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in her own group when she arrives, or we can arrange the ﬁrst n − 1 people into n−1 k groups, then pick a group for the last person to join. There are different k−1 ways to arrange the guests in the ﬁrst case, and k different possibilities in the second. Therefore, n−. (2.96) For example, to partition the set {a, b, c...
eger n > 2 where ,bn = 0. The ﬁrst few complementary Bell numbers are 1, −1, 0, 1, 1, −2, −9, −9, 50, 267, 413, −2180, −17731, −50533, and 110176. (a) Describe a combinatorial interpretation of ,bn. (b) Use (2.109) to determine a recurrence for the complementary Bell numbers. Then determine a closed form for their expo...
the offensive line, each of whom has a different jersey number, how many formations of linemen are possible? 2.9 Stable Marriage How do I love thee? Let me count the ways. — Elizabeth Barrett Browning, Sonnet 43, Sonnets from the Portuguese Most of the problems we have considered in this chapter are questions in enume...
e obtain a different stable matching: (Walda, Mort), (Wanda, Milt), (Wendy, Mack), (Wilma, Mark), and (Winny, Marv). Only Winny and Marv are paired together in both matchings; everyone else receives a higher-ranked partner precisely when he or she is among the proposers. Table 2.9 illustrates this for the two different...
bset of n men as potential partners, with the remaining men deemed unacceptable, and suppose each of the 2.9 Stable Marriage 257 men rank the women in the same way. Then there exists a subset X0 of the women and a subset Y0 of the men such that every stable matching of the n men and n women leaves precisely the members...
3. Using the reﬁnement obtained by replacing each = in these lists with >, the Gale-Shapley algorithm produces the following matching when the men propose: (Gatsby, Apolonia), (Hawkeye, Fayaway), (Ishmael, Cora), (Kino, Daisy). (2.131) However, if we reverse the order of Apolonia and Cora in the reﬁnement of Gatsby’s l...
ections of points in the plane. The latter problem leads us again to Ramsey’s theorem, and we prove this statement in a more general form than what we described in Section 1.8. (Ramsey theory is developed further in Chapter 3.) In particular, we establish some of the bounds on the Ramsey numbers R(p, q) that were cited...
o avoid degenerate cases, we will assume in this section that our given collection of points is in general position, which means that no three points lie on the same line, or, using the term from the previous section, that each line connecting two of the points is ordinary. Thus, the convex hull of a set of four points...
nd Szekeres [95] described a method for placing 2n−2 points in the plane in general position so that no convex n-gon appears. Their construction was later corrected by Kalbﬂeisch and Stanton [172]. Thus, certainly ES(n) ≥ 2n−2 + 1 for n ≥ 7. For additional information on this problem and many of its generalizations, se...
n [8]. 2.11 References 279 The history of Stirling numbers, the notations developed for them, and many interesting identities they satisfy are discussed by Knuth in [177]. Rhyming schemes, as in Exercise 7d of Section 2.8.3 and Exercise 8 of Section 2.8.4, are analyzed by Riordan [237]. Stirling set numbers arise in a ...
nstruct a path as follows. Let r be the ﬁrst element of the path. There are inﬁnitely many vertices in T above r. Each of these vertices is either in L1 or above a unique vertex in L1. Map each of the vertices above r to the vertex of L1 that it is equal to or above. We have mapped inﬁnitely many vertices (pigeons) to ...
3 and 3.4 can be modiﬁed to prove the following theorems. Theorem 3.5 (Inﬁnite Ramsey’s Theorem). For all n ∈ N and c ∈ N, ω → (ω)n c . 288 3. Inﬁnite Combinatorics and Graphs Proof. By induction on n. Exercise 2 gives hints. Theorem 3.6 (Finite Ramsey’s Theorem). For all k, n, c ∈ N, there is an m ∈ N such that m → (k...
these two approaches. 2. Empty set axiom: ∅ has no elements. Formally, ∀x(x /∈ ∅). The empty set has some unusual containment properties. For example, it is a subset of every set. Theorem 3.9. ∅ is a subset of every set. Formally, ∀t(∅ ⊂ t). 3.3 ZFC 293 Proof. The proof relies on the mathematical meaning of implicatio...
