Math 32A 2010.05.11 MATH 32A DISCUSSION 1. Higher Partial Derivatives
... 1.4. Exercise 15.3.87. Is it possible that there is a function f whose partial derivatives are fx (x, y) = x + 4y and fy (x, y) = 3x − y? Solution. Negative, since fxy = 4 and fyx = 3 are continuous but not equal, contardicting Clairaut’s theorem. ...
... 1.4. Exercise 15.3.87. Is it possible that there is a function f whose partial derivatives are fx (x, y) = x + 4y and fy (x, y) = 3x − y? Solution. Negative, since fxy = 4 and fyx = 3 are continuous but not equal, contardicting Clairaut’s theorem. ...
The weak and strong laws of large numbers
... sequence of random variables are the same, a strong law implies a weak law. We shall prove the weak law of large numbers for a sequence of independent identically distributed L1 random variables, and the strong law of large for the same hypotheses. We give separate proofs for these theorems as an oc ...
... sequence of random variables are the same, a strong law implies a weak law. We shall prove the weak law of large numbers for a sequence of independent identically distributed L1 random variables, and the strong law of large for the same hypotheses. We give separate proofs for these theorems as an oc ...
Diffuse Gaussian Multiple- Shot Patterns
... on the meaning of “kill” and the physical mechanism involved, the DG function can be attacked for being too sloppy or defended for being appropriately so. Regardless of its physical suitability as a model of damage, its analytical merits are ...
... on the meaning of “kill” and the physical mechanism involved, the DG function can be attacked for being too sloppy or defended for being appropriately so. Regardless of its physical suitability as a model of damage, its analytical merits are ...
This paper is a postprint of a paper submitted to and accepted
... these statistical quantities will be computed stochastically, using instantaneous gradient descent techniques that minimize the appropriate error functional. To the authors’ knowledge, besides the mean, and that only incidentally, for example, in self-organizing maps (SOMs) [2] and related algorithm ...
... these statistical quantities will be computed stochastically, using instantaneous gradient descent techniques that minimize the appropriate error functional. To the authors’ knowledge, besides the mean, and that only incidentally, for example, in self-organizing maps (SOMs) [2] and related algorithm ...
Answers to the regular section practice final
... e) F. Consider the matrix that gives the linear map from R3 to R5 that sends the domain to the 3-dimensional subspace where the coordinates x4 and x5 are zero. f) T. To see this, note that it has an even number of complex eigenvalues, and if it had four, they would come in complex conjugate pairs. T ...
... e) F. Consider the matrix that gives the linear map from R3 to R5 that sends the domain to the 3-dimensional subspace where the coordinates x4 and x5 are zero. f) T. To see this, note that it has an even number of complex eigenvalues, and if it had four, they would come in complex conjugate pairs. T ...
The Euler characteristic of the moduli space of curves
... with another edge b j, defined as follows. If the edge after vj is not labeled, pick it for bj; do this for all possible j. If any b~ remain unchosen, proceed to the edge third after v~ and if it is unlabeled, call it b j; again this should be done for all possible cases. Continue, selecting the fif ...
... with another edge b j, defined as follows. If the edge after vj is not labeled, pick it for bj; do this for all possible j. If any b~ remain unchosen, proceed to the edge third after v~ and if it is unlabeled, call it b j; again this should be done for all possible cases. Continue, selecting the fif ...
Lecture 5. Stochastic processes
... We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit ...
... We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit ...
Uniform pseudo-random number generators
... one realisation of each random variable Ui . For instance, we can plot the sample path (1, U1 ), . . . , (n, Un ) and study if there appears to be any systematic fluctuation. Let n = 1000 and produce such plots in R using the multiplicative congruential generator with first a = 3, m = 31, X0 = 2 and ...
... one realisation of each random variable Ui . For instance, we can plot the sample path (1, U1 ), . . . , (n, Un ) and study if there appears to be any systematic fluctuation. Let n = 1000 and produce such plots in R using the multiplicative congruential generator with first a = 3, m = 31, X0 = 2 and ...
SOLUTIONS TO HOMEWORK 6 1.
... (The definition of “indefinite” is given on p. 96 of the textbook.) 2(b). List the pairs of numbers appearing on successive edges of John Conway’s river for the quadratic form Q. (The sequence of pairs eventually repeats; you can list the pairs in one period, then put a bar over it.) Ans: The river ...
... (The definition of “indefinite” is given on p. 96 of the textbook.) 2(b). List the pairs of numbers appearing on successive edges of John Conway’s river for the quadratic form Q. (The sequence of pairs eventually repeats; you can list the pairs in one period, then put a bar over it.) Ans: The river ...
Random processes - basic concepts
... • The area under the normalized autocorrelation function for the fluctuating wind velocity measured at a point is a measure of the average time scale of the eddies being carried passed the measurement point, say T1 • If we assume that the eddies are being swept passed at the mean velocity, U.T1 is ...
... • The area under the normalized autocorrelation function for the fluctuating wind velocity measured at a point is a measure of the average time scale of the eddies being carried passed the measurement point, say T1 • If we assume that the eddies are being swept passed at the mean velocity, U.T1 is ...
Chapter 3
... The simplest random number generators are coins, dice and bags of colored balls. Thus in the RRT example above, the interviewee could be given a well-shaken bag of balls. : white balls with probability to answer N question. : Black balls with probability to answer E question. Other machines ar ...
... The simplest random number generators are coins, dice and bags of colored balls. Thus in the RRT example above, the interviewee could be given a well-shaken bag of balls. : white balls with probability to answer N question. : Black balls with probability to answer E question. Other machines ar ...
Binomial Theorem (Pascal`s Triangle)
... Use the pattern in Pascal’s triangle to write ( x y)7 in expanded form. ...
... Use the pattern in Pascal’s triangle to write ( x y)7 in expanded form. ...
1 C PROGRAMMING: SUMMING GEOMETRIC SERIES The Greek
... 1/2 the remaining distance, then 1/2 of what is left again. This process goes on indefinitely: You can keep dividing the remaining distance by two an infinite number of times. Hence the R.A. can never finish the race. Does anyone know what is wrong with Zeno’s argument? We can understand how to answ ...
... 1/2 the remaining distance, then 1/2 of what is left again. This process goes on indefinitely: You can keep dividing the remaining distance by two an infinite number of times. Hence the R.A. can never finish the race. Does anyone know what is wrong with Zeno’s argument? We can understand how to answ ...
a pdf file - The Citadel
... can not be factored as a product of two Gaussian Integers except by using a unit {±1, ±i}. For example, since 13 32 2 2 3 2i 3 2i we see 13 is not a Gaussian Prime even though it is a prime in Z. Definition: An element α in G = {a + bi : a,b Z}is prime iff α is not a unit and whenev ...
... can not be factored as a product of two Gaussian Integers except by using a unit {±1, ±i}. For example, since 13 32 2 2 3 2i 3 2i we see 13 is not a Gaussian Prime even though it is a prime in Z. Definition: An element α in G = {a + bi : a,b Z}is prime iff α is not a unit and whenev ...
Sequential Compressed Sensing
... The applications where compressed sensing is most beneficial (e.g., MRI) have a high cost of acquiring each additional sample. If this cost (in terms of time, power, etc.) is high as compared to the cost of computation, then it is suitable to use sophisticated recovery algorithms which include the - ...
... The applications where compressed sensing is most beneficial (e.g., MRI) have a high cost of acquiring each additional sample. If this cost (in terms of time, power, etc.) is high as compared to the cost of computation, then it is suitable to use sophisticated recovery algorithms which include the - ...