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Course: M 220, Fall 2009
School: E. Kentucky
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Word Count: 640

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of Systems first order linear equations (1) Fundamental Matrices x1 (t), . . . , xn (t) form a fundamental set of solutions of x = A(t)x. 1 x1 (t) xn (t) 1 (t) = 1 n xn (t) xn (t) Then , whose columns are the vectors x1 (t), . . . , xn (t), is a fundamental matrix. A fundamental matrix is nonsingular and = A(t). The general solution of the system is c1 x1 (t) + + cn xn (t) or x = (t)c...

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of Systems first order linear equations (1) Fundamental Matrices x1 (t), . . . , xn (t) form a fundamental set of solutions of x = A(t)x. 1 x1 (t) xn (t) 1 (t) = 1 n xn (t) xn (t) Then , whose columns are the vectors x1 (t), . . . , xn (t), is a fundamental matrix. A fundamental matrix is nonsingular and = A(t). The general solution of the system is c1 x1 (t) + + cn xn (t) or x = (t)c where c = (c1 , , cn )T . If the initial condition is x(t0 ) = x0 , (t0 )c = x0 and c = -1 (t0 )x0 since (t0 ) is nonsingular. Thus, the solution of the IVP becomes x = (t)-1 (t0 )x0 . The special fundamental matrix (t), whose columns are the vectors x1 (t), . . . , xn (t), with the initial conditions xj (t0 ) = ej , i.e, 1 0 0 0 1 0 (t0 ) = = I; 0 0 1 now the solution of the IVP becomes x = (t)x0 . Note (t) = (t)-1 (t0 ). exp(At) = (t). Transformation into a diagonal matrix: If 1 , . . . , n are all different, and 1 , . . . , n are linearly independent eigenvectors, form the matrix T as 1 n 1 1 T = . n 1 n n det T = 0, AT = T D where D= 1 0 0 0 2 0 0 0 n is a diagonal matrix whose diagonal entries are the eigenvalues of A. D = T -1 AT : similarity transformation, A: diagonalizable. If A has fewer than n linearly indeoendent eigenvectors, A is not diagonalizable. Consider x = Ax. Define x = T y and then we have y = T -1 AT y = Dy whose fundamental matrix is the diagonal matrix e1 t 0 0 0 e2 t 0 Q(t) = exp(Dt) = . 0 0 en t A fundamental matrix for x = Ax is = T Q. (2) Repeated Eigenvalues Suppose r = is a k-fold root of det(A - rI) = 0; is an eigenvalue of algebraic multiplicity k of A. If there are k linearly indenpendent eigenvectors, 1 , . . . , k , then x1 = 1 et , . . . , xk = k et are linearly indenpendent so solutions it makes no difference. If there are fewer than k linearly indenpendent eigenvectors, it is necessary to find other solutions of a different form. Suppose r = is a double eigenvalue of A and there is only one corresponding eigenvector . Then one solution is x1 = et , where (A - I) = 0 and a second solution is x2 = tet + et , where (A - I) = . is a generalized eigenvector. Jordan form: if A has repeated eigenvalues, A can be transformed into a nearly diagonal matrix (Jordan form), which has the eigenvalues of A on the mian diagonal, ones in certain positions on the diagonal above the main diagonal, and zeros elsewhere. (3) Nonhomogeneous Linear Systems x = A(t)x + g(t): the general solution x(t) = c1 x1 (t) + + cn xn (t) + v(t). Diagonalization: if A is diagonizable, use T defined above and x = T y. T y = AT y + g(t) y = T -1 AT y + T -1 g(t) = Dy + h(t), a system of n uncoupled equations, yj (t) = j yj (y) + hj (t), j = 1, . . . , n. t Thus, yj (t) = ej t t0 e-j s hj (s)ds + cj ej t , j = 1, . . . , n where cj are arbitrary constants. Undetermined coefficients If ...

