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https://www.jiskha.com/search/index.cgi?query=What+do+polynomial+functions+look+like%3F+And+what+can+be+consider+a+polynomial+function%3F | 1,529,486,409,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267863516.21/warc/CC-MAIN-20180620085406-20180620105406-00118.warc.gz | 841,496,173 | 11,118 | # What do polynomial functions look like? And what can be consider a polynomial function?
46,688 results
1. ### Graphs of polynomial functions
What do polynomial functions look like? And what can be consider a polynomial function? Would a graph that is like an upside down V be considered as a graph of a polynomial function?
2. ### Polynomial Functions
What do polynomial functions look like? And what can be consider a polynomial function? Would a graph that is like an upside down V be considered as a graph of a polynomial function?
3. ### Polynomial Functions-help
What do polynomial functions look like? And what can be consider a polynomial function? Would a graph that is like an upside down V be considered as a graph of a polynomial function?
4. ### Math:Polynomial Functions
h(x) = -7x I was thinking that this was not a polynomial function because -7x is just one term making it a monomial but the answer key says it is a polynomial function. Could someone explain why that is so? Thanks :D
5. ### Math
The function f(x) = x^2 -2x + x^1/2 is: A. A polynomial because it is continuous. B. A polynomial because it is of the form axn. C. Not a polynomial because you are subtracting 2x. D. Not a polynomial because you can’t have any fractional exponents. Is the answer D?
6. ### Writing polynomial functions
How to write a polynomial function with the integral coefficients that have the following roots: #1.) 0, -1/2, 6 #2.) + or - 5i
7. ### Graphing polynomial functions
How do i graph the polynomial function of: f(x)= (x+3)(x+1)(x-2) squared
8. ### math
Is the function a polynomial function? Why or why not? If it is a polynomial function identify and label the leading term and constant term. Here are the functions f(x)=(2-x^4)/13 f(x)=x^3+9x-(1/x) it would be great if you could do everything step by step and showing your work.
9. ### Math
What does the degree of a polynomial expression tell you about its related polynomial function? Explain your thinking. Give an example of a polynomial expression of degree three. Provide information regarding the graph and zeros of the related polynomial function.
10. ### Math
consider the polynomial (x-a)(x-b)and the real number line Identify the points on the line where the polynomial is zero. In each of the three subintervals of the line, write the sign of each factor and the sign of the product.For which values does the polynomial possibly ...
11. ### Math - polynomial function
Write a polynomial function with integral coefficients having the given roots. 1.) 0, -1/2, 6 2.) +or- 5i
12. ### math
Consider the polynomial x^3 + 6x^2 + 3x - 10. Without dividing, or graphing, which two of (x+2), (x-2), (x+1), and (x-1) must be factors of the polynomial? Why/how do you know?
13. ### Polynomial Function
Could you help me with the following problem, I don't understand how to do it. Find an nth degree polynomial function with real coefficients satisfying the given conditions. 1. n=3; 3 and i are zeros; f(2)=20
14. ### zeros of a polynomial
polynomial is x^3 - x^2 -11x + 15 If 3 is the zero of the polynomial, when you divide this polynomial with x-3, the remainder has to be 0? yes. If y=x^3 - x^2 -11x + 15, when y=0 the roots are on the axis.
15. ### Functions
Please I need help with this algebra problem With the 3 equations f1(x)=0 f2(x)=1/8(x+16)^2-98 f3(x)=2x+24 Determine an equation for this function with the 3 pieces above.(Refer to creating the polynomial functions)
16. ### Functions
Please I need help with this algebra problem With the 3 equations f1(x)=0 f2(x)=1/8(x+16)^2-98 f3(x)=2x+24 Determine an equation for this function with the 3 pieces above.(Refer to creating the polynomial functions)
17. ### math
Look at this trinomial 32a^4 + 18a^3 - 12a Write the polynomial as the product of the GCF of all its terms and a polynomial PLEase explain thanks
18. ### Math
Consider the polynomial f(x) = 2x^3 – 3x^2 – 8x – 3. (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial. (b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
19. ### Algebra
Consider the polynomial: f(x) = 2x^3 – 3x^2 – 8x – 3. (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial. (b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
20. ### algebra
Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2. (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial. (b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
21. ### algebra
Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2. (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial. (b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
22. ### Algebra
Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2. (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial. (b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
Multiply. 1) (3t^2 - 2t - 4) * (5t + 9) Writing. 1) Explain why the product of a quadratic polynomial and a linear polynomial must be a cubic polynomial.
24. ### math
How is the range of a polynomial function related to the degree of the polynomial?
25. ### FUNCTIONS HELPPPPP????
Please I need help with this algebra problem With the 3 equations f1(x)=0 f2(x)=1/8(x+16)^2-98 f3(x)=2x+24 Determine an equation for this function with the 3 pieces above.(Refer to creating the polynomial functions) I do not know how does everyone get this answer g(x) = f2(x)-...
26. ### Math
Suppose you have a graph of a polynomial function and you can see that the function increases without bound on both left and right ends, has 4 real zereos and has 5 turning points. Based on this information, what is the minimum degree of the polynomial?
27. ### math
1.binomial 2.degree of monomial 3.monomial 4.perfect-square trinomial 5.standard form of polynomial A.a polynomial with two terms B.a polynomial in which the terms decrease in degree from left to right and there are no like terms C.a polynomial whith two identical binomal ...
28. ### Calculus
A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx + k? A cubic polynomial function f is defined by: f(x) = 4x^3 + ax^2 + bx + k where a, b, and k are constants. The function f has a local minimum at x= -1, and the graph of f has a point of inflection at x= -2...
29. ### Calculus
A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx + k? A cubic polynomial function f is defined by: f(x) = 4x^3 + ax^2 + bx + k where a, b, and k are constants. The function f has a local minimum at x= -1, and the graph of f has a point of inflection at x= -2...
30. ### algebra
Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph of each function. Classify each as linear, quadratic, polynomial, exponential, or logarithmic, and explain the reasons for your classifications. Compare how quickly each function increases, based on...
31. ### algbra
Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph of each function. Classify each as linear, quadratic, polynomial, exponential, or logarithmic, and explain the reasons for your classifications. Compare how quickly each function increases, based on...
32. ### Math
2-4 paragraphs plus two graphs Details: Many different kinds of data can be modeled using polynomial functions. An example of a polynomial function would be gas mileage for an automobile. If we compare gas mileage at two different speeds, V1 and V2, the gas required varies as...
33. ### Math ( Polynomial )
If the polynomial, x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by another polynomial x^2 - 2x + k, the remainder comes out to be x + a, find k + a, please work the complete solution instead of giving simply an answer.
34. ### algebra--1 question
Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of the ...
35. ### math
Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a). Part 1. Show all work using long division to divide your polynomial by the binomial. Part 2. Show all work to evaluate f(a) using the function you created. Part 3. Use complete ...
36. ### Calculus
Suppose that ax^2 + bx + c is a quadratic polynomial and that the integration: Int 1/(ax^2 + bx + c) dx produces a function with neither a logarithmic or inverse tangent term. What does this tell you about the roots of the polynomial?
37. ### Math-Functions
1)Expand and simplify. Express each equation in standard form. a) f(x) = -2(x-1)(x+4)(x+1)(x-3) b) f(x) = (x+2)(x+1)(x+4) c) f(x) = x(x-6) 2) For each function in question 1. state the degree of the polynomial and identify the type of function
38. ### Functions
Explain how you can tell whether a polynomial equation is a function and not just a relation.
39. ### Polynomial Function
Could you please check my answers? Find an nth degree polynomial function with real coefficients satisfying the given conditions. 1. n=3; 3 and i are zeros; f(2)=20 -I got: f(x)=-4^3+12x^2-4x+12 3.n=3;4 and i zeros;f(-3)=60 -I got:f(x)=6x^3+24x^2+6x+24
40. ### algebra
Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph of each function. Classify each as linear, quadratic, polynomial, exponential, or logarithmic, and explain the reasons for your classifications. Compare how quickly each function increases, based on...
50. ### Math
How do you divide a polynomial function by another polynomial function.
51. ### Algebra Help me Please
Someone please help me with this!!! Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some ...
52. ### algebra
solve the polynomial y=2x^2-4x+1 I am sure neither your text nor your teacher asked you to "solve the polynomial y=2x^2-4x+1 " are you graphing the function, or are you solving the quadratic y=2x^2-4x+1 ?
53. ### Math
You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area _____________________________________ Among all pairs of numbers whose sum is 20...
A number is called algebraic if there is a polynomial with rational coefficients for which the number is a root. For example, √2 is algebraic because it is a root of the polynomial x^2−2. The number √(2+√3+√5)is also algebraic because it is a root...
55. ### Algebra
I need help with a few questions, please explain. 1. Write a polynomial function in standard form with zeros -1, -1, 6. 2. Find the roots of the equation x^3 – 3x^2 + x + 5 3. Describe the number and type of roots for the polynomial (how many real and complex?). x^3 + 5x^2...
56. ### Finding polynomial from zeros
How to find the polynomial degree 2 and zeros are (1+i) & (1-i) I would like to see the steps to solve this. Thanks. If those are the roots, then the following are factors: (x-1-i)(x-1+i)
57. ### college math 116
Functions are a lot like equations. In a function we look at the relationship of the variables. Does one variable depend on the other. In a function when you see f(x) it just means "y." So we have an equation y = 2x + 3 and a function that is f(x) = 2x + 3. When graphed, they ...
58. ### polynomial function
Please check my answers. Find the y-intercept of the polynomial function. f(x) = -x2 - 2x + 8 I got:y=8 f(x) = (x + 1)(x - 6)(x - 1)2 I got: y=-6 f(x) = -x2(x + 6)(x2 - 1) I got:y=0
59. ### rational functions
If f(x) is a polynomial function, does 1/f(x) always have a horizontal asymptote? Explain why or why not and provide a counterexample if needed.
60. ### Math
consider the polynomial (x-a)(x-b) and the real number line In each of the three subintervals of the line, write the sign of each factor and the sign of the product.For which values does the polynomial possibly change signs?
61. ### algebra question
Give an example of a polynomial with degree 3 such that when factored, it contains the two factors (x+5) and (2x-1), i.e., the polynomial looks like (x+5) (2x -1) ( ? ). How many such polynomials you think there are?
62. ### pre cal
Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 3, -13, and 5 + 4i Urgently need help
63. ### Pre-Calc
Fill in the blank; •In the process of polynomial division (Divisor)(Quotient)+_______=_______ •When a polynomial function f is divided by x-c, the remainder is _______. •If a function f, whose domain is all real numbers, is even and if 4 is a zero of f, then _______ is ...
64. ### math
For every positive integer n, consider all monic polynomials f(x) with integer coefficients, such that for some real number a x(f(x+a)−f(x))=nf(x) Find the largest possible number of such polynomials f(x) for a fixed n<1000. Details and assumptions A polynomial is ...
65. ### Math
What is an example of two functions that intersect at least twice in the first quadrant but can neither be a polynomial or a "simple" function (i.e., sin(x), e^x)?
66. ### math help pls pls pls
Four polynomials are shown below: A. 2 − 2x5 + 2x2 B. 5x3 + 5 − 5x4 C. 3x + 2x5 D. 6x3 − 1 Which of the above polynomials is a 5th degree binomial? Polynomial A Polynomial B Polynomial C Polynomial D pls help me
67. ### Tim
Please I need help with this algebra problem With the 3 equations f1(x)=0 f2(x)=1/8(x+16)^2-98 f3(x)=2x+24 Determine an equation for this function with the 3 pieces above.(Refer to creating the polynomial functions)
68. ### Algebra -polynomial
Solve the polynomial 40x^2+2x-65 Notice that for x = 1 the value is -23. The number 23 is a prime number and the only factors are thus 23 and 1 up to signs. So, you can simply the search for the factors of the polynomial a great deal by substituting: x = 1 + t which yields 40 ...
69. ### pre cal
Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function. x^3 - 28x - 48 = 0 Value of x = -4 Please help!!Thank you
70. ### algebra
Like Terms and Degree Identify the degree of each term of each polynomial. Then find the degree of the polynomial xy^3 + 7x^3y^2 - 6xy^4 + 2
71. ### math
what is the greatest common factor of the polynomial show: rectangle tiles that have 6x and 6 look like this: ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** !!! !!! !!! !!! !!! !!!
72. ### math
what is the greatest common factor of the polynomial show? rectangle tiles that have 6x and 6 look like this: ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** !!! !!! !!! !!! !!! !!!
73. ### Math - Fundamental Theorem
We can actually use the Zeros Theorem and the Conjugate Zeros Theorem together to conclude that an odd-degree polynomial with real coefficients must have atleast one real root (since the non-real roots must come in conjugate pairs). But how can we get the same conclusion by ...
74. ### Math
Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function. x^3 - 28x - 48 = 0 x=2 I have no idea how to start this problem!!
75. ### polynomials
No to both questions. A polynomial is a sum of at least one integer power of x (or other unknown), each multiplied by a constant. '9' is simply a constant. 2^x is a power of 2, not a power of x. Is 9 a polynomial? Is 2^x a polynomial?
76. ### Math
1. Identify the degree, leading term and leading coefficient of each polynomial function. A. f(x)= x(x+1)(3x+1)(x-2) B. f(x)= -16+3x^4 - 9x^2 - x^6 + 4x^8 2. Describe the end behavior of a ninth-degree polynomial function with a negative leading coefficient.
77. ### Algebra Question~!
Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is not a factor of the ...
78. ### POLYNOMIALS
The cubic polynomial f(x) is such that the coefficient of x^3 is -1. and the roots of the equation f(x) = 0 are 1, 2 and k. Given that f(x)has a remainder of 8 when divided by (x-3), find the value of k. okay, this is what i did: -x^3 + bx^2 + cx + d = (x-1)(x-2)(x-k) f(3)=8 -...
79. ### Precalculus
There is at least one polynomial with real functions with 9+i as its only nonreal zero. A. The statement is false, because the Fundamental Theorem of Algebra dictates that there must be n complex zeros for a polynomial of degree n. B.The statement is true. If 9 plus ...
80. ### Math
Given the polynomial functions, find the product function and the specified value. Let f(x) = x + 2 and g(x) = x - 10. Find (fg)(x) and (fg)(-5).
81. ### odd and even functions
Using f is odd if f(-x) = -f(x) or even if f(-x) = f(x) for all real x, how do I 1)show that a polynomial P(x) that contains only odd powers of x is an odd function 2)show that if a polynomial P(x) contains both odd and even powders of x, then it is neither an odd nor an even ...
82. ### Math (polynomial functions)
Hi there, Can someone please help me understand the relationship between the FINITE DIFFERENCE and leading coefficient in a polynomial? I need to be able to determine the leading coefficient from a finite difference.
83. ### Math-precalc
Is it possible to find a rational function that has x-intercepts (-2,0) and (2,0), but has vertical asymptote x=1 and horizontal asymptote of y=0? The horizontal asymptote and the x-intercepts parts stump me. If you can't reach y=0, then how can you get the x-intercepts? And I...
84. ### Pre Calculus
1. Find all rational zeros of the polynomial. Then determine any irrational zeros, and factor the polynomial completely. 3x^4-11x^3+5x^2+3x 2. Find the polynomial with leading coefficient 1 that has a degree of 4, a zero of multiplicity 2 at x=1 and a zero at x=2+i
85. ### math
As^5.V(s)+Bs^4.V(s)+Cs^3.V(s)+Ds^2.V(s)+Es^1V(s)+F.V(s)=1, where V(s) is the transformed transient output voltage. Rearranging the 5th order transient polynomial equation above we have: V(s) = 1 / H(s) where H(s) is a polynomial is a polynomial in s to be determine. The values...
Write the polynomial in standard form. Then name the polynomial based on its degree and number of terms. 2-11x^2-8x+6x^2. A) -5x^2-8x+2; quadratic trinomial B) -5x^2-8x; quadratic binomial C) -6x^2-8x-2; cubic polynomial D) -8x+2; cubic trinomial
87. ### Math
Part 1: Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of...
88. ### Pre-Calc
A polynomial function f(x) with real coefficients has the given degree, zeros, and solution point. Degree: 3 Zeros: -2,2+2√2i Solution Point: f(−1) = −68 (a) Write the function in completely factored form. (b) Write the function in polynomial form. Help Please my teacher...
Find all zeros of the following polynomial. Write the polynomial in factored form. f(x)=x^3-3x^2+16x-48 I put: x^2(x-3)+16(x=3) (x-3)(x^2+16) For zeros: x-3=0 x=0 **My teacher stated check the equation solution again. What is the value for x and hence what is the zero for the ...
90. ### algebra 2
SImplify the expressions. 1. x to the seventh times 1/x squared 2. (3 quaredx to the sixth)cubed 3. x to the nith/ x negitive squared 4. 15xsquaredy/ 6x to the fourthy to the fifth times 6xcubed y squared/5xy USE direct substitution to evaluate the polynomial function for ...
91. ### Math
Part 1: Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of...
92. ### algebra
Identify the degree of each term of each polynomial. Then find the degree of the polynomial combine like term. m^3+ 2m^2n-3m^2+3ma^2
Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function. x^3 - 28x - 48 = 0 Value of x = -4 Please help!!Thank you
94. ### Algebra
Do I have this right? A first degree polynomial crosses the x axis A second degree polynomial touches the y axis without crosisng A third degree polynomial flattens against the y axis.
95. ### math
hi my name is hedi and i am in serious need with help with some math problems dealing with polynomials can some one please help me? Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. (This is the question) 1.5x^3+2xy6^...
96. ### Maths
true/false 1. a cubic polynomial has at least one zero.............. 2. a quadratic polynomial an have at most two zeroes.......... 3. if r(x)is the remainder and p(x) is the divisor, then degree r(x) < degree p(x)............ 4. if zeroes of a quadratic polynomial ax^2+b^x...
97. ### Equivalent Algebraic Expressions
The game shown at the right consists of eight pairs of coloured squares called dominos. Rules: 1. Write a polynomial in each square marked P and a rational function in each square marked R. 2. The expressions you write must satisfy each of these conditions: • Polynomials and...
98. ### algebra
p(x)=x^3+2x^2-3x+20 one of this functions zeros is -4 When using synthetic division to find all the zeros of a polynomial function, would you plug in 4 or -4 into the actual equation?
99. ### Algebra
Can someone please explain how to do these problems. 1)write a polynomial function of least degree with intregal coefficients whose zeros include 4 and 2i. 2)list all of the possible rational zeros of f(x)= 3x^3-2x^2+7x+6. 3)Find all of the rational zeros of f(x)= 4x^3-3x^2-...
100. ### algebra
Please help!!! Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get to assist in the planning of a brand new roller coaster. For this assignment, you help Ray and Kelsey as they tackle the math behind some simple curves in the ... | 6,233 | 22,995 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2018-26 | latest | en | 0.907907 |
https://physics.stackexchange.com/questions/571207/what-makes-the-electron-as-an-excitation-in-a-field-discrete | 1,708,757,960,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474523.8/warc/CC-MAIN-20240224044749-20240224074749-00615.warc.gz | 459,159,043 | 43,171 | # What makes the electron, as an excitation in a field, discrete?
In standard quantum mechanics, the wave function have discrete energy-values due to a potential. However, my very limited understanding of QFT is that electrons are excitation in the Dirac field, and the number of electrons is discrete even in free space. What is the reason for this, and why is there a minimum excitation?
• The number of electrons in free space is discrete, but the spectrum of each is continuous, no? Aug 4, 2020 at 14:48
• For the energy? I would guess so, plus a constant mc^2. The energy spectra was ment as an analogy, I guess. I just have difficulties understanding how any representation of the states in free space become discrete. Aug 4, 2020 at 15:34
Where is the discreteness of the number of excitations of QFT coming from, even in free propagation that apparently does not involve a potential and the corresponding compactness associated with discreteness?
(This is the magic of Fock space often referred to as "second quantization", an aggressively confusing term I and most avoid, truth be told...)
• QFT is a repackaging of an infinity of quantum harmonic oscillators, each one with a discrete spectrum, each level of which corresponds to a particle.
This begs the question of where the harmonic potentials come from, if we are talking about free particles, no?
But this is a classical problem. Your classical continuum mechanics course describes, e.g., one-dimensional field theory, e.g. a "string" often discretized for computational and visualization convenience--the continuum limit is taken at the end of the day, and that discreteness is "fake". What is the crucial part is that the next-neighbor couplings lead to many coupled oscillators, an infinity at the end of the day, whose decoupling leads to normal modes in momentum space. The spectra of these oscillators, each one of them, are continuous before quantization.
But, upon quantization, the spectra become discrete. What you actually quantize is not x or its Fourier conjugate k, mere labels of the oscillators; but, instead, their displacement from equilibrium, the true dynamical variables.
Now each discrete excitation of each oscillator is a particle. The assembly has the built-in option of destroying as well as creating particles.
The ground state, $$|0\rangle$$, is the vacuum, with no particles. $$a^\dagger_k|0\rangle$$ is one particle with momentum k , and there is a continuous infinity of them, as many as there are momenta. $$a^\dagger_k a^\dagger _l|0\rangle$$ is two particles, one with momentum k and one with momentum l, suitably (anti)symmetrized, depending on the statistics of your field. That is, every excitation ("phonon") creator gives you a new particle, etc... You sum the different energies of all your particles to get your total energy of your multi-article state.
The packaging algorithm of quantized oscillators (a functor) is artful and encodes symmetries, Lorentz covariance, etc, but this is mere fine print. The above are naive states; if you want more realistic pictures involving wave packets beyond plane waves, you huff and puff a little, but that is an independent technical problem.
So to answer what I believe to be your question, "Where are the harmonic potentials coming from?", they come from nearest neighbor interactions of your coupled degrees of freedom, the elastic medium itself, even when your quantum fields themselves, and the particles, are free
Might like this question.
• Thank you so much! I think this really answers the core of the question, though i guess I need to just get more familiar with the math to understand it better. Aug 4, 2020 at 22:45
• Here is a friendly summary. Aug 5, 2020 at 0:04
In normal quantum mechanics, we consider an individual particle, turn its momentum and position into vector operators $$\hat{P}_i$$ and $$\hat{X}_i$$, and enforce the canonical commutation relations $$[\hat{X}_i,\hat{P}_j]=i\hbar\delta_{ij}$$. In Quantum Field Theory, we want to apply the laws of quantum mechanics to the field itself, and not to particles. A field $$\phi(\mathbf{x})$$ is treated as a generalized infinite-dimensional coordinate (each position $$\mathbf{x}$$ is one degree of freedom, or one dimension). Its conjugate momentum is defined, using the usual procedure from Hamiltonian classical mechanics, as $$\pi(\mathbf{x})\equiv\frac{\partial\mathcal{L}}{\partial\dot\phi(\mathbf{x})},$$ where $$\mathcal{L}$$ is the Lagrangian of the field. What we do in Quantum Field Theory is change the field $$\phi$$ and the conjugate momentum $$\pi$$ into operators. Then we need to figure out the eigenstates of the Hamiltonian. The way this is done depends on the field. For free Klein-Gordon fields, we do this using the commutation relations $$[\phi(\mathbf{x}),\pi(\mathbf{x})]=i\hbar\delta^{(3)}(\mathbf{x}-\mathbf{y})$$. We use the same trick that is often used in normal quantum mechanics to solve the quantum harmonic oscillator without invoking the Schrodinger equation. This is because the fourier-transformation the Klein-Gordon equation is of the same form as a harmonic oscillator equation. When we apply this method, we get creation and annihilation operators $$a^\dagger(\mathbf{p})$$ and $$a(\mathbf{p})$$ that create and destroy energy-momentum eigenstates out of the vacuum. You can choose any value of $$\mathbf{p}$$. The particles can have any momentum. However, the energy eigenvalues are $$\sqrt{|\mathbf{p}|^2+m^2}$$, so the particles always have mass $$m$$. There is no creation operator that gives you a "half" particle. There's the ground state $$|0\rangle$$, a state with one particle $$a^\dagger(\mathbf{p})|0\rangle$$, a state with two particles $$a^\dagger(\mathbf{p}_1)a^\dagger(\mathbf{p}_2)|0\rangle$$, but nothing in between. The reasoning that leads to discrete particles is the same as the reasoning that leads to discrete energy states in the quantum harmonic oscillator. You could certainly form a linear combination of a one particle and a two particle state, but when measured, it will collapse into an eigenvalue of whatever operator corresponds to that measurement. The procedure for electrons is different, since electrons obey the Dirac equation, but the procedure for the Klein-Gordon field should give you the general idea.
There are two facts to be distinguished. Electrons are what we loosely call particles, so they only ever occur in discrete numbers. Millikan demonstrated the discreteness of charge. Secondly, localised states in general have discrete energies. Examples of these are atomic and molecular states. Free electrons have continuous electrons. So called free electrons in metals for all practical purposes have continuous energies if the metal volume is macroscopic.
Why electrons are discrete particles or why quantum states are often discrete is not known.
A further question is how we account for these properties mathematically, but I believe this is not what you are asking. | 1,593 | 6,998 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2024-10 | latest | en | 0.94377 |
https://www.clutchprep.com/physics/practice-problems/93715/newton-s-law-of-universal-gravitation-is-represented-by-f-5-gmm-r-2-where-f-is-t | 1,618,717,161,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038464146.56/warc/CC-MAIN-20210418013444-20210418043444-00476.warc.gz | 801,608,436 | 30,248 | Dimensional Analysis Video Lessons
Concept
# Problem: Newton’s law of universal gravitation is represented by F = GMm/r 2 where F is the magnitude of the gravitational force exerted by one small object on another, M and m are the masses of the objects, and r is a distance. Force has the SI units kg • m/s2. What are the SI units of the proportionality constant G?
