BAND, IDAN OREN AND UZY SMILANSKY, 6. Summary and open questions In spite of the progress ahieved reently in the study of nodal domains on graphs, there are several outstanding open problems whih all for further study. We list here a few examples. Of fundamental importane is to find out whether there exists a trae formula for the nodal ount sequene of graphs, similar to the one derived in [6] for surfaes of revolution. The losest we reahed this goal is for the graph II in the previous setion, where (56) ould be expanded in a Fourier series. However (56) was dedued numerially but not proved. One a nodal trae formula is available, it ould be ompared to the spetral trae formula [41] and might show the way to prove or negate the onjeture ([7, 19]) that ounting nodal domains resolves isospetrality. The onjeture mentioned above an be addressed from a different angle. One may study the various systemati ways to onstrut isospetral pairs and investigate the relations between the onstrution method and the nodal ount sequene of the resulting graphs. Suh an approah worked suessfully for the dihedral graphs presented in the previous setion [19]. Another open question whih naturally arises in the present ontext: Can one find graphs whose Laplaians have different spetra but the nodal ount sequenes are the same? A positive answer is provided for tree graphs ([15]). Are there other less trivial examples of isonodal yet not isospetral domains? It follows from Berkolaiko s theorem [7] that the number of nodal domains (both metri and disrete) of the n th eigenfuntion is bounded in the interval [n r,n]. We an thus investigate the probability to have a nodal ount ν n = n r (for 0 r r). This probability, whih is defined with respet to a given ensemble of graphs, is denoted by P( r). It is defined for disrete graph laplaians as: (57) P( r) 1 V # { 1 n V : ν n = n r}. The orresponding quantity for metri Laplaians is: (58) N(K) # {n : k n K} P( r;k) 1 N(K) # { n N(K) : µ n = n r} P( r) lim K Here, stands for the expetation with respet to the ensemble. New questions arise from the investigation of the relation between the onnetivity of the graph and the nodal distribution P( r). Can one use the information stored in P( r) to gain information on the graphs e.g., the mean and the variane of the valeny (degree) distribution of the verties in the graphs? Many of the results we have presented, have analogues in Riemannian manifolds (whih in most ases, were disussed earlier) - for example, Courant s theorem was originally formulated for manifolds. One an searh for other analogues, and a good example is the Courant-Herrmann Conjeture (CHC). For manifolds the CHC onjeture states that any linear ombination of the first n eigenfuntions divide the domain, by means of its nodes, into no more than n nodal domains. Gladwell and Zhu [45] have shown that in general there is no disrete ounterpart to the CHC. However, we an still ask for whih lasses of graphs does the CHC still hold?

Channel Assignment Strategies for Cellular Phone Systems

Channel Assignment Strategies for Cellular Phone Systems Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious

More information

Chapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112

More information

1.3 Complex Numbers; Quadratic Equations in the Complex Number System*

1.3 Complex Numbers; Quadratic Equations in the Complex Number System* 04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x - 2x - 22 x - 2 2 ; x

More information

5.2 The Master Theorem

5.2 The Master Theorem 170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended

More information

chapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374

chapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374 CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems

More information

Sebastián Bravo López

Sebastián Bravo López Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that

More information

arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA

arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA LBNL-52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astro-ph/0304006v2 10 Jun 2003 Theory Group, MS 50A-5101 Lawrene Berkeley National Laboratory

More information

A Holistic Method for Selecting Web Services in Design of Composite Applications

A Holistic Method for Selecting Web Services in Design of Composite Applications A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,

More information

Hierarchical Clustering and Sampling Techniques for Network Monitoring

Hierarchical Clustering and Sampling Techniques for Network Monitoring S. Sindhuja Hierarhial Clustering and Sampling Tehniques for etwork Monitoring S. Sindhuja ME ABSTRACT: etwork monitoring appliations are used to monitor network traffi flows. Clustering tehniques are

More information

Chapter 1 Microeconomics of Consumer Theory

Chapter 1 Microeconomics of Consumer Theory Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deision-makers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some