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E. Kentucky - M - 220
MATH 220: Applied Differential Equations Lecture Lecture: 27884, Snow Hall 454, MWF 11:30AM12:20PM Lecturer: Myunghyun Oh Office Hours: MW 12:301:20PM, or by appointment Office: Snow Hall, Room #624, Phone: 864-5182 E-Mail: moh@math.ku.edu URL: htt
RIT - P - 08605
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Concept Level Project Plan P0860X (1-5) Eula Dozier's &quot;Tile Measurer&quot;Paul Gaylo (ME) Ryan Hellems (ME) Courtney Ross (ME) Jeremy Schiele (ME) Reid Williamson (ME)EDGETM Project Name Dozier Tile Layout Tool Project Number P0860(1-5) Pr
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Concept Level Project Plan P0860X (1-5) Eula Dozier's &quot;Tile Measurer&quot;Paul Gaylo (ME) Ryan Hellems (ME) Cortney Ross (ME) Jeremy Schiele (ME) Reid Williamson (ME)EDGETM EDGETMOriginal Prototype BackgroundDescriptionThis product is a device that
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Final Project PlanP0860X (1-5) Eula Dozier's &quot;Tile Measurer&quot;Paul Gaylo (ME) Ryan Hellems (ME) Cortney Ross (ME) Jeremy Schiele (ME) Reid Williamson (ME)EDGETM EDGETMEDGETMTHE VISION: May 2008 ProductsI. Eula Dozier Tile Layout Tool, ready fo
RIT - P - 08605
Eula Dozier Interview (9-19-2007 - 10AM) Hello, Mr. Dozier? I am Ryan Hellems, Well, before we get started with any questions I would like to just give you some background of what I got here. I'm sure you guys are going to have a lot of technical que
RIT - P - 08605
Linear drives DGP/DGPLnLow space requirement nHigh precision and load capacity nReliable service life of up to 40,000 km Specified types in accordance with ATEX directive for potentially explosive atmospheres www.festo.com/en/ex2003/10 Subject
RIT - P - 08605
Final Project PlanP0860X (1-5) Eula Dozier's &quot;Tile Measurer&quot;Paul Gaylo (ME) Ryan Hellems (ME) Cortney Ross (ME) Jeremy Schiele (ME) Reid Williamson (ME)EDGETM EDGETMEDGETMTHE VISION: May 2008 ProductsI. Eula Dozier Tile Layout Tool, ready fo
RIT - P - 08605
F A M I LY O V E R V I E WDistance MeasurementFrom micron to mile SICK covers the distanceDifferent operating principles adapted to your applicationOur distance sensors measure distances in difficult conditions with high precision at distan
RIT - P - 08605
Phone Interview: Tim @ Booher Tile Interview by Ryan Hellems (10/16/2007 - 4 pm) Q: Would you allow us to go on-site and observe an installation? A: Yes, upcoming job that we would be able to observe: Thursday 10/25 in Greece. Any time 7:30-4, 10 day
RIT - P - 08605
Concept Level Project Plan P0860X (1-5) Eula Dozier's &quot;Tile Measurer&quot;Paul Gaylo (ME) Ryan Hellems (ME) Cortney Ross (ME) Jeremy Schiele (ME) Reid Williamson (ME)EDGETM EDGETMOriginal Prototype BackgroundDescriptionThis product is a device that
RIT - P - 08605
SITE VISIT: Contact: Mike &quot;Tim&quot; Booher C: 585.278.8586 Location: Greece Date: Thurs. 10/25/07 Time: 2:45PM 3:30PM Interviewers: Ryan &amp; Reid BIO: Recommended contractor by the Tile Shop Learned to tile at 12 years old &quot;Family curse:&quot; Father &amp; grandfa
RIT - P - 08605
Sail-On Carpets- Meeting with Joe Rizzo &amp; Brad the Sales Man Ryan Hellems and Reid Williamson 10/3/2007 6:30 pm Sail-On Carpets- Meeting with Brad the Sales Man Ryan Hellems and Reid Williamson 10/3/2007 6:30 pm Summary of experiences/knowledge gaine