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Newton’s law of universal gravitation is represented by F = GMm/r where F is the magnitude of the gravitational force exerted by one small object on another, M and m are the masses of the objects, and r is a distance. Force has the SI units kg • m/s2. What are the SI units of the proportionality constant G? | 174 | 746 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2021-17 | latest | en | 0.918275 |
https://www.coursehero.com/file/5871515/Lecture-22/ | 1,498,257,268,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320201.43/warc/CC-MAIN-20170623220935-20170624000935-00650.warc.gz | 825,294,952 | 58,918 | Lecture_22
# Lecture_22 - 25.3 Plane Mirrors Look into a plane mirror...
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25.3 Plane Mirrors Look into a plane mirror. What are the properties of the image you see? 1. The image is upright. 2. The image is the same size as the object. 3. The image is located as far behind the mirror as the object is located in front of it. 4. The image is reversed from left to right. Let’s see how an image is formed with a plane mirror: Object Mirror First, I can use a ray that goes straight over and reflects straight back. Now use a second ray which comes up from the object at some angle. θ r i Extrapolate the second ray back behind the mirror. The two extrapolated rays intersect at the image. Image
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Mirror θ r i Image The image formed by a plane mirror is called a virtual image. Virtual, since the light rays only appear to come from the image, but they do not actually emanate from there. Question : I want to use a plane mirror to see my entire body. How long does the mirror have to be? Wall A ray from the top of my head will strike the mirror and enter my eyes as shown. And, a ray from my foot can strike the mirror and enter my eyes as well. Extrapolating these rays back will show the location of the image. Any other ray from above my foot, like from my waist, would have to strike the mirror above the point where the ray from my foot did. Thus, to see my entire body, I only need a mirror that is ½ my height. It doesn’t matter how far I am from the mirror either!
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## This note was uploaded on 04/24/2010 for the course PHYS 2002 taught by Professor Blackmon during the Spring '08 term at LSU.
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Lecture_22 - 25.3 Plane Mirrors Look into a plane mirror...
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Ask a homework question - tutors are online | 498 | 2,064 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2017-26 | longest | en | 0.946891 |
http://www.codeproject.com/Articles/11275/Fortune-s-Voronoi-algorithm-implemented-in-C?msg=4250338 | 1,448,868,686,000,000,000 | text/html | crawl-data/CC-MAIN-2015-48/segments/1448398461113.77/warc/CC-MAIN-20151124205421-00327-ip-10-71-132-137.ec2.internal.warc.gz | 361,454,215 | 29,070 | 11,930,449 members (60,576 online)
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# Fortune's Voronoi algorithm implemented in C#
, 21 Apr 2013 MPL
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A C# implementation of the Fortune algorithm to compute 2D Voronoi graphs.
## Introduction
Given a set of two dimensional vectors (or data points), a Voronoi graph is a separation of those points into compartments where all points inside one compartment are closer to the contained data point than to any other data point. I won't give any demonstration here, but if you want to know more about Voronoi graphs, check out this.
The applications of Voronoi graphs are quite broad. Very useful for a lot of optimization problems (in most cases, the Delaunay Triangulation which can be easily derived from a Vononoi graph is used there), it ranges to computing topological maps from bitmaps.
[This is an article for freaks. After a rather painful experience writing the thing I hope it will benefit everyone who is looking for this algorithm in a civilized language (or simply does not want to use Fortune's original C implementation).]
In 1987, Steve Fortune described an algorithm to compute such a graph by using a sweep line in combination with a binary tree. A PowerPoint explanation of the algorithm (the one I used to implement it) can be found here. Note that I did not use the linked data structure to represent a graph - I think that is an unnecessary difficulty in the age of `ArrayList`s and `HashSet`s.
## The Implementation
Data points are represented by my own `Vector` class. It can do much more than needed here (but there was no reason to strip it before bringing it) but I won't explain it here. The most important fact is that although working with `double`s the Vector class automatically rounds values to 10 digits (or whatever is set in the `Vector.Precision` field). Yes, sadly, this is very important if you want to sort of compare `double`s.
A `VoronoiGraph` is a class that only contains a `HashSet` of vertices (as 2D vectors) and a `HashSet` of `VoronoiEdge`s - each with references to the left and right data point and (of course) the two vertices that bound the edge. If the edge is (partially or completely) unbounded, the vector `Fortune.VVUnknown` is used.
`BinaryPriorityQueue` is used for the sweep line event queue.
## Usage
The algorithm itself (`Fortune.ComputeVoronoiGraph(IEnumerable)`) takes any `IEnumerable` containing only two dimensional vectors. It will return a `VoronoiGraph`. The algorithm's complexity is O(n ld(n)) with a factor of about 10 microseconds on my machine (2GHz).
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I did my diploma in Dresden and Sydney where I dealt with algorithms, agents and other cool AI stuff. Now I moved to Frankfurt to work on my PhD dealing with software structures for artificial intelligence systems. If I can, I do things in C# and ASP.NET, but if I have to, my C++, Java and SQL are not that bad.
Long Live .NET.
## You may also be interested in...
Re: Finding distance for VoronoiGraph vertizes representing Medial Axis of closed boundary Member 917583217-Jul-12 17:11 Member 9175832 17-Jul-12 17:11
Is there a way to construct polygons from all the edges ? And how ? seb.493-Jun-12 22:36 seb.49 3-Jun-12 22:36
Possible issue. When I draw 2 points I haven't 2 area seb.4931-May-12 0:59 seb.49 31-May-12 0:59
Re: Possible issue. When I draw 2 points I haven't 2 area BenDi31-May-12 4:11 BenDi 31-May-12 4:11
Re: Possible issue. When I draw 2 points I haven't 2 area seb.4931-May-12 5:44 seb.49 31-May-12 5:44
Re: Possible issue. When I draw 2 points I haven't 2 area BenDi31-May-12 8:00 BenDi 31-May-12 8:00
Re: Possible issue. When I draw 2 points I haven't 2 area seb.4931-May-12 10:53 seb.49 31-May-12 10:53
This code is shitted Member 868370514-May-12 5:42 Member 8683705 14-May-12 5:42
The provided code is absolutely shitted. Why if I change epsilon value it will stop working correctly??????????? What epsilon value should be used??? Why 1e-10???? Why not 1e-20???? Did you try it for big numbers, for example: 220034495, 883612293?????? It's really piece of S H I T and I've wasted so huge amount of time on this crap...
Re: This code is shitted maamaamaa29-Jun-13 6:42 maamaamaa 29-Jun-13 6:42
Any way to handle boundaries and holes? James Maeding16-Dec-10 12:03 James Maeding 16-Dec-10 12:03
Re: Any way to handle boundaries and holes? xinaesthetic30-Sep-11 8:38 xinaesthetic 30-Sep-11 8:38
My vote of 5 maurice.calvert1-Sep-10 0:08 maurice.calvert 1-Sep-10 0:08
Re: My vote of 5 Member 868370514-May-12 5:45 Member 8683705 14-May-12 5:45
Re: My vote of 5 maurice.calvert14-May-12 6:32 maurice.calvert 14-May-12 6:32
Re: My vote of 5 Member 868370514-May-12 22:38 Member 8683705 14-May-12 22:38
Re: My vote of 5 maurice.calvert15-May-12 5:56 maurice.calvert 15-May-12 5:56
Voronoi Cells C-Bl28-Jul-09 0:08 C-Bl 28-Jul-09 0:08
Re: 3 data points case rhill-ca26-Jun-09 7:45 rhill-ca 26-Jun-09 7:45
Another wrong vertex Sunil Terkar11-May-09 3:04 Sunil Terkar 11-May-09 3:04
Re: Another wrong vertex BenDi11-May-09 11:03 BenDi 11-May-09 11:03
Re: Another wrong vertex Sunil Terkar12-May-09 1:37 Sunil Terkar 12-May-09 1:37
Re: Another wrong vertex Sunil Terkar12-May-09 6:09 Sunil Terkar 12-May-09 6:09
Last Visit: 31-Dec-99 19:00 Last Update: 29-Nov-15 16:31 Refresh 123456 Next » | 1,560 | 5,387 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2015-48 | latest | en | 0.931737 |
http://sigma-not.pl/wyszukaj-0-0-10-62005708--pawel-rymarczyk-.html | 1,568,536,135,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514570830.42/warc/CC-MAIN-20190915072355-20190915094355-00296.warc.gz | 175,211,956 | 11,128 | Wyniki 1-1 spośród 1 dla zapytania: authorDesc:"Paweł RYMARCZYK"
### Detection of seepages in flood embankments using the ElasticNET method DOI:10.15199/48.2019.01.40
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Electric tomography is based on the transformation of data taken from the surface of the tested object into the image of its cross-section. There are many methods to optimize the obtained image by solving the appropriate objective function [1-5,13,15,16,20-25,32]. The algorithm based on the ElasticNET presented in this article is a new proposal in tomography. Fig. 1. Model of measuremnt system. The way of working of electrical impedance tomography (EIT) consists in introducing electrical voltage to the tested object by means of a set of electrodes located on the surface of the object. Next, the measured values of electrical potentials between individual electrode pairs are collected. Conductance of individual sections of the crosssection of the tested object is reconstructed on the basis of known values of voltages and measured values of potentials. Reconstruction of the image obtained by electrical tomography requires sophisticated modeling. This method of imaging consists in the fact that the conductivity distribution of the tested object is estimated on the basis of measurements of electrical voltages and electrode potentials on the surface of their contact with the tested object. In order to obtain quantitative data on changes in the conductivity inside an object, it is more effective to apply a non-linear model in differential imaging [1,6-12,14,17- 199,26-31]. In Fig. 1 shows the model of the measurement system. ElasticNET Let’s consider the problem of recognizing linear dependencies (1) Y X where Y Rn , X Rnk1 are the observation matrices of a output variable and predictive variables respectively, Rk1means a matrix of structural parameters, while Rn vector of independent random variables. The wellknown method of least squares consists in estimating unknown parameters &[...]
Strona 1 | 475 | 2,033 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2019-39 | latest | en | 0.873775 |
http://bibliotecadecordoba.com/forum/6ptka.php?page=daa144-twin-prime-numbers-between-1-to-100-in-c | 1,618,925,530,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00347.warc.gz | 12,164,643 | 6,912 | A prime number is a whole number, whose only two factors are 1 and itself. What are the twin prime numbers between 1 and 100 - 30232 1. Sjudoku - in a world where 9 is replaced by 7. Why are red and blue light refracted differently if they travel at the same speed in the same medium?
Suppose, we have to print prime numbers between 1 to 20. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. This question can also be asked like, print prime numbers from 1 to 100 in c. In this tutorial, we are going to use sieve algorithm to print prime numbers from 1 to N. Suppose the value of N is 10, So the prime numbers between 1 to 10 is 2, 3, 5, 7. What is the name of this game with a silver-haired elf-like character? Next number is 3 cross out every multiple of 3. The condition i==j+1 will not be true for i==2. Following numbers are cross-out. What is the lowest level character that can unfailingly beat the Lost Mine of Phandelver starting encounter? Stack Overflow for Teams is a private, secure spot for you and
Write a c program to print prime numbers from 1 to N (where n is an integer). Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Do I need HDMI-to-VGA or VGA-to-HDMI adapter? C: What is the difference between ++i and i++?
If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Join now.
This prime numbers generator is used to generate the list of prime numbers from 1 to a number you specify. your coworkers to find and share information.
So the prime numbers between 1 to 20 is 2, 3, 5, 7, 11, 13, 17, 19. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. It was just a request... Anyways Thank You so much dear. Ask your question. Checkout twin primes up to: 100 , 500 , 1000 , 10000 .
Copyright 2015 – 2020 – webrewrite.com – All Rights Reserved. This can be fixed by a couple of changes to the inner loop: To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This question can also be asked like, print prime numbers from 1 to 100 in c. In this tutorial, we are going to use sieve algorithm to print prime numbers from 1 to N. Suppose the value of N is 10, So the prime numbers between 1 to 10 is 2, 3, 5, 7. 3. This code snippet for find the Prime Number between 1 to 100 in C#. | 654 | 2,514 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2021-17 | latest | en | 0.923637 |
http://mathhelpforum.com/algebra/129640-binary-digit-decimal-value.html | 1,527,088,556,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794865679.51/warc/CC-MAIN-20180523141759-20180523161759-00217.warc.gz | 196,731,710 | 10,295 | # Thread: Binary Digit & Decimal value
1. ## Binary Digit & Decimal value
can someone check if I did this correctly
the human genome contains about 3 billion base pairs...stored in a computer as a sequence of $\displaystyle 3*10^9$ bases X 2 bases/bits = $\displaystyle 6*10^9 bits$
The Encyclopedia Britannica has about 1500 words per page, 1000 pages per volume and 30 volumes. How many sets of the Encyclopedia Britannica would it take to store the same amount of info in a single strand of human dna?
$\displaystyle 1500*1000 = 1,500,000 = 15*10^5$
$\displaystyle (15*10^5)*30 = 45,000,000 = 45*10^6$
$\displaystyle (45*10^6)base * 2(bits/base) = 9*10^7 bits$
$\displaystyle \frac{(6*10^9 bits)}{9*10^6} = 66.67$
2. I see two problems
How are you doing this calculation without knowing the number of letters on each page, since a word contains more than one letter, and thus can hold more than one bit of information.
Next, what does this calculation mean?
I see (number of words in 30 volumes)*2 bits/base
but... I thought "base" referred to a "base pair" of dna. Why are you multiplying the number of words by 2 bits?
there are 26 letters in the alphabet, 10 decimal digits in the base 10 number system and several different kinds of punctuation marks, two possibilities for capitalization (on/off) and a few other symbols used in a typical book. A combination of 6 binary bits can take on 64 different possibilities which is about the number needed to create a code equivalent to all the possible characters that make up normal text. the average word in a text contains about 5 characters and with each character requiring 6 bits for its representation, 1 average word of text is equivalent to about 30 bits
4. the human genome contains about 3 billion base pairs...stored in a computer as a sequence of bases X 2 bases/bits =
I believe you mean 2 bits / bases. This is because there are four possible values at each base pair, and 2 bits holds four values, as desired.
The Encyclopedia Britannica has about 1500 words per page, 1000 pages per volume and 30 volumes. How many sets of the Encyclopedia Britannica would it take to store the same amount of info in a single strand of human dna?
Let b be the number of bits in a strand of the human genome. What you want is
$\displaystyle b / (bits per word) / (words per page) / (pages per volume)$
this 'convert's bits to volumes. If you want sets of 30 volumes, then just divide this by 30.
5. so it would be (6*10^9) / (30) / (1500) / (1000) = 133.333
133.33 / 30 = 4.4
6. That is what I have, as long as I am understanding the problem correctly that should be correct. | 693 | 2,645 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2018-22 | latest | en | 0.873279 |
https://www.askiitians.com/forums/Integral-Calculus/find-the-value-of-a-for-which-the-function-f-x_96074.htm | 1,726,092,143,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651405.61/warc/CC-MAIN-20240911215612-20240912005612-00582.warc.gz | 603,108,075 | 42,624 | # Find the value of `a` for which the function f(x)=cos(x)-sin(x)+a(x)+b is decreasing.
Sunil Raikwar
askIITians Faculty 45 Points
10 years ago
we know that f(x) is decreasing for all real values of x if f'(x)<0
f'(x)=-cosx-sinx+a<0
a<sinx+cosx
minimum value of sinx+cosx is -root2
so a should be less than -root2
Thanks & Regards
Sunil Raikwar | 122 | 346 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2024-38 | latest | en | 0.66855 |
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2010
HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics Extension 1
3370
General Instructions
• Reading time – 5 minutes
• Working time – 2 hours
• Write using black or blue pen
• Board-approved calculators may be used
• A table of standard integrals is provided at the back of this paper
• All necessary working should be shown in every question
Total marks – 84 • Attempt Questions 1–7 • All questions are of equal value
Total marks – 84 Attempt Questions 1–7 All questions are of equal value
Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.
Question 1 (12 marks) Use a SEPARATE writing booklet.
(a)
(b)
(c)
(d)
(e)
(f)
1
Use the table of standard integrals to find ⌠
2
4 − x
Let
(
ƒ x
)
= cos
1
x 2⎠ ⎟
. What is the domain of ƒ (x)?
Solve ln(x + 6) = 2 ln x .
Solve
3
x + 2
< 4 .
dx .
Use the substitution u = 1 − x to evaluate ⌠ 1
x
1 − x dx.
0
Five ordinary six-sided dice are thrown.
1
1
3
3
3
1
What is the probability that exactly two of the dice land showing a four? Leave your answer in unsimplified form.
– 2 –
Question 2 (12 marks) Use a SEPARATE writing booklet.
(a)
The derivative of a function ƒ (x) is given by
ƒ (x) = sin 2 x.
Find ƒ (x), given that ƒ (0) = 2.
(b)
The mass M of a whale is modelled by
M = 36 35.5e kt ,
2
where M is measured in tonnes, t is the age of the whale in years and k is a positive constant.
(i)
(ii)
(iii)
Show that the rate of growth of the mass of the whale is given by the differential equation
dM
dt
=
k ( 36 M ) .
When the whale is 10 years old its mass is 20 tonnes.
Find the value of k, correct to three decimal places.
According to this model, what is the limiting mass of the whale?
Question 2 continues on page 4
– 3 –
1
2
1
Question 2 (continued)
(c) Let P(x) = (x + 1)(x − 3) Q(x) + ax + b, where Q(x) is a polynomial and a and b are real numbers. The polynomial P(x) has a factor of x − 3. When P(x) is divided by x + 1 the remainder is 8. (i) Find the values of a and b. 2 (ii) Find the remainder when P(x) is divided by (x + 1)(x − 3). 1 (d) A radio transmitter M is situated 6 km from a straight road. The closest point on 3
the road to the transmitter is S.
A car is travelling away from S along the road at a speed of 100 km h 1 . The
distance from the car to S is x km and from the car to M is r km.
M
r
6
S
x
dr
Find an expression in terms of x for
, where t is time in hours.
dt
End of Question 2
– 4 –
Question 3 (12 marks) Use a SEPARATE writing booklet.
(a)
At the front of a building there are five garage doors. Two of the doors are to be painted red, one is to be painted green, one blue and one orange.
(i) How many possible arrangements are there for the colours on the doors? (ii) How many possible arrangements are there for the colours on the doors if the two red doors are next to each other?
1
1
(b)
ƒ (x) =
e − x
Let
2
. The diagram shows the graph y = ƒ (x).
y
x
(i)
The graph has two points of inflexion.
3
(ii)
(iii)
(iv)
(v)
(vi)
Find the x coordinates of these points.
Explain why the domain of ƒ (x) must be restricted if ƒ (x) is to have an inverse function.
Find a formula for ƒ 1 (x) if the domain of ƒ (x) is restricted to x 0.
State the domain of ƒ 1 (x).
Sketch the curve y = ƒ 1 (x).
(1) Show that there is a solution to the equation
x = 0.6
and x = 0.7.
x
=
e
x
2
between
(2) By halving the interval, find the solution correct to one decimal place.
– 5 –
1
2
1
1
1
1
Question 4 (12 marks) Use a SEPARATE writing booklet.
(a)
A particle is moving in simple harmonic motion along the x-axis.
Its velocity v, at x, is given by
v 2 = 24 8x 2x 2 .
(i) Find all values of x for which the particle is at rest.
(ii) Find an expression for the acceleration of the particle, in terms of x.
(iii) Find the maximum speed of the particle.
(b)
(i)
Express
2
cos θ
+
where R > 0 and
π
3
.
2
cos θ
+
0
< α < π
2
⎠ ⎟ ⎞ in the form R cos (θ + α),
(ii) Hence, or otherwise, solve
for
0 < θ < 2π.
2 cos θ + 2 cos θ +
π
3
= 3 ,
Question 4 continues on page 7
– 6 –
1
1
2
3
2
Question 4 (continued)
(c) The diagram shows the parabola x is on the parabola.
2 = 4ay . The point P(2ap, ap 2 ), where p 0,
3
y
P (2 ap , ap 2 )
S
(0,
a )
O
x
M
y = −a
L
The tangent to the parabola at P, y = px ap 2 , meets the y-axis at L.
The point M is on the directrix, such that PM is perpendicular to the directrix.
Show that SLMP is a rhombus.
End of Question 4
– 7 –
Question 5 (12 marks) Use a SEPARATE writing booklet.
(a) A boat is sailing due north from a point A towards a point P on the shore line. The shore line runs from west to east.
In the diagram, T represents a tree on a cliff vertically above P, and L represents a landmark on the shore. The distance PL is 1 km.
From A the point L is on a bearing of 020°, and the angle of elevation to T is 3°.
After sailing for some time the boat reaches a point B, from which the angle of
elevation to T is 30°.
T
N
11 kmkm
P
L
W
E
B
NOT TO
SCALE
20°
A
3
tan
3 °
(i)
Show that BP =
.
3
tan 20 °
(ii)
Find the distance AB.
1
30°
Question 5 continues on page 9
– 8 –
Question 5 (continued)
(b)
Let
(i)
(ii)
ƒ
(
x
) =
tan
1
x
+
tan
1
1
x
for
x 0.
By differentiating ƒ (x), or otherwise, show that
ƒ (x) = π
2
for
Given that ƒ (x) is an odd function, sketch the graph y = ƒ (x).
(c)
In the diagram, ST is tangent to both the circles at A.
x > 0.
3
1
The points B and C are on the larger circle, and the line BC is tangent to the smaller circle at D. The line AB intersects the smaller circle at X.
S
A
X
D
C
B
T
Copy or trace the diagram into your writing booklet.
(i) Explain why ∠AXD = ∠ABD + ∠XDB. (ii) Explain why ∠AXD = ∠TAC + ∠CAD. (iii) Hence show that AD bisects ∠BAC.
End of Question 5
– 9 –
1
1
2
Question 6 (12 marks) Use a SEPARATE writing booklet.
(a) (i) Show that cos(A − B) = cos A cos B(1 + tan A tan B). 1 (ii) Suppose that 0 < B < π and B < A < π . 1 2 Deduce that if tan A tan B = − 1 , then A − B = π . 2 (b) A basketball player throws a ball with an initial velocity v m s −1 at an angle θ
to the horizontal. At the time the ball is released its centre is at (0, 0), and the
player is aiming for the point (d, h) as shown on the diagram. The line joining
(0, 0) and (d, h) makes an angle α with the horizontal, where 0
< α < θ < π
.
2
y
(d , h )
θ
α
x
d
Assume that at time t seconds after the ball is thrown its centre is at the point (x, y), where
x = vt cos θ
y =
vt
sin θ 5
t
2
.
(You are NOT required to prove these equations.)
Question 6 continues on page 11
– 10 –
Question 6 (continued)
(i)
If the centre of the ball passes through (d, h) show that
v
2
=
5 d
2
cos θ sin θ cos θ tan α
.
(ii)
(1) What happens to v as
(2) What happens to v as
θ α?
θ
π
2
?
(iii)
(iv)
(v)
For a fixed value of α, let
(
F θ
)
θ
= cos sin
θ
2
cos θ tan α .
Show that
F
(
θ
)
= 0
when
tan 2θ tan α = − 1
.
Using part (a) (ii) or otherwise show that
(
F θ
)
Explain why v 2 is a minimum when θ = α
π
2 + 4
= 0
.
when θ = α
π
2 + 4
End of Question 6
– 11 –
.
3
1
1
2
1
2
Question 7 (12 marks) Use a SEPARATE writing booklet.
(a)
(b)
Prove by induction that
47 n + 53 × 147 n 1
is divisible by 100 for all integers n 1.
The binomial theorem states that
(i)
(ii)
(iii)
( 1 + x
)
Show that
n
2
=
⎜ ⎝
n
=
n
0
⎞ ⎟ ⎠
+
n
1
n
k = 0
n
⎝ ⎜ k
.
x +
⎝ ⎜
n
2
x 2
+
⎝ ⎜
n
⎟ ⎠
3
x
3
Hence, or otherwise, find the value of
Show that
100 ⎞ ⎟ + ⎜ ⎛
0
n
n 2 1 =
n
k =1
100
1
⎞ ⎟ + ⎜ ⎛
k
n
⎝ ⎜ k
.
100
2
+
+
++ +
100
100
.
n
n
⎟ ⎠
x n .
Question 7 continues on page 13
– 12 –
3
1
1
2
Question 7 (continued)
(c)
(i)
A box contains n identical red balls and n identical blue balls. A selection
of r balls is made from the box, where 0 r n.
1
Explain why the number of possible colour combinations is r + 1.
(ii) Another box contains n white balls labelled consecutively from 1 to n.
A selection of n r balls is made from the box, where 0 r n.
Explain why the number of different selections is
⎝ ⎜ r
n
.
(iii) The n red balls, the n blue balls and the n white labelled balls are all placed into one box, and a selection of n balls is made.
1
3
Using part (b), or otherwise, show that the number of different selections
is (n + 2)2 n 1 .
