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37.4: Exercise 37 - Mathematics


37.4: Exercise 37 - Mathematics

37.4: Exercise 37 - Mathematics

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The Stacks project

where the first map sends a local section $f$ of $mathcal$ to the invertible section $1 + f$ of $mathcal_$. We also use the identification of the Picard group of a ringed space with the first cohomology group of the sheaf of invertible functions, see Cohomology, Lemma 20.6.1. $square$

Lemma 37.4.2 . Let $X subset X'$ be a thickening. Let $n$ be an integer invertible in $mathcal_ X$. Then the map $mathop> olimits (X')[n] o mathop> olimits (X)[n]$ is bijective.

Proof for a finite order thickening. By the general principle explained following Definition 37.2.1 this reduces to the case of a first order thickening. Then may use Lemma 37.4.1 to see that it suffices to show that $H^1(X, mathcal)[n]$, $H^1(X, mathcal)/n$, and $H^2(X, mathcal)[n]$ are zero. This follows as multiplication by $n$ on $mathcal$ is an isomorphism as it is an $mathcal_ X$-module. $square$

Proof in general. Let $mathcal subset mathcal_$ be the quasi-coherent ideal sheaf cutting out $X$. Then we have a short exact sequence of abelian groups

We obtain a long exact cohomology sequence as in the statement of Lemma 37.4.1 with $H^ i(X, mathcal)$ replaced by $H^ i(X, (1 + mathcal)^*)$. Thus it suffices to show that raising to the $n$th power is an isomorphism $(1 + mathcal)^* o (1 + mathcal)^*$. Taking sections over affine opens this follows from Algebra, Lemma 10.32.8. $square$


Python Math: - Exercises, Practice, Solution

1. Write a Python program to convert degree to radian. Go to the editor
Note : The radian is the standard unit of angular measure, used in many areas of mathematics. An angle's measurement in radians is numerically equal to the length of a corresponding arc of a unit circle one radian is just under 57.3 degrees (when the arc length is equal to the radius).
Test Data:
Degree : 15
Expected Result in radians: 0.2619047619047619
Click me to see the sample solution

2. Write a Python program to convert radian to degree. Go to the editor
Test Data:
Radian : .52
Expected Result : 29.781818181818185
Click me to see the sample solution

3. Write a Python program to calculate the area of a trapezoid. Go to the editor
Note : A trapezoid is a quadrilateral with two sides parallel. The trapezoid is equivalent to the British definition of the trapezium. An isosceles trapezoid is a trapezoid in which the base angles are equal so.
Test Data:
Height : 5
Base, first value : 5
Base, second value : 6
Expected Output: Area is : 27.5
Click me to see the sample solution

4. Write a Python program to calculate the area of a parallelogram. Go to the editor
Note : A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle.
Test Data:
Length of base : 5
Height of parallelogram : 6
Expected Output: Area is : 30.0
Click me to see the sample solution

5. Write a Python program to calculate surface volume and area of a cylinder. Go to the editor
Note: A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder.
Test Data:
volume : Height (4), Radius(6)
Expected Output:
Volume is : 452.57142857142856
Surface Area is : 377.1428571428571
Click me to see the sample solution

6. Write a Python program to calculate surface volume and area of a sphere. Go to the editor
Note: A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.
Test Data:
Radius of sphere : .75
Expected Output :
Surface Area is : 7.071428571428571
Volume is : 1.7678571428571428
Click me to see the sample solution

7. Write a Python program to calculate arc length of an angle. Go to the editor
Note: In a planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.
Test Data:
Diameter of a circle : 8
Angle measure : 45
Expected Output :
Arc Length is : 3.142857142857143
Click me to see the sample solution

8. Write a Python program to calculate the area of the sector. Go to the editor
Note: A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.
Test Data:
Radius of a circle : 4
Angle measure : 45
Expected Output:
Sector Area: 6.285714285714286
Click me to see the sample solution

9. Write a Python program to calculate the discriminant value. Go to the editor
Note: The discriminant is the name given to the expression that appears under the square root (radical) sign in the quadratic formula.
Test Data:
The x value : 4
The y value : 0
The z value : -4
Expected Output:
Two Solutions. Discriminant value is : 64.0
Click me to see the sample solution