More information

Classical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk

Classical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples

More information

Convergence of c k f(kx) and the Lip α class

Convergence of c k f(kx) and the Lip α class Convergene of and the Lip α lass Christoph Aistleitner Abstrat By Carleson s theorem a trigonometri series k osπkx or k sin πkx is ae onvergent if < (1) Gaposhkin generalized this result to series of the

More information

) ( )( ) ( ) ( )( ) ( ) ( ) (1)

) ( )( ) ( ) ( )( ) ( ) ( ) (1) OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples

More information

Special Relativity and Linear Algebra

Special Relativity and Linear Algebra peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory

More information

Weighting Methods in Survey Sampling

Weighting Methods in Survey Sampling Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiao-hih Chang Ferry Butar Butar Abstrat It is said that a well-designed survey an best prevent nonresponse. However, no matter

More information

Static Fairness Criteria in Telecommunications

Static Fairness Criteria in Telecommunications Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat-208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, e-mail: vesatimonen@hutfi

More information

Intelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds

Intelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds Intelligent Measurement Proesses in 3D Optial Metrology: Produing More Aurate Point Clouds Charles Mony, Ph.D. 1 President Creaform in. mony@reaform3d.om Daniel Brown, Eng. 1 Produt Manager Creaform in.

More information

User s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities

User s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear Least-Squares Fits with a Common Interept: Determination

More information

HEAT CONDUCTION. q A q T

HEAT CONDUCTION. q A q T HEAT CONDUCTION When a temperature gradient eist in a material, heat flows from the high temperature region to the low temperature region. The heat transfer mehanism is referred to as ondution and the

More information

cos t sin t sin t cos t

cos t sin t sin t cos t Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the

More information

Symmetric Subgame Perfect Equilibria in Resource Allocation

Symmetric Subgame Perfect Equilibria in Resource Allocation Proeedings of the Twenty-Sixth AAAI Conferene on Artifiial Intelligene Symmetri Subgame Perfet Equilibria in Resoure Alloation Ludek Cigler Eole Polytehnique Fédérale de Lausanne Artifiial Intelligene

More information

A Context-Aware Preference Database System

A Context-Aware Preference Database System J. PERVASIVE COMPUT. & COMM. (), MARCH 005. TROUBADOR PUBLISHING LTD) A Context-Aware Preferene Database System Kostas Stefanidis Department of Computer Siene, University of Ioannina,, kstef@s.uoi.gr Evaggelia

More information

Supply chain coordination; A Game Theory approach

Supply chain coordination; A Game Theory approach aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah Jean-Claude Hennet x and Yasemin Arda xx x LI CNR-UMR 668 Université

More information

Recovering Articulated Motion with a Hierarchical Factorization Method

Recovering Articulated Motion with a Hierarchical Factorization Method Reovering Artiulated Motion with a Hierarhial Fatorization Method Hanning Zhou and Thomas S Huang University of Illinois at Urbana-Champaign, 405 North Mathews Avenue, Urbana, IL 680, USA {hzhou, huang}@ifpuiuedu

More information

Computer Networks Framing

Computer Networks Framing Computer Networks Framing Saad Mneimneh Computer Siene Hunter College of CUNY New York Introdution Who framed Roger rabbit? A detetive, a woman, and a rabbit in a network of trouble We will skip the physial

More information

Capacity at Unsignalized Two-Stage Priority Intersections

Capacity at Unsignalized Two-Stage Priority Intersections Capaity at Unsignalized Two-Stage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minor-street traffi movements aross major divided four-lane roadways

More information

Improved SOM-Based High-Dimensional Data Visualization Algorithm

Improved SOM-Based High-Dimensional Data Visualization Algorithm Computer and Information Siene; Vol. 5, No. 4; 2012 ISSN 1913-8989 E-ISSN 1913-8997 Published by Canadian Center of Siene and Eduation Improved SOM-Based High-Dimensional Data Visualization Algorithm Wang

More information

Chapter 5 Single Phase Systems

Chapter 5 Single Phase Systems Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude

More information

Effects of Inter-Coaching Spacing on Aerodynamic Noise Generation Inside High-speed Trains

Effects of Inter-Coaching Spacing on Aerodynamic Noise Generation Inside High-speed Trains Effets of Inter-Coahing Spaing on Aerodynami Noise Generation Inside High-speed Trains 1 J. Ryu, 1 J. Park*, 2 C. Choi, 1 S. Song Hanyang University, Seoul, South Korea 1 ; Korea Railroad Researh Institute,