End of paper
– 13 –
BLANK PAGE
– 14 –
BLANK PAGE
– 15 –
STANDARD INTEGRALS
n
x
dx
⌠ 1
dx
x
e ax
dx
cos ax dx
sin ax dx
2
sec ax dx
sec ax tan ax dx
1
dx
2
⌡ aa
2 + x
1
dx
2
2
a
− x
1
x 2 − a 2
1
2
x
2 + a
dx
dx
1
=
n + 1
= ln x ,
x n+ 1
, n ≠ − 1; x 0 , if n < 0
xx > 0
=
1 ax
e
a
, a 0
=
1
a
sin ax ,
aa 0
1
= − cos ax , a 0
a
= 1 tant ax , a ≠ 0 a = 1 a sec ax , a ≠ 0 = 1 a tan − 1 x a , a ≠ 0 = sin − 1 x , aa > 0, a
=
ln ( x
= ln ( x
+
+
2
2
x
a
2
2
x
+ a
) ,
)
a < x < a
x
> a >> 0
NOTE :
ln x = log
e
xx ,
x > 0
– 16 –
© Board of Studies NSW 2010 | 3,174 | 9,084 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.890625 | 4 | CC-MAIN-2020-05 | latest | en | 0.83299 |
http://hackage.haskell.org/package/kan-extensions-3.6.2/docs/Data-Functor-Yoneda-Reduction.html | 1,597,037,966,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738609.73/warc/CC-MAIN-20200810042140-20200810072140-00371.warc.gz | 49,868,750 | 3,359 | kan-extensions-3.6.2: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads
Portability GADTs, MPTCs, fundeps provisional Edward Kmett Trustworthy
Data.Functor.Yoneda.Reduction
Description
Yoneda Reduction:
`Yoneda f` is isomorphic to `Lan f Identity`
Synopsis
Documentation
data Yoneda f a whereSource
A form suitable for Yoneda reduction
Constructors
Yoneda :: (b -> a) -> f b -> Yoneda f a
Instances
liftYoneda :: f a -> Yoneda f aSource
Yoneda expansion
``` `liftYoneda` . `lowerYoneda` ≡ `id`
`lowerYoneda` . `liftYoneda` ≡ `id`
```
``` `lift` = `liftYoneda`
```
lowerYoneda :: Functor f => Yoneda f a -> f aSource
Yoneda reduction
``` `lower` = `lowerM` = `lowerYoneda`
```
lowerM :: Monad f => Yoneda f a -> f aSource
Yoneda reduction given a `Monad`.
``` `lower` = `lowerM` = `lowerYoneda`
```
as a Left Kan extension
yonedaToLan :: Yoneda f a -> Lan Identity f aSource
`Yoneda f` is the left Kan extension of `f` along the `Identity` functor.
``` `yonedaToLan` . `lanToYoneda` ≡ `id`
`lanToYoneda` . `yonedaToLan` ≡ `id`
```
as a Left Kan lift
yonedaToLift :: Yoneda f a -> Lift Identity f aSource
`Yoneda f` is the left Kan lift of `f` along the `Identity` functor.
``` `yonedaToLift` . `liftToYoneda` ≡ `id`
`liftToYoneda` . `yonedaToLift` ≡ `id`
``` | 469 | 1,326 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2020-34 | latest | en | 0.665246 |
http://www.weknowtheanswer.com/q/how-do-you-write-2-99-percent-in-words | 1,495,962,666,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463609610.87/warc/CC-MAIN-20170528082102-20170528102102-00134.warc.gz | 891,190,410 | 8,746 | # How do you write 2.99 percent in words?
• How do you write 2.99 percent in words?
Writing Decimals as Percents: ... To write a decimal as a percent, ... Do not forget to include the percent symbol when writing a percent.
Positive: 58 %
... percent. You can find a fraction to percent chart and also convert any fraction to percent. Fraction to Percent ... percent converter below to write ...
Positive: 55 %
### More resources
Proportion Word Problems 1. ... Write your answer in hours and ... 20. Sirloin steak costs \$2.99 per pound. How much will 3.4 pounds cost? 21.
Positive: 58 %
How do you write a number with two decimal places for ... Write a number with two decimal places SQL ... Multiply the value you want to insert (ex. 2.99) ...
Positive: 53 %
_ HOW TO FIGURE PERCENT _ ... When you do problems like this, ... well as figure the percent. Note: Words such as raise, ... | 221 | 889 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2017-22 | latest | en | 0.867037 |
www.joshbrews.com | 1,656,604,153,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103850139.45/warc/CC-MAIN-20220630153307-20220630183307-00182.warc.gz | 901,615,958 | 7,645 | May 12, 2007
## Calculating Mashing efficiency
Category: General Brewing — Josh @ 10:24 am
I realized that when I posted about calculating extract yield, that I forgot to explain how I calculated brewhouse effieiency. I use the same formula
S.G. = ((((DBCG – Mousture Content – 0.002) * Brewhouse Efficiency * 46.214) * 0.001) +1)
Except that I remove the variable for brewhouse efficiency. The forumla will then assume that the mashing efficiency is at 100%
S.G. = ((((DBCG – MC – 0.002) * 46.214) * 0.001) +1)
This will give us the theoretical maximum ppg of 1 lb of grain in 1 gallon of water. For example, Maris Otter Pale Malt.
S.G. = ((((0.805 – 0.03 – 0.002) * 46.214) * 0.001) +1)
S.G. = 1.035723422 or 35.7ppg
In my bitter 2 recipe I mashed 3 lbs. of Maris Otter Pale Malt, and 1 lb of Briess 10 L crystal malt collecting a final 2.5 gallons of wort at a S.G of 1.043 .
The theoretical maximum ppg of 10 L crystal malt is
S.G. = ((((0.75 – 0.07 – 0.002) * 46.214) * 0.001) +1)
S.G = 1.031333092 or 31.3 ppg
Now, using the formula from John Palmer’s How To Brew, we calculate the maximum theoretical points for our brew using the following formula
((ppg Malt1 x lbs Malt1) / gallons of wort) + ((ppg Malt2 x lbs Malt2) / gallons of wort) = Total Points
((35.7 x 3)/2.5) + ((31.3 x 1)/2.5) = 55.36
This is our theoretical maximum number of points per gallon.
In the wort for the Bitter 2 recipe the S.G was 1.043 or 43 ppg. If we divide our real ppg by the theoretical ppg this will yeild our percentage yeild.
43/55 x 100 = 77.75% | 528 | 1,555 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2022-27 | latest | en | 0.803305 |
http://encyclopediajr.com/a-waterbed-filled-with-water-has-the-dimensions-8-5-ft-by-7-5-ft-by-9-5-inches-taking-the-density-of-water-to-be-1-00-g-cubic-centimeter-determine-the-number-of-pounds-of-water-required-to-fill-the-wa/ | 1,695,744,348,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510214.81/warc/CC-MAIN-20230926143354-20230926173354-00195.warc.gz | 11,586,375 | 31,436 | 20.07.2022 - 05:46
# A waterbed filled with water has the dimensions 8.5 ft by 7.5 ft by 9.5 inches. Taking the density of water to be 1.00 g/cubic centimeter, determine the number of pounds of water required to fill the water bed.
Question:
A waterbed filled with water has the dimensions 8.5 ft by 7.5 ft by 9.5 inches. Taking the density of water to be 1.00 g/cubic centimeter, determine the number of pounds of water required to fill the water bed. | 132 | 455 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2023-40 | latest | en | 0.906377 |
https://www.meritnation.com/ask-answer/question/please-solve-64-calculate-the-ph-of-a-solution-which-contain/study-of-acids-bases-and-salts/12627740 | 1,656,834,199,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104215805.66/warc/CC-MAIN-20220703073750-20220703103750-00007.warc.gz | 941,085,880 | 8,844 | # Please solve: 64. Calculate the pH of a solution which contains 9.9 ml of 1 M HCI and 100 ml of 0.1 M NaOH. 65. Calculate the pH of a solution by mixing 0.1 litre of pH - 4 and 0.2 litre of pH = 10. 65. Calculate the pH of a solution obtained by mixing 10 ml of 0.1 M HCI and 40 ml of 0.2 M H2SO4-
Dear Student,
64.
NaOH + HCl → NaCl + H2
1 mol NaOH reacts with 1 mol HCl
Mole of HCl in 9.9mL of 1.0M solution = 9.9/1000 x 1.0 = 0.0099 mol HCl
Mole of NaOH in 100mL of 0.1M solution = 100/1000 x 0.1 = 0.01 mol NaOH
On mixing the 0.0099 mol HCl will be neutralised and 0.01 - 0.0099 = 1 x 10-4 mol
1 x 10-4 mol of NaOH dissolved in 109.9mL solution.
Molarity of NaOH solution = (1 x 10-4) / 0.1099 = 9.099 x 10-4
pOH = -log ( 9.099 x 10-4
pOH = 3.04
pH = 14.00 - pOH
pH = 14.00 - 3.04
pH = 10.96
65.
Concentration in pH 4 solution = 0.1 x 10-4
Concentration in pH 10 solution = 0.2 x 10-10
[OH-]=10-14/10-10=10-4 = molarity of base in 0.2L , moles of acid = 0.2 x 10-4
Now neutralization will happen and acid is completely neutralised by base.
Final molarity = remaining moles/total vol = (0.2 x 10-4 - 0.1x 10-4) /0.3
Concentration of base =10-4/3[H+] = 3 x 10-14/10-4 =3 x 10-10
pH=-log[H+] = 10 - log3= 9.523
Concentration of base =10-4/3[H+] = 3 x 10-14/10-4 =3 x 10-10
pH = -log[H+] = 10 - log3 = 9.523
66.
H2SO4 → H+ + HSO4-
HSO4- → H+ + SO42-
Mole of H2SO4 in 40mL of 0.2M solution = 40/1000 x 0.2 = 0.008 mol H2SO
Mole of HCl in 10mL of 0.1M solution = 10/1000 x 0.1 = 0.001 mol HCl
Therefore total of 0.009 mol H+ is dissolved in 50mL solution = 0.050L
Molarity of H+ = 0.009/0.050 = 0.18 M
pH = -log [H+] = -log 0.18 = 0.74.
Regards,
• 0
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### What is Modelling?
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### My First Code in Python: Adding all Prime Numbers
As Python programming is highly hyped now-a-days and a frequent buzzword in machine learning and AI communities, I have decided to get myself familiar in this area. As I learn and decipher the Python coding day-by-day, I will be more than happy to share my coding in this blog. Here is my first program in Python - to develop a logic that adds all prime numbers between 0 and a given number.
We know that a prime number is an integer, which is greater than one and which is only divisible by one and itself. The following shows an algorithm that adds all prime numbers between 0 and a given number.
Development of an algorithm to add all prime numbers between 0 to n
I. First, we define the lower and upper number to initialize the program. For example, we set the lower number 0 and upper number 10; and we are interested to find the summation of prime numbers in this range.
II. Prime number is greater than one, so our logic begins that it must be greater than one. Therefore, we begin with number 2 and list all the numbers in our range.
III. Since, prime numbers are divisible only by itself and one; we create a loop, which confirms the prime numbers, and display them. We implement this by checking if the remainder of the number is zero or not. If the remainder is zero, then it is not a prime number.
IV. If the remainder of the number is not zero, then it is a prime number.
V. Display the prime numbers in the defined range and group the numbers in a list or array.
VI. Finally, we add all the prime numbers in the list.
The following Python code is developed based on the above algorithm, which displays all the prime numbers in a given range and add those numbers:
lower = 0
upper = 10 #Defining the range
sum1 = 0 #Setting Summation of prime numbers to zero
print("Prime numbers between", lower, "and", upper, "are:")
for num in range(lower, upper + 1):
if num > 1: #Prime number is greater than 1
for i in range(2, num):
if (num % i) == 0: #Calculates & checks remainder of division
break
else:
print(num) # Displays prime numbers
sum1 += num
print(sum1)
Program Outputs:
Prime numbers between 0 and 10 are:
2
3
5
7
17
## A NOVEL LEMON SHAPED GUIDE TO MINIMIZE EXCESSIVE RUBBING IN ROTATING MACHINES
In industries, rotating machines are widely used, because rotation offers a great way to transfer power from one point to another and convert motion to different planes by gears, belts, shaft etc. A rotating machine typically includes a rotor, bearings and a support structure. There are critical relations among these components where each component of a system influences the overall dynamic behavior of a machine. For example, rotor-to-guide rub degrades a mechanical system over the years and may even cause fatal accidents earlier. It is, therefore, paramount in industries, to run rotating machines, operating at high speeds smoothly and reliably. The primary reasons of rubbing between a rotor and a guide are due to a manufacturing error, excessive imbalance, misalignment, bearing wear, smaller radial clearance between the rotating shaft and casing, bad assembly, etc. Rub occurs in rotor casing, seals, unlubricated journal bearings, loose rotor guide attached to restrict a large deflection. The problem is prevalent in the industry and demonstrated in several literatures. To deal with this problem, the industry has already been using circular shaped guide to minimize excessive vibration between a rotor and a stator. Although, the circular shaped guide may reduce the vibration, but the rub, between rotor and guide, may still be present, which may lead to the permanent damage of a mechanical system. The lemon shaped guide is not only effective in minimizing rubbing between rotor and stator, but also it suppresses the excessive vibration similar to the circular shaped guide.
The following video shows an experiment on how to minimize rubbing between a rotor and a guide that typically happens in a high speed rotating machine with a newly developed lemon shaped bearing or guide. We see here as the speed goes high, the system enters into its resonance frequency where it vibrates excessively. The lemon/elliptical shaped guide helps prevent the excessive rubbing while the rotor-bearing assembly is in the natural frequency zone. It has been found from our research that the lemon shaped guide minimizes rubbing better than the circular shaped guide where there the rotational speed is very high (https://doi.org/10.1115/1.4043817).
The next video shows an experiment on how to minimize rubbing between a rotor and a guide that typically happens in a high speed rotating machine with a circular bearing or guide. This is a traditional approach, which has been in operation in industries for long. We see in the video, as the speed goes high, the system enters into its resonance frequency where it vibrates excessively. The circular guide helps prevent the excessive vibration while the rotor-bearing assembly is in the natural frequency zone. However, if we notice carefully, the rubbing is still present between the rotor and guide, which deteriorates the system gradually.
A US provisional utility patent (US Patent App. 62/956,833) has already been filed for this design. I am looking for industrial partners or investors if they are interested in investing this product.
### Learning Mathematica, Lesson 1: Plotting a Function
This is the very first lesson or tutorial on Mathematica. As I am in the process of learning this tool, I will gradually post more articles on this, ranging from basic to advanced level problems. So, the first question is, what is Mathematica? It is simply a tool for computing, but it has an advantage that the symbolic expression is much user-friendly and more interactive than MATLAB.
Although, MATLAB is a much bigger platform than Mathematica, because MATLAB has numerous toolboxes and libraries that are designed for specific fields. Nevertheless, Mathematica, a product from Wolfram Research, is great for symbolic and interactive computing with a very neat interface. You may try it for free here just to see its environment. Let's start, how Mathematica looks like. After you install it (which isn't complex, it's pretty straightforward, if you just follow the instructions), click on your desktop shortcut and it will look like the following:
Then, you need to click on the "Documentations" to proceed which will bring the following:
Now, if you like to start writing you very first code on Mathematica, then click File, and select Notebook.
Now, write you first code here, and execute it. Let's say, we like to plot a function which looks as follows,
To execute the program above, you need to click on Evaluation and then, select Evaluate Notebook. The variable X varies from 0 to 5.
#Mathematica #Matlab #NoteBook #MatlabvsMathematica #Blog #Blogger
### A SIMULINK Model to Solve a Simple Shaft-Disk Dynamics Problem
The following figure 1 represents pretty simple model where two circular disks of inertia J1 and J2 are mentioned. Torque (T) is applied to the disk 1. The shaft has its own stiffness which is K.
Fig.1: Showing a very simple model where two disks are connected by a shaft and torque is applied to one of them.
At first, let's find the state variables for this problem and then write the state equations.
State Variables and State Equations:
Next, we assume the following key parameters,
Initial Conditions:
Parameter Values:
J1 = 100, J2 = 100, K = 100, and T = 10000. (All values are considered unitless for simplicity)
The SIMULINK model is formed by implementing the state equations. And, the above parameter values are considered. The following simulation results are for the acceleration, velocity and position of the disks which are essentially the outputs from the SIMULINK block diagram.
Fig.2: SIMULINK model of the shaft-disk system.
Fig.3: Showings the results of angular acceleration, velocity and displacement respectively of disk 1.
Fig.4: Showings the results of angular acceleration, velocity and displacement respectively of disk 2.#Simulink #Matlab #ShaftDisk #BlockDiagram #Blog #Blogger
### Theory of Energy Conversion in Wind Turbine
In wind turbine, the wind energy is converted to first mechanical energy, and then this energy is converted to electrical energy. Damping is an essential part for a generator. However, it is not the key factor for energy conversion. If there were no damping or loss, we would have 100% efficient conversion. This is not possible in real life, as we would have certain losses during the energy conversion process. These losses are represented by non-conservative forces, such as friction, viscous damping etc. which are the essential parts to be considered in the energy conversion process. I want to show the fundamentals behind energy conversion in DC motor.
DC motor converts electrical energy (input voltage) to mechanical energy (shaft rotation). This electromechanical conversion involves Faraday’s law of induction and Ampere’s law for force generated on the conductor moving in a magnetic field. In ideal situation, the torque (T) developed on the motor shaft is proportional to the input current (I) and the induced electromotive force (EMF) (V) or back EMF is proportional to the speed (W) of the motor. This can be expressed as [1];
T = K1 I ..........................................................(1)
V = K2 W .......................................................(2)
Where, K1 and K2 are the proportionality constant.
The electrical power (Pe) input to the motor is the product of the induced EMF and current.
Pe = VI = K2 W T / K1 ..................................(3)
And, the mechanical power output (Pm) is the product of the speed of the motor and torque.
Pm = T W .......................................................(4)
Now, by comparing equation (3) and (4), the following relation is obtained.
Pe = (K2 /K1) Pm ...........................................(5)
From Ohm’s law, it is known that,
E - V = I R ..................................................(6)
Where, E is the input voltage to the motor, and R is the resistance of the motor armature.
Moreover, we also know that torque produced at the motor shaft is equal to the product of the inertia of the load (J) and rate of change of angular velocity or angular acceleration.
T = J (dW/dt) .............................................(7)
Now, from equations (1), (6) and (7), it is found that
J (dW/dt) = K1 I = K1 / R (E - V) ...........................................(8)
Using equation (2) further, the following expression can be established.
dW/dt = (K1 K2 / J R) W + (K1 / J R) E ..................................(9)
The above equation refers to the first order linear differential equation model where ‘W’ represents the state of the system and ‘E’ is the external control input. This first order equation is good enough to predict the output speed of the motor. However, in terms of measuring the position’ it is necessary to add the following equation.
W = dθ / dt ...........................................................................(10)
Where θ is the output position of the DC motor and refers to another state of the system. Therefore, the model has one control input and two state variables (position and velocity).
By the above mathematical analysis, I want to specify, that the electrical energy is converted to mechanical energy by a gyrator which has a constant ratio K. Here, resistance of the armature is the energy loss during the conversion of electrical to mechanical energy which is also mechanical equivalence of a damper. A damper not only signifies the energy dissipation/loss from a system, but also helps to make a system stable by removing oscillations.
Reference
[1] Control system design : an introduction to state-space methods / Bernard Friedland.—Dover ed. | 2,629 | 12,213 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2024-33 | latest | en | 0.917005 |
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Worthy James: more detail
1. Lebron James: Basketball Legend (Inspiring Lives) by Shanya Worthy, 2010-01
2. James Worthy: L.A. Lakers 1987 Basketball Championship Series by Jerry Carpenter, Steve Dimeglio, 1988-06
lists with details
1. Career Stats For Jim Kelly - Nba Basketball Score And Minnesota
career stats for jim kelly nba basketball score and minnesota high school In a heave of a massive sigh, worthy of destroying all sound barriers,
http://www.apogee-ccd.com/WinPoker-12-1-career_stats_for_jim_kelly.html
2. James Worthy Hand Signed Lay-up 8x10 Photograph
Drafted number 1 overall in 1982, worthy helped the Lakers to 3 NBA Championships. This 8x10 photograph has been hand signed by james worthy and it includes
http://www.steinersports.com/jawohasila8x.html
3. The Lantern Online
Cavaliers worthy of attention, , The Lantern, a newspaper of Ohio State. When it comes to basketball, I prefer college to professional.
http://www.thelantern.com/news/2004/04/16/Sports/Cavaliers.Worthy.Of.Attention-6
4. New Measurement Techniques And A Binomial Model Of The Game Of Basketball
worthy gets the ball from Scott and puts it on the floor going for the layup, Recording all this information during a fastpaced basketball game is not
http://www.rawbw.com/~deano/articles/bbalpyth.html
Extractions: The second goal of this paper is to check the accuracy of a binomial distribution in describing the game of basketball. A simple way to check this is to test a relationship known as the Pythagorean Method. The Pythagorean Method is a simple mathematical relationship between the number of points scored and allowed by a team and the team's winning percentage. First used by James (1984) in baseball work, we will empirically expand the Pythagorean Method to basketball and compare results with those predicted by a simple binomial model of the game. The new scoring system we developed is not designed to replace traditional scoring methods, which are quick and efficient for tabulating cumulative statistics. This scoring system, known as the Possession Scoring System, was designed to collect as much information as possible about the game, which means giving up the simple tabulating techniques. All that is really needed to score a game using this system is something to write with, four sheets of lined paper (both sides will likely be used), and a decent understanding of basketball scoring rules. The NBA usually has several people working to keep official stats, but this method only requires one person. That one person, however, must work fast. The System is very simple. It focuses on the player with the ball, following the ball from player to player until the ball is turned over to the opposition through a shot attempt or turnover. For example, a scoresheet for part of a Detroit-Los Angeles Lakers game might look something like this: [Editor's Note: In the original document, many of the following symbols were in subscripts or superscripts.]
5. Worthy Of Their Hire § Unqualified Offerings
worthy Of Their Hire. I remember the years of amateur Olympics Mark Spitz and In the amateur days, the US Olympic basketball team had a role as a
http://www.highclearing.com/index.php/archives/2002/02/22/375
Extractions: @import url( http://highclearing.com/wordpress/wp-content/themes/bricks/style.css ); Looking Sideways at Your World Since October 2001 Faced with that much crap, one hardly knows where to start. might And Russian hockey players lived like dukes. themselves. And the members of the IOC lived like bandit chiefs. And it was great for foreigners. Contrariwise, the NHL-vs.-USSR games of the 80s proved that the Russkies really were pretty good at hockey. Needless to say, as soon as Russian players got the chance to earn NHL money, they took it, too. Posted by Jim Henley @ 10:39 pm , Filed under: Best of Unqualified Offerings No comments yet. This entry comment RSS Sorry, the comment form is closed at this time. Thomas L. Knapp
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View Full Version : How to determine energy state of a plane
jada002
06-23-2004, 10:50 AM
Any tips on how to determine the energy state of an enemy plane ?
jada002
06-23-2004, 10:50 AM
Any tips on how to determine the energy state of an enemy plane ?
SeaFireLIV
06-23-2004, 10:52 AM
You ask a complex question. I would say it truly takes experience, but perhaps someone knows how to explain it... I know how, but explaining it`s a different story.
SeaFireLIV...
http://img12.photobucket.com/albums/v31/SeaFireLIV/dragon7.jpg
Aghh! Damn those LA7s!
jada002
06-23-2004, 11:06 AM
Give it a shoot SeaFire, if you say something thats wrong im sure someone will tell us http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif
SeaFireLIV
06-23-2004, 11:16 AM
Ok, let`s try:
A plane has a better energy state than you, if it`s
1, Higher altitude.
2, Moving at speed, but above you.
3, You are flying slow and/or low.
4, You are turning and not after a dive.
5, If he`s diving from high and going up his energy state will be better than if you`re flying in circles below.
Example: You`re flying along at 500meters above ground, he`s at 2000m - he has better potential for his energy state. He dives on you, you see him, but there`s very little you can do, because you`re slow and low, you have no energy state.
He dives and shoots and flies up easily because he still has lots of energy. He comes back and does it again and again. Eventually you get shot down.
He would lose his energy state if he did something dumb like fly to your level and try turning with you. That way he`s lost his Energy advantage and the situation is equal. If he`s in a 190 and you`re in a Spitfire then YOU should have the advantage...
That`s the best i can do. hope it helps..
SeaFireLIV...
http://img12.photobucket.com/albums/v31/SeaFireLIV/dragon7.jpg
Aghh! Damn those LA7s!
carguy_
06-23-2004, 11:17 AM
He`s right,it really takes experience.To simplify look if your opponent is faster at a moment.
http://carguy.w.interia.pl/tracki/sig23d.jpg
jada002
06-23-2004, 11:18 AM
Thanks for the reply http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif
I found this... and i would also like to know if there are any other ways to "profile" a target ?
http://mysite.verizon.net/res0l0yx/sustained%20turn%20technique.htm
From last picture ....
After the pass keep track of angular gains and losses by pointing a wingtip(profiling) and note how fast the bandit moves from a forward to a rearward hemisphere.
Btw i thought this was a verry good tutorial on how to gain advantages in energy
JG14_Josf
06-23-2004, 11:51 AM
From Here (http://people.ee.ethz.ch/~chapman/il2guide/tracks.htm)
is the track file titled:
Energy game (http://people.ee.ethz.ch/~chapman/il2guide/downloads/tracks/Energy_game.zip)
I think, my opinion is, that the sustained turn technique described by Robert Shaw in his book 'Fighter Combat' is the best tool for making those very important relative energy state judgments.
Note in the track file (can be viewed as a training file - instructions for loading training files are on Mike's page linked above)
after a sustained turn, after an oponent has obviously burned more energy in a turn trying to make up more angles for the shot (you lag turn he does not) after he makes a harder turn (presumably burning more energy) the time is ripe to begin relative energy judgment. Just after the sustained turn and at the begining of the extension note relative possition. As the extension progresses as both planes vector a similar path (the sustained turn was not similar since the opponent turned much harder) then note gains or losses in relative possition. Now you know who has more energy.
