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MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS.ORG - REAL ANALYSIS: 5.3. CONNECTED AND Note that the definition of disconnected set is easier for an open set S.In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected.. To show that a set is disconnected is generally easier than showing connectedness: if you can find a MATHCS.ORG - REAL ANALYSIS: EXAMPLES 5.1.2(A): The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x.Now let = min( 3 + x, 3 - x ).Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.; The interval is closed, because its complement consists of the two open MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS: INTRO TO STATISTICS WITH EXCEL We first need to open the Life Expectancy data file - click on the icon below for the data file. life.xls. When the spreadsheet opens up, mark all numeric data in column B (the Life Expectancy column) but not including the codlumn header and copy them to the clipboard (for example, press CTRL-C). MATHCS.ORG - REAL ANALYSIS: 6.4. TOPOLOGY AND CONTINUITY This is true in particular for function defined on all of R (which is both open and closed). Examples 6.4.2: Let f (x) = x 2. Show that f is continuous by proving. that the inverse image of an open interval is open. that the inverse image of a closed interval is closed. Let f (x) = 1 if x > 0 and f (x) = MATHCS.ORG - REAL ANALYSIS: 7.4. LEBESGUE INTEGRAL For simple functions we define the Lebesgue integral as follows: Definition 7.4.3: Lebesgue Integral for Simple Function. If s (x) = an XAn(x) is a simple function and m (An) is finite for all n, then the Lebesgue Integral of s is defined as. s (x) dx = an m (An) If E BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS: STATISTICS DATA SETS EXCEL A wealth of curated data sets, available in different formats (inluding CVS suitable for Excel), including " number of Prussian cavalry soldiers killed by horse kicks (1875 to 1894) ", " Global-mean monthly, seasonal, and annual temperatures since 1880 ", and many more. Kaggle is a platform for predictive modelling and analyticscompetitions
MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS.ORG - REAL ANALYSIS: 5.3. CONNECTED AND Note that the definition of disconnected set is easier for an open set S.In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected.. To show that a set is disconnected is generally easier than showing connectedness: if you can find a MATHCS.ORG - REAL ANALYSIS: EXAMPLES 5.1.2(A): The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x.Now let = min( 3 + x, 3 - x ).Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.; The interval is closed, because its complement consists of the two open MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS: INTRO TO STATISTICS WITH EXCEL We first need to open the Life Expectancy data file - click on the icon below for the data file. life.xls. When the spreadsheet opens up, mark all numeric data in column B (the Life Expectancy column) but not including the codlumn header and copy them to the clipboard (for example, press CTRL-C). MATHCS.ORG - REAL ANALYSIS: 6.4. TOPOLOGY AND CONTINUITY This is true in particular for function defined on all of R (which is both open and closed). Examples 6.4.2: Let f (x) = x 2. Show that f is continuous by proving. that the inverse image of an open interval is open. that the inverse image of a closed interval is closed. Let f (x) = 1 if x > 0 and f (x) = MATHCS.ORG - REAL ANALYSIS: 7.4. LEBESGUE INTEGRAL For simple functions we define the Lebesgue integral as follows: Definition 7.4.3: Lebesgue Integral for Simple Function. If s (x) = an XAn(x) is a simple function and m (An) is finite for all n, then the Lebesgue Integral of s is defined as. s (x) dx = an m (An) If E BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - REAL ANALYSIS: 5.3. CONNECTED AND Note that the definition of disconnected set is easier for an open set S.In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected.. To show that a set is disconnected is generally easier than showing connectedness: if you can find a MATHCS.ORG - REAL ANALYSIS: 7.1. RIEMANN INTEGRAL Examples 7.1.11: Is the function f(x) = x 2 Riemann integrable on the interval ?If so, find the value of the Riemann integral. Do the same for the interval (since this is the same example as before, using