More information

A novel active mass damper for vibration control of bridges

A novel active mass damper for vibration control of bridges IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural

More information

An integrated optimization model of a Closed- Loop Supply Chain under uncertainty

An integrated optimization model of a Closed- Loop Supply Chain under uncertainty ISSN 1816-6075 (Print), 1818-0523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 9-17 An integrated optimization model of a Closed- Loop Supply Chain under unertainty Xiaoxia

More information

Programming Basics - FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html

Programming Basics - FORTRAN 77 http://www.physics.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html CWCS Workshop May 2005 Programming Basis - FORTRAN 77 http://www.physis.nau.edu/~bowman/phy520/f77tutor/tutorial_77.html Program Organization A FORTRAN program is just a sequene of lines of plain text.

More information

THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES

THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES Proeedings of FEDSM 98 998 ASME Fluids Engineering Division Summer Meeting June 2-25, 998 Washington DC FEDSM98-529 THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES John D. Wright Proess

More information

Fixed-income Securities Lecture 2: Basic Terminology and Concepts. Present value (fixed interest rate) Present value (fixed interest rate): the arb

Fixed-income Securities Lecture 2: Basic Terminology and Concepts. Present value (fixed interest rate) Present value (fixed interest rate): the arb Fixed-inome Seurities Leture 2: Basi Terminology and Conepts Philip H. Dybvig Washington University in Saint Louis Various interest rates Present value (PV) and arbitrage Forward and spot interest rates

More information

USA Mathematical Talent Search. PROBLEMS / SOLUTIONS / COMMENTS Round 3 - Year 12 - Academic Year 2000-2001

USA Mathematical Talent Search. PROBLEMS / SOLUTIONS / COMMENTS Round 3 - Year 12 - Academic Year 2000-2001 USA Mathematial Talent Searh PROBLEMS / SOLUTIONS / COMMENTS Round 3 - Year - Aademi Year 000-00 Gene A. Berg, Editor /3/. Find the smallest positive integer with the property that it has divisors ending

More information

A Three-Hybrid Treatment Method of the Compressor's Characteristic Line in Performance Prediction of Power Systems

A Three-Hybrid Treatment Method of the Compressor's Characteristic Line in Performance Prediction of Power Systems A Three-Hybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition of Power Systems A Three-Hybrid Treatment Method of the Compressor's Charateristi Line in Performane Predition

More information

3 Game Theory: Basic Concepts

3 Game Theory: Basic Concepts 3 Game Theory: Basi Conepts Eah disipline of the soial sienes rules omfortably ithin its on hosen domain: : : so long as it stays largely oblivious of the others. Edard O. Wilson (1998):191 3.1 and and

More information

Pattern Recognition Techniques in Microarray Data Analysis

Pattern Recognition Techniques in Microarray Data Analysis Pattern Reognition Tehniques in Miroarray Data Analysis Miao Li, Biao Wang, Zohreh Momeni, and Faramarz Valafar Department of Computer Siene San Diego State University San Diego, California, USA faramarz@sienes.sdsu.edu

More information

Asymmetric Error Correction and Flash-Memory Rewriting using Polar Codes

Asymmetric Error Correction and Flash-Memory Rewriting using Polar Codes 1 Asymmetri Error Corretion and Flash-Memory Rewriting using Polar Codes Eyal En Gad, Yue Li, Joerg Kliewer, Mihael Langberg, Anxiao (Andrew) Jiang and Jehoshua Bruk Abstrat We propose effiient oding shemes

More information

Impact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities

Impact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary

More information

Robust Classification and Tracking of Vehicles in Traffic Video Streams

Robust Classification and Tracking of Vehicles in Traffic Video Streams Proeedings of the IEEE ITSC 2006 2006 IEEE Intelligent Transportation Systems Conferene Toronto, Canada, September 17-20, 2006 TC1.4 Robust Classifiation and Traking of Vehiles in Traffi Video Streams

