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Index 1201 RL circuit, 338–339 RLC analog, 1022, 1022f Rodrigues formula, 551–554, 720–721 for Hermite ODE, 554 Laguerre polynomials, 889–890 associated, 895 Rodrigues representation, Hermite polynomials, 874 root diagram, 857 root test, Cauchy, 4 rotations groups SO(2) and SO(3), 849–851 in R3, 139–142, 140f exercises... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,201 | 7th_Mathematical_Methods_for_Physicists | 0 |
, 411–413 classes of, 409–411 exercises, 414 nonlinear, 413–414 second-order Sturm-Liouville ordinary differential equations (ODEs), 551 second-rank tensor, 207–208 secular determinant, 302 secular equation, 302, 306 self-adjoint matrices, 108 self-adjoint ODEs boundary conditions, 381 deuteron, 391–393 eigenvalues, 38... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,201 | 7th_Mathematical_Methods_for_Physicists | 0 |
1202 Index simple pendulum, 927–928, 1113–1114, 1113f simple pole, 498 simultaneous diagonalization, 314–315 sine infinite products, 575 integrals in asymptotic series, 580–582 single-electron wave function, 396 single-slit diffraction pattern, 972 singular points, 343–345, 345t essential (irregular), 344 irregular (es... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,202 | 7th_Mathematical_Methods_for_Physicists | 0 |
–801 spherical harmonics, 445, 473, 756 addition theorem for, 797–798 Cartesian representations, 758 Condon-Shortley phase, 758, 760t exercises, 765–766 integrals of three, 803–805 ladder, 779 Laplace expansion, 760–761, 799–801 Laplace series–gravity fields, 762 properties of, 764–765 symmetry of solutions, 762–763 ve... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,202 | 7th_Mathematical_Methods_for_Physicists | 0 |
70–1174 confidence interval, 1176–1178 error propagation, 1165–1168 exercises, 1178–1179 fitting curves to data, 1168–1170 student t distribution, 1174–1176 steepest descent method of, 585 asymptotic form of gamma function, 588–589 exercises, 590–591 factorial function, 588 saddle points, 585–588 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,202 | 7th_Mathematical_Methods_for_Physicists | 0 |
Index 1203 step function, 1013–1014, 1014f Stirling’s expansion, 589 Stirling’s series derivation from Euler-Maclaurin integration formula, 623–624 exercises, 625–626 Stirling’s series, 624 Stirlings formula, 567 Stokes’ theorem, 167–168, 167f , 168f , 193–194 on differential forms, 245–248 STOs, see Slater-type orbita... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,203 | 7th_Mathematical_Methods_for_Physicists | 0 |
equilateral triangle, 817f , 817, 818f of solutions, 762–763 relations, 593–594 T Taylor expansion, 492–494, 493f Taylor series, 560 Taylor’s expansion, 653 binomial theorem, relativistic energy, 35–36 Maclaurin theorem exponential function, 27–28 logarithm, 28–29 tensor analysis, 205–213 addition and subtraction of, 2... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,203 | 7th_Mathematical_Methods_for_Physicists | 0 |
1204 Index triangular symmetry, quantum mechanics of, 829–830 trigonometric form, 904–905 trigonometric functions exploiting periodicity of, 537–538 trigonometric integrals, 69, 522–524 triple scalar product, 128–130, 129f triple vector product, 130 triplet state, 259 Two and three dimension problems, Green’s function,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,204 | 7th_Mathematical_Methods_for_Physicists | 0 |
method, 395–397 exercises, 397 vector analysis reciprocal lattice, 130 rotation of coordinate transformations, 133–135 vector fields, 46, 143 vector integration exercises, 163–164 line integrals, 159–160, 160f surface integrals, 161–162, 161f , 162f volume integrals, 162–163 vector Laplacian, 155–156 vector model, 786–... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,204 | 7th_Mathematical_Methods_for_Physicists | 0 |