. 9. Regularity axiom: Every nonempty set x contains an element y such that x ∩ y = ∅. Formally, ∀x(x = ∅ → ∃y(y ∈ x ∧ x ∩ y = ∅)). 3.3 ZFC 297 The idea here is that ∈ can be viewed as a partial ordering on any set by letting x < y mean x ∈ y. The regularity axiom says that every set has a minimal element in this order...
element of T . Since the elements of T are disjoint, Y is an SDR for T . To prove the converse, suppose T is a family of disjoint nonempty sets. Let Y be an SDR for T . Then Y ⊂ ∪T , and Y has exactly one element in common with each element of T , as required by AC. What if T is not disjoint? For ﬁnite families of set...
We just prove that Z is countable. Not every inﬁnite set is countable, as shown by the following theorem of Cantor. Theorem 3.18 (Cantor’s Theorem). For any set X, X ≺ P(X). In particular, N ≺ P(N). Proof. Deﬁne f : X → P(X) by setting f (t) = {t} for each t ∈ X. Since f is one-to-one, it witnesses that X P(X). It rem...
this property as part of the deﬁnition of an ordinal number. A set X is transitive if for all y ∈ X, if x ∈ y then x ∈ X. A transitive set that is well-ordered by ≤∈ is called an ordinal. The ordinals have some interesting properties. Theorem 3.25. Suppose X is a set of ordinals and α and β are ordinals. Then the foll...
if there is a cardinal λ < κ and a function f : λ → κ such that ∪α<λf (α) = κ. (Remember, κ is transitive, so if f (α) ∈ κ, then f (α) ⊂ κ.) As an example, if we deﬁne f : ℵ0 → ℵω by f (n) = ℵn, then ∪α<ℵ0 f (α) = ℵω, showing that ℵω is singular. Any inﬁnite cardinal number that is not singular is called regular. One ...
n C, let xC be an upper bound of C that is not an element of C. Call C a γ-chain if for every p ∈ C, x{y∈C \| y<p} = p. Use the union of the γ-chains to derive a contradiction.) (b) Prove that 2 implies 3. (Hint: Fix a set X to well-order. Let P be the set of all one-to-one maps from ordinals to subsets of X. P is parti...
riﬁcation of the inﬁnity axiom relies on κ being larger than ℵ0, since every set in Lℵ0 is ﬁnite. The veriﬁcation of the power set axiom is particularly tricky and relies on the assumption that κ is a regular limit cardinal. Now we can ﬁnish our proof of the unprovability of the existence of a weakly inaccessible cardi...
ue. Thus G contains no monochromatic subgraph of size ℵ1. To prove the last statement in the theorem, we note that by Theorem 3.18 and Corollary 3.23, N ≺ R. Thus ℵ0 < \|R\|, and so \|R\| ≥ ℵ1. Since there is a way to color a graph with \|R\| vertices so that no ℵ1-sized monochromatic subgraphs exist, we can certainly do the...
as of T will appear in U , and every formula of T will be true of the objects in U . Suppose that (M, W ) is a society satisfying item 1 in the theorem. For each m ∈ M , if W (m) = {w1, . . . , wn} is the set of women on m’s list, add the formula f (m) = w1 ∨ · · · ∨ f (m) = wn to T . For every pair m1, m2 ∈ M with m1 ...
to cases where β is a successor ordinal and where β is a limit ordinal. Like the standard induction scheme, transﬁnite induction can be viewed 3.8 Inﬁnite Marriage Problems 333 as a consequence of a least element principle. If θ(β) fails for some β ≤ α, then the set {β ≤ α \| ¬θ(β)} exists. This set is well ordered, and...
) less than ω1, and each man’s list consists of those women whose indices are strictly less than his own. Since mω is the man of lowest index (poor guy), he has the shortest list. Even he has an inﬁnitely long list of potential wives. In fact, every man’s list contains exactly ℵ0 women. We will show that (M, W ) is not...
ble” means that a result very like Theorem 3.41 holds for all societies of size κ. There is some variation from the theorem. The third clause requires espousability of all small subsocieties where Theorem 3.41 just speciﬁes that there is a sufﬁciently large number of women. This modiﬁcation is necessary, since the unes...
