Temperature and precipitation probability density functions in ENSEMBLES regional scenarios
|
|
|
- Έρις Δουμπιώτης
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Temperature and probability density functions in ENSEMBLES regional scenarios ENSEMBLES Technical Report No. M. Déqué April 9 ISSN 7-8
2
3 Contents Description of the method. Experiments used Pdf of individual models From weights to pdfs Results 9. Organization of the gures Overview of the capitals
4 CONTENTS
5 Chapter Description of the method. Experiments used At the time of writing this report 6 simulations are available in the daily database of RTB with AB scenarios at km resolution: CIR, CNRM, DMI, ETHZ, KNMI, ICTP, METN, METO-Q, METO-Q, METO-Q6, MPI, OURA, SMHI (driven by dierent GCMs) and UCLM In this study we will restrict to seasonal means (DJF, MAM, JJA and SON) of m and. The scenario period is - and the reference period is Pdf of individual models In studying the probability density functions we are interested both in the mean and in the extremes. In this context extreme means the tail of the distribution, i.e. the last deciles; it does not mean the very rare and catastrophic events for which our approach is not suited. It is however important to work with model grid point data, if we want to extend the study to daily pdfs. The calculation, and moreover the display, of all grid points is a huge unnecessary work. In addition models do not share the same grid. If we had to work only with means, considering spatial averages per country could have been a good approach. In the case of pdf, we will consider the capital of each European country, more precisely the closest model grid point to each capital. When including non-eu countries, we have capitals in Europe (tiny countries like Vatican, San Marino, Andorra, Monaco... are not considered here). Malta is an EU country, but this island is not resolved by the km meshes, so the nearest grid point is a sea point, with a dierent climate. We have also disregarded Nicosia and Reykjavik, because some models provide
6 6 CHAPTER. DESCRIPTION OF THE METHOD there unrealistic due to the too close lateral boundary. Finally, we have selected model locations, which will be referred to as the nearest European capital. Each model provides an uncertain result because the model seasonal response for a variable ( or ) is the average of two -year means. This sampling uncertainty is named the natural variability, which is a non-negligible part of the uncertainty. The model response is: R = 99 n=96 (X n+6 X n ) where X n is the seasonal mean of year n. If we assume that each year is independent of the others in terms of anomaly, R is the mean of independent values. As we work at the scale of a grid point, this assumption is widely veried. Otherwise, we would be able to predict the local seasonal weather one year in advance. As X is a seasonal mean, it is nearly gaussian (even for ). According to the limit central theorem, R is gaussian with the same mean as the yearly dierences and a standard deviation divided by. In the bivariate context ( and ), we get a gaussian couple with the same correlation coecient as the yearly dierences. It is thus easy to produce for each model a pdf (monovariate or bivariate) which takes into account the only uncertainty at this stage, which is the sample variability.. From weights to pdfs As we have, like in PRUDENCE, several models, a new source of uncertainty can be considered, which is the choice of a model. The above pdfs are in fact conditional pdfs if the selected model is considered to be true. The models are a combination of a driving GCM and a driven RCM. We will consider that all GCMs are equiprobable. The weights of the RCMs have been calculated in RT with a complex procedure based on ERA driven simulations. The closest the simulation to the 96- observations, the largest the weight. We chose the rescaled weight, which make a ratio of. in each of the 6 subscores between the largest and the lowest RCM. CI w=.69 CNRM w=.6 DMI w=.6