Riemann's Lemma will hopefully simplify the solution).; Suppose f is Riemann integrable over an interval and { P n} is a sequence of partitions whose mesh converges to zero. MATHCS.ORG - REAL ANALYSIS: INTEGRAL TEST Suppose that f(x) is positive, continuous, decreasing function on the interval [N, ).Let a n = f(n).Then converges if and only if converges MATHCS.ORG - REAL ANALYSIS: EXAMPLES 4.1.7(B): Take a look at the following inequality: 2 n + j < 2 n+1 for j = 1, 3, , 2 n-1 (which makes for 2 n-1 terms). Therefore: (2 n + 1)-1 + (2 n + 3)-1 + + (2 n+1 - 1)-1 > 2 n-1 / 2 n+1 = 1 / 4 But now we can rearrange the terms of the alternating harmonic series as follows: MATHCS.ORG - REAL ANALYSIS: 6.4. TOPOLOGY AND CONTINUITY This is true in particular for function defined on all of R (which is both open and closed). Examples 6.4.2: Let f (x) = x 2. Show that f is continuous by proving. that the inverse image of an open interval is open. that the inverse image of a closed interval is closed. Let f (x) = 1 if x > 0 and f (x) = MATHCS.ORG - REAL ANALYSIS: 6.2. CONTINUOUS FUNCTIONS 6. Limits, Continuity, and Differentiation 6.2. Continuous Functions If one looks up continuity in a thesaurus, one finds synonyms like perpetuity or lack of interruption.Descartes said that a function is continuous if its graph can be drawn without lifting the pencil fromthe paper.
MATHCS.ORG - REAL ANALYSIS: 8.2. UNIFORM CONVERGENCE The good news is that uniform convergence preserves at least some properties of a sequence. Theorem 8.2.3: Uniform Convergence preserves Continuity. If a sequence of functions fn(x) defined on D converges uniformly to a function f (x), and if each fn(x) is continuous on D, then the limit function f (x) is also continuous on D. Proof. BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - REAL ANALYSIS: EXAMPLE 8.3.6 (B): POWER Here we need to determine the general term a n so that we can rewrite the series in sigma notation as: a n (x - c) n = a 0 + a 1 (x-c) + a 2 (x-c) 2 + For 1 + 2x MATHCS.ORG - REAL ANALYSIS: EXAMPLE 4.1.10: THE LEANING Example 4.1.10: The Leaning Tower of Lire. Jillian, a diligent but overworked student, fell asleep in the library and got locked in for the night. When she awoke, the room was dimly lit and she was alone. To pass the time (and to annoy the librarian in the morning) she decided to stack books on a table so that they would overhang the edgeof
MATHCS: STATISTICS DATA SETS EXCEL A wealth of curated data sets, available in different formats (inluding CVS suitable for Excel), including " number of Prussian cavalry soldiers killed by horse kicks (1875 to 1894) ", " Global-mean monthly, seasonal, and annual temperatures since 1880 ", and many more. Kaggle is a platform for predictive modelling and analyticscompetitions
MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS.ORG - REAL ANALYSIS: 5.3. CONNECTED AND Note that the definition of disconnected set is easier for an open set S.In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected.. To show that a set is disconnected is generally easier than showing connectedness: if you can find a MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG - REAL ANALYSIS: THEOREM 5.2.6: HEINE-BOREL THEOREM C = { comp ( ), n > 0 } Then each set in C is open, being the complement of closed sets. They also cover S, because the only point that this collection is missing is the point s, which is not part of S. By assumption, a finite subcover already covers S. If N is the largest index of that finite subcovering, then aN+1 is not BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - REAL ANALYSIS: DEFINITION 1.2.6: ONE-ONE Definition 1.2.6: One-one, Onto, Bijection. A function f from A to B is called one to one (or one-one) if whenever f (a) = f (b) then a = b. Such functions are also called injections. A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. Such functions are also called surjections. A function f from A MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.2.8 Proof: Each A j is compact, hence closed and bounded. Therefore, A is closed and bounded as well, and hence A is compact. Pick an a j A j for each j. Then the sequence { a j} is contained in A 1.Since that set is compact, there exists a convergent subsequence { a j k} with limit in A 1. But that subsequence, except the first number, is alsocontained in A 2.