kk regressive values on [E]k. In the statement of the proposition, p, n, and k represent natural numbers. Here, the notation [n]k denotes the set of all ordered k-tuples with coordinates in {0, 1, 2, . . . , n − 1}. If x is a k-tuple, then \|x\| is the maximum coordinate of x and min(x) is the minimum coordinate. For exa...
ious way to handle functions on Nk and subsets of Nk. The computable functions are sometimes called Turing computable functions or general recursive functions. The study of these functions is called computability theory or recursion theory. Good introductory texts include [31] and [259], and [120] is a survey of comput...
htful article [173]. For the ﬁnal word in many matters of logic, one can consult Kleene’s blue bible [176] or the more accessible text of Mendelson [201]. Finally, other treatments of inﬁnite graphs and combinatorics can be found in the books of Ore [218] and Diestel [74]. References [1] [2] R. Aharoni, C. St. J. A. Na...
. Combin. Theory Ser. A 114 (2007), no. 7, 1332–1349. [72] W. Degen, Pigeonhole and choice principles, MLQ Math. Log. Q. 46 (2000), no. 3, 313–334. [73] K. J. Devlin, Constructibility, Perspect. Math. Logic, Springer-Verlag, Berlin, 1984. [74] R. Diestel, Graph Decompositions, Oxford Sci. Publ., Oxford Univ. Press, New...
no. 1, 31–38. [142] J. Guare, Six Degrees of Separation: A Play, Random House, New York, 1990. [143] D. Gusﬁeld and R. W. Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989. [144] H. Hadwiger and H. Debrunner, Combinatorial Geometry in the Plane, Holt, Rinehart & Win- ston, Ne...
ective choice functions for countable families, J. Combin. Theory Ser. B 21 (1976), no. 1, 40–46. [224] G. P´olya, Kombinatorische Anzahlbestimmungen f¨ur Gruppen, Graphen, und chemische Verbindungen, Acta Math. 68 (1937), 145–254; English transl. in G. P´olya and R. C. Read, Combinatorial Enumeration of Groups, Graphs...
127, 277, 278 alephs, 308 ℵ0, ℵ1, . . . , 312 Alexander the Great, 52 Alice in Wonderland, 1 Alma, Alabama, 144, 148 Amarillo, Texas, 137 Anacreontea, 336 analytical sets, 351 Anastasia, 111 Anderson, Poul, 202 Andrews, George E., 225, 278 Anquetil, Jacques, 200 Anthony and Cleopatra, 83 anthracene, 207 anvil salesman,...
n, Paul, 127 Hollywood Graph, 27 hopscotch, 181 Horton, Joseph D., 277 Houston, Whitney, 343 humuhumunukunukuapua’a, 150 hungry fraternity brother, 235 math major, 177 Hunting of the Snark, The, 109 hydroxyl group, 206 hypergraph, 4 ichthyologists, 150 icosahedron, 81 Icosian Game, The, 60 incidence matrix, 48 of verte...
, 264, 274, 279, 280 Tarjan, Robert E., 277 Tarski, Alfred, 325 Tarsy, Michael, 275 tennis, 231 termination argument, 265 Tesman, Barry, 126 tetrahedron, 81 tetramethylnaphthalene, 206 tetraphenylmethane, 208 Texas cities, 137, 150, 249, 254 lottery, see lottery thistle, 217 Thomas, Robin, 95 Thompson, Emma, 27 Thornhi...
= 0 − δS δx + ˙x · d dt ∂L ∂ ˙x + ¨x · ˙x · d dt ∂L ∂ ˙x ∂L ∂ ˙x So we have If ∂L ∂t = 0, then d dt L − ˙x · ∂L ∂ ˙x = ∂L ∂t . ˙x · ∂L ∂ ˙x − L = E for some constant E. For example, for one particle, E = m\| ˙x\|2 − 1 2 m\| ˙x\|2 + V = T + V = total energy. Example. Consider a central force ﬁeld F = −∇V , where V = V (r) ...
(Minimal surfaces in E3). This is a natural generalization of geodesics. A minimal surface is a surface of least area subject to some boundary conditions. Suppose that (x, y) are good coordinates for a surface S, where (x, y) takes values in the domain D ⊆ R2. Then the surface is deﬁned by z = h(x, y), where h is the ...