7 .. FROM WEIGHTS TO PDFS 7 ETHZ w=.67 ICTP w=.76 KNMI w=.9 METN w=.6 METO-Q w=.6 METO-Q w=. METO-Q6 w=.69 MPI w=.7 OURA w=.6 SMHI w=.79 UCLM w=.9 The nal range is, between METO-Q and KNMI, which allows each RCM to contribute signicantly to the nal pdf. With the unscaled weights, KNMI and ICTP would have a contribution equivalent to all the other RCMs. Then we make the assumption that the probability of an RCM is proportional to its weight. After multiplicative combination (we assume GCM and RCM are independent) and normalization, we have 6 positive numbers p i, the sum of which is. The compounded probability, e.g. for is: 6 pdf(t) = p i pdf(t model = i) i= and similarly for or for the couple (, ). The continuous pdf is a mathematical concept that we cannot handle with computers. So we have to dene bins. Here we have considered bins for each variable, from - to +6 C and from - to +6 mm/day, according to the range of responses of the models. As the dierence between the model means is often greater than the standard deviation due to their natural variability, the nal pdf may have several modes which correspond to individual models. This is the consequence of the underlying probabilistic model: the true response is given by one of the model. To articially introduce the possibility that the response may be between two models, we have applied a gaussian lter, aka gaussian kernel method. We replace the probability in each bin by a gaussian distribution with the mean as the center of the bin and a standard deviation of σ =. K and. mm/day. These values have been visually
8 8 CHAPTER. DESCRIPTION OF THE METHOD adjusted to cancel out as far as possible the multimodal eect, but to avoid a pure gaussian shape for the smoothed pdf (except when all models are in agreement). The extension to the bivariate pdf is easy, but computationally expensive with bins in dimensions. For, the ltered pdf pdf(t) reads: pdf(t) = T pdf(t ) T πσ exp ( T T σ ) where T is the width of each bin.
9 Chapter Results. Organization of the gures Figs.. to.8 present the pdf of changes (multiplied by ) for each season (blue for winter, green for spring, red for summer and yellow for autumn). Figs..9 to.6 show similar quantities for change. There are capitals per page (in alphabetic order). Finally, since: for a given model, there is an interannual correlation between and change (e.g. the warmest summers are often the driest) models which have a larger response in may also have a larger response in and responses are not independent. Figs..7 to.8 show the bivariate density (multiplied by ). Here it is not possible to superimpose the four seasons in a same panel, so there is one capital per page.. Overview of the capitals Let us examine rst the mean of the and pdf for each capital. Note that for the sign of the response is never signicant, since there is always at least % probability that the sign is positive and at least % probability that it is negative, whatever the capital and the season. So, when one reads mean increases by. mm/day, this does not means that this increase is certain. 9
10 CHAPTER. RESULTS Athens: mean decreases by. mm/day in all seasons. The maximum warming is in summer and autumn (.7 K). Belgrade: mean increases by. mm/day in winter and decreases by. mm/day in summer. The maximum warming is in summer (. K). Berlin: mean increases in all seasons, with a maximum in winter (. mm/day). The maximum warming is in winter (.8 K). Bern: mean increases in autumn and winter (. mm/day). It decreases in summer (. mm/day). The maximum warming is in winter (.7 K). Bratislava: mean increases in all season except summer (. mm/day). The maximum warming is in winter (.8 K). Brussels: mean increases in autumn (. mm/day) and winter (. mm/day). The maximum warming is in winter (. K). Bucharest: mean increases in autumn (. mm/day) and decreases in spring and summer (. mm/day). The maximum warming is in summer (. K). Budapest: mean increases in autumn and winter (. mm/day). The maximum warming is in summer and winter (.8 K). Copenhagen: mean increases in all seasons with a maximum in autumn and winter (. mm/day). The maximum warming is in winter (.7 K). Dublin: mean increases in winter and autumn (. mm/day) and decreases in spring (. mm/day). The maximum warming is in autumn (. K). Helsinki: mean increases in all seasons with a maximum in autumn (. mm/day). The maximum warming is also in winter (. K). Lisbon: mean increases in winter (. mm/day) and decreases in the other seasons (maximum. mm/day in autumn). The maximum warming is in summer and autumn (.6 K) Ljubljana: mean increases in winter (. mm/day) and decreases in summer (. mm/day). The maximum warming is both in winter and summer (.8 K).