RIT - P - 08605
The Tile Shop- Meeting with Saleswoman Ryan Hellems and Reid Williamson (10/3/2007 5PM) Summary of experiences/knowledge gained: -Lots of flooring options, mostly ceramic tiles -Use a diamond cutter on an angle grinder to cut smooth corners -Classes
RIT - P - 08605
United Carpet Broker Interview with Kevin Nasser Ryan Hellems and Reid Williamson (10/11/2007 - 3pm) Hello, we are Engineering Students at RIT; we are doing a project involving an improvement on the flooring process. Would you mind if we asked you a
RIT - P - 08605
Dual Speed 10&quot; Tile SawRips tile 18&quot; square, 12&quot; diagonally# 60010PRODUCT INFORMATION Capacitor start 2HP motor for extra power Automatic thermal overload protection Permanently attached rail assembly to saw frame for improved, precise cuttin
Berkeley - C - 043
+-+-+-+-+-+-+-+-+-+| name | sos_total | sv_total | sr_total | diff_sv_sr | diff_sr_sos | diff_sv_sos | election | county |+-+-+--+-+-+-+-+-+-+| ABSVOTE | 0 | 0 | 0 | 0 | 0 | 0 | s
NJIT - CIS - 786
Alex Gerbessiotis CIS 786-107:Parallel Computation Sep 1, 2004 Course InformationFall 2004 Page 1A course on aspects of parallel computing involving clusters and networks of PC workstations. Parallel algorithms are introduced in the context of si
NJIT - CIS - 786
%!PS-Adobe-2.0 %Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %Title: hand0.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %DocumentFonts: CMCSC10 CMR12 CMTT12 CMSY8 %EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSComm
Ohio State - FIN - 726
Finance 726 MidtermProfessor Helwege Spring 2003You have the entire class period to finish this quiz. You may use a calculator, scrap paper, and a writing tool to complete the test. You may hand in the scrap paper (with your name on it) if you th
Colorado State - M - 331
M34331Drug Model Project1. (a) Solve the two initial value problems for x t and y t and plot them on the same graph in the following three cases: k 1 k 2 , k 1 k 2 , and k 1 k 2 . In each case, describe what the plot tells you about the drug l
Colorado State - M - 331
Absorption of Medications When you take a pill to obtain medication, the pill first goes into your stomach and the medication passes into your GI tract. From there the medication is absorbed into your bloodstream and circulated through your body befo
Colorado State - M - 331
&gt;restart :First, load the library that contains the LP software&gt;with Optimization :y = Dark crudeNow we'll do the crude oil production problem: x = Light crude Minimize the cost function Subject to : C=25\$x C 17\$.21\$x C .55\$ R 3 y .50\$x
Colorado State - M - 331
M 331Linear Programming ProblemsThese problems will be due Monday Oct 16. You should have all the computing done by Friday Oct 13 but the Monday due date will give you the weekend to think over what you want to say in problem 4. I am especially i
Colorado State - M - 331
Solutions to LP Problems &gt;M331with Optimization ;ImportMPS, Interactive, LPSolve, LSSolve, Maximize, Minimize, NLPSolve, QPSolveProblem 1 MinimizeCost = 30(x11+x12+x13+x14) + 50(x21+x22+x23+x24) + 80(x31+x32+x33+x34) x11+x12+x13+x14 &lt;= 50 x2
Colorado State - M - 331