ρ(t) ˙ξ2 + σ(t)ξ2 dt, where ρ(t) = ∂2f ∂ ˙x2 x=x0 , σ(t) = ∂2f ∂x2 − d dt ∂2f ∂x∂ ˙x . x=x0 Assume ρ(t) > 0 for α < t < β (the strong Legendre condition) and assume boundary conditions ξ(α) = ξ(β) = 0. When is this suﬃcient for δ2F > 0? First of all, notice that for any smooth function w(x), we have 0 = β α (wξ2) dt si...
x x 3.2 Linear Maps Deﬁnition (Domain, codomain and image of map). Consider sets A and B and mapping T : A → B such that each x ∈ A is mapped into a unique x = T (x) ∈ B. A is the domain of T and B is the co-domain of T . Typically, we have T : Rn → Rm or T : Cn → Cm. Deﬁnition (Linear map). Let V, W be real (or comple...
s of the decomposition will correspond to diﬀerent types of stresses. 3.4.4 Matrix inverse Deﬁnition (Inverse of matrix). Consider an m × n matrix A and n × m matrices B and C. If BA = I, then we say B is the left inverse of A. If AC = I, then we say C is the right inverse of A. If A is square (n × n), then B = B(AC) =...
f determinants with a sensible choice of i to evaluate det(A). Example. Consider 1 a a2 b2 b 1 c2 c 1 . Row 1 - row 2 gives 0 a − b a2 − b2 1 1 b2 c2 b c = (a − b b2 c2 . Do row 2 - row 3. We obtain (a − b)(b − c c2 c . Row 1 - row 2 gives (a − b)(b − c)(a − c c2 = (a − b)(b − c)(a − c). 37 4 Matrices and linear equati...
e kernel now is a plane. n(A) = 2. (We also have the trivial case where r(A) = 0, we have the zero mapping and the kernel is R3) 4.5.2 Linear mapping view of Ax = 0 In the general case, consider a linear map α : Rn → Rn x → x = Ax. The kernel k(A) = {x ∈ Rn : Ax = 0} has dimension n(A). (i) If n(A) = 0, then A(e1), A(e...
envalues since rotation changes the direction of any other vector. The other eigenvalues turn out to be e±iθ. If θ = 0, there are 3 distinct eigenvalues and the eigenvectors form a basis of C3. 49 5 Eigenvalues and eigenvectors IA Vectors and Matrices (vi) Consider a shear 1 µ 0 1 The characteristic equation is (1 − λ)...
are −2, −2, −2 and the eigenvectors are w =   1 0 . Write u = (A − λI)w =  0   −1 −1 −1 −1 −2 − 56   1 0 . Pick  1 . Note that    ,  −1 1 0  −1 −1 −2   5 Eigenvalues and eigenvectors IA Vectors and Matrices Au = −2u. We also have Aw = u − 2w. Form a basis {u, w, v}, where v is another eigenvector lin...
bT x + c = 0 with S symmetric. Since S is real and symmetric, we can diagonalize it using S = QDQT with D diagonal. We write x = QT x and b = QT b. So we have (x)T Dx + (b)T x + c = 0. If S is invertible, i.e. with no zero eigenvalues, then write x = x + 1 2 D−1b which shifts the origin to eliminate the linear term (b...
). We write r(u, v) for the point labelled by (u, v). Example. Let S be part of a sphere of radius a with 0 ≤ θ ≤ α. α We can then label the points on the spheres by the angles θ, ϕ, with r(θ, ϕ) = (a cos ϕ sin θ, a sin θ sin ϕ, a cos θ) = aer. We restrict the values of θ, ϕ by 0 ≤ θ ≤ α, 0 ≤ ϕ ≤ 2π, so that each point...
curvature of a curve through P , denoted κmin and κmax respectively. Deﬁnition (Gaussian curvature). The Gaussian curvature of a surface at a point P is K = κminκmax. Theorem (Theorema Egregium). K is intrinsic to the surface S. It can be expressed in terms of lengths, angles etc. which are measured entirely on the sur...
P ∂y dA. Proposition. Green’s theorem ⇒ Stokes’ theorem. Proof. Green’s theorem describes a 2D region, while Stokes’ theorem describes a 3D surface r(u, v). Hence to use Green’s to derive Stokes’ we need ﬁnd some 2D thing to act on. The natural choice is the parameter space, u, v. Consider a parametrised surface S = r(...