MATHCS.ORG - REAL ANALYSIS: 3.2. CAUCHY SEQUENCES 3. Sequences of Numbers. 3.2. Cauchy Sequences. What is slightly annoying for the mathematician (in theory and in praxis) is that we refer to the limit of a sequence in the definition of a convergent sequence when that limit may not be known at all. In fact, more often then not it is quite hard to determine the actual limit of a sequence. MATHCS: STATISTICS DATA SETS EXCEL A wealth of curated data sets, available in different formats (inluding CVS suitable for Excel), including " number of Prussian cavalry soldiers killed by horse kicks (1875 to 1894) ", " Global-mean monthly, seasonal, and annual temperatures since 1880 ", and many more. Kaggle is a platform for predictive modelling and analyticscompetitions
MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS.ORG - REAL ANALYSIS: 5.3. CONNECTED AND Note that the definition of disconnected set is easier for an open set S.In principle, however, the idea is the same: If a set S can be separated into two open, disjoint sets in such a way that neither set is empty and both sets combined give the original set S, then S is called disconnected.. To show that a set is disconnected is generally easier than showing connectedness: if you can find a MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG - REAL ANALYSIS: THEOREM 5.2.6: HEINE-BOREL THEOREM C = { comp ( ), n > 0 } Then each set in C is open, being the complement of closed sets. They also cover S, because the only point that this collection is missing is the point s, which is not part of S. By assumption, a finite subcover already covers S. If N is the largest index of that finite subcovering, then aN+1 is not BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - REAL ANALYSIS: DEFINITION 1.2.6: ONE-ONE Definition 1.2.6: One-one, Onto, Bijection. A function f from A to B is called one to one (or one-one) if whenever f (a) = f (b) then a = b. Such functions are also called injections. A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. Such functions are also called surjections. A function f from A MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.2.8 Proof: Each A j is compact, hence closed and bounded. Therefore, A is closed and bounded as well, and hence A is compact. Pick an a j A j for each j. Then the sequence { a j} is contained in A 1.Since that set is compact, there exists a convergent subsequence { a j k} with limit in A 1. But that subsequence, except the first number, is alsocontained in A 2.
MATHCS.ORG - REAL ANALYSIS: 3.2. CAUCHY SEQUENCES 3. Sequences of Numbers. 3.2. Cauchy Sequences. What is slightly annoying for the mathematician (in theory and in praxis) is that we refer to the limit of a sequence in the definition of a convergent sequence when that limit may not be known at all. In fact, more often then not it is quite hard to determine the actual limit of a sequence. MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS: INTRO TO STATISTICS WITH EXCEL We first need to open the Life Expectancy data file - click on the icon below for the data file. life.xls. When the spreadsheet opens up, mark all numeric data in column B (the Life Expectancy column) but not including the codlumn header and copy them to the clipboard (for example, press CTRL-C). MATHCS.ORG - REAL ANALYSIS: 7.1. RIEMANN INTEGRAL Examples 7.1.11: Is the function f(x) = x 2 Riemann integrable on the interval ?If so, find the value of the Riemann integral. Do the same for the interval (since this is the same example as before, using Riemann's Lemma will hopefully simplify the solution).; Suppose f is Riemann integrable over an interval and { P n} is a sequence of partitions whose mesh converges to zero. MATHCS.ORG - REAL ANALYSIS: EXAMPLES 4.1.7(B): Take a look at the following inequality: 2 n + j < 2 n+1 for j = 1, 3, , 2 n-1 (which makes for 2 n-1 terms). Therefore: (2 n + 1)-1 + (2 n + 3)-1 + + (2 n+1 - 1)-1 > 2 n-1 / 2 n+1 = 1 / 4 But now we can rearrange the terms of the alternating harmonic series as follows: MATHCS.ORG - REAL ANALYSIS: INTEGRAL TEST Suppose that f(x) is positive, continuous, decreasing function on the interval [N, ).Let a n = f(n).Then converges if and only if converges MATHCS.ORG - REAL ANALYSIS: EXAMPLES 5.1.2(A): The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x.Now let = min( 3 + x, 3 - x ).Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.; The interval is closed, because its complement consists of the two open MATHCS.ORG - REAL ANALYSIS: 6.5. DIFFERENTIABLE FUNCTIONS Definition 6.5.1: Derivative : Let f be a function with domain D in R, and D is an open set in R.Then the derivative of f at the point c is defined as . f'(c) = If that limit exits, the function is called differentiable at c.If f is differentiable at every point in D then f is called differentiable in D.. Other notations for the derivative off are or f(x).