More information

Optimal Online Buffer Scheduling for Block Devices *

Optimal Online Buffer Scheduling for Block Devices * Optimal Online Buffer Sheduling for Blok Devies * ABSTRACT Anna Adamaszek Department of Computer Siene and Centre for Disrete Mathematis and its Appliations (DIMAP) University of Warwik, Coventry, UK A.M.Adamaszek@warwik.a.uk

More information

10.1 The Lorentz force law

10.1 The Lorentz force law Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been

More information

An Enhanced Critical Path Method for Multiple Resource Constraints

An Enhanced Critical Path Method for Multiple Resource Constraints An Enhaned Critial Path Method for Multiple Resoure Constraints Chang-Pin Lin, Hung-Lin Tai, and Shih-Yan Hu Abstrat Traditional Critial Path Method onsiders only logial dependenies between related ativities

More information

Improved Vehicle Classification in Long Traffic Video by Cooperating Tracker and Classifier Modules

Improved Vehicle Classification in Long Traffic Video by Cooperating Tracker and Classifier Modules Improved Vehile Classifiation in Long Traffi Video by Cooperating Traker and Classifier Modules Brendan Morris and Mohan Trivedi University of California, San Diego San Diego, CA 92093 {b1morris, trivedi}@usd.edu

More information

FOOD FOR THOUGHT Topical Insights from our Subject Matter Experts

FOOD FOR THOUGHT Topical Insights from our Subject Matter Experts FOOD FOR THOUGHT Topial Insights from our Sujet Matter Experts DEGREE OF DIFFERENCE TESTING: AN ALTERNATIVE TO TRADITIONAL APPROACHES The NFL White Paper Series Volume 14, June 2014 Overview Differene

More information

DSP-I DSP-I DSP-I DSP-I

DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I DSP-I Digital Signal Proessing I (8-79) Fall Semester, 005 IIR FILER DESIG EXAMPLE hese notes summarize the design proedure for IIR filters as disussed in lass on ovember. Introdution:

More information

Big Data Analysis and Reporting with Decision Tree Induction

Big Data Analysis and Reporting with Decision Tree Induction Big Data Analysis and Reporting with Deision Tree Indution PETRA PERNER Institute of Computer Vision and Applied Computer Sienes, IBaI Postbox 30 11 14, 04251 Leipzig GERMANY pperner@ibai-institut.de,

More information

TEACHING FUNDAMENTALS OF ELECTRICAL ENGINEERING: NODAL ANALYSIS

TEACHING FUNDAMENTALS OF ELECTRICAL ENGINEERING: NODAL ANALYSIS TACHING FUNDAMNTALS OF LCTICAL NGINING: NODAL ANALYSIS Đorđević, A..; Božilović, G.N.; Olćan, D.I.; Sh. of letr. ng., Univ. of Belgrade, Belgrade, Serbia 00 I. Personal use of this material is permitted.

More information

Relativity in the Global Positioning System

Relativity in the Global Positioning System Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 80309-00390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop

More information

Neural network-based Load Balancing and Reactive Power Control by Static VAR Compensator

Neural network-based Load Balancing and Reactive Power Control by Static VAR Compensator nternational Journal of Computer and Eletrial Engineering, Vol. 1, No. 1, April 2009 Neural network-based Load Balaning and Reative Power Control by Stati VAR Compensator smail K. Said and Marouf Pirouti

More information

Procurement auctions are sometimes plagued with a chosen supplier s failing to accomplish a project successfully.

Procurement auctions are sometimes plagued with a chosen supplier s failing to accomplish a project successfully. Deision Analysis Vol. 7, No. 1, Marh 2010, pp. 23 39 issn 1545-8490 eissn 1545-8504 10 0701 0023 informs doi 10.1287/dea.1090.0155 2010 INFORMS Managing Projet Failure Risk Through Contingent Contrats

More information

In order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method

In order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method BEAM DESIGN In order to be able to design beams, we need both moments and shears. 1. Moment a) From diret design method or equivalent frame method b) From loads applied diretly to beams inluding beam weight

More information

A Keyword Filters Method for Spam via Maximum Independent Sets

A Keyword Filters Method for Spam via Maximum Independent Sets Vol. 7, No. 3, May, 213 A Keyword Filters Method for Spam via Maximum Independent Sets HaiLong Wang 1, FanJun Meng 1, HaiPeng Jia 2, JinHong Cheng 3 and Jiong Xie 3 1 Inner Mongolia Normal University 2