10. Write a Python program to find the smallest multiple of the first n numbers. Also, display the factors. Go to the editor
Test Data:
If n = (13)
Expected Output :
[13, 12, 11, 10, 9, 8, 7]
360360
Click me to see the sample solution

11. Write a Python program to calculate the difference between the squared sum of first n natural numbers and the sum of squared first n natural numbers.(default value of number=2). Go to the editor
Test Data:
If sum_difference(12)
Expected Output :
5434
Click me to see the sample solution

12. Write a Python program to calculate the sum of all digits of the base to the specified power. Go to the editor
Test Data:
If power_base_sum(2, 100)
Expected Output :
115
Click me to see the sample solution

13. Write a Python program to find out, if the given number is abundant. Go to the editor
Note: In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16.
Test Data:
If is_abundant(12)
If is_abundant(13)
Expected Output:
True
False
Click me to see the sample solution

14. Write a Python program to sum all amicable numbers from 1 to specified numbers. Go to the editor
Note: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.)
Test Data:
If amicable_numbers_sum(9999)
If amicable_numbers_sum(999)
If amicable_numbers_sum(99)
Expected Output:
31626
504
0
Click me to see the sample solution

15. Write a Python program to returns sum of all divisors of a number. Go to the editor
Test Data:
If number = 8
If number = 12
Expected Output:
7
16
Click me to see the sample solution

16. Write a Python program to print all permutations of a given string (including duplicates). Go to the editor
Click me to see the sample solution

17. Write a Python program to print the first n Lucky Numbers. Go to the editor
Lucky numbers are defined via a sieve as follows.
Begin with a list of integers starting with 1 :
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, . . . .
Now eliminate every second number :
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, .
The second remaining number is 3, so remove every 3rd number:
1, 3, 7, 9, 13, 15, 19, 21, 25, .
The next remaining number is 7, so remove every 7th number:
1, 3, 7, 9, 13, 15, 21, 25, .
Next, remove every 9th number and so on.
Finally, the resulting sequence is the lucky numbers.
Click me to see the sample solution

18. Write a Python program to computing square roots using the Babylonian method. Go to the editor
Perhaps the first algorithm used for approximating &radicS is known as the Babylonian method, named after the Babylonians, or "Hero's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method. It can be derived from (but predates by 16 centuries) Newton's method. The basic idea is that if x is an overestimate to the square root of a non-negative real number S then S / x will be an underestimate and so the average of these two numbers may reasonably be expected to provide a better approximation.
Click me to see the sample solution

19. Write a Python program to multiply two integers without using the * operator in python. Go to the editor
Click me to see the sample solution

20. Write a Python program to calculate magic square. Go to the editor
A magic square is an arrangement of distinct numbers (i.e., each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number, called the "magic constant." A magic square has the same number of rows as it has columns, and in conventional math notation, "n" stands for the number of rows (and columns) it has. Thus, a magic square always contains n2 numbers, and its size (the number of rows [and columns] it has) is described as being "of order n".

Click me to see the sample solution

21. Write a Python program to print all primes (Sieve_of_Eratosthenes) smaller than or equal to a specified number. Go to the editor
In mathematics, the sieve of Eratosthenes, one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the multiples of 2.
Click me to see the sample solution

22. Write a python program to find the next smallest palindrome of a specified number. Go to the editor
Click me to see the sample solution

23. Write a python program to find the next previous palindrome of a specified number. Go to the editor
Click me to see the sample solution

24. Write a Python program to convert a float to ratio. Go to the editor

25. Write a Python program for nth Catalan Number. Go to the editor
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. They are named after the Belgian mathematician Eugène Charles Catalan (1814 &ndash1894).
Click me to see the sample solution

26. Write a Python program to print number with commas as thousands separators(from right side). Go to the editor
Click me to see the sample solution