11 .. OVERVIEW OF THE CAPITALS London: mean increases in winter and autumn (. mm/day in winter) and decreases in spring (. mm/day). The maximum warming is winter and autumn (. K). Luxembourg: mean increases in winter and autumn (. mm/day in winter). The maximum warming is winter (.6 K). Madrid: mean increases in winter (. mm/day) and decreases in the other seasons (maximum. mm/day in autumn). The maximum warming is in summer (. K). Oslo: mean increases in all seasons, with a maximum in autumn and winter (. mm/day). The maximum warming is also in autumn and winter (.8 K). Paris: mean increases in winter (. mm/day) and decreases in spring (. mm/day) and summer (. mm/day). The warming is uniform (. K) except in spring (. K). Prague: mean increases in all seasons with a maximum in summer and autumn (. mm/day). The maximum warming is in winter (.9 K). Riga: mean increases in all seasons (. mm/day). The maximum warming is in winter (. K). Rome: mean decreases in all seasons with a maximum in autumn (. mm/day). The maximum warming is in summer (. K). Sarajevo: mean increases in winter (. mm/day) and decreases in summer (. mm/day). The maximum warming is in summer (. K). Skopje: mean decreases in spring and summer (. mm/day). The maximum warming is in summer (. K). Soa: mean increases in autumn and winter (. mm/day) and decreases in spring and summer (. mm/day). The maximum warming is in summer (. K). Stockholm: mean increases in all seasons (. mm/day). The maximum warming is in winter (.9 K). Tallinn: mean increases in all seasons (. mm/day). maximum warming is in winter (. K) The The Hague: mean increases in autumn (. mm/day) and winter (. mm/day). The maximum warming is also in autumn and winter (. K).
12 CHAPTER. RESULTS Tirana: mean increases in winter (. mm/day) and decreases in spring and summer (. mm/day). The maximum warming is in summer (. K). Vienna: mean increases in all seasons (. mm/day). maximum warming is in winter (.8 K). The Vilnius: mean increases in all seasons (. mm/day in autumn and winter). The maximum warming is in winter (. K). Warsaw: mean increases in all seasons (. mm/day in autumn and winter). The maximum warming is in winter (. K). Zagreb: mean increases in autumn and winter (. mm/day) and decreases in summer (. mm/day). The maximum warming is in summer (.9 K).
13 .. OVERVIEW OF THE CAPITALS Figure.: Probability density function ( C ) for response ( C) in Athens (top left), Belgrade (top right), Berlin (bottom left) and Bern (bottom right) for DJF (blue), MAM (green), JJA (red) and SON (yellow).
14 CHAPTER. RESULTS Figure.: As Figure. for Bratislava (top left), Brussels (top right), Bucharest (bottom left) and Budapest (bottom right).
15 .. OVERVIEW OF THE CAPITALS Figure.: As Figure. for Copenhagen (top left), Dublin (top right), Helsinki (bottom left) and Lisbon (bottom right).
16 6 CHAPTER. RESULTS Figure.: As Figure. for Ljubljana (top left), London (top right), Luxembourg (bottom left) and Madrid (bottom right).
17 .. OVERVIEW OF THE CAPITALS Figure.: As Figure. for Oslo (top left), Paris (top right), Prague (bottom left) and Riga (bottom right).
18 8 CHAPTER. RESULTS Figure.6: As Figure. for Rome (top left), Sarajevo (top right), Skopje (bottom left) and Soa (bottom right).
19 .. OVERVIEW OF THE CAPITALS Figure.7: As Figure. for Stockholm (top left), Tallinn (top right), The Hague (bottom left) and Tirana (bottom right).
20 CHAPTER. RESULTS Figure.8: As Figure. for Vienna (top left), Vilnius (top right), Warsaw (bottom left) and Zagreb (bottom right).
21 .. OVERVIEW OF THE CAPITALS Figure.9: Probability density function ( mm day) for response (mm/day) in Athens (top left), Belgrade (top right), Berlin (bottom left) and Bern (bottom right) for DJF (blue), MAM (green), JJA (red) and SON (yellow).
22 CHAPTER. RESULTS Figure.: As Figure.9 for Bratislava (top left), Brussels (top right), Bucharest (bottom left) and Budapest (bottom right).