Index 1205 overview, 251–253 scalar product, 254–255, 260–261 Schwarz inequality, 257 irrotational, 154–155 matrix representation of, 106–107 multiplication of, 252 orthogonality, 51 physical, 272–273 radius vector, 48 Stokes’ theorem, 167–168, 167f , 168f successive applications of ∇, 153–154 triple product, 130 tripl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/7th_Mathematical_Methods_for_Physicists.pdf | 1,205 | 7th_Mathematical_Methods_for_Physicists | 0 |
Midterm Exam, Advanced Algorithms 2017-2018 • You are only allowed to have a handwritten A4 page written on both sides. • Communication, calculators, cell phones, computers, etc... are not allowed. • Your explanations should be clear enough and in sufficient detail that a fellow student can understand them. In particul... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 0 | MidTerm2018sol | 0 |
1 (consisting of subproblems a-b, 20 pts) Basic questions. This problem consists of two subprob- lems each worth 10 points. 1a (10 pts) Suppose we use the Simplex method to solve the following linear program: maximize 4x1 −x2 −2x3 subject to x1 −x3 + s1 = 1 x1 + s2 = 4 −3x2 + 2x3 + s3 = 4 x1, x2, x3, s1, s2, s3 ≥0 At t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 1 | MidTerm2018sol | 0 |
1b (10 pts) Chef Baker Buttersweet just took over his family business - baking tasty cakes! He notices that he has m different ingredients in various quantities. In particular, he has bi ≥0 kilograms of ingredient i for i = 1, . . . , m. His family cookbook has recipes for n types of mouthwatering cakes. A kilogram of ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 2 | MidTerm2018sol | 0 |
and to iteratively update those weights in a multiplicative manner based on the cost function at each step. Initially, the Hedge method will give a weight w(1) i = 1 for every constraint/expert i = 1, . . . , m (the number m of constraints now equals the number of experts). And at each step t, it will maintain a convex... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 2 | MidTerm2018sol | 0 |
As explained, we associate an expert to each constraint of the covering LP. In addition, we wish to increase the weight of unsatisfied constraints and decrease the weight of satisfied constraints (in a smooth manner depending on the size of the violation or the slack). The Hedge algorithm for covering LPs thus becomes:... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 3 | MidTerm2018sol | 0 |
(Primal) LP Relaxation minimize X S∈T c(S)xS subject to X S∈T :e∈S xS ≥1 for e ∈U xS ≥0 for S ∈T (Dual) maximize X e∈U ye subject to X e∈S ye ≤c(S) for S ∈T ye ≥0 for e ∈U Figure 1. The standard LP relaxation of set cover and its dual. 2 (20 pts) Recall that a set cover instance is specified by a universe U = {e1, . . ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 4 | MidTerm2018sol | 0 |
C = {S ∈T : P e∈S ye = c(S)} is not a set cover (i.e., ∪S∈CS <unk>= U): • Select an element e ∈U that is not covered by any set in C (i.e., choose an element e ∈U \ (∪S∈CS). • Increase ye until one of the dual constraints becomes tight (i.e., P e∈S ye = c(S) for some S /∈C). 3. Return C = {S ∈T : P e∈S ye = c(S)}. This... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 4 | MidTerm2018sol | 0 |
following: X S∈C c(S) = X S∈C X e∈S ye (by the definition of set C) ≤ X S∈T X e∈S ye (because ye’s are non-negative and C ⊆T ) = X e∈U X S<unk>e ye (by rearranging the summations) = X e∈U |{S ∈T : e ∈S}| · ye ≤f · X e∈U ye (f = max e∈U |{S ∈T : e ∈S}|) ≤f · LPOPT (by the weak-duality theorem) ≤f · OPT. (because the LP ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 5 | MidTerm2018sol | 0 |