MATHCS.ORG: STATCRUNCH MANUAL StatCrunch Overview. Version 0.9 Bert G. Wachsmuth . Contents. Introduction.. How to start StatCrunch.. Getting around in StatCrunch.. How to enter Data.. To enter your own data. MATHCS.ORG - REAL ANALYSIS: 5.2. COMPACT AND PERFECT SETS Another, rather peculiar example of a closed, compact, and perfect set is the Cantor set. Definition 5.2.12: Cantor Middle Third Set. Start with the unit interval. S0 = Remove from that set the middle third and set. S1 = S0 \ (1/3, 2/3) Remove from that set the two middle thirds and set. MATHCS.ORG - REAL ANALYSIS: EXAMPLE 4.1.10: THE LEANING Example 4.1.10: The Leaning Tower of Lire. Jillian, a diligent but overworked student, fell asleep in the library and got locked in for the night. When she awoke, the room was dimly lit and she was alone. To pass the time (and to annoy the librarian in the morning) she decided to stack books on a table so that they would overhang the edgeof
MATHCS: STATISTICS DATA SETS EXCEL A wealth of curated data sets, available in different formats (inluding CVS suitable for Excel), including " number of Prussian cavalry soldiers killed by horse kicks (1875 to 1894) ", " Global-mean monthly, seasonal, and annual temperatures since 1880 ", and many more. Kaggle is a platform for predictive modelling and analyticscompetitions
MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS: INTRO TO STATISTICS WITH EXCEL We first need to open the Life Expectancy data file - click on the icon below for the data file. life.xls. When the spreadsheet opens up, mark all numeric data in column B (the Life Expectancy column) but not including the codlumn header and copy them to the clipboard (for example, press CTRL-C). MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG - REAL ANALYSIS: 2.3. THE PRINCIPLE OF INDUCTION Examples 2.3.2: Determine which of the following sets and their ordering relations are partially ordered, ordered, or well-ordered: S is any set. Define a b if a = b; S is any set, and P(S) the power set of S.Define A B if A B; S is the set of real numbers between . Define a b if a is less than or equal to b (i.e. the 'usual' interpretation of the symbol ) MATHCS.ORG - REAL ANALYSIS: EXAMPLES 5.1.2(A): The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x.Now let = min( 3 + x, 3 - x ).Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.; The interval is closed, because its complement consists of the two open MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.2.3: COMPACT Proof: First, suppose S is closed and bounded. Take a sequence in S.Because S is bounded, the sequence is bounded also, and by the Bolzano-Weierstrass theorem we can extract a convergent subsequence from .Using the theorem about closed sets, accumulation points and sequences, we know that the limit of the subsequence is again in S. CHAPTER 1: FOUNDATIONS Java by Definition Chapter 1: Foundations Page 3 of 60 Bert G. Wachsmuth DRAFT April 2009 } Every statement whose next statement is not a separate group must end in a semicolon. A Java program containing the above framework must be saved using the filename Name.java, where Name (including correct upper and lower cases) is theword that follows
BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - WELCOMEWEB VIEW Sheet3 Sheet2 Sheet1 ID Age Weight Height NumCDs TVHours ExerciseHours SleepHours 55107.00 18.00 128.00 64.00 70.00 24.00 1.00 7.00 55108.00 22.00 240.00 74.00 300.00 MATHCS: STATISTICS DATA SETS EXCEL A wealth of curated data sets, available in different formats (inluding CVS suitable for Excel), including " number of Prussian cavalry soldiers killed by horse kicks (1875 to 1894) ", " Global-mean monthly, seasonal, and annual temperatures since 1880 ", and many more. Kaggle is a platform for predictive modelling and analyticscompetitions