23 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.9 for Copenhagen (top left), Dublin (top right), Helsinki (bottom left) and Lisbon (bottom right).
24 CHAPTER. RESULTS Figure.: As Figure.9 for Ljubljana (top left), London (top right), Luxembourg (bottom left) and Madrid (bottom right).
25 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.9 for Oslo (top left), Paris (top right), Prague (bottom left) and Riga (bottom right).
26 6 CHAPTER. RESULTS Figure.: As Figure.9 for Rome (top left), Sarajevo (top right), Skopje (bottom left) and Soa (bottom right).
27 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.9 for Stockholm (top left), Tallinn (top right), The Hague (bottom left) and Tirana (bottom right).
28 8 CHAPTER. RESULTS Figure.6: As Figure.9 for Vienna (top left), Vilnius (top right), Warsaw (bottom left) and Zagreb (bottom right).
29 .. OVERVIEW OF THE CAPITALS Figure.7: Bivariate probability density function for and response in Athens for DJF (top left), MAM (top right), JJA (bottom left) and SON (bottom right). Contours are,,, 6, 8 and C mm day.
30 CHAPTER. RESULTS Figure.8: As Figure.7 for Belgrade.
31 .. OVERVIEW OF THE CAPITALS Figure.9: As Figure.7 for Berlin.
32 CHAPTER. RESULTS Figure.: As Figure.7 for Bern.
33 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Bratislava.
34 CHAPTER. RESULTS Figure.: As Figure.7 for Brussels.
35 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Bucharest.
36 6 CHAPTER. RESULTS Figure.: As Figure.7 for Budapest.
37 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Copenhagen.
38 8 CHAPTER. RESULTS Figure.6: As Figure.7 for Dublin.
39 .. OVERVIEW OF THE CAPITALS Figure.7: As Figure.7 for Helsinki.
40 CHAPTER. RESULTS Figure.8: As Figure.7 for Lisbon.
41 .. OVERVIEW OF THE CAPITALS Figure.9: As Figure.7 for Ljubljana.
42 CHAPTER. RESULTS Figure.: As Figure.7 for London.
43 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Luxembourg.
44 CHAPTER. RESULTS Figure.: As Figure.7 for Madrid.
45 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Oslo.
46 6 CHAPTER. RESULTS Figure.: As Figure.7 for Paris.
47 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Prague.
48 8 CHAPTER. RESULTS Figure.6: As Figure.7 for Riga.
49 .. OVERVIEW OF THE CAPITALS Figure.7: As Figure.7 for Rome.
50 CHAPTER. RESULTS Figure.8: As Figure.7 for Sarajevo.
51 .. OVERVIEW OF THE CAPITALS Figure.9: As Figure.7 for Skopje.
52 CHAPTER. RESULTS Figure.: As Figure.7 for Soa.
53 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Stockholm.
54 CHAPTER. RESULTS Figure.: As Figure.7 for Tallinn.
55 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for The Hague.
56 6 CHAPTER. RESULTS Figure.: As Figure.7 for Tirana.
57 .. OVERVIEW OF THE CAPITALS Figure.: As Figure.7 for Vienna.
58 8 CHAPTER. RESULTS Figure.6: As Figure.7 for Vilnius.
59 .. OVERVIEW OF THE CAPITALS Figure.7: As Figure.7 for Warsaw.
60 6 CHAPTER. RESULTS Figure.8: As Figure.7 for Zagreb.