27. Write a Python program to calculate the distance between two points using latitude and longitude. Go to the editor

28. Write a Python program to calculate the area of regular polygon. Go to the editor

29. Write a Python program to calculate wind chill index. Go to the editor

30. Write a Python program to find the roots of a quadratic function. Go to the editor

31. Write a Python program to convert a binary number to decimal number. Go to the editor

32. Write a Python program to print a complex number and its real and imaginary parts. Go to the editor

33. Write a Python program to add, subtract, multiply and division of two complex numbers. Go to the editor

34. Write a Python program to get the length and the angle of a complex number. Go to the editor

35. Write a Python program to convert Polar coordinates to rectangular coordinates. Go to the editor

36. Write a Python program to find the maximum and minimum numbers from the specified decimal numbers. Go to the editor

Decimal numbers : 2.45, 2.69, 2.45, 3.45, 2.00, 0.04, 7.25

37. Write a Python program to find the sum of the following decimal numbers and display the numbers in sorted order. Go to the editor

Decimal numbers : 2.45, 2.69, 2.45, 3.45, 2.00, 0.04, 7.25

38. Write a Python program to get the square root and exponential of a given decimal number. Go to the editor

39. Write a Python program to retrieve the current global context (public properties) for all decimal. Go to the editor

40. Write a Python program to round a specified decimal by setting precision (between 1 and 4). Go to the editor

Sample Number : 0.26598
Original Number : 0.26598
Precision- 1 : 0.3
Precision- 2 : 0.27
Precision- 3 : 0.266
Precision- 4 : 0.2660

41. Write a Python program to round a specified number upwards towards infinity and down towards negative infinity of precision 4. Go to the editor

42. Write a Python program to get the local and default precision. Go to the editor

43. Write a Python program to display the fraction instances of the string representation of a number. Go to the editor

Sample data : '0.7', '2.5', '9.32', '7e-1'

44. Write a Python program to create the fraction instances of float numbers. Go to the editor

Sample numbers: 0.2, 0.7, 6.5, 6.0

45. Write a Python program to create the fraction instances of decimal numbers. Go to the editor

Sample decimal.2' number: Decimal('0), Decimal('0.7'), Decimal('2.5'), Decimal('3.0')

46. Write a Python program to add, subtract, multiply and divide two fractions. Go to the editor

47. Write a Python program to convert a floating point number (PI) to an approximate rational value on the various denominator. Go to the editor

48. Write a Python program to generate random float numbers in a specific numerical range. Go to the editor

49. Write a Python program to generate random integers in a specific numerical range. Go to the editor

50. Write a Python program to generate random even integers in a specific numerical range. Go to the editor

51. Write a Python program to get a single random element from a specified string. Go to the editor

52. Write a Python program to shuffle the following elements randomly. Go to the editor

Sample elements : [1, 2, 3, 4, 5, 6, 7]

53. Write a Python program to flip a coin 1000 times and count heads and tails. Go to the editor

54. Write a Python program to print a random sample of words from the system dictionary. Go to the editor

55. Write a Python program to randomly select an item from a list. Go to the editor

56. Write a Python program to calculate the absolute value of a floating point number. Go to the editor

57. Write a Python program to calculate the standard deviation of the following data. Go to the editor

58. Write a Python program to print the floating point from mantissa, exponent pair. Go to the editor

59. Write a Python program to split fractional and integer parts of a floating point number. Go to the editor

60. Write a Python program to parse math formulas and put parentheses around multiplication and division. Go to the editor

61. Write a Python program to describe linear regression. Go to the editor

Note : A linear regression line has an equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable. The slope of the line is b, and a is the intercept (the value of y when x = 0).