If the opponent gains possition after a sustained turn then he has an energy advantage. If he gains possition after many sustained turns where his plane must turn harder than that plane is a better energy fighter compared to your plane.
To illustrate the same principle yet for a situation where one plane has a much higher energy state (more obvious difference) consider a typical head on merge.
You fly straight.
The opponent makes a hard 180 degree turn and then catches up with you.
Note that the opponent must have had more energy, more speed, more power, less drag, etc.
Take that same plane match-up where the one plane could turn 180 and still catch up and now bounce it from high altitude while it is stalling out in a climb. It will not be able to catch you after the pass for a long time even though past experience suggests that it is a better energy fighter. You have time to make a few more passes before it catches up in the energy game.
Note in the track file that the plane match-up is a P-51 and a 109. The 109 starts with more energy. The P-51 took off from the base. The 109 dove in to attack.
Those two planes are close, in my opinion, in energy fighting ability. Close enough to make it difficult to judge which one is better from patch to patch. Close enough to make it unwise to fight P-51s with 109s on an equal footing. The other guy may be very good.
[This message was edited by JG14_Josf on Wed June 23 2004 at 11:16 AM.]
F19_Ob
06-23-2004, 11:59 AM
Another thing........ is to determine what plane u fight against. Is it a slow turnfighter or fast turnfighter and so on.
Example:
If u are sitting in a 109 and see a hurricane aproach from behind (depending on how close and your own speed) U might want to try the spiraling climb, since the hurri looses its energy fast in a climb, especially if he is trying to draw deflection on U, and the 109 climbs very well even at slow speeds.
Ofcourse the hurriflyer can be an ace and he might risk hanging on his prop to get the proper deflection on U....there is always risks as u know.
The whole idea with the spiral climb is to force the enemy behind to draw more and more deflection....(dont evade if he gets a few hits on u , just tighten the spiral carefully) soon the attacker must level out because he is out of speed, Then its your turn, so u dive on him. This doesnt always work like in textbooks and it depends much on the experience of the flyers....aces makes mistakes too...but they often recover fast to lethal state.
So take a look at the speed specs on the planes u might run in to.....and dont forget to record tracks. http://ubbxforums.ubi.com/images/smiley/16x16_smiley-wink.gif
Oh, and the spiral will take some practice to master.
james8325
06-23-2004, 12:01 PM
seafire can u give me a link to your drawings? i saw them when u posted a link once before, but id like to take another look. i like the one in your sig.
jada002
06-23-2004, 12:16 PM
Thx JG14_Josf and F19_Ob. I learned a lot from that track Josf. You should have shot the guy down 3 times in that track I think, but you fired to late http://ubbxforums.ubi.com/infopop/emoticons/icon_wink.gif I never hit anything so i know what that looks likehttp://ubbxforums.ubi.com/infopop/emoticons/icon_wink.gif It makes me a dead man to often, even tho I have angles and energy at the start of a fight http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif
SeaFireLIV
06-23-2004, 12:27 PM
<BLOCKQUOTE class="ip-ubbcode-quote"><font size="-1">quote:</font><HR>Originally posted by james8325:
seafire can u give me a link to your drawings? i saw them when u posted a link once before, but id like to take another look. i like the one in your sig.<HR></BLOCKQUOTE>
I`ve got a completely new website coming up at the weekend which is better than the old one (no dumb pop-ups). I`ll put a link up then. http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif
JG14_Josf
06-23-2004, 12:27 PM
jada002,
I'm getting better:
http://mysite.verizon.net/res0l0yx/Cut%20the%20corner.jpg http://mysite.verizon.net/res0l0yx/Looks%20high.jpg
http://mysite.verizon.net/res0l0yx/Just%20a%20little%20late.jpg
http://mysite.verizon.net/res0l0yx/30mm%20love.jpg
james8325
06-23-2004, 12:28 PM
<BLOCKQUOTE class="ip-ubbcode-quote"><font size="-1">quote:</font><HR>Originally posted by SeaFireLIV:
<BLOCKQUOTE class="ip-ubbcode-quote"><font size="-1">quote:</font><HR>Originally posted by james8325:
seafire can u give me a link to your drawings? i saw them when u posted a link once before, but id like to take another look. i like the one in your sig.<HR></BLOCKQUOTE>
I`ve got a completely new website coming up at the weekend which is better than the old one (no dumb pop-ups). I`ll put a link up then. http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif<HR></BLOCKQUOTE>
alright thanks.
carguy_
06-23-2004, 12:29 PM
LOL Josf good one,the P40 is still in one piece http://ubbxforums.ubi.com/images/smiley/16x16_smiley-very-happy.gif
http://carguy.w.interia.pl/tracki/sig23d.jpg
gates123
06-23-2004, 12:56 PM
When the plane starts to buffet wildly you can assume you have alot of E and no ones gonna catch you in the next 20 seconds http://ubbxforums.ubi.com/images/smiley/35.gif
http://www.flightjournal.com/images/index_photos/gunslinging.jpg
Did anyone see that or was it just me?
jada002
06-23-2004, 12:59 PM
I see, i actually didnt doubt that http://ubbxforums.ubi.com/infopop/emoticons/icon_smile.gif I just had to remark on it since some of the shots where pretty late.
JG14_Josf
06-23-2004, 01:13 PM
Jada002,
If you really want to get better then I suggest that you use the single most important tool you have at your disposal toward that end i.e. the track recording feature in IL2.
What you will find during the proccess of debriefing your flights is that the tracks do not reproduce exactly the same situation each time you run the track file.
On my energy game track file, during the actual flight, I hit the P-51 on the left wing. The shot was high not late. The hit does not show up often when replaying the track file but it does show up occasionally.
That P-51 went down after my 3rd pass.
The P-40 went down due to control damage. It looks warm in the cockpit. The P-40 shot shows up on the left wing root sometimes during the replay. Other times the explosion shows up on the tail. I think the guy flying the P-40 registered a hit on his tail. My wingman and I both saw a hit on the cockpit as the snap shot illustrates. We didn't follow the P-40 down, it looked like a done deal.
AirBot
06-23-2004, 01:32 PM
The answer is rather simple really:
E = mgh + (m * v^2)/2
http://ubbxforums.ubi.com/images/smiley/16x16_smiley-very-happy.gif
Magister__Ludi
06-23-2004, 04:08 PM
<BLOCKQUOTE class="ip-ubbcode-quote"><font size="-1">quote:</font><HR>Originally posted by AirBot:
The answer is rather simple really:
E = mgh + (m * v^2)/2
http://ubbxforums.ubi.com/images/smiley/16x16_smiley-very-happy.gif<HR></BLOCKQUOTE>
Yes, it is. The above formula applied to aerial combat says that you have to maintain a speed or height advantage over the enemy (better be both).
PikeBishop
06-24-2004, 02:55 AM
Excellent attempts to explain very difficult concepts.........in the end one does it without thinking with lots and lots of practice. Everyone...3 housemarks.
regards,
SLP.
Fehler
06-24-2004, 04:24 AM
Recognizing an energy state can be difficult at times.
If you find yourself in the bounce, sometimes it is more important to force a slower energy state from your opponant.
The scissors move is great for this. I think the best descriptioon of the scissors I have ever heard was simply put, "Attempting to cover the greatest distance in the smallest amount of space." Basically weaving back and forth, keeping the other person from getting in your 4-8 o'clock line. All the while, lessening his energy state while he attempting to pull lead shot on you. (Bleeding him dry!)
The problem with any 2D screen game is that judging distance is not done in a natural way. Pixels change size to fool the eyes, but your brain does not calculate distance in that manner. It uses two points of reference (Each eye) to calculate distance.
http://webpages.charter.net/cuda70/FehlerSig.gif
http://webpages.charter.net/cuda70/9JG54.html
SeaFireLIV
06-24-2004, 04:27 AM
It took me a long while to understand the scissors and other manouevers, but once I finally understood- wow! Many times Online I have shaken and confused guys on my tail with this process.
And when I actually get on THEIR tail, you can almost hear them thinking, "What the heck just happened?"
SeaFireLIV...
http://img12.photobucket.com/albums/v31/SeaFireLIV/dragon7.jpg
Aghh! Damn those LA7s!
Maple_Tiger
06-24-2004, 11:39 AM
Don't they have energy state meters that you can by now? I think they do. You just point it at the target plane and it will give you the energy state.
Capt. 361stMapleTiger.
http://img52.photobucket.com/albums/v158/Maple_Tiger/FBAA2.gif
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INSURANCE CENTER: Life
What factors might supplement replacement income?
Life Insurance Discussion Boards
After you've figured out your replacement income and you've picked yourself up off the floor, you can stop gathering items for a garage sale and really start to chip away at that figure. (Besides, you're only going to get a few bucks for that dusty Ronco Toe Hair Curler.)
There are three ways for the surviving members of your household to supplement replacement income: Social Security, income, and current savings.
1. Social Security for survivors
You may not realize it, but Uncle Sam is one of the largest life insurers in the land. If you have 10 years or more of employment earnings under your belt, your surviving dependents will probably qualify for a piece of this action. Even with fewer than 10 years of work experience, your family may still get something.
To help you estimate this benefit, the Social Security Administration (SSA) provides an online calculator. In the output from the calculator, scroll down to the table labeled "Survivors." Note that the monthly payment per family is capped. If you have more than one dependent, be sure to use the "Family maximum" figure in place of the per-child amount.
Once you get this monthly income amount, things get a little rocky. Hang on!
First, we're talking here about an adjustment to annual replacement income. In other words, it will send you to a new row in the replacement income table. The problem, however, is that the table is laid out in terms of annual income and the Social Security calculator gives us a monthly amount. So, to adjust Uncle Sammy's contribution to match the table, multiply the monthly amount by 12. This gives you annual replacement income provided by Social Security. Subtract this amount from your current salary, and go back into the replacement income table at this lower annual amount.
Let's extend the example in which your current salary is \$40,000 and you are 25 years away from retirement. Without considering Social Security payments, we came up with a table value of \$803,000, the amount required to pay your family \$40,000 every year for 25 years. Now, tack on the assumption that you have two or more dependent children. (There were a few cold winters back in the '90s, you know.) Given these numbers, the Social Security calculator spits out a rough estimate of \$2,245 in pre-tax monthly income (the family maximum). On an annual basis, this equates to roughly \$27,000, before taxes.
We then subtract the \$27,000 in Social Security benefits from the required \$40,000 in annual income, we get a new figure of just \$13,000 in replacement income from life insurance. Go to the replacement income table with this new number and see that you are now looking at between \$201,000 and \$402,000 in term life benefits, less than half as much!
As in this example, adjusting for Social Security survivor payments can substantially reduce the amount of life insurance you need. If you have serious concerns about the reliability of Social Security income, you may want to skip this adjustment or water it down. It's up to you.
If you want a more precise estimate of Social Security survivor benefits, our friends at the Social Security Administration provide additional calculators on their website. | 687 | 3,340 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2017-26 | latest | en | 0.933551 |
https://forum.freecodecamp.org/t/intermediate-algorithm-scripting-diff-two-arrays/375454 | 1,652,883,031,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662522270.37/warc/CC-MAIN-20220518115411-20220518145411-00089.warc.gz | 330,254,924 | 8,444 | # *Intermediate Algorithm Scripting: Diff Two Arrays
How may I compare every element of an array with every element of another array?
function diffArray(arr1, arr2) {
let sortedArr1=arr1.sort();
let sortedArr2=arr2.sort();
let a=arr1.length;
let b=arr2.length;
var newArr = [];
//console.log(a)
//console.log(b)
//console.log(arr1)
if(a>=b){
//console.log(arr1)
for(let i=0;i<=arr1.length;i++){
if(sortedArr1[i]!=sortedArr2[i]){
//console.log(arr1)
newArr.push(arr1[i])
}
}
}
console.log(sortedArr1);
console.log(sortedArr2)
//console.log(newArr)
return newArr;
}
diffArray(["andesite", "grass", "dirt", "pink wool", "dead shrub"],["diorite", "andesite", "grass", "dirt", "dead shrub"]);
How may I solve this test: [1, "calf", 3, "piglet"], [1, "calf", 3, 4] should return ["piglet", 4] .
function diffArray(arr1, arr2) {
var newArr = [];
var arr=[];
//var arr = [1,2,3,4],
//brr = [2,4],
let arr1Length=arr1.length;
let arr2Length=arr2.length;
if(arr2Length>arr1Length){
let filteredArr2 = arr2.filter(element => ! arr1.includes(element));
//console.log(newArr);
if( ){
}
return newArr;
}else{
let filteredArr1 = arr1.filter(element => ! arr2.includes(element));
//console.log(newArr)
return newArr;
}
}
diffArray([1, "calf", 3, "piglet"], [7, "filly"]);
I can write out the logic or the pseudo code for how I approached this but I don’t want to rob you of the learning experience either.
I will also say that at first it seems like you would want to check the longer array against the shorter array because there will always be extra elements in the larger array that you want to check for and not skip but the shorter array might have a unique element so that logic would only work if we knew the shorter array did not contain any unique elements from the longer array.
Just wanted to mention that because in the code from your first post I see you sorting them and then getting the length and then looping through the longer array to compare against the shorter. I see how you are thinking about it.
Let me lay out my logic below now:
• Take the first element of the first array and check it against each element in the second array until we find a match or reach the end of the second array.
• If the element we are checking from the first array is found in the second array then we will stop going through the second array and move on to the next element from the first array, repeating the same process.
• If the element from the first array is not found in the second array, because we’ve reached the end of the second array, then we can push the element from the first array into a new array that holds only the elements not found in the second.
• Now repeat the same process but flip the arrays around starting with the first element from the second array and iterating through each element of the first array to find the elements only present in the second array and store them in a new array.
• Now we have two new arrays, one which has the unique elements from arr1 and another withe the unique elements from arr2.
• The last step is simple to concatenate these two arrays and return the result.
Let me know if this is helpful or not.
How may I compare every element of an array with every element of another array?
To answer your original question, this can be accomplished with a nested for loop or with the built in methods suggested in the help. Those being filter and includes.
Here is a simple nested for loop that will check all the elements of one array against another.
const arr1 = [1, 2, 3];
const arr2 = [4, 2, 1];
const arr1_Unique = [];
for (let i = 0; i < arr1.length; i++) {
for (let j = 0; j < arr2.length; j++) {
if (arr1[i] === arr2[j]) {
// If we find a match we can break
// out of loop and check next element
break;
} else if (j === arr2.length - 1) {
// If no match is found and we are at the
// end of the array, push element to unique array
arr1_Unique.push(arr1[i]);
}
}
}
The methods filter and includes could replace this and I’m sure many other solutions could work.
1 Like
I am thankful for your time!
I tried to use filters, but hot can I extract the result from them? I used for, slice and they bring me brackets back again.
function diffArray(arr1, arr2) {
var newArr = [];
//var arr = [1,2,3,4],
// brr = [2,4],
let filteredArr2 = arr2.filter(element => ! arr1.includes(element));
//console.log(filteredArr2)
//console.log(slicedArr2)
let filteredArr1 = arr1.filter(element => ! arr2.includes(element))
//console.log(filteredArr1)
newArr.push(filteredArr2)
newArr.push(filteredArr1)
console.log(newArr)
}
diffArray([1, "calf", 3, "piglet"], [1, "calf", 3, 4]);
Instead of push I think you want concat.
Make sure you are returning the final result instead of just console logging it.
Also, even though what you wrote still works, you should instead write !arr2.includes(element) with no space between the exclamation point and the statement.
1 Like
I am thankful for your time
function diffArray(arr1, arr2) {
var newArr = [];
//var arr = [1,2,3,4],
// brr = [2,4],
let filteredArr2 = arr2.filter(element => !arr1.includes(element));
//console.log(filteredArr2)
//console.log(slicedArr2)
let filteredArr1 = arr1.filter(element => !arr2.includes(element))
return filteredArr1.concat(filteredArr2);
}
diffArray([1, "calf", 3, "piglet"], [1, "calf", 3, 4]);
Yep, and you can get rid of
filteredArr1.concat(filteredArr2)
and just keep
return filteredArr1.concat(filteredArr2)
As you might already have figured out because, from the docs on concat…
“This method does not change the existing arrays, but instead returns a new array.”
1 Like
Vielen Dank, mein Freund!
function diffArray(arr1, arr2) {
var newArr = ;
//var arr = [1,2,3,4],
// brr = [2,4],
let filteredArr2 = arr2.filter(element => !arr1.includes(element));
//console.log(filteredArr2)
//console.log(slicedArr2)
let filteredArr1 = arr1.filter(element => !arr2.includes(element))
return filteredArr1.concat(filteredArr2);
}
diffArray([1, “calf”, 3, “piglet”], [1, “calf”, 3, 4]); | 1,609 | 6,109 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2022-21 | latest | en | 0.774801 |
http://enhtech.com/margin-of/help-relative-standard-error-and-margin-of-error.php | 1,544,894,910,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376826892.78/warc/CC-MAIN-20181215152912-20181215174912-00342.warc.gz | 92,960,551 | 4,849 | Home > Margin Of > Relative Standard Error And Margin Of Error
# Relative Standard Error And Margin Of Error
It can only be calculated if standard deviation of the Student t-distribution. As a random variable, it has a (sampling) distribution The mean of all possible sampleHomoskedastic A statistics term indicating that relative
sampling distribution of a statistic,[1] most commonly of the mean. How to search for flights for standard navigate here how big your sample needs to be to come within your desired margin of error. error Based On Sample Data, What Do We Call Our Best Guess Of A Population Parameter? American standard 18 years and over in New South Wales who are current daily smokers is 16.3%.
The smaller standard deviation for age at first marriage pp.63–67. Follow us on... Browse other questions tagged definition and But, for now, let's assume you can count with 100% claims made by clients for inadequate work or negligent actions.
Other statistics Confidence intervals can be calculated, and so can margins of error, for why can it be created by dividing two numbers? Browse other questions tagged definitionError Example Further reading Why do we have Standard Errors? Margin Of Error Formula The standard error can be computed from a knowledge of Zelda-like map in custom game engine?For example, the sample mean is
Related 9Meaning of based on a quantitative measure of uncertainty: the standard error. Less variety in the data see Margin for error (film).This can be contrasted to the standard deviation (SD)
mathematician wouldn't say that Candidate A has a two-point lead in the actual race.See also Engineering tolerance Key relevance Measurement Margin Of Error Calculator The top portion charts probability density against actual percentage, showing the relative 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. That's because many reporters have no ideaThat's called a leading question, and it's a big no-no in surveying.
Often, however, the distinction is not explicitly margin mean of a sample may be from the true population mean.The standard error is margin there are more than two possible poll responses.Because of random variation in sampling, the proportion or mean calculated using the his comment is here
More variety is likely to Promoter Score) result? 8 How are margins of error related to confidence Intervals?The standard error is an estimateis off the mark will decrease as you add more people to your sample. Logical && statement with null their explanation x 37,600) = 10,773,500. 95% Confidence = 10,773,500 to 10,924,000.The ages in one such sample are 23, 27, 28, 29, 31, relative confidence interval as being equal to two standard deviations in your polling sample.
Standard Error: The standard error (SE) of the computed from known sample statistics. In other words, the more people you ask, thethe 95% confidence level, you would have $z_{0.975}=1.96$ resulting in a margin of error $0.0081\cdot1.96=0.0158$.Find out the size and causes of ETF of Principle if time and space are continuous?Retrieved February 15, 2007. ^ Braiker, Brian. "The Race is On: With voters widely it with one of these being the zero weight.
Ecology 76(2): 628 – 639. ^ Klein, RJ. error that can be characterized by mean, variance, distribution function, etc.The standard error (0.016 or 1.6%) helps to give 20,000 samples of size n=16 from the age at first marriage population. Before we answer your question, let's Margin Of Error Definition the statistic has a confidence interval radius of 5 people.If you want to get a more accurate picture of who's as the sampling fraction grows, lest sampling bias be introduced.
The records in the group that this contact form be calculated as: $$p\,\pm\,Z_{\alpha/2}\,\text{SE}$$ Given that $Z_{\alpha/2}=Z_{0.975}=1.959964\sim1.96$, the CI will be: $$p\,\pm\,1.96\,\sqrt{\frac{p\,(1-p)}{n}}$$.It holds that the FPC approaches zero as the sample size (n) approaches the estimates is itself an estimate, and is therefore subject to sampling variability.Roman letters indicate that error that takes into account that spread of possible σ's.The margin of error is the half-width of the associated confidence interval, so for error and proportion estimates have been calculated in the publications released for the AHS survey.
Understand how the standard error is will focus on the standard error of the mean. The Relative Standard Error (RSE) is the standard error expressed as Margin Of Error Excel is 23.44, and the standard deviation of the 20,000 sample means is 1.18. support p = 0.47 and n = 1,013.
Margin of error is commonly expressed as error In media reports of poll results, the term usually refers tomore likely you are to get a representative sample.The notation for standard error can be any one of'replicate estimates'. 28 There are various ways of creating replicate sub-samples from the full sample.
One example is the percent of people weblink And Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p.Print some JSON Would it be ok to is the standard deviation of the sampling distribution. Wikipedia® is a registered trademark of Acceptable Margin Of Error
33.87, and the population standard deviation is 9.27. Political Animal, Washingtonthe sample standard deviation is 2.56.Are there to the Investing Basics newsletter Thanks for signing up to Investing Basics. The survey with the lower relative standard error can be said to have
techniques for data from complex sample designs. error standard Margin Of Error Sample Size probably unhealthy food Are there other Pokemon with higher spawn rates right now? error The formula for relative standard errorSee also unbiased estimation of standard deviation for more discussion.
off camera before switching auto-focus on/off? Learn about the "newRights Reserved. The size of the sample was 1,013.[2] Unless otherwise stated, Standard Error Formula new drug lowers cholesterol by an average of 20 units (mg/dL).And the same goes for youngGuide to Questionnaire Design.
Since this is a sample for people with substandard credit scores or limited credit histories. The margin of error for a particular individual percentage will usually error samples tend to have approximately normal distributions and low sample errors. A practical result: Decreasing the uncertainty in a mean value estimate by a and Bradburn, Norman (1982).
The survey results also often provide strong information for each replicate group (i.e. Linearization and resampling are widely used error of your proportion estimate is $\sqrt{0.07\cdot0.93/1000}$ $=0.0081$. Ultimately each record had 60 replicate weights attached to it can be expressed as a percentage.
Associates. ^ Drum, Kevin.
Now that's true in this poll, but given the likely margin of error, a is what statisticians call a confidence interval. | 1,527 | 6,848 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2018-51 | latest | en | 0.884309 |
https://donghiadigest.com/ask-842 | 1,669,599,263,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710462.59/warc/CC-MAIN-20221128002256-20221128032256-00550.warc.gz | 252,680,749 | 5,124 | # Apps to help you with math
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Help with algebra Web math Finite math help Solve the compound inequality solver Angle addition postulate solver How to solve fractions over fractions | 724 | 3,691 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2022-49 | latest | en | 0.956209 |
https://security.stackexchange.com/questions/118466/how-do-rainbow-tables-solve-collisions | 1,716,378,665,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058542.48/warc/CC-MAIN-20240522101617-20240522131617-00557.warc.gz | 455,179,193 | 39,554 | # How do rainbow tables solve collisions?
I get the gist of it. It's like a middle ground between brute force attack and lookup table, it stores the starting plaintext and ending hash for each chain where a chain is made by reduction and hash.
What I don't get is:
1. It's said that rainbow tables solve collisions, but why are collisions such a big deal to begin with?
2. It's said that rainbow tables solve collisions by using a different reduction function for each column in the chain, but how does this prevent collisions? Aren't reduction functions just random characters you take from the hash? So what difference does it make if you take the first 8 characters instead of the last 8?
• does this answer your question: security.stackexchange.com/questions/379/… Mar 24, 2016 at 16:32
• Nope. It doesn't even mention collisions. Mar 24, 2016 at 16:38
• So your question is focused on collisions? If so, can you edit your title? You might get better attention and higher quality answers. Mar 24, 2016 at 16:43
• Yes but collisions are the very reason rainbow tables were created... to prevent collisions. Mar 24, 2016 at 16:46
The current highest voted answer doesn't really seem to give a proper response to your question. I'll try to answer both your questions simultaneously.
### Same reduction function in every column
Say, you use the same reduction function for every column and have a basic table with 2 rows and 3 columns.
``````P1 --R--> P1' --R--> P1'' --R--> P1'''
P2 --R--> P2' --R--> P1' --R--> P1''
``````
Here, an `--R-->` represents a hashing followed by a reduction. And `P1, P2, P1', ...` represent passwords. As you can see, there was a collision in the second chain. The value `P1'` has already been encountered in the first chain.
Notice what happens afterwards.
Since the hashing followed by the reduction of `P1'` is exactly the same as in the first chain, we get a value that has already been computed. If we continue this even further, the part of the second chain starting from `P1'` becomes an exact copy of the part of the first chain starting from `P1'`.
So in effect, this is why collisions are bad. The second chain merged into the first. We have duplicate results in our table, resulting in wasted storage space and computation time.
### Different reduction function in every column
This time, let's see what happens if we use a different reduction for every column. A reduction is represented by `--RX-->` where `X` is the column number.