= V ρ(r, t) dV Then the rate of change of amount of Q in V is dQ dt = ∂ρ ∂t V dV = − ∇ · j dV = − V S j · ds. by divergence theorem. So this states that the rate of change of the quantity Q in V is the ﬂux of the stuﬀ ﬂowing out of the surface. ie Q cannot just disappear but must smoothly ﬂow out. In particular, if V i...
aps do not contribute to the ﬂux since the ﬁeld lines are perpendicular to the normal. Also, the curved surface has area 2πrL. Then by Gauss’ law in integral form, So S E · dS = E(r)2πrL = σL ε0 . E(r) = σ 2πε0r ˆr. Note that the ﬁeld varies as 1/r, not 1/r2. Intuitively, this is because we have one more dimension of “...
ave the same sign. This clearly cannot happen if the sum is 0. Therefore we can only have saddle points. (note we ignored the case where all λi = 0, where this analysis is inconclusive) 11.3 Integral solutions of Poisson’s equations 11.3.1 Statement and informal derivation We want to ﬁnd a solution to Poisson’s equatio...
pRjqRkrεpqr = (det R)εijk = εijk, using results from Vectors and Matrices. (iii) (Physical example) In some substances, an applied electric ﬁeld E gives rise to a current density j, according to the linear relation ji = εijEj, where εij is the conductivity tensor. Note that this relation entails that the resulting curr...
+ Aij, where Sij = 1 2 (Tij + Tji), Aij = 1 2 (Tij − Tji). Here Tij has 9 independent components, whereas Sij and Aij have 6 and 3 independent components, since they must be of the form (SijAija −b −c 0   . The symmetric part can be be further reduced to a traceless part plus an isotropic (i.e. multiple of δij) part:...
(1, 2) + (3, 4, 5) = (1, 2, 3, 4, 5). While this notation might give a valid expression in some computer languages, it is not standard mathematical notation, and we will not use it in this book. In general, it is very important to distinguish between mathematical notation for vectors (which we use) and the syntax of sp...
eral examples. • • • • • Unit vector. eT unit vector gives (or ‘picks out’) the ith element a. i a = ai. The inner product of a vector with the ith standard Sum. 1T a = a1 + ones gives the sum of the elements of the vector. · · · + an. The inner product of a vector with the vector of Average. (1/n)T a = (a1 + + an)/n. ...
xi for i = 1, . . . , n. Vector addition x + y of two n-vectors takes n additions, i.e., xi + yi for i = 1, . . . , n. Computing the inner product xT y = x1y1 + 1 ﬂops, n scalar 1 scalar additions. So scalar multiplication, vector addition, multiplications and n and the inner product of n-vectors require n, n, and 2n 1...
positive. If you must choose just one supplier, how would you do it? Your answer should use vector notation. A (highly paid) consultant tells you that you might do better (i.e., get a better total cost) by splitting your order into two, by choosing two suppliers and ordering (1/2)q (i.e., half the quantities) from each...
, 1/n) = 1/n. Maximum. The maximum element of an n-vector x, f (x) = max , x1, . . . , xn} { is not a linear function (except when n = 1). We can show this by a coun1, 1), α = 1/2, β = 1/2. terexample for n = 2. Take x = (1, Then 1), y = ( − − f (αx + βy) = 0 = αf (x) + βf (y) = 1. Aﬃne functions. A linear function plu...
the n-vector x represents a feature vector. The aﬃne function of x given by ˆy = xT β + v, (2.7) where β is an n-vector and v is a scalar, is called a regression model. In this context, the entries of x are called the regressors, and ˆy is called the prediction, since the regression model is typically an approximation...