MATHCS.ORG: INTRO TO STATISTICS We would like to know whether there is a linear relationship between the high school GPA and the freshmen GPA, and we would like to be able to predict the freshmen GPA, if we know that the high school GPA of another student is, say, 3.4. MATHCS: INTRO TO STATISTICS WITH EXCEL We first need to open the Life Expectancy data file - click on the icon below for the data file. life.xls. When the spreadsheet opens up, mark all numeric data in column B (the Life Expectancy column) but not including the codlumn header and copy them to the clipboard (for example, press CTRL-C). MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG - REAL ANALYSIS: 2.3. THE PRINCIPLE OF INDUCTION Examples 2.3.2: Determine which of the following sets and their ordering relations are partially ordered, ordered, or well-ordered: S is any set. Define a b if a = b; S is any set, and P(S) the power set of S.Define A B if A B; S is the set of real numbers between . Define a b if a is less than or equal to b (i.e. the 'usual' interpretation of the symbol ) MATHCS.ORG - REAL ANALYSIS: EXAMPLES 5.1.2(A): The interval (-3, 3) is open, because if x is any number in (-3, 3), then -3 < x < 3. or equivalently, -3 - x < 0 < 3 - x.Now let = min( 3 + x, 3 - x ).Then > 0, and the interval (x - / 2, x + / 2) is contained in (-3, 3). In fact, the same argument works for any interval of the form (a, b) with a, b real numbers.; The interval is closed, because its complement consists of the two open MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.2.3: COMPACT Proof: First, suppose S is closed and bounded. Take a sequence in S.Because S is bounded, the sequence is bounded also, and by the Bolzano-Weierstrass theorem we can extract a convergent subsequence from .Using the theorem about closed sets, accumulation points and sequences, we know that the limit of the subsequence is again in S. CHAPTER 1: FOUNDATIONS Java by Definition Chapter 1: Foundations Page 3 of 60 Bert G. Wachsmuth DRAFT April 2009 } Every statement whose next statement is not a separate group must end in a semicolon. A Java program containing the above framework must be saved using the filename Name.java, where Name (including correct upper and lower cases) is theword that follows
BUILDING A (DIFFERENTIAL DRIVE) ROBOT A differential drive robot is a robot with two motorized wheels that can be controlled independently and that allow your robot to: go straight by rotating both motors (wheels) at the same speed in the same direction. go in a curve by rotating the wheels at different speeds in the same direction. turn "on the spot" by rotating thewheels at the
MATHCS.ORG - WELCOMEWEB VIEW Sheet3 Sheet2 Sheet1 ID Age Weight Height NumCDs TVHours ExerciseHours SleepHours 55107.00 18.00 128.00 64.00 70.00 24.00 1.00 7.00 55108.00 22.00 240.00 74.00 300.00 MATHCS.ORG - REAL ANALYSIS: REAL ANALYSIS MathCS.org - Real Analysis: Real Analysis. Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more. MATHCS.ORG - REAL ANALYSIS: 5.1. OPEN AND CLOSED SETS Definition 5.1.1: Open and Closed Sets. A set U R is called open, if for each x U there exists an > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x. A set F is called closed if MATHCS.ORG - REAL ANALYSIS: 7.1. RIEMANN INTEGRAL Examples 7.1.11: Is the function f(x) = x 2 Riemann integrable on the interval ?If so, find the value of the Riemann integral. Do the same for the interval (since this is the same example as before, using Riemann's Lemma will hopefully simplify the solution).; Suppose f is Riemann integrable over an interval and { P n} is a sequence of partitions whose mesh converges to zero. MATHCS.ORG - REAL ANALYSIS: THEOREM 5.2.6: HEINE-BOREL THEOREM Proof: First, assume that every open cover C of S can be reduced to a finite subcovering. We will show that S must then be closed and bounded, which means by the previous result that S is compact.. S must be bounded: Take the collection C = { : S}, where = (- 1, + 1). Then this collection is an open cover of S, and by assumption can be reduced to a finite subcovering of S. MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.1.7: BOUNDARY Proposition 5.1.7: Boundary, Accumulation, Interior, and Isolated Points. Let S R. Then each point of S is either an interior point or a boundary point. Let S R. Then bd ( S) = bd ( R \ S ). A closed set contains all of its boundary points. An open set contains none of itsboundary points.