``````P1 --R1--> P1' --R2--> P1'' --R3--> P1'''
P2 --R1--> P2' --R2--> P1' --R3--> P2''
``````
Again, `P1'` was encountered in both chains. However, since the reduction functions are different, the value calculated after `P1'` in the second chain won't result in `P1''`, as it does in the first chain. This effectively solves the merge issue from the first example.
Note that this doesn't solve every chain merge though. Watch what happens in the following example:
``````P1 --R1--> P1' --R2--> P1'' --R3--> P1'''
P2 --R1--> P1' --R2--> P1'' --R3--> P1'''
``````
This time a collision happens in the first column of both chains. Since it happens in the same column, every next reduction function will be the same and the chains are merged once again. The probability of this happening is lower though.
Collisions are the only problem with Rainbow Tables. Ironically collisions are seen as a bad thing for hashing algorithms, but in the case of Rainbow Tables a hashing algorithm which generates collisions fairly regularly will be more secure.
It's said that rainbow tables solve collisions, but why are collisions such a big deal to begin with?
A given hash may be generated by multiple plaintexts (this is called a collision), which is a big problem for chains because it causes chains which start different to converge into one. Also you get loops, which are caused when a hash is reduced to a plaintext that was hashed at a previous point in the chain.
It's said that rainbow tables solve collisions by using a different reduction function for each column in the chain, but how does this prevent collisions? Aren't reduction functions just random characters you take from the hash? So what difference does it make if you take the first 8 characters instead of the last 8? The way collisions are handled is what sets Rainbow Tables apart from its predecessor which was developed in 1980.
The predecessor solved the problem of certain plaintexts never being reduced to by using many small tables. Each small table uses a different reduction function. This doesn't solve the problem completely, but it does help.
To solve chain merges and loops each chain ended at a "distinct point"; a hash which was unique in some way, eg hashes where the first 4 characters are 0. The chains keep on going until it reaches a distinct point. If two chains end up at the same distinct point then there has been a collision somewhere in the chain, and one of the chains is discarded. If a chain is generated for an unusually long time without reaching a distinct point a loop is suspected (where a chain of hashes ends up reducing and hashing to a previous hash in the chain). The problem with this is that if there is a collision there is potentially a whole branch which has to be cut off and won't make it into the chains, and a loop will cause all the hashes which came before the loop in the chain to be discarded. Also all the time spend generating that chain will be wasted, and by ending only at distinct points you have chains of variable length. This means that you may have to keep checking for a hash within especially long chains long after the other chains have ended.
I got the proper inspiration from here: http://kestas.kuliukas.com/RainbowTables/
• I already read that link. It doesn't explain anything. Mar 24, 2016 at 16:23
• can you describe what you did not understand? :) Mar 24, 2016 at 16:28
• "A given hash may be generated by multiple plaintexts (this is called a collision), which is a big problem for chains because it causes chains which start different to converge into one." This explains what a collision is, not why it's a problem. Mar 24, 2016 at 16:32
• Ok so: you have A, B, C as starting points. A starts first and has the following value (poetically example) 3123212553633474373473734734737 and B starts with 489676236 and ends with "33474373473734734737" the same "value" the C starts with 49186481624 and ends with "33474373473734734737" and you waste time generating because yo have common points with A. Mar 24, 2016 at 16:42
• Is there something specific you are trying to understand? Mar 24, 2016 at 16:45 | 1,603 | 6,671 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2024-22 | latest | en | 0.947695 |
https://help.hcltechsw.com/unica/Campaign/en/12.1.4/Macros/users_guide_topics/inverse.html | 1,719,186,312,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198864968.52/warc/CC-MAIN-20240623225845-20240624015845-00793.warc.gz | 261,520,380 | 12,971 | # INVERSE macro
The `INVERSE` macro is available only in Unica Campaign.
## Syntax
`INVERSE(data)`
## Parameters
`data`
The numerical values to compute the inverse of. This can be a constant value, a column, a cell range, or an expression evaluating to any of the above. For the format definition of `data`, see the "Macro Function Parameters" section in the chapter in this guide for your product.
## Description
`INVERSE` calculates the negative of the values in the specified data range. It returns -x (that is, negative values are returned as positive values, and positive values are returned as negative values). `INVERSE` returns one new column for each input column, each containing the inverse of the values in the corresponding input column.
Note: To invert a value or a column, precede it with a minus sign (`-`). For example, `V2` `=` `-V1` is the same as `V2` `=` `INVERSE(V1)`.
## Examples
`TEMP = INVERSE(3.2)` Creates a new column named `TEMP` containing the value `-3.2`. `TEMP = INVERSE(V1)` Creates a new column named `TEMP`, where each value is the negative of the values in column `V1`. `TEMP = INVERSE(V1:V3)` Creates three new columns named `TEMP`, `VX`, and `VY`. The values in the `TEMP` column are the negatives of values in column `V1`, the values of the `VX` column are the negatives of the values in column `V2`, and the values of the `VY` column are the negatives of the values in column `V3`. `TEMP = INVERSE(V1[10:20])` Creates a new column named `TEMP`, where the first 11 cells contain the negatives of the values of the values in rows 10-20 of column `V1`. The other cells in `TEMP` are empty. `TEMP = INVERSE(V1[1:5]:V2)` Creates two new columns named `TEMP` and `VX`, each with values in rows 1-5 (the other cells are empty). The values in column `TEMP` are the negatives of the values of the corresponding rows of column `V1`, and the values in column `VX` are the negatives of the values of the corresponding rows of column `V2`.
## Related functions
Function Description
`ABS` Computes the absolute value of the contents of the specified data range
`NOT` Computes the logical NOT of the contents of the specified data range
`SIGN` Computes the sign (positive or negative) of the values in the specified data range | 572 | 2,266 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2024-26 | latest | en | 0.577989 |
https://yellowcomic.com/what-percent-of-150-is-45/ | 1,638,311,465,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964359073.63/warc/CC-MAIN-20211130201935-20211130231935-00502.warc.gz | 1,197,298,294 | 4,893 | If you desire to discover more, then please save reading, and also you won"t be disappointed.
You are watching: What percent of 150 is 45
## Step by step an approach for calculating what percent the 150 is 45
We currently have our first value 150 and the 2nd value 45. Let"s assume the unknown value is Y i m sorry answer us will discover out.
As we have actually all the forced values us need, currently we have the right to put castle in a basic mathematical formula together below:
STEP 1Y = 45/150
By multiply both numerator and denominator through 100 we will get:
STEP 2Y = 45/150 × 100/100 = 30/100
STEP 3Y = 30
Finally, we have discovered the worth of Y i m sorry is 30 and also that is ours answer.
You can use a calculator to uncover what percent of 150 is 45, just enter 45 ÷ 150 × 100 and also you will acquire your answer i m sorry is 30
Here is a calculator come solve percentage calculations such together what percent the 150 is 45. You have the right to solve this kind of calculation with your values by entering them right into the calculator"s fields, and also click "Calculate" to get the result and explanation.
What percent of
is
Calculate
## Sample questions, answers, and also how to
Question: your uncle had actually 150 share of his own agency a couple of years earlier, and now he has 45 of them. What percent of the shares of his company he has now?
Answer: He has 30 percent of shares of his firm now.
How To: The an essential words in this difficulty are "What Percent" since they permit us understand that it"s the Percent that is missing. For this reason the two numbers that it offers us have to be the "Total" and the "Part" us have.
Part/Total = Percent
In this case, it"s the full that our uncle owned. So we put 150 ~ above the bottom that the fraction and 45 ~ above top. Now we"re prepared to number out the part we don"t know; the Percent.
See more: Get Mew In Pokémon Fire Red How To Get Mew In Pokemon Leafgreen Fire Red?
45/150 = Percent
To find the percent, all we have to do is transform the portion into the percent type by multiplying both top and bottom part by 100 and here is the means to figure out what the Percent is:
45/150 × 100/100 = 30/100
30 = Percent
And that way he has 30 percent the the shares of his agency now.
## Another step by action method
Step 1: Let"s deal with the equation for Y by first rewriting that as: 100% / 150 = Y% / 45
Step 2: fall the percent marks to leveling your calculations: 100 / 150 = Y / 45
Step 3: main point both political parties by 45 to isolate Y on the best side that the equation: 45 ( 100 / 150 ) = Y | 665 | 2,627 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2021-49 | longest | en | 0.960288 |
https://www.airmilescalculator.com/distance/kgk-to-mll/ | 1,620,545,263,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988961.17/warc/CC-MAIN-20210509062621-20210509092621-00251.warc.gz | 634,709,665 | 7,795 | Distance between Koliganek, AK (KGK) and Marshall, AK (MLL)
Flight distance from Koliganek to Marshall (Koliganek Airport – Marshall Don Hunter Sr. Airport) is 219 miles / 352 kilometers / 190 nautical miles. Estimated flight time is 54 minutes.
Map of flight path from Koliganek to Marshall.
Shortest flight path between Koliganek Airport (KGK) and Marshall Don Hunter Sr. Airport (MLL).
How far is Marshall from Koliganek?
There are several ways to calculate distances between Koliganek and Marshall. Here are two common methods:
Vincenty's formula (applied above)
• 218.816 miles
• 352.150 kilometers
• 190.146 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth.
Haversine formula
• 218.173 miles
• 351.116 kilometers
• 189.587 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
Airport information
A Koliganek Airport
City: Koliganek, AK
Country: United States
IATA Code: KGK
ICAO Code: PAJZ
Coordinates: 59°43′35″N, 157°15′32″W
B Marshall Don Hunter Sr. Airport
City: Marshall, AK
Country: United States
IATA Code: MLL
Coordinates: 61°51′51″N, 162°1′33″W
Time difference and current local times
There is no time difference between Koliganek and Marshall.
AKDT
AKDT
Carbon dioxide emissions
Estimated CO2 emissions per passenger is 57 kg (126 pounds).
Frequent Flyer Miles Calculator
Koliganek (KGK) → Marshall (MLL).
Distance:
219
Elite level bonus:
0
Booking class bonus:
0
In total
Total frequent flyer miles:
219
Round trip? | 442 | 1,700 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2021-21 | latest | en | 0.784874 |
http://www.bankersadda.com/2015/10/night-class-reasoning-quiz_7.html | 1,513,449,908,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948588420.68/warc/CC-MAIN-20171216181940-20171216203940-00438.warc.gz | 301,175,072 | 28,813 | # Night Class: Reasoning Quiz
Directions: Study the following information carefully and answer the questions given below:
Six friends P, Q, R, S, T and U work in six different companies Walmart, Toyota, Apple, Petronas, Cargill and Honda each wears different coloured shirt, viz. Black, Golden, Blue, Orange, Green and White but not necessarily in the same order.
• The one who is wearing the Black shirt works in Petronas, and one who is wearing the Golden shirt works in Walmart.
• U does not work for Apple or Cargill.
• P wears Blue coloured shirt and works for Toyota.
• S does not work for Cargill and the green coloured shirt is not wear by any of the Apple's worker.
• T works for Honda and neither S nor Q works for Petronas.
• No Cargill worker wears either Green or Orange shirt.
• R works in Walmart.
1. Which coloured shirt does Apple's worker wear?
1) Golden
2) Black
3) Green
4) Orange
5) White
2. Which pair is correctly matched?
1) White–Apple–P
2) White–Cargil–Q
3) Golden–Apple–R
4) Cannot be determined
5) Other than given options
3. If Apple's and Petronas decide to interchange the colour of their employees' shirt, then which two person have to exchange their shirt?
1) S and U
2) P and R
3) S and T
4) Q and S
5) Other than given options
4. T works for which company?
1) Walmart
2) Petronas
3) Honda
4) Cargill
5) Toyota
5. Which of the following is true?
1) Honda – Golden
2) S – Cargill
3) T – White
4) White – Cargill
5) Other than given options
Directions (Q. 6 –10): In these questions, a relationship between different elements is shown in the statements. The statements are followed by two conclusions. Give answer:
1) If only conclusion I is true.
2) If only conclusion II is true.
3) If either conclusion I or II is true.
4) If neither conclusion I nor II is true.
5) If both conclusions I and II are true.
6-7:
Statements:
I > J
L < O
N >= M
P >= N
M = L
6.
Conclusions:
I. I > L
II. N >= L
7.
Conclusions:
I. J >= L
II. O > N
8.
Statements:
B = Y >= C < N
P > Y
Conclusions:
I. P > C
II. Y < N
9.
Statement:
R <= K < N > Y >= Z
Conclusions:
I. Y < K
II. Z >= R
10.
Statements:
B < L <= S
B > N
K = N
Conclusions:
I. K < S
II. S < N
(1–5):
Name - Company - Colour
P - Toyota - Blue
Q - Cargill - White
R - Walmart - Golden
S - Apple - Orange
T - Honda - Green
U - Petronas - Black
1. 4
2. 2
3. 1
4. 3
5. 4
6. 2; Given statement:
I > J - (i)
L < O - (ii)
N >= M - (iii)
P >= N - (iv)
M = L - (v)
Combining (i), (ii), (iii) and (iv), we get
We cannot compare I and L. Thus conclusion I is not true.
Again,
Comparing N and L,
N >= L. Thus conclusion II is true.
7. 4;
From above,
Here, we cannot compare J and L. Thus conclusion I is not true.
Again,
O and N cannot be compared. Thus conclusion II is not true.
8. 1
Given statements:
B =Y >= C < N - (i)
P > Y - (ii)
Combining (i) and (ii), we get
Comparing P and C, we get
P > C. Thus conclusion I is true.
Again,
We cannot compare Y and N.
Thus, conclusion II is not true.
9. 4
Given statements:
We cannot compare Y and K. Thus conclusion I is not true.
Again,
We cannot compare Z and R. Thus conclusion II is not true.
10. 1
Given statement:
B < L <= S - (i)
B > N - (ii)
K = N - (iii)
Combining (i), (ii) and (iii), we get
Comparing K and S,
K < S, Thus conclusion I is true.
Again,
Comparing N and S,
N < S. Thus conclusion II (S < N) is not true. | 1,056 | 3,363 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.953125 | 4 | CC-MAIN-2017-51 | latest | en | 0.876367 |
https://books.google.com.jm/books?id=9s0PAAAAYAAJ&qtid=3f7e78dd&source=gbs_quotes_r&cad=5 | 1,660,663,539,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572408.31/warc/CC-MAIN-20220816151008-20220816181008-00186.warc.gz | 160,709,134 | 5,335 | Books Books
The three angles of any triangle taken together are equal to the three angles of any other triangle taken together. From whence it follows, 2.
Elements of Geometry, Briefly, Yet Plainly Demonstrated by Edmund Stone - Page 31
by Euclid - 1728
Euclide's Elements ... compendiously demonstrated, by I. Barrow. Transl
Euclides - 1660 - 368 pages
...whence it follows, i. That if ia one triangle , two angle? ("taken fevcrnlly , or together) be equall to two angles of another triangle (taken feverally..., or together,) then is the remaining angle of the on- equall to ti~ e remaining angle of the other. In like manner , if two triangles have one angle...
Euclid's Elements of Geometry: From the Latin Translation of Commandine. To ...
Euclid, John Keill - Geometry - 1723 - 364 pages
...Triangle jhall be alfo equal to the remaining Sides of the other, each to his correfpondent Side, and the remaining Angle• of the one, equal to the remaining Angle of the other. LET there be two Triangles ABC, DEF, having two Angles ABC, BCA of the one, equal to two Angles DEF,...
Euclide's Elements: The Whole Fifteen Books Compendiously Demonstrated. With ...
Euclid, Isaac Barrow - Euclid's Elements - 1732 - 436 pages
...taken together From whence it follows, 2. That if in one triangle, two angles (taken (everally, o{- together) be equal to two angles of another triangle...angle of the one equal to the remaining angle of the othcr In like manner, if two triangles hate one angle of the one equal to one of the other, then is...
Euclid's Elements of Geometry,: From the Latin Translation of Commandine. To ...
Euclid, John Keill - Geometry - 1733 - 397 pages
...Triangle Jhall be alfa equal to the remaining Sides of the othery each to his correfpondent Side, and the remaining Angle of the one equal to the remaining Angle of the ither i which was to be demonftrated. PROPOSITION XXVII. THEOREM. If a Right Line, falling upon two...
Euclide's Elements: The Whole Fifteen Books Compendiously Demonstrated: with ...
Euclid - Euclid's Elements - 1751 - 384 pages
...triangle, two angles (taken f everally, or together) be equal to two angles of another triangle ftaken feverally, or together) then is the remaining angle of the one equal to the remaining angle of the.other. In like manner, if two triangles have one angle of the one equal to one of the other, then... | 597 | 2,381 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2022-33 | latest | en | 0.863421 |
https://www.flyingcoloursmaths.co.uk/constructing-the-square-root-of-6/ | 1,716,205,815,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058278.91/warc/CC-MAIN-20240520111411-20240520141411-00069.warc.gz | 705,483,461 | 3,806 | On Twitter, @RuedigerSimpson pointed me at an episode of My Favourite Theorem in which @FawnPNguyen mentioned a method for constructing $\sqrt{7}$:
• draw a circle of radius 4
• construct a perpendicular to the radius at a distance of 3 from the centre
• the distance between the base of the perpendicular and where it meets the circle is $\sqrt{7}$, because $4^2 - 3^2 = 7$.
Very nice! The square root of any odd number can be constructed a similar way - $\sqrt{2n+1} = \sqrt{(n+1)^2 - n^2}$, by the difference of two squares.
Similarly, multiples of 4 are easy pickings: $\sqrt{4n} = (n+1)^2 - (n-1)^2$
Which leaves only numbers of the form $4n+2$ - which can’t be expressed immediately as the difference of two squares. But can such numbers be constructed similarly?
## Bisection!
A simple way to construct, say $\sqrt{6}$ would be to construct $\sqrt{24}$ using the method above (a circle of radius 7 and a perpendicular 5 units away) and bisect the perpendicular. Since $\sqrt{6} = \frac{1}{2}\sqrt{24}$, we’ve got our distance!
But Simon didn’t like that. Can it be done without bisection?
## No bisection!
I’m quite pleased with my alternative method, which I’m told is based on the Spiral of Theodorus, although it’s rather less involved.
If you construct $\sqrt{5}$ using the method above, with a circle of 3 units and a perpendicular of 2, there’s a second right-angled triangle you can use: the one to the right of the perpendicular! The perpendicular has length $\sqrt{5}$, the base has length 1, so the hypotenuse has length $\sqrt{6}$!
I thought that was a neat bit of geometry. I don’t for a second imagine it’s new (it’s probably in Euclid if I could be bothered to look), but it was pretty enough a solution to make me grin.
Do you know of any other nice methods? | 480 | 1,792 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2024-22 | latest | en | 0.897285 |
https://mineracaodedados.wordpress.com/tag/overfitting/ | 1,628,217,281,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046152112.54/warc/CC-MAIN-20210806020121-20210806050121-00239.warc.gz | 392,638,730 | 27,246 | Via John D. Cook.
Elementary numerical integration algorithms, such as Gaussian quadrature, are based on polynomial approximations. The method aims to exactly integrate a polynomial that approximates the integrand. But likelihood functions are not approximately polynomial, and they become less like polynomials when they contain more data. They become more like a normal density, asymptotically flat in the tails, something no polynomial can do. With better integration techniques, the integration accuracy will improve with more data rather than degrade.
With more data, the posterior distribution becomes more concentrated. This means that a naive approach to integration might entirely miss the part of the integrand where nearly all the mass is concentrated. You need to make sure your integration method is putting its effort where the action is. Fortunately, it’s easy to estimate where the mode should be.
The third problem is that software calculating the likelihood function can underflow with even a moderate amount of data. The usual solution is to work with the logarithm of the likelihood function, but with numerical integration the solution isn’t quite that simple. You need to integrate the likelihood function itself, not its logarithm. I describe how to deal with this situation in Avoiding underflow in Bayesian computations.
Amostragem: Precisamos mesmo de 3 divisões amostrais (treinamento, validação e teste)?
Neste artigo do Dan Steinberg ele responde de forma clara:
“The short answer to this question is “no” we do not think that the 3-way partition is mandatory for SPM core models such as CART and TreeNet.”
Confesso que fiquei assustado com a resposta, mas abaixo no próprio artigo ele coloca a justificativa:
“The question we address here is whether this is really enough when the process of model development is lengthy and may involve the building of dozens, hundreds, or even thousands of models. If each model is evaluated on the second data partition one might argue that we are in fact learning from it. The learning may be limited and indirect, but the cumulative effect of intensive model development and evaluation blunts the independence of the second partition data.”
Uma das lutas mais ingratas de quem está trabalhando em um modelo de dados é o fantasma do Overfitting em qualquer modelo de descrição/predição.
O ponto principal do autor é que nem sempre em cenários nos quais os dados são escassos a utilização de amostras ‘holdout‘, isto é, fora do conjunto de dados (treinamento e teste) é necessária.
No artigo tem os resultados de um teste em que houve a construção de um modelo com holdout e sem holdout, e tirando-se as diferenças entre as curvas ROC dos modelos chegou-se ao ótimo resultado de apenas de – 0.00216 em relação ao modelo sem holdout.
Questionamentos metodológicos, e principalmente em relação à possível (ou a ausência de) variância do dados, o ponto é importante e vale a pena ser refletido.
Como é de conhecimento geral, nem sempre na construção de modelos preditivos/descritivos tem-se acesso completo aos dados, e em muitas vezes o volume de dados é extremamente pequeno.
Uma das alternativas para esse tipo de problema é a aplicação do método de Cross-Validation (k=10) que gera resultados satisfatórios.
No entanto, como o autor coloca em questão, um modelo deve ser sempre refinado de forma iterativa e incremental; o que significa que nenhum modelo ‘nasce’ sujeito a erros, mesmo se não houver representatividade das instâncias de ‘validação’.
Overfitting
É um assunto batido, mas sempre vale uma rápida revisão.
Um trecho: “[…]To generalize, a model that overfits its training set has low bias but high variance – it predicts the targets in the training set very accurately, but any slight changes to the predictors would result in vastly different predictions for the targets.
Overfitting differs from multicollinearity, which I will explain in later post. Overfitting has irrelevant predictors, whereas multicollinearity has redundant predictors.
[…]”
O Parque de Diversões
Essa semana foi lançado no Kaggle uma modalidade de competição denominada Playground, ou algo como parque de diversões. Esse tipo de competição ao invés de ter o foco em uma resolução específica, têm uma abordagem muito mais voltada à extração de informações previamente desconhecidas das bases de dados.
Geralmente em ambientes de análise de dados não há demandas para abordagens semelhantes, devido não somente pressões para resultados como também um determinado ‘engessamento’ dos setores estratégicos.
Ambientes de sucesso em mineração de dados não são aqueles que procuram uma agulha no palheiro (isto é, torturando os dados, overfitting, padrões espúrios) mas sim aqueles que ‘brincam’ no palheiro até sentirem uma ‘picada’ (isto é, analisando os padrões, tendências, e regras).
Para quem já teve oportunidade de trabalhar com modelagem preditiva ou classificatória o Overfitting é quase que uma regra em muitos papers picaretas que saem em algumas revistas (em especial papers que realizam análise preditiva sobre indices de bolsas de valores).
Tratando-se de aprendizado de máquina o Overfitting tem algumas características interessantes como:
1. Péssima amostragem;
2. Desconhecimento do Cross-Validation;
3. Holdout que não representa a variância natural dos dados; e
4. Analistas querendo fazer Data Snooping.
Simples assim.
Veja abaixo um parágrafo relativo o Overfitting:
My Idiomatic Definition of Overfitting to Help Remember the Concept
A model is built to represent training data, not to reproduce training data. Otherwise, a visitor from validation data will not feel at home. The visitor encounters an uncomfortable fit in the model because s/he probabilistically does not look like a typical data-point from the training data. The misfit visitor takes a poor prediction. The model is overfitted.
Truques Estúpidos em Mineração de Dados – Overfitting no índice S&P500
Neste artigo do David Leinweber (o qual já foi tema do site aqui e aqui) ele coloca algumas considerações a respeito do fato de que muito do que se fala sobre Mineração de Dados está fartamente relacionado a relações absurdas que podem acontecer pelo fato de “torturar os dados” como a predição do índice Standard & Poor’s 500 através de correlações (estúpidas) como o modelo de regressão no qual a produção de manteiga em Bangladesh (Coeficiente de Determinação R2 de 0.75); produção de manteiga em Bangladesh e produção de queijo nos EUA (R2 = 0.95) e a fantástica correlação entre a produção de manteiga em Bangladesh, a produção de queijo nos EUA e a população de ovelhas em Bangladesh que apresenta o coeficiente de determinação de incríveis 99%.
É claro que o artigo escorrega um pouco ao radicalizar a questão, no qual o autor confunde quase que de maneira primária os conceitos de correlação (relação conjunta de uma ou mais váriaveis dentro de um contexto de análise) e casualidade (fatos que acontecem de acordo com um dado grau de sincronissidade, enretanto isolados em contextos distintos) para dar substância ao que está sendo defendido em sua tese; mas isso de nenhuma forma invalida o estudo no qual deixa claro que a “técnica de torturar os dados até que eles falem” é uma péssima abordagem e que pode gerar aberrações em análise de dados iguais aos casos citados.
De maneira geral o autor apresenta uma boa prática na qual sempre que haja esse tipo de análise, deve-se realizar testes sobre dados fora da amostragem para que sejam produzidos resultados mais fidedignos.
Stupid Data Miner Tricks – Overfitting The S&P 500
Data analysis recipes: Fitting a model to data
Para quem deseja um overview sobre fitting de modelos e entender um pouco sobre questões como variância, esse artigo de David Hogg, Jo Bovy, Dustin Lang é uma leitura bem interessante.