2, − 2), or state that − Justify your answers. 2.6 Questionnaire scoring. A questionnaire in a magazine has 30 questions, broken into two sets of 15 questions. Someone taking the questionnaire answers each question with ‘Rarely’, ‘Sometimes’, or ‘Often’. The answers are recorded as a 30-vector a, with ai = 1, 2, 3 if q...
or that are at least a in absolute 48 3 Norm and distance Figure 3.1 The norm of the displacement b points with coordinates a and b. − a is the distance between the value. The right-hand side is the inverse square of the ratio of a to rms(x). It says, for example, that no more than 1/25 = 4% of the entries of a vector ...
me. The standard deviation of an n-vector x is deﬁned as the RMS value of the de-meaned vector x − avg(x)1, i.e., std(x) = (x1 − avg(x))2 + + (xn − avg(x))2 . · · · n 3.3 Standard deviation 53 This is the same as the RMS deviation between a vector x and the vector all of whose entries are avg(x). It can be written usin...
th dimension 100. The angle is a symmetric function of a and b: We have (a, b) = (b, a). The angle is not aﬀected by scaling each of the vectors by a positive scalar: We have, for any vectors a and b, and any positive numbers α and β, (αa, βb) = (a, b). Acute and obtuse angles. Angles are classiﬁed according to the sig...
er term can be ignored, so the ﬂop count is 3kn. The order of ﬁnding the nearest neighbor in a collection of k n-vectors is kn. x − − 64 3 Norm and distance Exercises 3.1 Distance between Boolean vectors. Suppose that x and y are Boolean n-vectors, which means that each of their entries is either 0 or 1. What is their ...
n by the T vector r. This asset has mean return µ and risk σ, which we assume is positive. We also consider cash as an asset, with return vector µrf 1, where µrf is the cash interest rate per period. Thus, we model cash as an asset with return µrf and zero risk. (The superscript in µrf stands for ‘risk-free’.) We will ...
the rainfall data, so they have a typical range between vector xi summarizes the annual weather pattern in county i. A clustering algorithm can be used to cluster the counties into k groups that have similar weather patterns, called weather zones. This clustering can be shown on a map, and used to recommend landscape ...
earest representative. 2. Update representatives. For each group j = 1, . . . , k, set zj to be the mean of the vectors in group j. One iteration of the k-means algorithm is illustrated in ﬁgure 4.2. Comments and clariﬁcations. • • Ties in step 1 can be broken by assigning xi to the group associated with one of the clo...
elease 0.015 song 0.014 music 0.011 single 0.010 record 0.009 band perform 0.007 0.007 0.007 tour chart 0.023 game season 0.020 team 0.018 0.017 win 0.014 player 0.013 play 0.010 league 0.010 ﬁnal 0.008 score 0.007 record series season episode character ﬁlm television cast announce release appear 0.029 0.027 0.013 0.01...
ortant role in the sequel. 5.1 Linear dependence A collection or list of n-vectors a1, . . . , ak (with k if ≥ 1) is called linearly dependent β1a1 + · · · + βkak = 0 holds for some β1, . . . , βk that are not all zero. In other words, we can form the zero vector as a linear combination of the vectors, with coeﬃcients ...
ly independent collection a1, . . . , ak of 1-vectors. = 0. This means that every element ai of the collection can be We must have a1 expressed as a multiple ai = (ai/a1)a1 of the ﬁrst element a1. This contradicts linear independence unless k = 1. Next suppose n 2 and the independence-dimension inequality holds for di1...
qj = 0 for ˜qj for j < i.) Since qi = (1/ 1. So assertion 2 holds for i. )˜qi, we have qT ˜qi It is immediate that ai is a linear combination of q1, . . . , qi: − ai = ˜qi + (qT = (qT 1 ai)q1 + 1 ai)q1 + + (qT i − 1ai)qi 1ai)qi qT i − qi. ˜qi · · · From step 1 of the algorithm, we see that ˜qi is a linear combination ...
    1 1 1 ... 1 (The vector ai has its ﬁrst i entries equal to one, and the remaining entries zero.) Describe what happens when you run the Gram–Schmidt algorithm on this list of vectors, i.e., say what q1, . . . , qn are. Is a1, . . . , an a basis? 104 5 Linear independence 5.7 Running Gram–Schmidt algorithm twic...
number of objects with ﬁrst attribute i and second attribute j. (This is the analog of a count n-vector, that records the counts of one attribute in a collection.) For example, a population of college students can be described by a 4 50 matrix, with the i, j entry the number of students in year i of their studies, fro...
trix are also interesting. The jth column of A, which is an N -vector, gives the number of times word j appears in the corpus of N documents. × Data matrix. A collection of N n-vectors, for example feature n-vectors associated with N objects, can be given as an n N matrix whose N columns are the vectors, as described o...