MATHCS.ORG: STATCRUNCH MANUAL StatCrunch Overview. Version 0.9 Bert G. Wachsmuth . Contents. Introduction.. How to start StatCrunch.. Getting around in StatCrunch.. How to enter Data.. To enter your own data. MATHCS.ORG - REAL ANALYSIS: 5.2. COMPACT AND PERFECT SETS Another, rather peculiar example of a closed, compact, and perfect set is the Cantor set. Definition 5.2.12: Cantor Middle Third Set. Start with the unit interval. S0 = Remove from that set the middle third and set. S1 = S0 \ (1/3, 2/3) Remove from that set the two middle thirds and set. MATHCS.ORG - REAL ANALYSIS: DEFINITION 1.2.6: ONE-ONE Definition 1.2.6: One-one, Onto, Bijection. A function f from A to B is called one to one (or one-one) if whenever f (a) = f (b) then a = b. Such functions are also called injections. A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. Such functions are also called surjections. A function f from A MATHCS.ORG - REAL ANALYSIS: PROPOSITION 5.2.8 Proof: Each A j is compact, hence closed and bounded. Therefore, A is closed and bounded as well, and hence A is compact. Pick an a j A j for each j. Then the sequence { a j} is contained in A 1.Since that set is compact, there exists a convergent subsequence { a j k} with limit in A 1. But that subsequence, except the first number, is alsocontained in A 2.
MATHCS.ORG - REAL ANALYSIS: EXAMPLE 4.1.10: THE LEANING Example 4.1.10: The Leaning Tower of Lire. Jillian, a diligent but overworked student, fell asleep in the library and got locked in for the night. When she awoke, the room was dimly lit and she was alone. To pass the time (and to annoy the librarian in the morning) she decided to stack books on a table so that they would overhang the edgeof
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Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus. It deals with sets, sequences, series, continuity, differentiability, integrability, topology, etc. Android Tutorial is an online, self-contained tutorial to create programs for the Android operating system for small mobile devices such as cell phones. It allows anyone to develop applications for the Android system, and - if they are worthwhile - upload them to the Android marketplace. Java , a programming language introduced by Sun, is getting more and more accepted as a teaching language. Here you can find resources, applets , source code, as well as a complete online textbook Java by Definition to learn Java fromthe ground up.
Statistics is the science of making sense of data - quite a useful skill. Even more useful is this online introductory stats course using Excel . If you know some stats already, you will appreciate learning how to apply Excel's power to statistics. and if you know Excel already, you will learn about some of its hidden functionality and pick up some useful statistics knowledge along the way - as we statisticians say: _ "may all your statistics besignificant"_.
Resources for an introduction to robotics . This resource will include information on how to program the Lego NXT Brick using Java (via LeJOS), including many sample programs, an introduction to Microsoft Robotics Studio, how to program a Corobot vehicle with a six-degrees-of-freedom arm, and some videos from a robotics course I am teaching periodically. Interactive Complex Analysisis an online,
interactive textbook for Complex Analysis. It deals with complex numbers and topology, complex functions, holomorphic functions and properties, singularities, residues, conformal maps, harmonicfunctions, etc.
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