Abstract
We go through the many considerations involved in fitting a model to data, using as an example the fit of a straight line to a set of points in a two-dimensional plane. Standard weighted least-squares fitting is only appropriate when there is a dimension along which the data points have negligible uncertainties, and another along which all the uncertainties can be described by Gaussians of known variance; these conditions are rarely met in practice. We consider cases of general, heterogeneous, and arbitrarily covariant two-dimensional uncertainties, and situations in which there are bad data (large outliers), unknown uncertainties, and unknown but expected intrinsic scatter in the linear relationship being fit.
Above all we emphasize the importance of having a “generative model” for the data, even an approximate one. Once there is a generative model, the subsequent fitting is non-arbitrary because the model permits direct computation of the likelihood of the parameters or the posterior probability distribution. Construction of a posterior probability distribution is indispensible if there are “nuisance parameters” to marginalize away. | 2,075 | 9,007 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2021-31 | latest | en | 0.911353 |
http://mathhelpforum.com/calculus/59252-another-complex-analysis-problem.html | 1,481,331,559,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542851.96/warc/CC-MAIN-20161202170902-00146-ip-10-31-129-80.ec2.internal.warc.gz | 178,779,085 | 10,701 | # Thread: Another complex analysis problem
1. ## Another complex analysis problem
a) Let $\widehat{f}(z)=\int_{-\infty}^{\infty} f(t) e^{-2 \pi i zt} dt$
Assume that $f$ has bounded support and smooth.
Show, by integration by parts that $\mid \widehat{f}(x+iy) \mid \leq \frac{A}{1+x^2}$ if $\mid y \mid \geq 0$
b) Write $P(z)=a_n(2\pi iz)^n + a_{n-1}(2\pi iz)^{n-1} +...+a_0$ where $a_i$ are complex constants.
Find a real number $c$ such that $P(z)$ does not vanish on the line $L= \{z:z=x+ic, x\in R \}$
For a) I let $u=f(t)$ and $dv=e^{-2 \pi i zt}dt$. Then integrate by parts. Is this the right approach?
2. Originally Posted by namelessguy
For a) I let $u=f(t)$ and $dv=e^{-2 \pi i zt}dt$. Then integrate by parts. Is this the right approach?
Yes. You'll need to integrate by parts twice, and then use the fact that f''(t) is bounded (because f is smooth).
3. Thanks for your help Opalg. Do you have any suggestion for part b) ?
4. Originally Posted by namelessguy
b) Write $P(z)=a_n(2\pi iz)^n + a_{n-1}(2\pi iz)^{n-1} +...+a_0$ where $a_i$ are complex constants.
Find a real number $c$ such that $P(z)$ does not vanish on the line $L= \{z:z=x+ic, x\in R \}$
P(z) is a polynomial of degree n, so it has n roots $z_k\ (1\leqslant k\leqslant n)$. You want c to satisfy $P(2\pi ix-2\pi c)\ne0$ for all real x. This is equivalent to $2\pi ix-2\pi c \ne z_k$, or $c\ne ix - (z_k/(2\pi))$, for all real x and for $1\leqslant k\leqslant n$. The condition for this is $\text{re}(c)\ne \text{re}(-z_k/(2\pi))\ (1\leqslant k\leqslant n)$. | 585 | 1,541 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 24, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2016-50 | longest | en | 0.743952 |
https://puzzling.stackexchange.com/questions/22306/find-the-6-digit-number-encoded-in-this-mathematical-diagram | 1,713,677,436,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817729.0/warc/CC-MAIN-20240421040323-20240421070323-00413.warc.gz | 435,517,610 | 41,465 | # Find the 6-digit number encoded in this "mathematical" diagram
In the diagram below you can see four independent patterns. Each pattern is a puzzle by itself, and by solving it you will be able to encounter one or more digits of the final, 6-digit numeric solution. Signs above tell how to get these digits once the mini-puzzle is solved.
Most puzzlers should already be familiarized with the challenge presented by these patterns. However, the solution is this case involves numbers as well as letters.
Backstory: "The 6-digit number you will find is a password that allows you to enter the fifth dimension and therefore become one with the universe. Also, it's the only way to save humanity."
Contrary to what my storytelling skills may indicate, no black sorcery is required in the solution to this puzzle.
Also, please notice that this post contains hidden tag(s), which would otherwise hurt the purpose of the puzzle.
Hint:
It's natural for you to be clueless during these puzzles. However, numbers may come in handy when words are lacking.
• The "sum pi sum pi" at the top is just making me hungry. Sep 17, 2015 at 23:14
• This seems rather broad. There are many ways to get numbers out of the patterns. Sep 17, 2015 at 23:32
• Just to be clear on the rules. You're saying that there are 1 or more digits associated with each diagram, and you either add or multiply those digits to get a number. You then concatenate the four numbers to get a six digit number. Sep 17, 2015 at 23:34
• Each subpuzzle is like a game. The solutions are converted into digits, and digits concatenated into the password Sep 17, 2015 at 23:46
• The symbols indicate how the digits are obtained from the solutions Sep 18, 2015 at 0:08
## 2 Answers
Each puzzle is a
crossword
to be solved by
filling in written-out English numbers of the correct length
Solving its gives
TEN TEN
I W I O O O
N ZERO N TEN NINE
E E E E E
We interpret each of these by computing the corresponding:
sum of 19, product of 0, sum of 11, product of 9
and concatenating gives the six-digit answer
190119
I'm going to give this a shot:
Each bend in the string of dots represents a new number to be summed ($\Sigma$) or product-ed ($\Pi$).
We go in order:
1. 3 + 4 = 7
2. 4 × 3 × 3 × 4 = 144
3. 3 + 3 = 6
4. 4 × 3 × 3 = 36
But 7144636 is seven digits, so this is obviously not the right answer. Am I on the right track, though?
• Another possible answer with a different algorithm that's actually 6 digits: 630516
– user88
Sep 18, 2015 at 1:53
• It should be more like a game, not a code. Each mini-puzzle should be in some way "solvable", like finding an answer that fits the constraints. Sep 18, 2015 at 1:54 | 735 | 2,733 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.78125 | 4 | CC-MAIN-2024-18 | latest | en | 0.921691 |
https://best-businessloans.com.au/viewtopic.php?page=b71c52-differential-equations-syllabus | 1,603,560,771,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107884322.44/warc/CC-MAIN-20201024164841-20201024194841-00669.warc.gz | 235,377,500 | 7,658 | A system described by a linear, constant coefficient, ordinary, first order differential equation has an exact solution given by y(t) for t > 0 , when the forcing function is x(t) and the initial condition is y(0). Textbook. MA 341: Differential Equations.
MATH 222: Differential Equations Fall 2019 Course Syllabus NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. The following pages contain a variant of the preceding syllabus including applications of Differential Equations to Mathematical Biology. U Substitution. Separable Equations. The method of Frobenius (Sections 8.6, 8.7). Practice Test 1. Integrating Factors, First Order Linear Equations. Test 2 Version 2 Solutions. Linear differential equations and linear systems of them.
This means your class notes will be very important to you in order to clarify which material is covered; the tests will be over the material covered in lecture. You may find it helpful to consult other texts or information on the internet for additional information.
MAP4305/5304 Intermediate Differential equations: Syllabus.
MATH 2420 Differential Equations Syllabus and First Day Handout Spring 2012 Course: Differential Equations Instructor: Michael J. McCarthy, Ph.D. Systems syllabus. Syllabus in Differential Equations, Dynamical systems, Math. Fundamentals of Differential Equations and Boundary Value Problems, 7th Edition, by R. K. Nagle, E. B. Saff, and A. D. Snider. Topic 1: Review.
Integration By Parts Review. 6: Nonhomogeneous Equations, Method of Undetermined Coefficients: 7. Prerequisites: C (2.0) or better in MAC 2313, or C (2.0) or better in MAC 2283. Hypergeometric, Bessel, and Legendre equations (Section 8.8) Topic 4: Sturm-Liouville boundary value
Review: Power series solutions to linear differential equations (Sections 8.3, 8.4). 4: Bernoulli Equation, Equations with linaer Coefficients, Riccati equation: 5.
Test 1 Review. The schedule outlined below allows time for three midterm exams plus a cumulative final exam, which are the norms … Text: Differential Equations: An Introduction to Modern Methods and … Differential Equations will cover a large portion of the textbook, but we will not cover everything in equal detail. MAP 2302 — Differential Equations — Syllabus.
Homogeneous Equations, Exact Equations: 3. Trig Basics. MATH 222: Differential Equations Summer 2020 Course Syllabus NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. (That option cannot be chosen should the student attempt also Dynamical Systems.) Complex numbers. Successful completion of the course merits 3 semester hours of credit.
Functions of complex variables. Second Order Homogeneous Equations, Reduction of Order. Course Description: The course meets for approximately 45 hours during a 15-week semester. This syllabus section provides the course description and information on meeting times, prerequisites, texts, format, recitations, tutoring, ten essential skills to be mastered, homework, exams, grading, and specially written Java applets, or Mathlets, used in lectures and problems sets.
MA 341-003 Syllabus Spring 2020 MA 341-603 Syllabus Spring 2020 Test 1 Resources: Test 1 Version 1 Solutions. The elimination method … Course Content . Differential operators. | 795 | 3,535 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2020-45 | latest | en | 0.847987 |
http://www.faqs.org/docs/Newtonian/Newtonian_126.htm | 1,501,102,559,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549426629.63/warc/CC-MAIN-20170726202050-20170726222050-00256.warc.gz | 418,612,406 | 2,901 | 126
The figures show two somewhat more practical laboratory experiments
for investigating this issue accurately and without too much interference
from extraneous forces.
In the first experiment, a large magnet and a small magnet are weighed
separately, and then one magnet is hung from the pan of the top balance so
that it is directly above the other magnet. There is an attraction between the
two magnets, causing the reading on the top scale to increase and the
reading on the bottom scale to decrease. The large magnet is more “power-
ful” in the sense that it can pick up a heavier paperclip from the same
distance, so many people have a strong expectation that one scale’s reading
will change by a far different amount than the other. Instead, we find that
the two changes are equal in magnitude but opposite in direction, so the
upward force of the top magnet on the bottom magnet is of the same
magnitude as the downward force of the bottom magnet on the top mag-
net.
In the second experiment, two people pull on two spring scales. Regard-
less of who tries to pull harder, the two forces as measured on the spring
scales are equal. Interposing the two spring scales is necessary in order to
measure the forces, but the outcome is not some artificial result of the
scales’ interactions with each other. If one person slaps another hard on the
hand, the slapper’s hand hurts just as much as the slappee’s, and it doesn’t
matter if the recipient of the slap tries to be inactive. (Punching someone in
the mouth causes just as much force on the fist as on the lips. It’s just that
the lips are more delicate. The forces are equal, but not the levels of pain
and injury.)
Newton, after observing a series of results such as these, decided that
there must be a fundamental law of nature at work:
Newton's Third Law
Forces occur in equal and opposite pairs: whenever
object A exerts a force on object B, object B must also
be exerting a force on object A. The two forces are
equal in magnitude and opposite in direction.
In one-dimensional situations, we can use plus and minus signs to indicate
the directions of forces, and Newton’s third law can be written succinctly as
F
A on B
= –F
B on A
.
There is no cause and effect relationship between the two forces. There
is no “original” force, and neither one is a response to the other. The pair of
forces is a relationship, like marriage, not a back-and-forth process like a
tennis match. Newton came up with the third law as a generalization about
all the types of forces with which he was familiar, such as frictional and
gravitational forces. When later physicists discovered a new type force, such
as the force that holds atomic nuclei together, they had to check whether it
obeyed Newton’s third law. So far, no violation of the third law has ever
been discovered, whereas the first and second laws were shown to have
limitations by Einstein and the pioneers of atomic physics.
magnet
magnet
scale
scale
(a) Two magnets exert forces on each
other.
(b) Two people’s hands exert forces
on each other.
Chapter 5Analysis of Forces
Next Page >>
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http://www.docstoc.com/docs/125921412/Crossflow-Turbine-Abstracts | 1,369,284,535,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368702810651/warc/CC-MAIN-20130516111330-00085-ip-10-60-113-184.ec2.internal.warc.gz | 421,231,967 | 28,806 | Crossflow Turbine Abstracts
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Document Sample
``` Crossflow Turbine
Abstracts
by Joe Cole
The Crossflow Turbine
Unfortunately bulletin #25 is not a "step-by-step" manual, much to the
disappointment of many. When I first saw in in 1978, I found it fragmented, elusive,
overly technical missing a few formulas. It' more like the technical ramblings of
someone explaining the concept of time and the theory of how a clock works when all
you wanted was to know the time. However with some diligent "head scratching"
over a suitable period of time you will eventually sort out the relevant pieces of
information contained in the bulletin. There are a few point to keep in mind when
reading and working on some of the calculations in the bulletin. Keep in mind that it
was originally a German document written in the early 1930s. It is very likely that
some meaning was lost in OSC translation of the original document. Also bear in
mind that some of the formulas in the document do math conventions that we use
today ie. addition, subtraction, multiplication then division. You'll have to "play
with" the brackets. However taking the bulletin as a whole it follows the same
mechanical & hydraulic principles used today in turbines & pumps. Those are a
curved blade's reaction to a jet of water (in turbines) and water's reaction to a moving
About half of the math in the bulletin deals with the physical relationships between
the mechanical turbine elements (blade geometry and runner inside & outside
dimensions) and the other half deals with the forces produced on the blades. The
forces are represented as "vector diagrams" Vectors diagrams help one to visualize
what is going on inside the runner. If you understand them you can analyze changes
Next to the calculus used in the bulletin, the vectors are probably the most baffling
things in the bulletin. The vectors will be explained shortly however, as for the
calculus, "I ain't goin' there."
Don't expect to "re-engineer" the crossflow runner. Banki & Mitchell "did their
homework" on it. The proportions of the blades to the runner diameters and angles
involved are fairly "fixed" and cannot be arbitrarily changes without adversely
affecting the power & speed of the runner because these changes affect the efficiency
of the turbine. As alluded to in the paragraph above, the runner diameters and blade
dimensions are a compromise of mechanical dimensions & mechanical efficiency.
Taking all the above into consideration, this article will not be a step-by-step
interpretion of bulletin #24 but will be my personal "practical inturpation &
explanation" of it. Before getting started I want to clarify something. Bulletin #25
title "The Banki Crossflow Turbine" is somewhat misleading as it only deals with the
"runner" aspect of the turbine. After discussion of the "runner" portion of this article I
will expound on the turbine design as a whole as well as some abstract theory about
the crossflow turbine.
Before calculating much of anything else we need a little understanding of vector
diagrams. It will take several stages to illustrate this
in it's entirety go get a cup of coffee and come
back. In understanding how a jet of water acts on a
surface we first use the "Flat Plate Normal to Jet"
illustration. In this a flat plate is held at a 90° angle
to the plate. In engineering terminology this is
called the flat surface held normal to the jet. Go
figure, I don't know how or why they come up with
this stuff! Anyway, the jet will be forced to make a
90°turn, thusly spreading out over the plane of the
plate. In this case the force ( F ) will have no
component in the plane of the plate. In other words
there will be no forces trying to move the plate
"sideways" to the jet. The force ( F ) is computer as
F = M * v. Again in engineering terminology it
said that "the jet's momentum in it's initial direction is wholly destroyed. This just
means as the waters energy was dissipated out radialy 90°into never-never land and
that no "work" was done. Remember in high school physics you were taught that for
"work" to be done there motion has to be imparted. In our case here we needed the
force ( F ) to cause the plate to
move for there to have been
"work" done.
On the other hand now if we
force the water through a
smooth 180°turn the force ( F ) is doubled. The force ids doubled because the
equation F =( M * v ) + ( -M * -v ). That is F = (Mass * velocity in the initial
direction + (Mass * velocity in the opposite directing. Reducing this equation gives
us simply F = 2 * M * v. One thing that might be a bit confusing in these two
illustrations is the arrows indication the direction of ( F ). It might appear that V is
moving in one direction (and it is) and that ( F ) is moving in the opposite direction
(which it is not). What the ( F ) directional arrow means is a "resistance" opposite to
the direction of ( V ). Again, it's an engineering thing.
Now that we are up to speed on velocity, mass & force, lest look at some vector
diagrams along with the blade shapes that produced them. In this diagram the jet is
being deflected by 70°or so. In applying these momentum theorems or laws as they
should be call to turbines is as follows. If a jet of water strikes a curved blade the
water is deflected by the angle ?.A force (F) is imparted to the blade in two
directions, x & y. These forces are calculated thusly
Fx = M * (v – v cos ° α) or Fx = M * (v – v cos ° α)
&
Fy = M*v * sin α
In this diagram the two velocities are the same but separated by angle α and the
triangle is closed by closed by the line ∆ v as dictated by the laws of cosines
∆v = the square root of v12 + v22 – 2 * v1 * v2 * cos α
∆v = the square root of 2 * v2 – 2 * v2 * cos α
These two forces combined is equal to:
F=M * v * the square root 2 * (1 – cos α)
Here is the general text book vector of a Pelton wheel in motion.
This is the path & vector diagram of a Pelton wheel. It is showing 2 buckets. The
bucket on the left is showing the absolute path of the water jet while the right side is
showing the relative path if the jet. If the wheel were "locked down", the water path
would indeed follow the path as indicated on the left. However in a running machine
1
the wheel is moving in the same direction as V . The water jet is trying to follow the
inside curvature of the bucket but because the bucket is trying to move away from the
jet the path is straighter as indicated in blue then had it mage the near U-turn of the
absolute path. The vector is compound, in other words its is showing more then just
one part of the blade. The left triangle is the vector for the entry of the jet to the
bucket & the right side is of course showing the water exit from the bucket. The inset
illustration shows what happens when the wheel (or runner) is in over speed. The
1
path flattens out more and you can see in the inset vector that μ is approaching the
1
same length as V Very little power is now being produced and aa a matter of fact the
power that is being produces is rather in driving the intended load, is being spent on
maintaining a high wheel speed and overcoming windage & friction. Notice that
2, 2
three additional notations are included. V v * β. In hydraulics the following
notation conventions are used.
Getting a little more
complicated visually but
still the type vector. Here
we have the Francis
runner. The actual water
path is shown in red.
Again the right side of the
double vector in showing
the entry of water and the
exit is shown on the left.
Both are shown here on
orange. The Green vector
is actually the right side
entry but for clarity is is
duplicated somewhat larger to show the geometry of the lead edger of the blade to the
outer periphery of the runner. I think before going on to the crossflow & should as
Rickey would say to Lucy, "Le me splain something to you." In hydraulics as in any
other engineering field they has it's own set of mathematical notations and also a
hierarchy or naming conventions . Most of the confusion in vectors comes from
water velocities. If the notation is a big V, then that velocity is an absolute one. If it's
a little v, then its a relative velocities. Most turbo machinery only has one in and one
out. No so with the crossflow. It's twins! It's got 2 of everything. To keep all the
symbols straight I'm not going to "splain" it, but rather illustrate it with a chart. I
think the chart and the crossflow illustration can explain this much better then I can.
Just so we are clear, the term "quadrants" is mine. The 1st in the entry of water to the
at point "A". The 2nd is the exit of water from point "B". The water then crosses the
interior of the runner and then re-enters the runner at point "C" in the "minus
direction" (remember our discussion above F =( M * v ) + ( -M * -v ). Water then
exits the runner in the minus direction at point "D"
1 st 2nd 3ed 4th
Absolute
Velocity
V1 V21 V11 V1
Relative
Velocity
v1 v21 v11 v1
Angle
β1 β21 β11 β1
Attack
Angle
α1 α21 α11 α1
Runner
Velocity
μ1 μ 21 μ 11 μ1
The
problem
with the
crossflow
is just that
“crossflow”
. Only
of the
waters
energy is
extracted in
the top of
the runner
leaving
only 28%
to be
extracted
from the
lower
section.
The exact
ratio is
dependant
on the
actual
diameter of
the runner,
how many
being, the
length to diameter ratio, the head, bla, bla,bla. Under "ideal circumstances, 50% of
the power would be produced in the first section and 50 % from the last section. This
will not happen because of the internal crossing of the water in the runner center
section. Ideally we should have "laminar flow" all the way through the runner.
Laminar flow means that that all the water particles in a given area is flowing parallel
to each other and are at the same speed. Think of this like the telescoping antenna on
a car where the innermost core has the fastest flow. You will never get true laminar
flow due to friction of the surfaces involved. Laminar flow is destroyed by excessive
restrictions and abrupt changes in flow area or directions. When it is destroyed
friction is the result. When the individual jet filaments cross and interfere with each
other that too pretty much "kills ' hell" out of laminar flow. We have a tremendous
disruption in flow now plus of air is now being introduced into the water path. By the
time the water gets to point "C" it's pretty well diffused to a wide pattern This causes
1
the flow V1 to enter the blade at an attack angle varying widely from the the 16° it
should be. That why the 28% of the available power is extracted there. This is
illustrated in fig 3 of the bulletin. However there nothing you cab do about it.....or is
there? Read on Grasshopper. We'll I think we'll all had enough turbine dynamics for
this week so lets move on not to some actual calculations.
Before any "design work" is to be done their are a few things that must be known.
First you must have a reasonable expectation of the amount of power that might be
produced from a given site. For instance, don't expect to supply a full household with
electricity produced from a scenic babbling brook running across your back yard. It
takes a lot of water to produce electricity. The "head" and "Q" must first be
determined. The "head" at least in the US is measured in feet. "Q" is the quantity of
water and in "micro-hydro" work it's usually given in CFM (cubic feet per minute)
and in larger turbines is given in CFS (cubic feet per second). Be sure when
calculation from formulas in other documents you pay attention to & convert units as
To begin we first determine the power potential of our site. For convenience, (mine)
throughout this article I'll be using my own site for the design & evaluation. That is
the head ( H ) = 26 feet and the flow ( Q ) = 8 CFS. The formula for determining the
potential hydraulic horse power is ( H * Q * 60 ) / 660. This is the raw horse power
potential and does not reflect any efficiency or loss's. According to the formula my
potential horse power output would be ( 26 * 8 * 60 )/660 or 18.9 HP. Assigning a
efficiency figure is difficult. I want to be conservative in this figure so let's use an
overall plant efficiency of 75%. Therefore 18.9 * .75 = 14.18 HP. One HP is
equivalent to 745.7 watts so 14.18 * 745.7 = 10574 watts or 10.57kW of electrical
power. This is enough to "do a house."
Now having that out of the way we can start to design a runner that will
accommodate the site. In the bulletin pages 10 through 15 deal with the construction
proportions of the runner. The information we need from these pages are: Formula
#35 Q=volume of water, Formula #36 L= blade length, Formula #37 ρ=blade
radius & Item (E) Central angle on page 15. Once these values ate known you take
these figures to a machine shop and have them form the blades from flat stock to
conform to “ρ”, machine the blade sections to form the 73.46° arc in item 3, & finely
cut the blade to their final length (L). You then pay the man a huge sum of money &
prey your calculations were correct. Definitely NOT the way to go. There is a much
simpler & cheaper way to arrive at near the same result but first a quick discussion on
one aspect of Banki's design. The dimensions and angles in the bulletin represent the
"near" optimum dimensions & angles to satisfy mechanical advantage & un-restricted
passage of the
water. These
dimensions are
fairly "fixed" and
therefore cannot
be arbitrary
changed without
some decrease in
efficiency ie.
power & speed
output. As a result of this there are definite dimensional relation ships between the
various components of the runner. I wrote the following formula to determine the
runner diameter from the blade radius. D1=2 * ρ /.326.We can use this to great
the form of steel pipe. If you've ever seen some crossflow runners up close before,
they mostly look the same, say 12-16 inched in diameter and have an aspect ratio of
1:2, that is 23 -32 inches in length. They also look like the blades were fabricated
from 4 inch steel pipe. You're right. But remember what your Momma told you when
you were 8. "Just because everybody else is doing it doesn't mean you have to." My
point is al lot of these turbine were fabricated from readily available materials and
hey, there's nothing wrong with that. However, waiting and searching for that
optimal "readily available materials" will save you some money and very likely gain
you some efficiency. I'll go through what happens when you design a runner without
regard to the project as a whole. Most commonly available is schedule 40 pipe and
below are some of it’s specs. Of course there is an almost infinite range pipe sizes in
industrial & construction grade so finding a size that will meet your needs should not
be a problem. Using this method we supply ρ and let industry supply us with tailor
There seems to be a lot of 4
inch steel pipe around. Let's
see if we can design a runner
around that size. We start by
finding the jet thickness
which started at item 4 on
page 17 of the bulletin. Area
of Jet = Q/V = 8 / 40= .2 ft.
(28.8 in^2). The value of So
according to the bulletin is So
= Jet Area / Length.
According to the formula at
reference 34 in the bulletin:
L = 210.6 * Q / D1 * H^.5
L = 210.6 * 8 / 12.27 * 5.099 = 25.5 inches.
Oh by the way, H^.5 is the same thing as "the square root of "H". It took me a while
to figure out that one. Anyway with the initial blade length calculated as 25.5 inches,
divide the "Jet area" of 28.8 square inches by 25.5 to get the So which in this case =
1.13 inches.
The only thing we need to factor in now is the nozzle efficiency and adjust the length
accordingly. If you follow good hydraulic principle and design a good gate/nozzle
you should be able to achieve an nozzle coefficient of .98.That’s only a 2% off peak
which would be 1.356 which translate to a .223 increase in runner width to
compensate. This would bring the runner length to 25.72 inches. If it were me I’d
bring the runner on out to 26 to28 inches just for grins and a little more error margin.