More information

Trade Information, Not Spectrum: A Novel TV White Space Information Market Model

Trade Information, Not Spectrum: A Novel TV White Space Information Market Model Trade Information, Not Spetrum: A Novel TV White Spae Information Market Model Yuan Luo, Lin Gao, and Jianwei Huang 1 Abstrat In this paper, we propose a novel information market for TV white spae networks,

More information

Outline. Planning. Search vs. Planning. Search vs. Planning Cont d. Search vs. planning. STRIPS operators Partial-order planning.

Outline. Planning. Search vs. Planning. Search vs. Planning Cont d. Search vs. planning. STRIPS operators Partial-order planning. Outline Searh vs. planning Planning STRIPS operators Partial-order planning Chapter 11 Artifiial Intelligene, lp4 2005/06, Reiner Hähnle, partly based on AIMA Slides Stuart Russell and Peter Norvig, 1998

More information

Magnetic Materials and Magnetic Circuit Analysis

Magnetic Materials and Magnetic Circuit Analysis Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent

More information

REDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION

REDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION C I E 17 th International Conferene on Eletriity istriution Barelona, 1-15 May 003 EUCTION FACTO OF FEEING LINES THAT HAVE A CABLE AN AN OVEHEA SECTION Ljuivoje opovi J.. Elektrodistriuija - Belgrade -

More information

Conversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory

Conversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory PHYSICAL REVIEW B 76, 35114 007 Conversion of short optial pulses to terahertz radiation in a nonlinear medium: Experiment and theory N. N. Zinov ev* Department of Physis, University of Durham, Durham

More information

Henley Business School at Univ of Reading. Pre-Experience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD)

Henley Business School at Univ of Reading. Pre-Experience Postgraduate Programmes Chartered Institute of Personnel and Development (CIPD) MS in International Human Resoure Management For students entering in 2012/3 Awarding Institution: Teahing Institution: Relevant QAA subjet Benhmarking group(s): Faulty: Programme length: Date of speifiation:

More information

GABOR AND WEBER LOCAL DESCRIPTORS PERFORMANCE IN MULTISPECTRAL EARTH OBSERVATION IMAGE DATA ANALYSIS

GABOR AND WEBER LOCAL DESCRIPTORS PERFORMANCE IN MULTISPECTRAL EARTH OBSERVATION IMAGE DATA ANALYSIS HENRI COANDA AIR FORCE ACADEMY ROMANIA INTERNATIONAL CONFERENCE of SCIENTIFIC PAPER AFASES 015 Brasov, 8-30 May 015 GENERAL M.R. STEFANIK ARMED FORCES ACADEMY SLOVAK REPUBLIC GABOR AND WEBER LOCAL DESCRIPTORS

More information

Chapter 1: Introduction

Chapter 1: Introduction Chapter 1: Introdution 1.1 Pratial olumn base details in steel strutures 1.1.1 Pratial olumn base details Every struture must transfer vertial and lateral loads to the supports. In some ases, beams or

More information

On Some Mathematics for Visualizing High Dimensional Data

On Some Mathematics for Visualizing High Dimensional Data On Some Mathematis for Visualizing High Dimensional Data Edward J. Wegman Jeffrey L. Solka Center for Computational Statistis George Mason University Fairfax, VA 22030 This paper is dediated to Professor

More information

AUDITING COST OVERRUN CLAIMS *

AUDITING COST OVERRUN CLAIMS * AUDITING COST OVERRUN CLAIMS * David Pérez-Castrillo # University of Copenhagen & Universitat Autònoma de Barelona Niolas Riedinger ENSAE, Paris Abstrat: We onsider a ost-reimbursement or a ost-sharing

More information

Exploiting Relative Addressing and Virtual Overlays in Ad Hoc Networks with Bandwidth and Processing Constraints

Exploiting Relative Addressing and Virtual Overlays in Ad Hoc Networks with Bandwidth and Processing Constraints Exploiting Relative Addressing and Virtual Overlays in Ad Ho Networks with Bandwidth and Proessing Constraints Maro Aurélio Spohn Computer Siene Department University of California at Santa Cruz Santa