M 331Exam on Nonlinear Optimization.1. A beer company wants to make their cans so that the upper half is a cylinder and the lower half is a rectangular box having a square base. The circular base of the upper cylinder is perfectly inscribed in th
Colorado State - M - 331
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Colorado State - M - 331
`331bo 2005/8/ page 14A P P E N D I XAMatlab Code for Data FittingA.1 MAMMALIAN HEART RATE PROBLEM File: ls_mammals.m -Start of actual file contents-%LEAST SQUARES ANALYSIS OF MAMMALIAN HEART RATE %w body weights %r corresponding heart rates %
Colorado State - M - 331
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C H A P T E R6Modeling with Discrete Dynamical Systems6.1 INTRODUCTION One of the most exciting areas of modeling concerns predicting temporal evolution. The main question that is posed in this setting is how do variables of interest change over
Colorado State - M - 331
&quot;331boo 2005/8/ page 140C H A P T E R7Simulation ModelingIt is not unusual that the complexity of a phenomenon or system makes a direct mathematical attack time-consuming, or worse, intractable. An alternative modeling approach consists of the
ASU - MAT - 343
MAT 343SUMMER 2009LEC: 45810*Important Note: All items on this syllabus are subject to change. Any in-class announcement, verbal or written, is considered official addendum to this syllabus. Time and place: 1:15-2:55 pm ( MTWF) in PSA 108 &amp; TH i
Berkeley - ASTRO - 00146970
-19.000000 -3.6399999 558.97861 169.77610 -3.6399999 -1.7200000 2876.7947 536.13983 -1.7200000 0.19999999 4746.4555 550.14303 0.20000011 2.1200001 5133.2623
Berkeley - ASTRO - 00146970
-19.000000 -3.6399999 558.97861 169.77610 -3.6399999 -1.7200000 2876.7947 536.13983 -1.7200000 0.19999999 4746.4555 550.14303 0.20000011 2.1200001 5133.2623
Berkeley - ASTRO - 00146970
# Time [days] Mag Magerr Band Uplim Ref 0.00427 15 -0.2 I no GCN3655 0.01720 -0.2 R no GCN3657 0.01736 -0.2 R no GCN3662 0.0
Berkeley - ASTRO - 00146970
1486.08 -0.2 0 no GCN3657 1499.9 -0.2 0 no GCN3662 3371.33 18.6 0.2 R no
Berkeley - ASTRO - 00146970
1486.08 7.200302e+03 0.000000e+00 no GCN3657 1499.9 7.200302e+03 0.000000e+00 no GCN3662 3371.33 2.174456e-04 4.005497e-05 R no
Berkeley - ASTRO - 00146970
-19 1 1
Berkeley - ASTRO - 00146970
15.360001 21.120001 5.5570212 0.94341827 728.65890 21.120001 23.040001 8.4765915 0.0000000 480.10577 23.040001 120.96000 10.555121 -0.92309061 2433.7325
Berkeley - ASTRO - 00146970
15.3600 21.1200 2.36748 0.983370 23.0400 120.960 -1.34270 0.187139 120.960 213.416 -6.42014 1.01643 213.416 440.889 -2.41914 0.160156 440.889 23194.7 -1.2977
Berkeley - ASTRO - 00146970
chi^2/nu= 235.05327 / 180The fit is rejectable at 99.635965 % Confidence -3.64000 -1.72000 3609.8236 -1.72000 0.200001 4009.1988 0.200001 2.12000 4406.1966 2.12000 4.0400
Berkeley - ASTRO - 00146970
Berkeley - ASTRO - 00146970
193.91 194.416 19.3653 6.4551194.416 194.928 19.1384 6.37945194.928 196.038 18.2948 4.28912196.038 197.305 16.0278 3.75764197.305 197.72 23.6117 7.87056197.72 198.337 15.8814 5.29381198.337 199.009 16.6933 5.43247199.009 199.538 19.8646 6.5478
Berkeley - ASTRO - 00146970
#ra dec hmag dhmag253.434774 -28.630537 13.948 0.050253.434527 -28.629358 11.313 0.029253.436668 -28.625603 14.008 0.021253.410792 -28.626026 16.606 0.141253.407508 -28.627878 16.193 0.101253.409090 -28.627268 14.739 0.047253.416261 -28.626024