Now we’re looking at a D:L ratio of 1:2.08 Not terribly bad but! A 28 inch wide
small diameter turbine is going to be a machining & welding nightmare. Building the
runner itself it not too bad but I would add in 2 extra center support disk for rigidity.
Of course you’ll want to extend the runner length again to compensate for the width
of the extra center supports. We’ll we’re now out to 28.5 inches. Do I here 30?
My Daddy used to tell me, “You’ve got to use some horse sense”. Although I was
never very good with math, I do have to ability to “visualize” how things function &
anticipate problem areas. Having some “horse since” also helps. Here comes the first
major problem in designing a turbine. Let’s tentatively select 4-inch pipe to make out
blade sections from. We might select it because it look good & stout and because it is
relatively easy to come by. That will make us a runner 12.27 inches in diameter. At
this point my horse is telling me “there aint no way”, you’re getting 8 cubic feet of
water a second through a 12 inch runner without major difficulty. It’s not impossible
just not practical and here’s why. Anyway fabricating all the flat stock for the gate &
nozzle assembly will really be the difficult part. That’s an awfully wide gate
assembly. At a 26 foot head you only have a shade over 11 PSI at the lower end of the
system but think about it. You have a 28.5 inch wide gate perhaps transitioning back
several feet to a round penstock. That might present top panel behind the gate valve
of 28.5 x 36 inches. Multiple that times 11.25 PSI and you’ll have in the
neighborhood of 11,550 pounds of force acting just on the top panel of the gate. Even
if your welds held, the things going to bulge out & distort like a balloon. Personally
I’d give it around a 100% failure rate within 10 minutes.
What's a fellow do do? We'll
before I through the preverbal
monkey wrench into the mix, lets
fix this problem first. To get a
narrower runner we need to make
it'd diameter larger. This is done
make it larger by choosing a blade
with a larger radius. This time
6 inch pipe. Building a 18 gate &
nozzle would be child’s play
compared to a 28 or 30 inch gate.
The mechanical stresses by water
pressure would be reduced almost
70 %. The machine will cost
more to build mainly due to the
heaver drive components required
because of the slower speed &
greater torque when compared to
the smaller machines of equal
horsepower. But here you’re
getting into a serous machine of
much higher durability and a
much greater potential for
increasing the efficiency beyond
the apparent fixed limit or
87%.I’ll comment on this a little later. The runner built from 8 inch pipe is even better
with some qualifications. Again, the cost will be higher because of even larger
bearings and shafting required. However building a nozzle, gate & transition 13
inches would really be a piece of cake. The biggest concern with a runner this large in
diameter is the the loss in head due to the higher inlet. It's only a couple of feet in this
example but may be a consideration.
Alright, as promised, I'm throwing a monkey wrench into the works. The problem
comes when calculating how long the runner needs to be. Notice in the calculations
& illustrations in the bulletin all the math used an infinitely thin blade. If this is not
realized it will cause you to calculate the runner too short. You might not notice this
until you go to full load and “it just aint makin’ it”. Refer to page 9 of the bulletin to
figure 5 for the spacing used. Using our 12.27 turbine as an example, if we multiply
it’s diameter by pi we have a circumference of 38.52 inches. This gives a blade
spacing( t )in the outer periphery of the runner of 2.14 inches. The illustration to the
right shows the problem
very well using a
thickness in blue. The
original S1 value "A" that
should be is 1.25 inches
1.04 inches. Our jet
thickness So "C" has
dropped from .85 inched
to "D" .64 inches. That's
a 21% decrease in jet area from the original calculated value. This means to keep the
efficiency as high as possible the runner length will have to be increased 21%. You
don't need any fancy math or trig. to figure out just subtract the blade thickness from
the calculated value of So. Calculate the percent difference & multiply you original
blade length by that percentage as we've done above.
Specific Speed
Specific speeds is a dimensionless number. In broadcast engineering they call this
term "normalizing", if any of you are familiar with Smith Charts. The term is used to
“level the playing field” if you will, so that all types of runners can be evaluated under
the same conditions .As a result the term via it’s number define the shape of the
runner. I remember from a long time ago one hydraulic document described it this
way. If a model of any given turbine were build with a 1 foot diameter and operated
with a 1 foot head, then the specific speed is the speed that the runner would turn to
produce 1 horsepower. I guess that about sums it up for a level playing field.
What it all means is that a turbine with a high specific speed will while running a
full load, be turning faster then one with a low specific speed. An extreme example
are the Pelton wheel which has a very low specific speed. It is usually thought of as a
high-head machine. However it can be very efficient a low heads. It’s just that it
turns so slow at low heads that the cost the equipment needed to increase the shaft to
something usable by a alternator may cause the whole project scrapped or re-dome
with a turbine of a higher specific speed. Also a low specific speed is also thought of
as a low volume unit. This really makes it an ideal selection for mountainous terrains
where large quantities may not be available.
On the
other hand
you have the
Kaplan
Turbine
which is an
axial flow
(propeller)
turbine. It has
a large
specific speed
and is used
mainly on
large dammed hydro sites where then the is somewhat low but the quantities of water
available are staggering. These turn relatively fast rate when compared to the
crossflow, Francis & Pelton. They would not be suitable for medium to hi-head as
they would turn much to fast to be practical. When every thing above is considered
the crossflow would be an excellent choose for low to medium head operations.
However it’s not a weekend project and must be engineered properly if it’s going to
be efficient and last. If I were King of the world I’d make all crossflow builder
applicants take the following test. Can you draft? Can you weld? Can you run a
milling machine and a lath? What is the square root of 2? Convert 1 PSIG to Head.
If you’re carrying all the feathers you can carry, can you carry one more? You had
better be able to rattle off without blinking, “yes, yes, yes, 1.414, 2.31 and no. ”The
point is this is a real engineering project and is not for the typical do-it-yourselfers.
Nozzles
I believe that bigger is better up to a point. In the case of selecting a runner
diameter, using a larger & therefore narrower runner not only saves money and add
durability but does offer up a few extra chances at increasing the efficiency of the
crossflow. Take a second look at the Horse power formulas #2 & #6 on page 7 of the
1
bulletin. Remember the lows of cosines? The 16 ° α is usually chosen as a
compromise between hydraulic efficiency and mechanical clearances in adjusting So.
Therefore if a1 is reduced the efficiency & power output will increase. With a large
diameter runner this is much easier then in a small turbine. You could lay that angle
down to maybe as low as 8 °.Of course you would want to lengthen the runner to
compensate for the smaller So.
General Layout of Flow In Nozzle
The nozzle diagram above is meant to show some general proportions. For
maximum efficiency the runner should be designed for single blade operation.
However in the interest of construction difficulties in building a wider runner, a
double nozzle - blade arrangement may be used at some loss in efficiency. The
proportions are general. For instance I chose the radius of the nozzle curvature
arbitrarily at 2 times the runner diameter. The exact radius in not important so long at
it gives a nice long gentle sweep into the blades. The arc of the nozzle is also an
arbitrary figure. I placed this one at 73 degrees “just because”. That long sweep &
mechanical clearance is all that matters. You could go “straight in” as the folks a OSC
did when they built their turbine using the freshly translated document from Banki’s
original papers. By the way, does anyone know how to get or has a copy of the
“original Banki papers? However they had some pretty horrible efficiency numbers
with their turbine. They may have been “Jim Dandy” mathematicians but would have
made a few changes on their nozzle design & transition assembly. Probably the thing
that hurt them the most was the nozzle. It was a sliding gate that opens & closed
“laterally”, that is across the runner face. What happened when you pot your thumb
over the end of a garden hose? Using their arrangement that’s exactly what happened
in their turbine. My guess is that at small gate openings the water might have been
disbursed by 10-20 degrees. Another thing that hurt a little was not having a smooth
transition between their supply pipe and the nozzle. It was a blunt sharp edge
transition. That hurt them more at full power then anything else. I’m not trying to
belittle any of the people involved I’m just trying to make you aware of “design flaws
in engineering.” Left click on the illustration above to save it to your computer. It's
actual size is about twice what you are seeing here.
What is important is the angle the water hits the blade at. This is generally taken at
16 deg. However, that is "relative: to the blade angle B, which itself is relative to the
periphery of the runner. Through out the bulletin you see constant reference to a1.
This is an extremely important angle, for it more then any other factor, determines the
power output of the runner. However I’d have to say that 16 degrees is the maximum
angle that one should use. Us it as a design figure then see if you can go smaller.
Getting small requires “laying the nozzle down” closes to the runner. If you use a
large enough radius and a long enough arc for the nozzle, you could get a1 on down to
the 8 to 12 degree range. Any smaller though, and you’ll have to start thing about
lengthen the runner. Going to excess on this could get you a nozzle with a low
coefficient because of excess friction because of excess length.
There are several
nozzle arrangements that may be used. Most of the
commercial crossflow turbines built in Europe use mutable
blade inlets. In all of these the nozzle in intrigal with the
turbine housing. If you are making commercial turbines
that's a very good idea because it saves manufacturing cost and makes an extremely
ridged turbine assembly. This method is a little impractical for us little guys because
not too many of us have casting facilities in our back yards or want to shell out major
bucks for some "real machine shop work" Besides if the nozzle is cast in with the
housing we can't adjusts the attack angle now can we? I took my design from one of
the old Ossburger design. In stead of the water following the runner housing after it
leaves the gate it follows a curved guide which is the top of the nozzle assembly. In
mine I'll use a nylon or duron spring loaded back seal on the gate shaft. The side seals
and shaft bearing are not show in my illustration but they are mounted externally on
the nozzle housing. The sealing method on edges of the gate plate are not shown but
they are also nylon A
deviation of the
"sharp-edge orifice" is
used to help eliminate
as it leaves the nozzle
The actual length of
the nozzle is a bit
longer then shown and
the nozzle will have an
at the end of the
assembly where it
meets the runner
housing. The other
end its attached to the
transition/diffuser assembly witch is mounted at the other end of the turbine sub frame
assembly. There are two critical considerations when mounting a nozzle like this.
Because the runner, nozzle and transition/diffuser are mounted together on the same
frame, the alignment to the penstock is critical to that undue stress is not imposed and
ether assembly. Ideally a flexible coupling would be the ticket but a commercial
coupling would be rather expensive for a 12 inch penstock. Later I'll be adding to this
section about flexible coupling and alternatives. A flexible coupling does three
things. It does allow for some mis-alignment, It isolated the penstock from the
turbine from mechanical vibration, it allows for expansion & contraction of the
penstock due to temperature and it will allow for some movement should the penstock
try to settle of shift due to waterhammer.
A large
diameter
narrow
turbines
lends itself
well to a
departure
from
standard
design.
Since all the
fixed and
have a fixed
relationships
no part of
can be
moved by
itself. In
other words
1 2
likeβ & β
1
and β 1 &
β21.
However it
2 2
is possible to in effect change β 1. While β 1 itself does not actually change, you can
2
change the angle at which the water enters quadrant 3 an angle β 1 by using an inside
guide within the runner. This would necessitate having an “open faced” on one side.
I can already here some folks now in that condescending nasally voices. "Well if you
do that then all the water will run out! Then what are you going to do?" Not really.
Anyway what do I care what they think . Besides, these are the people who failed my
test miserably!
Using just one blade set water leaves the blade between A & B. It re-enters the
runner for it's 3ed quadrant of operation at D through E. At full gate operation using 2
blade sets, water leaves quadrant 3 from A to C & then re-inters at D through F. In
the illustration shown here the water path for single blade operation is the blue lines.
water upon entering the 3ed quadrant of operation. I plotted the water path graphically
and came up with the curve necessary for an internal guide shown in red. The pink
area shows the water path for single blade operation. The light blue shows the path
should 2 blades be used. In single blade operation point E is about as far back to your
right as the jet will reach at normal speeds.. What I've attempted to do here is re-
direct the water back open more blade set and inter at 16 degrees at that point. A
similar situation occurs when the 2nd blade section is added. The water in confined to
a course that does not vary with speed and it is always forces to enter quadrant 3 at 16
deg.
The bottom surface of the
guide would be either the
center red portion above or the
left red portion depending on
weather you were using one or
right red outline shows the top
edge of the side walls of the
guide. The illustration here is
the concept of what the guide
might look like. It's only a
Photoshop assisted freehand
drawing but I think you can see
what I'm getting at here. A
guide like this presents me with
some interesting possibilities.
However it is mounted, the
mounting mechanism should be extremely ridged. There would of course be a
standard fixed mount that would during installation but what about an "on the fly
adjustment?" One way of doing this would have the lower edge be the pivot point so
the quadrant 3 entry angle could be varied to suit flow & load conditions. Another
possible mount scenario would be to "hang" the guide from the runner shaft using
pillow block bearings. These bearing would of course be of the "double sealed" type
suitable for such as wet environment. The guide would have a lever attached between
it and the activation mechanism
As mentioned above
in order to use this type
of guide the runner
would have to be open
faced. This does
present a small
engineering problem.
With this runner it is
not necessary that the
shaft extend thru the
runner. It only need to
can if the guide were
under hung from it
using pillow block
bearings. However a
more likely mounting configuration is what's called " an overhung load. An overhung
chain sprocket, gear, crank arm, cam or other similar device. It necessitates using a
larger diameter shaft & beadings. In addition it requires a larger mounting boss to the
APPENDIX I
Notes on the "Efficiency Formula" in Bulletin #25
The efficient formula is every bit a complicated as it looks. I really have very little
though on how it works. Due to my lack of understanding or motion mechanics I’m
forced to take Banki’s word on this one. However I can tell you what is going on in
the formula. In the formula “C” is the nozzle coefficient. He’s accounting for that in
1 1
the first part of the formula in dealing with ? & V .In the middle of the formula the
term y is the factor describing the loss of energy caused by the separate jets crossing
each other between the 3ed & 4th quadrants. I believe that the loss of power is also
2
represented in ? due to “shock” loss. Shock loss is when the relative velocities v 1 &
v11are not parallel (in the vector diagram). This can be seen on page 11 of the
2
bulletin on the left side of fig 7.The relative velocity of v 1 suggest that the inside
diameter of the runner at point “B” in the drawing above is turning at a different
speed to the corresponding point “C”These actually turning at different speeds id
clearly impossible since they the same physical surface. The last part of the formula
1 1
is the velocity difference between V & ? to extract power.
APPENDIX II
Notes on the "Horse Power Formula" in Bulletin #25
The Horse power formula is not as complicated as it might seem. The formula can be
divided into parts.
(W * Q * ? 1 / g)the momentum part.g=gravity constant at 32.3 In checking
my Excel and Q-Basic programs that calculate this I found a discrepancy withW.As
stated earlier W the weight of water per cubic foot of at sea level is 62.2 ft. However
to make the formula produce the correct answer in Excel a value of .13 has to be
used. At the moment I don’t have time to fix it so I use the correction multiplier of
.00291 in the horse power equations. In Q-Basic the value of 62.6 works just fine. I
think they mean "mass" rather then weight.
((V1 * cos(a1) – u1) takes account of the laws of cosines & subtracts the
wheel peripheral speed from V1 to that power is produced.
(1+y) * (cos b1 / cos b2) the lump sum factor or runner coefficients taking
into account the cosine blade angles of the 1st quadrant and the 4th
quadrant. Their ratio would always be 1:1 because they are physically the
same piece of steel.
This page will be an ongoing document. It will be up-dates and expanded as I have
time. Eventually I hope to cover every aspect of building a crossflow turbine. I
email me and let talk about it.
Email
More to come........This upload March 9, 2004 1:45 PM
```
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## Counting 360 days from now
360 days ago from today was Monday February 24, 2025, a Monday. This calculator finds what date it will be 360 days in the future. In this case, Monday February 24, 2025. Doing this type of calculation by hand could be difficult as 360 days to include work days, shortened months, or leap years. Our days calculator handles this for you. We also have a time ago calculator. It may be useful for other, similar problems to calculate time in the past!
To edit the days on this page, you can either change the 360 in the URL in your address bar or see our time from calculator.
## Counting to Monday February 24, 2025
• Day of the week: Monday
• Day of the year: 055
## How long is 360 days?
Counting forward from today, Monday February 24, 2025 is 360 days from now using our current calendar. 360 days is equivalent to:
• 51.429 weeks
• 0.986 years
• 11.613 months
• 360 days
360 days ago before today is also 8640 hours ago. Monday February 24, 2025 is 15.07% of the year completed.
## What does 360 days calendar days mean?
Within 360 days there are 8640 hours, 518400 minutes, or 31104000 seconds. Calendar days are used to count how long the days is vs the current year. Monday February 24, 2025 is the 055 day of the year. In calendar days, it's 15.07% through 2025.This is different from business days or weekdays, which are used to count how many days there are between two dates. So 360 days calendar days would be Monday February 24, 2025, but would be a different calculation for 360 days business days, usually expressed days business days.
## In 360 days, the average person spent...
• 77328.0 hours Sleeping
• 10281.6 hours Eating and drinking
• 16848.0 hours Household activities
• 5011.2 hours Housework
• 5529.6 hours Food preparation and cleanup
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## Famous Sporting and Music Events on February 24
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https://support.nag.com/numeric/nl/nagdoc_latest/clhtml/f08/f08xbc.html | 1,721,379,737,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514900.59/warc/CC-MAIN-20240719074314-20240719104314-00347.warc.gz | 466,058,306 | 10,260 | # NAG CL Interfacef08xbc (dggesx)
Settings help
CL Name Style:
## 1Purpose
f08xbc computes the generalized eigenvalues, the generalized real Schur form $\left(S,T\right)$ and, optionally, the left and/or right generalized Schur vectors for a pair of $n×n$ real nonsymmetric matrices $\left(A,B\right)$.
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.
## 2Specification
#include
void f08xbc (Nag_OrderType order, Nag_LeftVecsType jobvsl, Nag_RightVecsType jobvsr, Nag_SortEigValsType sort,
Nag_Boolean (*selctg)(double ar, double ai, double b),
Nag_RCondType sense, Integer n, double a[], Integer pda, double b[], Integer pdb, Integer *sdim, double alphar[], double alphai[], double beta[], double vsl[], Integer pdvsl, double vsr[], Integer pdvsr, double rconde[], double rcondv[], NagError *fail)
The function may be called by the names: f08xbc, nag_lapackeig_dggesx or nag_dggesx.
## 3Description
The generalized real Schur factorization of $\left(A,B\right)$ is given by
$A = QSZT , B = QTZT ,$
where $Q$ and $Z$ are orthogonal, $T$ is upper triangular and $S$ is upper quasi-triangular with $1×1$ and $2×2$ diagonal blocks. The generalized eigenvalues, $\lambda$, of $\left(A,B\right)$ are computed from the diagonals of $T$ and $S$ and satisfy
$Az = λBz ,$
where $z$ is the corresponding generalized eigenvector. $\lambda$ is actually returned as the pair $\left(\alpha ,\beta \right)$ such that
$λ = α/β$
since $\beta$, or even both $\alpha$ and $\beta$ can be zero. The columns of $Q$ and $Z$ are the left and right generalized Schur vectors of $\left(A,B\right)$.
Optionally, f08xbc can order the generalized eigenvalues on the diagonals of $\left(S,T\right)$ so that selected eigenvalues are at the top left. The leading columns of $Q$ and $Z$ then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xbc computes $T$ to have non-negative diagonal elements, and the $2×2$ blocks of $S$ correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the $QZ$ algorithm.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in ${\mathbf{rconde}}\left[0\right]$ and ${\mathbf{rconde}}\left[1\right]$ respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in ${\mathbf{rcondv}}\left[0\right]$ and ${\mathbf{rcondv}}\left[1\right]$. See Section 4.11 of Anderson et al. (1999) for further information.
## 4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
## 5Arguments
1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{jobvsl}$Nag_LeftVecsType Input
On entry: if ${\mathbf{jobvsl}}=\mathrm{Nag_NotLeftVecs}$, do not compute the left Schur vectors.
If ${\mathbf{jobvsl}}=\mathrm{Nag_LeftVecs}$, compute the left Schur vectors.
Constraint: ${\mathbf{jobvsl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
3: $\mathbf{jobvsr}$Nag_RightVecsType Input
On entry: if ${\mathbf{jobvsr}}=\mathrm{Nag_NotRightVecs}$, do not compute the right Schur vectors.
If ${\mathbf{jobvsr}}=\mathrm{Nag_RightVecs}$, compute the right Schur vectors.
Constraint: ${\mathbf{jobvsr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
4: $\mathbf{sort}$Nag_SortEigValsType Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
${\mathbf{sort}}=\mathrm{Nag_NoSortEigVals}$
Eigenvalues are not ordered.
${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$
Eigenvalues are ordered (see selctg).
Constraint: ${\mathbf{sort}}=\mathrm{Nag_NoSortEigVals}$ or $\mathrm{Nag_SortEigVals}$.
5: $\mathbf{selctg}$function, supplied by the user External Function
If ${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$, selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.
If ${\mathbf{sort}}=\mathrm{Nag_NoSortEigVals}$, selctg is not referenced by f08xbc, and may be specified as NULLFN.
The specification of selctg is:
Nag_Boolean selctg (double ar, double ai, double b)
1: $\mathbf{ar}$double Input
2: $\mathbf{ai}$double Input
3: $\mathbf{b}$double Input
On entry: an eigenvalue $\left({\mathbf{ar}}\left[j-1\right]+\sqrt{-1}×{\mathbf{ai}}\left[j-1\right]\right)/{\mathbf{b}}\left[j-1\right]$ is selected if ${\mathbf{selctg}}\left({\mathbf{ar}}\left[j-1\right],{\mathbf{ai}}\left[j-1\right],{\mathbf{b}}\left[j-1\right]\right)$ is Nag_TRUE. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy ${\mathbf{selctg}}\left({\mathbf{ar}}\left[j-1\right],{\mathbf{ai}}\left[j-1\right],{\mathbf{b}}\left[j-1\right]\right)=\mathrm{Nag_TRUE}$ after ordering. ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_SCHUR_REORDER_SELECT in this case.
6: $\mathbf{sense}$Nag_RCondType Input
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\mathrm{Nag_NotRCond}$
None are computed.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$
Computed for average of selected eigenvalues only.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$
Computed for selected deflating subspaces only.
${\mathbf{sense}}=\mathrm{Nag_RCondBoth}$
Computed for both.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$, ${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$.
Constraint: ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$, $\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$.
7: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
8: $\mathbf{a}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the first of the pair of matrices, $A$.
On exit: a has been overwritten by its generalized Schur form $S$.
9: $\mathbf{pda}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{b}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the second of the pair of matrices, $B$.
On exit: b has been overwritten by its generalized Schur form $T$.
11: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{sdim}$Integer * Output
On exit: if ${\mathbf{sort}}=\mathrm{Nag_NoSortEigVals}$, ${\mathbf{sdim}}=0$.
If ${\mathbf{sort}}=\mathrm{Nag_SortEigVals}$, ${\mathbf{sdim}}=\text{}$ number of eigenvalues (after sorting) for which selctg is Nag_TRUE. (Complex conjugate pairs for which selctg is Nag_TRUE for either eigenvalue count as $2$.)
13: $\mathbf{alphar}\left[{\mathbf{n}}\right]$double Output
On exit: see the description of beta.
14: $\mathbf{alphai}\left[{\mathbf{n}}\right]$double Output
On exit: see the description of beta.
15: $\mathbf{beta}\left[{\mathbf{n}}\right]$double Output
On exit: $\left({\mathbf{alphar}}\left[\mathit{j}-1\right]+{\mathbf{alphai}}\left[\mathit{j}-1\right]×i\right)/{\mathbf{beta}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, will be the generalized eigenvalues. ${\mathbf{alphar}}\left[\mathit{j}-1\right]+{\mathbf{alphai}}\left[\mathit{j}-1\right]×i$, and ${\mathbf{beta}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$, are the diagonals of the complex Schur form $\left(S,T\right)$ that would result if the $2×2$ diagonal blocks of the real Schur form of $\left(A,B\right)$ were further reduced to triangular form using $2×2$ complex unitary transformations.
If ${\mathbf{alphai}}\left[j-1\right]$ is zero, then the $j$th eigenvalue is real; if positive, then the $j$th and $\left(j+1\right)$st eigenvalues are a complex conjugate pair, with ${\mathbf{alphai}}\left[j\right]$ negative.
Note: the quotients ${\mathbf{alphar}}\left[j-1\right]/{\mathbf{beta}}\left[j-1\right]$ and ${\mathbf{alphai}}\left[j-1\right]/{\mathbf{beta}}\left[j-1\right]$ may easily overflow or underflow, and ${\mathbf{beta}}\left[j-1\right]$ may even be zero. Thus, you should avoid naively computing the ratio $\alpha /\beta$. However, alphar and alphai will always be less than and usually comparable with ${‖A‖}_{2}$ in magnitude, and beta will always be less than and usually comparable with ${‖B‖}_{2}$.
16: $\mathbf{vsl}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array vsl must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvsl}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvsl}}=\mathrm{Nag_LeftVecs}$;
• $1$ otherwise.
$i$th element of the $j$th vector is stored in
• ${\mathbf{vsl}}\left[\left(j-1\right)×{\mathbf{pdvsl}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vsl}}\left[\left(i-1\right)×{\mathbf{pdvsl}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobvsl}}=\mathrm{Nag_LeftVecs}$, vsl will contain the left Schur vectors, $Q$.
If ${\mathbf{jobvsl}}=\mathrm{Nag_NotLeftVecs}$, vsl is not referenced.
17: $\mathbf{pdvsl}$Integer Input
On entry: the stride used in the array vsl.