More information

State of Maryland Participation Agreement for Pre-Tax and Roth Retirement Savings Accounts

State of Maryland Participation Agreement for Pre-Tax and Roth Retirement Savings Accounts State of Maryland Partiipation Agreement for Pre-Tax and Roth Retirement Savings Aounts DC-4531 (08/2015) For help, please all 1-800-966-6355 www.marylandd.om 1 Things to Remember Complete all of the setions

More information

i_~f e 1 then e 2 else e 3

i_~f e 1 then e 2 else e 3 A PROCEDURE MECHANISM FOR BACKTRACK PROGRAMMING* David R. HANSON + Department o Computer Siene, The University of Arizona Tuson, Arizona 85721 One of the diffiulties in using nondeterministi algorithms

More information

In this chapter, we ll see state diagrams, an example of a different way to use directed graphs.

In this chapter, we ll see state diagrams, an example of a different way to use directed graphs. Chapter 19 State Diagrams In this hapter, we ll see state diagrams, an example of a different way to use direted graphs. 19.1 Introdution State diagrams are a type of direted graph, in whih the graph nodes

More information

TECHNOLOGY-ENHANCED LEARNING FOR MUSIC WITH I-MAESTRO FRAMEWORK AND TOOLS

TECHNOLOGY-ENHANCED LEARNING FOR MUSIC WITH I-MAESTRO FRAMEWORK AND TOOLS TECHNOLOGY-ENHANCED LEARNING FOR MUSIC WITH I-MAESTRO FRAMEWORK AND TOOLS ICSRiM - University of Leeds Shool of Computing & Shool of Musi Leeds LS2 9JT, UK +44-113-343-2583 kia@i-maestro.org www.i-maestro.org,

More information

An Efficient Network Traffic Classification Based on Unknown and Anomaly Flow Detection Mechanism

An Efficient Network Traffic Classification Based on Unknown and Anomaly Flow Detection Mechanism An Effiient Network Traffi Classifiation Based on Unknown and Anomaly Flow Detetion Mehanism G.Suganya.M.s.,B.Ed 1 1 Mphil.Sholar, Department of Computer Siene, KG College of Arts and Siene,Coimbatore.

More information

Revista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, 1999 469. Surface Charges and Electric Field in a Two-Wire

Revista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, 1999 469. Surface Charges and Electric Field in a Two-Wire Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, 999 469 Surfae Charges and Eletri Field in a Two-Wire Resistive Transmission Line A. K. T.Assis and A. J. Mania Instituto de Fsia Gleb Wataghin'

More information

A Game Theoretical Approach to Gateway Selections in Multi-domain Wireless Networks

A Game Theoretical Approach to Gateway Selections in Multi-domain Wireless Networks 1 A Game Theoretial Approah to Gateway Seletions in Multi-domain Wireless Networks Yang Song, Starsky H.Y. Wong and Kang-Won Lee IBM Researh, Hawthorne, NY Email: {yangsong, hwong, kangwon}@us.ibm.om Abstrat

More information

UNIVERSITY AND WORK-STUDY EMPLOYERS WEB SITE USER S GUIDE

UNIVERSITY AND WORK-STUDY EMPLOYERS WEB SITE USER S GUIDE UNIVERSITY AND WORK-STUDY EMPLOYERS WEB SITE USER S GUIDE September 8, 2009 Table of Contents 1 Home 2 University 3 Your 4 Add 5 Managing 6 How 7 Viewing 8 Closing 9 Reposting Page 1 and Work-Study Employers

More information

A Comparison of Default and Reduced Bandwidth MR Imaging of the Spine at 1.5 T

A Comparison of Default and Reduced Bandwidth MR Imaging of the Spine at 1.5 T 9 A Comparison of efault and Redued Bandwidth MR Imaging of the Spine at 1.5 T L. Ketonen 1 S. Totterman 1 J. H. Simon 1 T. H. Foster 2. K. Kido 1 J. Szumowski 1 S. E. Joy1 The value of a redued bandwidth