62. Write a Python program to calculate a grid of hexagon coordinates of the given radius given lower-left and upper-right coordinates. The function will return a list of lists containing 6 tuples of x, y point coordinates. These can be used to construct valid regular hexagonal polygons. Go to the editor

63. Write a Python program to create a simple math quiz. Go to the editor

64. Write a Python program to calculate the volume of a tetrahedron. Go to the editor

Note: In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

65. Write a Python program to compute the value of e(2.718281827. ) using infinite series. Go to the editor

66. Write a Python program to create an ASCII waveform. Go to the editor

67. Write a Python program to create a dot string. Go to the editor

68. Write a Python program to create a Pythagorean theorem calculator. Go to the editor

Note : In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

69. Write a Python function to round up a number to specified digits. Go to the editor

70. Write a Python program for casino simulation. Go to the editor

71. Write a Python program to reverse a range. Go to the editor

72. Write a Python program to create a range for floating numbers. Go to the editor

73. Write a Python program to generate (given an integer n) a square matrix filled with elements from 1 to n raised to the power of 2 in spiral order. Go to the editor

74. Write a Python program to select a random date in the current year. Go to the editor

75. Write a Python program to calculate clusters using Hierarchical Clustering method. Go to the editor

76. Write a Python program to implement Euclidean Algorithm to compute the greatest common divisor (gcd). Go to the editor

77. Write a Python program to convert RGB color to HSV color. Go to the editor

78. Write a Python program to find perfect squares between two given numbers. Go to the editor

79. Write a Python program to compute Euclidean distance. Go to the editor

Note: In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" (i.e. straight-line) distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm.

80. Write a Python program to convert an integer to a 2 byte Hex value. Go to the editor

81. Write a Python program to generate a series of unique random numbers. Go to the editor

82. Write a Python program to convert a given float value to ratio. Go to the editor

83. Write a Python program to calculate the aliquot sum of an given integer. Go to the editor

84. Write a Python program to get the nth tetrahedral number from a given integer(n) value. Go to the editor

85. Write a Python program to get the sum of the powers of all the numbers from start to end (both inclusive). Go to the editor

86. Write a Python program to calculate the Hamming distance between two given values. Go to the editor

87. Write a Python program to cap a number within the inclusive range specified by the given boundary values x and y. Go to the editor

More to Come !

Do not submit any solution of the above exercises at here, if you want to contribute go to the appropriate exercise page.

Test your Python skills with w3resource's quiz


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Math 5051

Basic Information

Instructor: Ari Stern
Email: [email protected]
Office: Cupples I, 211B
Office Hours: MWTh 11am-12pm

Lectures

Lectures will be held MWF 10-11am in Cupples I, 199. The first class will be on Wednesday, August 29, and the last will be on Friday, December 7. Class will be canceled for Labor Day (Monday, September 3), Fall Break (Friday, October 19), and Thanksgiving Break (Wednesday, November 21 through Friday, November 23).

Homework Assignments

Problem sets will be posted here weekly, on Wednesdays, and will be collected the following Wednesday at the beginning of class. You are encouraged to discuss the homework with your fellow students, and to collaborate on problems, but your final write-up must be your own. Please make sure that your solutions are written clearly and legibly. (Typing up solutions in LaTeX is encouraged, and is valuable practice for mathematical writing later in your career.)

Will Ward ([email protected]) is responsible for homework grades and solutions.

    . Due Wednesday, September 5. Solution [pdf]
  • HW2: Folland, Chapter 1, Exercises 7, 8, 13, 14, 17, 18, 19. Due Wednesday, September 12. Solution [pdf]
  • HW3: Folland, Chapter 1, Exercises 22a, 26, 27, 29, 30, 31, 33. Due Wednesday, September 19. Solution [pdf]
  • HW4: Folland, Chapter 2, Exercises 3, 4, 8, 9, 10, 12, 13. Due Wednesday, September 26. Solution [pdf]
  • HW5: Folland, Chapter 2, Exercises 19, 20, 21, 26, 34, 36. (Hint for Exercise 34: Use the fact that a sequence of real numbers converges iff every subsequence has a further subsequence converging to the same limit.) Due Wednesday, October 3. Solution [pdf]
  • HW6: Folland, Chapter 2, Exercises 39, 42, 44, 46, 50, 56, 59. Due Wednesday, October 10. Solution [pdf]
  • HW7: Folland, Chapter 3, Exercises 2, 4, 5, 9, 13, 16, 17. Due Wednesday, October 17. Solution [pdf]
  • HW8: Folland, Chapter 5, Exercises 3, 5, 6, 7, 12ab, 13, 17, 19. Due Wednesday, November 7. Solution [pdf]
  • HW9: Folland, Chapter 5, Exercises 31, 32, 37, 44, 46, 47, 55, 56 . Due Wednesday, November 14. Solution [pdf]
  • HW10: Folland, Chapter 5, Exercises 55, 56, 57, 58, 59, 63, 67. Due Wednesday, November 28. Solution [pdf]
  • HW11: Folland, Chapter 6, Exercises 7, 9, 11, 12, 18, 19, 21. Due Wednesday, December 5 Friday, December 7. Solution [pdf]