Berkeley - ASTRO - 00146970
;instrument XRT;exposure 43935.123;xunit kev;bintype counts 0.0000000 0.0049999999 13.699265 1.00000 0.0049999999 0.0099999998 13.748119 1.00000 0.0099999998 0.015000000 13.796972 1.0
Berkeley - ASTRO - 00146970
;instrument XRT;exposure 205.02987;xunit kev;bintype counts 0.0000000 0.0049999999 14.544516 1.00000 0.0049999999 0.0099999998 14.596416 1.00000 0.0099999998 0.015000000 14.648316 1.0
Berkeley - ASTRO - 00146970
#ra dec rmag drmag253.719301-28.63056718.1380.154253.533132-28.63073817.7910.112253.446271-28.63072617.6680.100253.159740-28.63031717.6240.096253.157735-28.63031517.7370.106253.689486-28.63053417.3120.076253.600854-28.630578
Berkeley - ASTRO - 00146970
chi^2/nu= 76.473382 / 322.000The fit is rejectable at 2.5688205e-47 % Confidence#index t1 t2 fade_index delta_mag_pk hindex dhindex rate1 drate1 rate2 drate2 logr dlogr 0 0.1939 0.2051 -2.97 0.0 -2.95 5.95 6.47E+0
Berkeley - ASTRO - 00146970
# t1 t2 hardness error 0.19391000 0.19603800 0.45432251 0.14750030 0.19603800 0.19772000 0.20438850 0.18719565 0.19772000 0.19953800 0.12611560 0.18855479 0.19953800 0.20173200
Berkeley - ASTRO - 00146970
output00146970000_999/sw00146970000xpcw4po_cl.evtoutput00146970001_999/sw00146970001xpcw4po_cl.evtoutput00146970003_999/sw00146970003xpcw3po_cl.evtoutput00146970003_999/sw00146970003xpcw4po_cl.evtoutput00146970004_999/sw00146970004xpcw4po_cl.evt
Berkeley - ASTRO - 00146970
# t1 t2 dt rad_min rad_max cts err scl bg bg_rat wt 0.193910 0.194416 0.000506 0. 16. 9.00 3.00 0.918476 0.000000 0.348314 1 0.194416 0.194928 0.000512 0. 16. 9.00
Berkeley - ASTRO - 00146970
tmin 128.60045tmin 1031.2218 2064.0582 33128.471 0.21688333 0.023000630 8tmax 46869.069
Berkeley - ASTRO - 00146970
# t1 t2 dt rad_min rad_max cts err scl bg bg_rat wt 0.193910 0.194416 0.000506 0. 16. 9.00 3.00 0.918476 0.000000 0.348314 1 0.194416 0.194928 0.000512 0. 16. 9.00
Berkeley - ASTRO - 00146970
# tmin tmax 0.324104 3385.42 [ksec];instrument XRT;exposure 41436.321;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
Berkeley - ASTRO - 00146970
# tmin tmax 0.324104 3385.42 [ksec];instrument XRT;exposure 41436.321;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
Berkeley - ASTRO - 00146970
# tmin tmax 0.193910 4.75645 [ksec];instrument XRT;exposure 196.86733;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
Berkeley - ASTRO - 00146970
# tmin tmax 0.193910 4.75645 [ksec];instrument XRT;exposure 196.86733;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
Berkeley - ASTRO - 00146970
Wavdetect Sources with S/N&gt;3: # ra dec err [&quot;] signif counts steady? -log10(Prob_steady) 0253.435698-28.3811270.16580.91127.3 0-531.6 1253.633184-28.5347640.44018.155.5 0-31.6 2253.670469-28.4859820.67611.327.7 0-23.2
Berkeley - ASTRO - 00146970
output00146970000_999/sw00146970000xwtw2po_cl.evtoutput00146970001_999/sw00146970001xwtw2po_cl.evtoutput00146970003_999/sw00146970003xwtw2po_cl.evtoutput00146970004_999/sw00146970004xwtw2po_cl.evtoutput00146970005_999/sw00146970005xwtw2po_cl.evt
Berkeley - ASTRO - 00146970
SIMPLE = T / file does conform to FITS standardBITPIX = 8 / number of bits per data pixelNAXIS = 0 / number of data axesEXTEND = T / FITS dataset may contain extensio