More information

tr(a + B) = tr(a) + tr(b) tr(ca) = c tr(a)

tr(a + B) = tr(a) + tr(b) tr(ca) = c tr(a) Chapter 3 Determinant 31 The Determinant Funtion We follow an intuitive approah to introue the efinition of eterminant We alreay have a funtion efine on ertain matries: the trae The trae assigns a numer

More information

Customer Efficiency, Channel Usage and Firm Performance in Retail Banking

Customer Efficiency, Channel Usage and Firm Performance in Retail Banking Customer Effiieny, Channel Usage and Firm Performane in Retail Banking Mei Xue Operations and Strategi Management Department The Wallae E. Carroll Shool of Management Boston College 350 Fulton Hall, 140

More information

Learning Curves and Stochastic Models for Pricing and Provisioning Cloud Computing Services

Learning Curves and Stochastic Models for Pricing and Provisioning Cloud Computing Services T Learning Curves and Stohasti Models for Priing and Provisioning Cloud Computing Servies Amit Gera, Cathy H. Xia Dept. of Integrated Systems Engineering Ohio State University, Columbus, OH 4310 {gera.,

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

International Journal of Supply and Operations Management. Mathematical modeling for EOQ inventory system with advance payment and fuzzy Parameters

International Journal of Supply and Operations Management. Mathematical modeling for EOQ inventory system with advance payment and fuzzy Parameters nternational Journal of Supply and Operations Management JSOM November 0, Volume, ssue 3, pp. 60-78 SSN-Print: 383-359 SSN-Online: 383-55 www.ijsom.om Mathematial modeling for EOQ inventory system with

More information

Parametric model of IP-networks in the form of colored Petri net

Parametric model of IP-networks in the form of colored Petri net Parametri model of IP-networks in the form of olored Petri net Shmeleva T.R. Abstrat A parametri model of IP-networks in the form of olored Petri net was developed; it onsists of a fixed number of Petri

More information

Draft Notes ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer

Draft Notes ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Draft Notes ME 608 Numerial Methods in Heat, Mass, and Momentum Transfer Instrutor: Jayathi Y. Murthy Shool of Mehanial Engineering Purdue University Spring 2002 1998 J.Y. Murthy and S.R. Mathur. All Rights

More information

WORKFLOW CONTROL-FLOW PATTERNS A Revised View

WORKFLOW CONTROL-FLOW PATTERNS A Revised View WORKFLOW CONTROL-FLOW PATTERNS A Revised View Nik Russell 1, Arthur H.M. ter Hofstede 1, 1 BPM Group, Queensland University of Tehnology GPO Box 2434, Brisbane QLD 4001, Australia {n.russell,a.terhofstede}@qut.edu.au

More information

JEFFREY ALLAN ROBBINS. Bachelor of Science. Blacksburg, Virginia

JEFFREY ALLAN ROBBINS. Bachelor of Science. Blacksburg, Virginia A PROGRAM FOR SOLUtiON OF LARGE SCALE VEHICLE ROUTING PROBLEMS By JEFFREY ALLAN ROBBINS Bahelor of Siene Virginia Polytehni Institute and State University Blaksburg, Virginia 1974 II Submitted to the Faulty

More information

Behavior Analysis-Based Learning Framework for Host Level Intrusion Detection

Behavior Analysis-Based Learning Framework for Host Level Intrusion Detection Behavior Analysis-Based Learning Framework for Host Level Intrusion Detetion Haiyan Qiao, Jianfeng Peng, Chuan Feng, Jerzy W. Rozenblit Eletrial and Computer Engineering Department University of Arizona

More information

EE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits

EE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits EE 201 EECTRIC CIUITS ECTURE 25 The material overed in this leture will be as follows: Natural, Fored and Complete Responses of First Order Ciruits Step Response of First Order Ciruits Step Response of

More information

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders A Robust Optimization Approah to Dynami Priing and Inventory Control with no Bakorders Elodie Adida and Georgia Perakis July 24 revised July 25 Abstrat In this paper, we present a robust optimization formulation

More information

VOLUME 13, ARTICLE 5, PAGES 117-142 PUBLISHED 05 OCTOBER 2005 DOI: 10.4054/DemRes.2005.13.