61 List of Figures. Probability density function ( C ) for response ( C) in Athens (top left), Belgrade (top right), Berlin (bottom left) and Bern (bottom right) for DJF (blue), MAM (green), JJA (red) and SON (yellow) As Figure. for Bratislava (top left), Brussels (top right), Bucharest (bottom left) and Budapest (bottom right) As Figure. for Copenhagen (top left), Dublin (top right), Helsinki (bottom left) and Lisbon (bottom right) As Figure. for Ljubljana (top left), London (top right), Luxembourg (bottom left) and Madrid (bottom right) As Figure. for Oslo (top left), Paris (top right), Prague (bottom left) and Riga (bottom right) As Figure. for Rome (top left), Sarajevo (top right), Skopje (bottom left) and Soa (bottom right) As Figure. for Stockholm (top left), Tallinn (top right), The Hague (bottom left) and Tirana (bottom right) As Figure. for Vienna (top left), Vilnius (top right), Warsaw (bottom left) and Zagreb (bottom right) Probability density function ( mm day) for response (mm/day) in Athens (top left), Belgrade (top right), Berlin (bottom left) and Bern (bottom right) for DJF (blue), MAM (green), JJA (red) and SON (yellow) As Figure.9 for Bratislava (top left), Brussels (top right), Bucharest (bottom left) and Budapest (bottom right) As Figure.9 for Copenhagen (top left), Dublin (top right), Helsinki (bottom left) and Lisbon (bottom right) As Figure.9 for Ljubljana (top left), London (top right), Luxembourg (bottom left) and Madrid (bottom right).... 6
62 6 LIST OF FIGURES. As Figure.9 for Oslo (top left), Paris (top right), Prague (bottom left) and Riga (bottom right) As Figure.9 for Rome (top left), Sarajevo (top right), Skopje (bottom left) and Soa (bottom right) As Figure.9 for Stockholm (top left), Tallinn (top right), The Hague (bottom left) and Tirana (bottom right) As Figure.9 for Vienna (top left), Vilnius (top right), Warsaw (bottom left) and Zagreb (bottom right) Bivariate probability density function for and response in Athens for DJF (top left), MAM (top right), JJA (bottom left) and SON (bottom right). Contours are,,, 6, 8 and C mm day As Figure.7 for Belgrade As Figure.7 for Berlin As Figure.7 for Bern As Figure.7 for Bratislava As Figure.7 for Brussels As Figure.7 for Bucharest As Figure.7 for Budapest As Figure.7 for Copenhagen As Figure.7 for Dublin As Figure.7 for Helsinki As Figure.7 for Lisbon As Figure.7 for Ljubljana As Figure.7 for London As Figure.7 for Luxembourg As Figure.7 for Madrid As Figure.7 for Oslo As Figure.7 for Paris As Figure.7 for Prague As Figure.7 for Riga As Figure.7 for Rome As Figure.7 for Sarajevo
63 LIST OF FIGURES 6.9 As Figure.7 for Skopje As Figure.7 for Soa As Figure.7 for Stockholm As Figure.7 for Tallinn As Figure.7 for The Hague As Figure.7 for Tirana As Figure.7 for Vienna As Figure.7 for Vilnius As Figure.7 for Warsaw As Figure.7 for Zagreb
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Math221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Every set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Finite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Math 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Instruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Μηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games
Problem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
TMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Potential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Right Rear Door. Let's now finish the door hinge saga with the right rear door
Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents
Concrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality
The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,
Lecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
Srednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Solution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Démographie spatiale/spatial Demography
ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης
CRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Section 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
1, +,*+* + +-,, -*, * : Key words: global warming, snowfall, snowmelt, snow water equivalent. Ohmura,,**0,**
1, +,*+* + +-,, + : /+* m,1+ m, -*, * +3132* : Key words: global warming, snowfall, snowmelt, snow water equivalent + IPCC,,**+ Inoue and,**2 Yokoyama,**- Ohmura,,**0,**0 +331 +332 + +2- **+, ++,* 14 1,
A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Figure A.2: MPC and MPCP Age Profiles (estimating ρ, ρ = 2, φ = 0.03)..
Supplemental Material (not for publication) Persistent vs. Permanent Income Shocks in the Buffer-Stock Model Jeppe Druedahl Thomas H. Jørgensen May, A Additional Figures and Tables Figure A.: Wealth and
Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
Lecture 2. Soundness and completeness of propositional logic
Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness
ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ
ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ Γηπισκαηηθή Δξγαζία ηνπ Φνηηεηή ηνπ ηκήκαηνο Ζιεθηξνιόγσλ Μεραληθώλ θαη Σερλνινγίαο Ζιεθηξνληθώλ
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? What is the 50 th percentile for the cigarette histogram?
HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? The point on the horizontal axis such that of the area under the histogram lies to the left of that point (and to the right) What
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Second Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
Assalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Μηχανισμοί πρόβλεψης προσήμων σε προσημασμένα μοντέλα κοινωνικών δικτύων ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΕΠΙΚΟΙΝΩΝΙΩΝ, ΗΛΕΚΤΡΟΝΙΚΗΣ ΚΑΙ ΣΥΣΤΗΜΑΤΩΝ ΠΛΗΡΟΦΟΡΙΚΗΣ Μηχανισμοί πρόβλεψης προσήμων σε προσημασμένα μοντέλα κοινωνικών
CE 530 Molecular Simulation
C 53 olecular Siulation Lecture Histogra Reweighting ethods David. Kofke Departent of Cheical ngineering SUNY uffalo kofke@eng.buffalo.edu Histogra Reweighting ethod to cobine results taken at different
Δθμόςια Αςφρματα Δίκτυα και το παράδειγμα του athenswifi. Νεθηάρηος Κοδύρες ΔΠΙΣΔΥ/ΔΜΠ & e-tee
Δθμόςια Αςφρματα Δίκτυα και το παράδειγμα του athenswifi Νεθηάρηος Κοδύρες ΔΠΙΣΔΥ/ΔΜΠ & e-tee 1ο μζροσ Athenswifi ΕΠΙΧΕΙΡΗΙΑΚΟ ΠΡΟΓΡΑΜΜΑ ΚτΠ Ζργο: «Ανάπτυξθ Αςφρματων Ευρυηωνικών Τποδομών και Προώκθςθ
Tanzania. General Climate. UNDP Climate Change Country Profiles. C. McSweeney 1, M. New 1,2 and G. Lizcano 1
UNDPClimateChangeCountryProfiles Tanzania C.McSweeney 1,M.New 1,2 andg.lizcano 1 1.SchoolofGeographyandEnvironment,UniversityofOxford. 2.TyndallCentreforClimateChangeResearch http://country-profiles.geog.ox.ac.uk
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ Μελέτη των υλικών των προετοιμασιών σε υφασμάτινο υπόστρωμα, φορητών έργων τέχνης (17ος-20ος αιώνας). Διερεύνηση της χρήσης της τεχνικής της Ηλεκτρονικής Μικροσκοπίας
EE101: Resonance in RLC circuits
EE11: Resonance in RLC circuits M. B. Patil mbatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Deartment of Electrical Engineering Indian Institute of Technology Bombay I V R V L V C I = I m = R + jωl + 1/jωC
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
The challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Tridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ
ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται
Statistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
Capacitors - Capacitance, Charge and Potential Difference
Capacitors - Capacitance, Charge and Potential Difference Capacitors store electric charge. This ability to store electric charge is known as capacitance. A simple capacitor consists of 2 parallel metal
HOMEWORK#1. t E(x) = 1 λ = (b) Find the median lifetime of a randomly selected light bulb. Answer:
HOMEWORK# 52258 李亞晟 Eercise 2. The lifetime of light bulbs follows an eponential distribution with a hazard rate of. failures per hour of use (a) Find the mean lifetime of a randomly selected light bulb.
ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΛΛΗΝΙΔΕΣ
ΕΛΛΗΝΙΔΕΣ ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΚΔΟΣΗ ΤΗΣ ΣΟΡΟΠΤΙΜΙΣΤΙΚΗΣ ΕΝΩΣΗΣ ΕΛΛΑΔΟΣ - ΤΕΥΧΟΣ Νο 110 - Δ ΤΡΙΜΗΝΟ 2014 Το πρώτο βραβείο κέρδισε η Ελλάδα για την φωτογραφία Blue + Yellow = Green στον διαγωνισμό 2014 του
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Probability and Random Processes (Part II)
Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation
Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
On a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response
Strain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
A Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Partial Trace and Partial Transpose
Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This