Exams

There was one in-class midterm exam, held on Friday, October 26. Solution [pdf]

The final exam was held on Monday, December 17, from 10:30am-12:30pm.

Grading

Grades will be based on a weighted average of homework (40%, lowest two scores dropped), midterm exam (20%), and final exam (40%).

Required and Supplemental Texts

The required textbook for this course is Real Analysis: Modern Techniques and Their Applications, by Gerald B. Folland (second edition, Wiley, 1999). This book has more than a few typographical errors, so it's a good idea to check the list of errata on Folland's homepage.

In addition, I have asked the library to place the following supplemental texts on reserve:

  • H. L. Royden, Real Analysis (3rd edition).
  • E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces.
  • T. Tao, An Introduction to Measure Theory (based on his freely-available course notes).
  • R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis.

Do not feel obligated to purchase any of these non-required books (although each one is excellent in its own way). I am making them available simply because it can be helpful to see alternative treatments of the same material.

Course Outline

I plan to cover the topics discussed in Folland chapters 1-3, 5, and part of 6:

  • Chapter 1: Measure
  • Chapter 2: Integration
  • Chapter 3: Signed Measures and Differentiation
  • Chapter 5: Elements of Functional Analysis
  • Chapter 6: L p Spaces

The topics in chapters 0 and 4 are assumed to be prerequisites, as they are typically covered in undergraduate real analysis, and you are encouraged to review them on your own, as needed.


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The elimination method of solving systems of equations is also called the addition method. To solve a system of equations by elimination we transform the system such that one variable "cancels out".

Example 1: Solve the system of equations by elimination

$ egin 3x - y &= 5 x + y &= 3 end $

In this example we will "cancel out" the y term. To do so, we can add the equations together.

Now we can find: $x = 2$

In order to solve for y, take the value for x and substitute it back into either one of the original equations.

The solution is $(x, y) = (2, 1)$.

Example 2: Solve the system using elimination

$ egin x + 3y &= -5 4x - y &= 6 end $

Look at the x - coefficients. Multiply the first equation by -4, to set up the x-coefficients to cancel.

Now we can find: $y = -2$

Take the value for y and substitute it back into either one of the original equations.

$ egin x + 3y &= -5 x + 3cdot(color<-2>) &= -5 x - 6 &= -5 x &= 1 end $

The solution is $(x, y) = (1, -2)$.

Example 3: Solve the system using elimination method

$ egin 2x - 5y &= 11 3x + 2y &= 7 end $

In this example, we will multiply the first row by -3 and the second row by 2 then we will add down as before.


37.4: Exercise 37 - Mathematics

1989 Volume 37 Issue 4 Pages 1790-1795

  • Published: February 25, 1989 Received: - Available on J-STAGE: February 25, 2010 Accepted: - Advance online publication: - Revised: -

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We are carring out therapeutic exercise in a heated swimming pool for motor functuion disease. This time we report results of the effect we conducted an investigation. The cases were 7 patients with Rheumatoid arthritis, 21 osteoarthritis, 7 shoulder-arm-neck syndrome (cervical disc herniation, periarthritis of the shoulder, etc&hellip) and 21 low back pain. We prescribed for them the exercise for three months, and valued, about each muscular strength, R. O. M., range of pain, etc&hellip.
The results were maintenance of the present situation or some improvements as to each patient. Effects of therapeutic exercise in a heated swimming pool are to reduce excessive pain of patients by heat, and more than anything else to improve motivation. We hope we will take in therapeutic exercise in a heated swimming pool.


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