VOLUME 13, ARTICLE 5, PAGES 117-142 PUBLISHED 05 OCTOBER 2005  DOI: 10.4054/DemRes.2005.13. Demographi Researh a free, expedited, online journal of peer-reviewed researh and ommentary in the population sienes published by the Max Plank Institute for Demographi Researh Konrad-Zuse Str. 1, D-157

More information

Algorithm of Removing Thin Cloud-fog Cover from Single Remote Sensing Image

Algorithm of Removing Thin Cloud-fog Cover from Single Remote Sensing Image Journal of Information & Computational Siene 11:3 (2014 817 824 February 10, 2014 Available at http://www.jois.om Algorithm of Removing Thin Cloud-fog Cover from Single Remote Sensing Image Yinqi Xiong,

More information

CHAPTER 14 Chemical Equilibrium: Equal but Opposite Reaction Rates

CHAPTER 14 Chemical Equilibrium: Equal but Opposite Reaction Rates CHATER 14 Chemial Equilibrium: Equal but Opposite Reation Rates 14.1. Collet and Organize For two reversible reations, we are given the reation profiles (Figure 14.1). The profile for the onversion of

More information

There are only finitely many Diophantine quintuples

There are only finitely many Diophantine quintuples There are only finitely many Diophantine quintuples Andrej Dujella Department of Mathematis, University of Zagreb, Bijenička esta 30 10000 Zagreb, Croatia E-mail: duje@math.hr Abstrat A set of m positive

More information

Electrician'sMathand BasicElectricalFormulas

Electrician'sMathand BasicElectricalFormulas Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In. 1.888.NEC.CODE www.mikeholt.om Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial

More information

Heat Generation and Removal in Solid State Lasers

Heat Generation and Removal in Solid State Lasers Chapter 1 Heat Generation and Removal in Solid State Lasers V. Ashoori, M. Shayganmanesh and S. Radmard Additional information is available at the end of the hapter http://dx.doi.org/10.577/63 1. Introdution

More information

To the Graduate Council:

To the Graduate Council: o the Graduate Counil: I am submitting herewith a dissertation written by Yan Xu entitled A Generalized Instantaneous Nonative Power heory for Parallel Nonative Power Compensation. I have examined the

More information

On the Notion of the Measure of Inertia in the Special Relativity Theory

On the Notion of the Measure of Inertia in the Special Relativity Theory www.senet.org/apr Applied Physis Researh Vol. 4, No. ; 1 On the Notion of the Measure of Inertia in the Speial Relativity Theory Sergey A. Vasiliev 1 1 Sientifi Researh Institute of Exploration Geophysis

More information

Chapter 38A - Relativity. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 38A - Relativity. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 38A - Relativity A PowerPoint Presentation by Paul E. Tippens, Professor of Physis Southern Polytehni State University 27 Objetives: After ompleting this module, you should be able to: State and

More information

Granular Problem Solving and Software Engineering

Granular Problem Solving and Software Engineering Granular Problem Solving and Software Engineering Haibin Zhu, Senior Member, IEEE Department of Computer Siene and Mathematis, Nipissing University, 100 College Drive, North Bay, Ontario, P1B 8L7, Canada

More information

Another Look at Gaussian CGS Units

Another Look at Gaussian CGS Units Another Look at Gaussian CGS Units or, Why CGS Units Make You Cool Prashanth S. Venkataram February 24, 202 Abstrat In this paper, I ompare the merits of Gaussian CGS and SI units in a variety of different

More information

Computational Analysis of Two Arrangements of a Central Ground-Source Heat Pump System for Residential Buildings

Computational Analysis of Two Arrangements of a Central Ground-Source Heat Pump System for Residential Buildings Computational Analysis of Two Arrangements of a Central Ground-Soure Heat Pump System for Residential Buildings Abstrat Ehab Foda, Ala Hasan, Kai Sirén Helsinki University of Tehnology, HVAC Tehnology,

More information

A Survey of Usability Evaluation in Virtual Environments: Classi cation and Comparison of Methods

A Survey of Usability Evaluation in Virtual Environments: Classi cation and Comparison of Methods Doug A. Bowman bowman@vt.edu Department of Computer Siene Virginia Teh Joseph L. Gabbard Deborah Hix [ jgabbard, hix]@vt.edu Systems Researh Center Virginia Teh A Survey of Usability Evaluation in Virtual

More information