gree n 1, at the points t1, . . . , tm), named for the mathematician Alexandre-Th´eophile Vandermonde. − • • Total price from multiple suppliers. Suppose the m n matrix P gives the prices of n goods from m suppliers (or in m diﬀerent locations). If q is an n-vector of quantities of the n goods (sometimes called a baske...
n. Potential customers are divided into m market segments, which are groups of customers with similar demographics, e.g., college educated women aged 25–29. A company markets its products by purchasing advertising in a set of n channels, i.e., speciﬁc TV or radio shows, magazines, web sites, blogs, direct mail, and so ...
nd a reﬂection through a line that makes an angle of π/4 radians with the horizontal line. Projection onto a line. The projection of the point x onto a set is the point in the set that is closest to x. Suppose y is the projection of x onto the line that passes through the origin, inclined θ radians with respect to hori...
ghtward ﬂow on edge 2 passes through node 4, where ﬂow is conserved, and proceeds up on edge 5 to node 3. The one unit of excess ﬂow arriving at node 3 is removed as external ﬂow. Node potentials. A graph is also useful when we focus on the values of some quantity at each graph vertex or node. Let v be an n-vector, oft...
computed far faster, using a so-called fast convolution algorithm. By exploiting the special structure of the convolution equations, this algorithm can compute the convolution of an n-vector and an m-vector in around 5(m + n) log2(m + n) ﬂops, and with no additional memory requirement beyond the original m + n numbers...
rn to the normal height once the rain stops. 7.15 Channel equalization. We suppose that u1, . . . , um is a signal (time series) that is transmitted (for example by radio). A receiver receives the signal y = c u, where the n-vector c is called the channel impulse response. (See page 138.) In most applications n is smal...
rl Gustav Jacob Jacobi. As in the scalar-valued case, Taylor approximation is sometimes written with a second argument as ˆf (x; z) to show the point z around which the approximation is made. Evidently the Taylor series approximation ˆf is an aﬃne function of x. (It is often called a linear approximation of f , even th...
ple, a pollutant). We also have a source (or exogenous) ﬂow si at each node, with si > 0 meaning that an exogenous ﬂow is injected into node i, and si < 0 means that an exogenous ﬂow is removed from node i. (In some contexts, a node where ﬂow is removed is called a sink.) In a thermal system, the sources represent ther...
that gives its concentration of n diﬀerent constituents, such as speciﬁc hydrocarbons. Find a set of linear equations on the blending coeﬃcients, Aθ = b, that expresses the requirement that the blended crude oil achieves a target set of constituent concentrations, given by the n-vector ctar. (Include the condition that...
re they are called (vector) auto-regressive models. When K = 1, the Markov model (9.3) is the same as a linear dynamical system (9.1). When K > 1, the Markov model (9.3) can be reduced to a standard linear dynamical system (9.1), with an appropriately chosen state; see exercise 9.4. Simulation. If we know the dynamics ...
he form (9.2), with input fk and dynamics and input matrices A = 1 0 1 − , h hη/m B = . 0 h/m This linear dynamical system gives an approximation of the true motion, due to our approximation (9.5) of the derivatives. But for h small enough, it is accurate. This linear dynamical system can be used to simulate the motion...
t + ut · · · − − − − − − 9.7 Complexity of linear dynamical system simulation. Consider the time-invariant linear dynamical system with n-vector state xt and m-vector input ut, and dynamics xt+1 = Axt + But, t = 1, 2, . . .. You are given the matrices A and B, the initial state x1, and the 1. What is the complexity of ...
er products of pairs of columns of A. Note that a Gram matrix is symmetric, since aT j ai. This can also be seen using the transpose-of-product rule: aT 1 an aT 2 an ... aT n an G = AT A = i aj = aT · · · · · · . . .         · · ·  . GT = (AT A)T = (AT )(AT )T = AT A = G. The Gram matrix will play an important...
he idea that the state is measured, and then (after multiplying by K) fed back into the system, via the input. This leads to a loop, where the state aﬀects the input, and the input aﬀects the (next) state. State feedback is very widely used 17.2.3 we will see methods for choosing or designing an in many applications. (...