3 (20 pts) LP-based algorithm for packing knapsacks. Homer, Marge, and Lisa Simpson have decided to go for a hike in the beautiful Swiss Alps. Homer has greatly surpassed Marge’s expectations and carefully prepared to bring n items whose total size equals the capacity of his and his wife Marge’s two knapsacks. Lisa doe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 6 | MidTerm2018sol | 0 |
feasible solution, xiH + xiM = 1 for all i = 1, . . . , n. Otherwise, if xjH + xjM < 1, we would have that (since si > 0 for every item i) 2 · C = n X i=1 sixiH + n X i=1 sixiM = sj(xjH + xjM | {z } <1 ) + n X i=1,i<unk>=j si(xiH + xiM | {z } ≤1 ) < n X i=1 si = 2 · C, which is a contradiction. Now suppose that x∗is an... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 6 | MidTerm2018sol | 0 |
We select ε > 0 to be small enough so that all the values x∗ iH±ε, x∗ iM±ε, x∗ jH±ε si sj , x∗ jM±ε si sj stay in the range [0, 1] (note that, since si > 0 and sj > 0, 0 < si sj < ∞). As shown below, we can verify that the solutions x(1) and x(2) both satisfy the LP constraints, and hence are feasible solutions. For x(... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 7 | MidTerm2018sol | 0 |
4 (20 pts) Balancing degrees. A beautiful result by the Swiss mathematician Leonhard Euler (1707 - 1783) can be stated as follows: Let G = (V, E) be an undirected graph. If every vertex has an even degree, then we can orient the edges in E to obtain a directed graph where the in-degree of each vertex equals its out-deg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 8 | MidTerm2018sol | 0 |
a The solution A′ has value |A′ \ A| = 1 (the number of edges for which the orientation was flipped). Solution: Consider the directed graph G′ = (V, E′) obtained from G by replacing every edge {u, v} ∈E by the two arcs e1 = (u, v) and e2 = (v, u). If e ∈A′, we assign weight we = n2 + 1 to it, otherwise we set we = n2. ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 8 | MidTerm2018sol | 0 |
• To be independent in M2, one can take at most 1 2deg(v) arcs among the set δ+(v) of outgoing arcs for every v: I2 = {F ⊆E′ : |F ∩δ+(v)| ≤1 2deg(v), for all v ∈V } . Let solution S be the maximum weight independent set in the intersection of the two matroids M1, and M2. Now we prove that a solution S is feasible if an... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 9 | MidTerm2018sol | 0 |
5 (20 pts) Comparing algorithms with little communication. Two excellent students, Alice from EPFL and Bob from MIT, have both built their own spam filters. A spam filter is an algorithm that takes as input an email and outputs 1 if the email is spam and 0 otherwise. Alice and Bob now want to compare their two spam fil... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 10 | MidTerm2018sol | 0 |
shared randomness to devise a randomized protocol of the following type: • As a function of a1, a2, . . . , an and the random bits r1, r2, . . ., Alice computes a message m that consists of only 2 bits. She then transmits this 2-bit message m to Bob. • Bob then, as a function of b1, b2, . . . , bn, the message m, and t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 10 | MidTerm2018sol | 0 |
⟨a, r2⟩mod 2, and transmits (x1, x2) to Bob. Bob uses the shared random bits to generate the same vectors r1 and r2, and computes y1 = ⟨b, r1⟩mod 2 and y2 = ⟨b, r2⟩mod 2. If x1 = y1 and x2 = y2, Bob outputs Equal. Otherwise, Bob outputs Not Equal. We prove that the above protocol succeeds with probability at least 2/3.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 10 | MidTerm2018sol | 0 |
a <unk>= b. We first show that Pr[x1 = y1|a <unk>= b] = 1/2. Notice that Pr[x1 = y1|a <unk>= b] = Pr[⟨a, r1⟩mod 2 = ⟨b, r1⟩mod 2|a <unk>= b] = Pr[⟨a −b, r1⟩mod 2 = 0|a <unk>= b]. Let c = a −b. Since a <unk>= b, we have that c <unk>= 0. This means that for at least one index j, cj = ±1. Now fix such j, and suppose that ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/MidTerm2018sol.pdf | 11 | MidTerm2018sol | 0 |
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