s for QR factorization that eﬃciently handle the case when the matrix A is sparse. In this case the matrix Q is stored in a special format that requires much less memory than if it were stored k matrix, i.e., nk numbers. The ﬂop count for these sparse QR as a generic n factorizations is also much smaller than 2nk2. × E...
trix T . 10.21 Integral of product of polynomials. Let p and q be two quadratic polynomials, given by p(x) = c1 + c2x + c3x2, q(x) = d1 + d2x + d3x2. Express the integral J = 1 Give the entries of G (as numbers). 0 p(x)q(x) dx in the form J = cT Gd, where G is a 3 3 matrix. × 10.22 Composition of linear dynamical syste...
or i = 1, . . . , k. Using index range notation, we have i matrix Ri as the submatrix of R containing its ﬁrst i × Ai = A1:n,1:i, Qi = A1:n,1:i, Ri = R1:i,1:i. Show that Ai = QiRi is the QR factorization of Ai. This means that when you compute the QR factorization of A, you are also computing the QR factorization of al...
the inverse. Consider the square system of n linear equations with n variables, Ax = b. If A is invertible, then for any n-vector b, x = A− 1b (11.1) is a solution of the equations. (This follows since A− Moreover, it is the only solution of Ax = b. (This follows since A− of A.) We summarize this very important result ...
re are n/2 such pairs, the total is (n/2)(2n) = n2. Solving linear equations using the QR factorization. The formula (11.3) for the inverse of a matrix in terms of its QR factorization suggests a method for solving a square system of linear equations Ax = b with A invertible. The solution x = A− 1b = R− 1QT b (11.4) ca...
iables (f, s, e):        0 A I R 0 AT The matrix A is the incidence matrix of the graph, and R is the resistance matrix; see page 155.) Assuming the coeﬃcient matrix is invertible, we have  AT This is illustrated with an example in ﬁgure 11.3. The graph is a 100 100 grid, with 10000 nodes, and edges connecting ...
in the (n-vector) state in period t, and A is the n formula gives the state in the next period as a function of the current state. We want to derive a recursion of the form × n dynamics matrix. This 1 = Arevxt, xt − which gives the previous state as a function of the current state. We call this the reverse time linear ...
Ay = c, and Az = d (using your method) for n = 3000 on a computer capable of carrying out 1 Gﬂop/s. Part III Least squares Chapter 12 Least squares In this chapter we look at the powerful idea of ﬁnding approximate solutions of over-determined systems of linear equations by minimizing the sum of the squares of the err...
identity u + v 2 = (u + v)T (u + v) = 2 + u v 2 + 2uT v. The third term in (12.7) is zero: (Ax − Aˆx)T (Aˆx − b) = (x = (x = (x = 0, ˆx)T AT (Aˆx ˆx)T (AT Aˆx ˆx)T 0 − − b) AT b) − − − where we use (AT A)ˆx = AT b (the normal equations) in the third line. With this simpliﬁcation, (12.7) reduces to Ax b − 2 = A(x ˆx) 2...
r 132, or 13.2% of the target values. The views vector is shown in ﬁgure 12.4. Illumination. A set of n lamps illuminates an area that we divide into m regions or pixels. We let li denote the lighting level in region i, so the m-vector l gives the illumination levels across all regions. We let pi denote the power at wh...
and only if the columns of A are linearly independent. × 12.10 Numerical check of the least squares approximate solution. Generate a random 30 10 matrix A and a random 30-vector b. Compute the least squares approximate solution 2. (There may be several ˆx = A†b and the associated residual norm squared ways to do this,...
model. We will focus on a speciﬁc form for the model, which has the form ˆf (x) = θ1f1(x) + + θpfp(x), · · · → where fi : Rn R are basis functions or feature mappings that we choose, and θi are the model parameters that we choose. This form of model is called linear in the parameters, since for each x, ˆf (x) is a lin...
mputational advantage to doing so. The Gram matrix is The 2-vector AT yd is AT A = N 1T xd 1T xd (xd)T xd . AT y = 1T yd (xd)T yd , so we have (using the formula for the inverse of a 2 ˆθ1 ˆθ2 = 1 N (xd)T xd − (1T xd)2 (xd)T xd 1T xd − 2 matrix) × 1T xd N − 1T yd (xd)T yd . Multiplying the scalar term by N 2, and divid...

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