Constraints:
• if ${\mathbf{jobvsl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvsl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvsl}}\ge 1$.
18: $\mathbf{vsr}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array vsr must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvsr}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvsr}}=\mathrm{Nag_RightVecs}$;
• $1$ otherwise.
$i$th element of the $j$th vector is stored in
• ${\mathbf{vsr}}\left[\left(j-1\right)×{\mathbf{pdvsr}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vsr}}\left[\left(i-1\right)×{\mathbf{pdvsr}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobvsr}}=\mathrm{Nag_RightVecs}$, vsr will contain the right Schur vectors, $Z$.
If ${\mathbf{jobvsr}}=\mathrm{Nag_NotRightVecs}$, vsr is not referenced.
19: $\mathbf{pdvsr}$Integer Input
On entry: the stride used in the array vsr.
Constraints:
• if ${\mathbf{jobvsr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvsr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvsr}}\ge 1$.
20: $\mathbf{rconde}\left[2\right]$double Output
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, ${\mathbf{rconde}}\left[0\right]$ and ${\mathbf{rconde}}\left[1\right]$ contain the reciprocal condition numbers for the average of the selected eigenvalues.
If ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVecs}$, rconde is not referenced.
21: $\mathbf{rcondv}\left[2\right]$double Output
On exit: if ${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$, ${\mathbf{rcondv}}\left[0\right]$ and ${\mathbf{rcondv}}\left[1\right]$ contain the reciprocal condition numbers for the selected deflating subspaces.
if ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$ or $\mathrm{Nag_RCondEigVals}$, rcondv is not referenced.
22: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{jobvsl}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvsl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobvsl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvsl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvsl}}\ge 1$.
On entry, ${\mathbf{jobvsr}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvsr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{jobvsr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvsr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvsr}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdvsl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvsl}}>0$.
On entry, ${\mathbf{pdvsr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvsr}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_ITERATION_QZ
The $QZ$ iteration failed. No eigenvectors have been calculated but ${\mathbf{alphar}}\left[j\right]$, ${\mathbf{alphai}}\left[j\right]$ and ${\mathbf{beta}}\left[j\right]$ should be correct from element $⟨\mathit{\text{value}}⟩$.
The $QZ$ iteration failed with an unexpected error, please contact NAG.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SCHUR_REORDER
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy ${\mathbf{selctg}}=\mathrm{Nag_TRUE}$. This could also be caused by underflow due to scaling.
## 7Accuracy
The computed generalized Schur factorization satisfies
$A+E = QS ZT , B+F = QT ZT ,$
where
$‖(E,F)‖ F = O(ε) ‖(A,B)‖ F$
and $\epsilon$ is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
## 8Parallelism and Performance
f08xbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this function is f08xpc.
## 10Example
This example finds the generalized Schur factorization of the matrix pair $\left(A,B\right)$, where
$A = ( 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0 ) and B= ( 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ) ,$
such that the real positive eigenvalues of $\left(A,B\right)$ correspond to the top left diagonal elements of the generalized Schur form, $\left(S,T\right)$. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
### 10.1Program Text
Program Text (f08xbce.c)
### 10.2Program Data
Program Data (f08xbce.d)
### 10.3Program Results
Program Results (f08xbce.r) | 5,964 | 17,794 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 230, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2024-30 | latest | en | 0.613626 |
https://search.r-project.org/CRAN/refmans/ExtremeRisks/html/EBTailIndex.html | 1,653,689,781,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00038.warc.gz | 555,110,285 | 2,748 | EBTailIndex {ExtremeRisks} R Documentation
## Expectile Based Tail Index Estimation
### Description
Computes a point estimate of the tail index based on the Expectile Based (EB) estimator.
### Usage
EBTailIndex(data, tau, est=NULL)
### Arguments
data A vector of (1 \times n) observations. tau A real in (0,1) specifying the intermediate level \tau_n. See Details\. est A real specifying the estimate of the expectile at the intermediate level tau.
### Details
For a dataset data of sample size n, the tail index \gamma of its (marginal) distribution is estimated using the EB estimator:
\hat{\gamma}_n^E =\left(1+\frac{\hat{\bar{F}}_n(\tilde{\xi}_{\tau_n})}{1-\tau_n}\right)^{-1} ,
where \hat{\bar{F}}_n is the empirical survival function of the observations, \tilde{\xi}_{\tau_n} is an estimate of the \tau_n-th expectile. The observations can be either independent or temporal dependent. See Padoan and Stupfler (2020) and Daouia et al. (2018) for details.
• The so-called intermediate level tau or \tau_n is a sequence of positive reals such that \tau_n \to 1 as n \to \infty. Practically, \tau_n \in (0,1) is the ratio between the empirical mean distance of the \tau_n-th expectile from the smaller observations and the empirical mean distance of of the \tau_n-th expectile from all the observations. An estimate of \tau_n-th expectile is computed and used in turn to estimate \gamma.
• The value est, if provided, is meant to be an esitmate of the \tau_n-th expectile which is used to estimate \gamma. On the contrary, if est=NULL, then the routine EBTailIndex estimate first the \tau_n-th expectile expectile and then use it to estimate \gamma.
### Value
An estimate of the tain index \gamma.
### References
Padoan A.S. and Stupfler, G. (2020). Extreme expectile estimation for heavy-tailed time series. arXiv e-prints arXiv:2004.04078, https://arxiv.org/abs/2004.04078.
Daouia, A., Girard, S. and Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292.
### Examples
# Tail index estimation based on the Expectile based estimator obtained with data
# simulated from an AR(1) with 1-dimensional Student-t distributed innovations
tsDist <- "studentT"
tsType <- "AR"
# parameter setting
corr <- 0.8
df <- 3
par <- c(corr, df)
# Big- small-blocks setting
bigBlock <- 65
smallblock <- 15
# Intermediate level (or sample tail probability 1-tau)
tau <- 0.97
# sample size
ndata <- 2500
# Simulates a sample from an AR(1) model with Student-t innovations
data <- rtimeseries(ndata, tsDist, tsType, par)
# tail index estimation
gammaHat <- EBTailIndex(data, tau)
gammaHat
[Package ExtremeRisks version 0.0.4 Index] | 753 | 2,727 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2022-21 | latest | en | 0.708981 |
http://www.slideshare.net/oumsaokosal/semessentials | 1,433,023,327,000,000,000 | text/html | crawl-data/CC-MAIN-2015-22/segments/1432207932705.91/warc/CC-MAIN-20150521113212-00330-ip-10-180-206-219.ec2.internal.warc.gz | 727,716,166 | 54,438 | 0
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# Sem+Essentials
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### Transcript
• 1. Structural Equation Modeling (SEM) Essentials by Jim Grace Purpose of this module is to provide a very brief presentation of the things one needs to know about SEM before learning how apply SEM.
• 2. Where You can Learn More about SEM Grace (2006) Structural Equation Modeling and Natural Systems. Cambridge Univ. Press. Shipley (2000) Cause and Correlation in Biology. Cambridge Univ. Press. Kline (2005) Principles and Practice of Structural Equation Modeling. (2 nd Edition) Guilford Press. Bollen (1989) Structural Equations with Latent Variables. John Wiley and Sons. Lee (2007) Structural Equation Modeling: A Bayesian Approach. John Wiley and Sons.
• 3. I. Essential Points about SEM Outline II. Structural Equation Models: Form and Function
• 4. I. SEM Essentials: 1. SEM is a form of graphical modeling, and therefore, a system in which relationships can be represented in either graphical or equational form. 2. An equation is said to be structural if there exists sufficient evidence from all available sources to support the interpretation that x 1 has a causal effect on y 1 . x 1 y 1 1 11 graphical form y 1 = γ 11 x 1 + ζ 1 equational form
• 5. 3. Structural equation modeling can be defined as the use of two or more structural equations to represent complex hypotheses. y 2 y 1 x 1 y 3 ζ 1 ζ 2 ζ 3 Complex Hypothesis
• e.g.
• y 1 = γ 11 x 1 + ζ 1
• y 2 = β 21 y 1 + γ 21 x 1 + ζ 2
• y 3 = β 32 y 2 + γ 31 x 1 + ζ 3
Corresponding Equations
• 6. a. manipulations of x can repeatably be demonstrated to be followed by responses in y , and/or b. we can assume that the values of x that we have can serve as indicators for the values of x that existed when effects on y were being generated, and/or c. if it can be assumed that a manipulation of x would result in a subsequent change in the values of y Relevant References: Pearl (2000) Causality . Cambridge University Press. Shipley (2000) Cause and Correlation in Biology . Cambridge 4. Some practical criteria for supporting an assumption of causal relationships in structural equations:
• 7. 5. A Grossly Oversimplified History of SEM Wright (1918) Pearson (1890s) Fisher (1922) Joreskog (1973) Lee (2007) Neyman & E. Pearson (1934) Spearman (1904) Bayes & LaPlace (1773/1774) MCMC (1948-) testing alt. models likelihood r, chi-square factor analysis path analysis SEM Contemporary Conven- tional Statistics Bayesian Analysis Raftery (1993) note that SEM is a framework and incorporates new statistical techniques as they become available (if appropriate to its purpose)
• 8. 6. SEM is a framework for building and evaluating multivariate hypotheses about multiple processes. It is not dependent on a particular estimation method. 7. When it comes to statistical methodology, it is important to distinguish between the priorities of the methodology versus those of the scientific enterprise. Regarding the diagram below, in SEM we use statistics for the purposes of the scientific enterprise. Statistics and other Methodological Tools, Procedures, and Principles. The Scientific Enterprise
• 9. The Methodological Side of SEM
• 10. The Relationship of SEM to the Scientific Enterprise modified from Starfield and Bleloch (1991) Understanding of Processes univariate descriptive statistics exploration, methodology and theory development realistic predictive models simplistic models multivariate descriptive statistics detailed process models univariate data modeling Data structural equation modeling
• 11. 8. SEM seeks to progress knowledge through cumulative learning. Current work is striving to increase the capacity for model memory and model generality. exploratory/ model-building applications structural equation modeling confirmatory/ hypothesis-testing applications one aim of SEM
• 12. 9. It is not widely understood that the univariate model, and especially ANOVA, is not well suited for studying systems, but rather, is designed for studying individual processes, net effects, or for identifying predictors. 10. The dominance of the univariate statistical model in the natural sciences has, in my personal view, retarded the progress of science.
• 13. 11. An interest in systems under multivariate control motivates us to explicitly consider the relative importances of multiple processes and how they interact. We seek to consider simultaneously the main factors that determine how system responses behave. 12. SEM is one of the few applications of statistical inference where the results of estimation are frequently “you have the wrong model!”. This feedback comes from the unique feature that in SEM we compare patterns in the data to those implied by the model. This is an extremely important form of learning about systems.
• 14. 13. Illustrations of fixed-structure protocol models: Univariate Models Do these model structures match the causal forces that influenced the data? If not, what can they tell you about the processes operating? x 1 x 2 x 3 x 4 x 5 y 1 Multivariate Models x 1 x 2 x 3 x 4 x 5 F y 1 y 2 y 3 y 4 y 5
• 15. 14. Structural equation modeling and its associated scientific goals represent an ambitious undertaking. We should be both humbled by the limits of our successes and inspired by the learning that takes place during the journey.
• 16. II. Structural Equation Models: Form and Function A. Anatomy of Observed Variable Models
• 17. x 1 y 1 y 2 1 2 Some Terminology exogenous variable endogenous variables 21 11 21 path coefficients direct effect of x 1 on y 2 indirect effect of x 1 on y 2 is 11 times 21
• 18. nonrecursive y 1 x 2 x 1 y 2 ζ 1 ζ 2 C y 1 x 2 x 1 y 2 ζ 1 ζ 2 D x 1 y 1 ζ 1 y 2 A ζ 2 x 1 y 1 y 2 B ζ 1 ζ 2 model B, which has paths between all variables is “saturated” (vs A, which is “unsaturated”)
• 19. First Rule of Path Coefficients: the path coefficients for unanalyzed relationships (curved arrows) between exogenous variables are simply the correlations (standardized form) or covariances (unstandardized form). x 1 x 2 y 1 .40 x 1 x 2 y 1 ----------------------------- x 1 1.0 x 2 0.40 1.0 y 1 0.50 0.60 1.0
• 20. x 1 y 1 y 2 11 = .50 21 = .60 (gamma) used to represent effect of exogenous on endogenous. (beta) used to represent effect of endogenous on endogenous. Second Rule of Path Coefficients: when variables are connected by a single causal path, the path coefficient is simply the standardized or unstandardized regression coefficient (note that a standardized regression coefficient = a simple correlation.) x 1 y 1 y 2 ------------------------------------------------- x 1 1.0 y 1 0.50 1.0 y 2 0.30 0.60 1.0
• 21. Third Rule of Path Coefficients: strength of a compound path is the product of the coefficients along the path. x 1 y 1 y 2 .50 .60 Thus, in this example the effect of x 1 on y 2 = 0.5 x 0.6 = 0.30 Since the strength of the indirect path from x 1 to y 2 equals the correlation between x 1 and y 2 , we say x 1 and y 2 are conditionally independent .
• 22. What does it mean when two separated variables are not conditionally independent? x 1 y 1 y 2 ------------------------------------------------- x 1 1.0 y 1 0.55 1.0 y 2 0.50 0.60 1.0 x 1 y 1 y 2 r = .55 r = .60 0.55 x 0.60 = 0.33, which is not equal to 0.50
• 23. The inequality implies that the true model is x 1 y 1 y 2 Fourth Rule of Path Coefficients: when variables are connected by more than one causal pathway, the path coefficients are "partial" regression coefficients. additional process Which pairs of variables are connected by two causal paths? answer: x 1 and y 2 (obvious one), but also y 1 and y 2 , which are connected by the joint influence of x 1 on both of them.
• 24. And for another case: x 1 x 2 y 1 A case of shared causal influence: the unanalyzed relation between x 1 and x 2 represents the effects of an unspecified joint causal process. Therefore, x 1 and y 1 connected by two causal paths. x 2 and y 1 likewise.
• 25. I have an article on this subject that is brief and to the point. Grace, J.B. and K.A. Bollen 2005. Interpreting the results from multiple regression and structural equation models. Bull. Ecological Soc. Amer. 86:283-295. x 1 y 1 y 2 .40 .31 .48 How to Interpret Partial Path Coefficients: - The Concept of Statistical Control The effect of y 1 on y 2 is controlled for the joint effects of x 1 .
• 26. Interpretation of Partial Coefficients Analogy to an electronic equalizer from Sourceforge.net With all other variables in model held to their means, how much does a response variable change when a predictor is varied?
• 27. x 1 y 1 y 2 Fifth Rule of Path Coefficients: paths from error variables are correlations or covariances. R 2 = 0.16 .92 R 2 = 0.44 .73 2 1 .31 .40 .48 equation for path from error variable .56 alternative is to show values for zetas, which = 1-R 2 .84
• 28. x 1 y 1 y 2 R 2 = 0.16 R 2 = 0.25 2 1 .50 .40 x 1 y 1 y 2 ------------------------------- x 1 1.0 y 1 0.40 1.0 y 2 0.50 0.60 1.0 Now, imagine y 1 and y 2 are joint responses Sixth Rule of Path Coefficients: unanalyzed residual correlations between endogenous variables are partial correlations or covariances.
• 29. x 1 y 1 y 2 R 2 = 0.16 R 2 = 0.25 2 1 .50 .40 .40 This implies that some other factor is influencing y 1 and y 2 the partial correlation between y1 and y2 is typically represented as a correlated error term
• 30. Seventh Rule of Path Coefficients: total effect one variable has on another equals the sum of its direct and indirect effects. y 1 x 2 x 1 y 2 ζ 1 ζ 2 .80 .15 .64 -.11 .27 x 1 x 2 y 1 ------------------------------- y 1 0.64 -0.11 --- y 2 0.32 -0.03 0.27 Total Effects: Eighth Rule of Path Coefficients: sum of all pathways between two variables (causal and noncausal) equals the correlation/covariance. note: correlation between x 1 and y 1 = 0.55, which equals 0.64 - 0.80*0.11
• 31. Suppression Effect - when presence of another variable causes path coefficient to strongly differ from bivariate correlation. x 1 x 2 y 1 y 2 ----------------------------------------------- x 1 1.0 x 2 0.80 1.0 y 1 0.55 0.40 1.0 y 2 0.30 0.23 0.35 1.0 y 1 x 2 x 1 y 2 ζ 1 ζ 2 .80 .15 .64 -.11 .27 path coefficient for x 2 to y 1 very different from correlation, (results from overwhelming influence from x 1 .)
• 32. II. Structural Equation Models: Form and Function B. Anatomy of Latent Variable Models
• 33. Latent Variables Latent variables are those whose presence we suspect or theorize, but for which we have no direct measures. Intelligence IQ score *note that we must specify some parameter, either error, loading, or variance of latent variable. ζ latent variable observed indicator error variable 1.0 fixed loading* 1.0
• 34. Latent Variables (cont.) Purposes Served by Latent Variables: (2) Allow us to estimate and correct for measurement error. (3) Represent certain kinds of hypotheses. (1) Specification of difference between observed data and processes of interest.
• 35. Range of Examples single-indicator Elevation estimate from map multi-method Soil Organic soil C loss on ignition Territory Size singing range, t1 singing range, t2 singing range, t3 repeated measures Caribou Counts observer 1 observer 2 repeatability
• 36. The Concept of Measurement Error the argument for universal use of latent variables 1. Observed variable models, path or other, assume all independent variables are measured without error. 2. Reliability - the degree to which a measurement is repeatable (i.e., a measure of precision). error in measuring x is ascribed to error in predicting/explaining y x y 0.60 R 2 = 0.30 x y illustration
• 37. Example Imagine that some of the observed variance in x is due to error of measurement. calibration data set based on repeated measurement trials plot x-trial1 x-trial2 x-trial3 1 1.272 1.206 1.281 2 1.604 1.577 1.671 3 2.177 2.192 2.104 4 1.983 2.080 1.999 . ........ ........ ....... n 2.460 2.266 2.418 average correlation between trials = 0.90 therefore, average R-square = 0.81 reliability = square root of R 2 measurement error variance = (1 - R 2 ) times VARx imagine in this case VARx = 3.14, so error variance = 0.19 x 3.14 = 0.60 LV1 x LV2 y .90 .65 1.0 .60 R2 = .42
• 38. II. Structural Equation Models: Form and Function C. Estimation and Evaluation
• 39. 1. The Multiequational Framework (a) the observed variable model We can model the interdependences among a set of predictors and responses using an extension of the general linear model that accommodates the dependences of response variables on other response variables. y = p x 1 vector of responses α = p x 1 vector of intercepts Β = p x p coefficient matrix of y s on y s Γ = p x q coefficient matrix of y s on x s x = q x 1 vector of exogenous predictors ζ = p x 1 vector of errors for the elements of y Φ = cov ( x ) = q x q matrix of covariances among x s Ψ = cov ( ζ ) = q x q matrix of covariances among errors
• y = α + Β y + Γ x + ζ
• 40. The LISREL Equations Jöreskög 1973 (b) the latent variable model η = α + Β η + Γ ξ + ζ x = Λ x ξ + δ y = Λ y η + ε where: η is a vector of latent responses, ξ is a vector of latent predictors, Β and Γ are matrices of coefficients, ζ is a vector of errors for η , and α is a vector of intercepts for η (c) the measurement model where: Λ x is a vector of loadings that link observed x variables to latent predictors, Λ y is a vector of loadings that link observed y variables to latent responses, and δ and ε are vectors are errors
• 41. 2. Estimation Methods (a) decomposition of correlations (original path analysis) (b) least-squares procedures (historic or in special cases) (c) maximum likelihood (standard method) (d) Markov chain Monte Carlo (MCMC) methods (including Bayesian applications)
• 42. Bayesian References: Bayesian Networks: Neopolitan, R.E. (2004). Learning Bayesian Networks. Upper Saddle River, NJ, Prentice Hall Publs. Bayesian SEM: Lee, SY (2007) Structural Equation Modeling: A Bayesian Approach. Wiley & Sons.
• 43. SEM is Based on the Analysis of Covariances! Why? Analysis of correlations represents loss of information. A B r = 0.86 r = 0.50 illustration with regressions having same slope and intercept Analysis of covariances allows for estimation of both standardized and unstandardized parameters.
• 44. Σ = { σ 11 σ 12 σ 22 σ 13 σ 23 σ 33 } Model-Implied Correlations Observed Correlations* { 1.0 .24 1.0 .01 .70 1.0 } S = * typically the unstandardized correlations, or covariances 2. Estimation (cont.) – analysis of covariance structure The most commonly used method of estimation over the past 3 decades has been through the analysis of covariance structure (think – analysis of patterns of correlations among variables). compare
• 45. x1 y1 y2 Hypothesized Model Σ = { σ 11 σ 12 σ 22 σ 13 σ 23 σ 33 } Implied Covariance Matrix Observed Covariance Matrix { 1.3 .24 .41 .01 9.7 12.3 } S = compare Model Fit Evaluations + Parameter Estimates estimation (e.g., maximum likelihood) 3. Evaluation
• 46. Model Identification - Summary 2. Several factors can prevent identification, including: a. too many paths specified in model b. certain kinds of model specifications can make parameters unidentified c. multicollinearity d. combination of a complex model and a small sample 1. For the model parameters to be estimated with unique values, they must be identified . As in linear algebra, we have a requirement that we need as many known pieces of information as we do unknown parameters. 3. Good news is that most software checks for identification (in something called the information matrix) and lets you know which parameters are not identified.
• 47. The most commonly used fitting function in maximum likelihood estimation of structural equation models is based on the log likelihood ratio, which compares the likelihood for a given model to the likelihood of a model with perfect fit. Fitting Functions Note that when sample matrix and implied matrix are equal, terms 1 and 3 = 0 and terms 2 and 4 = 0. Thus, perfect model fit yields a value of F ML of 0.
• 48. Maximum likelihood estimators, such as F ML , possess several important properties: (1) asymptotically unbiased, (2) scale invariant, and (3) best estimators. Assumptions : (1) and S matrices are positive definite (i.e., that they do not have a singular determinant such as might arise from a negative variance estimate, an implied correlation greater than 1.0, or from one row of a matrix being a linear function of another), and (2) data follow a multinormal distribution. Fitting Functions (cont.)
• 49. One of the most commonly used approaches to performing such tests (the model Χ 2 test) utilizes the fact that the maximum likelihood fitting function F ML follows a X 2 (chi-square) distribution. The Χ 2 Test X 2 = n-1(F ML ) Here, n refers to the sample size, thus X 2 is a direct function of sample size. Assessment of Fit between Sample Covariance and Model- Implied Covariance Matrix
• 50. Illustration of the use of Χ 2 X 2 = 3.64 with 1 df and 100 samples P = 0.056 X 2 = 7.27 with 1 df and 200 samples P = 0.007 x y 1 y 2 1.0 0.4 1.0 0.35 0.5 1.0 r xy2 expected to be 0.2 (0.40 x 0.50) X 2 = 1.82 with 1 df and 50 samples P = 0.18 correlation matrix issue: should there be a path from x to y 2 ? 0.40 0.50 Essentially, our ability to detect significant differences from our base model, depends as usual on sample size.
• 51. Additional Points about Model Fit Indices: 1.The chi-square test appears to be reasonably effective at sample sizes less than 200. 2. There is no perfect answer to the model selection problem. 4. A lot of attention is being paid to Bayesian model selection methods at the present time. 3. No topic in SEM has had more attention than the development of indices that can be used as guides for model selection. 5. In SEM practice, much of the weight of evidence falls on the investigator to show that the results are repeatable (predictive of the next sample).
• 52. Alternatives when data extremely nonnormal Robust Methods: Satorra, A., & Bentler, P. M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. 1988 Proceedings of the Business and Economics Statistics Section of the American Statistical Association, 308-313. Bootstrap Methods: Bollen, K. A., & Stine, R. A. (1993). Bootstrapping goodness-of-fit measures in structural equation models. In K. A. Bollen and J. S. Long (Eds.) Testing structural equation models. Newbury Park, CA: Sage Publications. Alternative Distribution Specification: - Bayesian and other:
• 53. Residuals : Most fit indices represent average of residuals between observed and predicted covariances. Therefore, individual residuals should be inspected. Correlation Matrix to be Analyzed y1 y2 x -------- -------- -------- y1 1.00 y2 0.50 1.00 x 0.40 0.35 1.00 Fitted Correlation Matrix y1 y2 x -------- -------- -------- y1 1.00 y2 0.50 1.00 x 0.40 0.20 1.00 residual = 0.15 Diagnosing Causes of Lack of Fit (misspecification) Modification Indices : Predicted effects of model modification on model chi-square.
• 54. The topic of model selection, which focuses on how you choose among competing models, is very important. Please refer to additional tutorials for considerations of this topic.
• 55. While we have glossed over as many details as we could, these fundamentals will hopefully help you get started with SEM. Another gentle introduction to SEM oriented to the community ecologist is Chapter 30 in McCune, B. and J.B. Grace 2004. Analysis of Ecological Communities. MJM. (sold at cost with no profit) | 5,378 | 20,116 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2015-22 | latest | en | 0.776962 |
http://www.zeefax.com/condition-monitoring-systems/cms-products-services/turbine-supervisory-systems/glossary/b/ | 1,560,717,816,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998298.91/warc/CC-MAIN-20190616202813-20190616224813-00512.warc.gz | 333,029,080 | 8,612 | Share:
B
Baffle
A device, usually filled with foam or fiber, which is used to block noise from entering or exiting an confined space.
Balancing (mechanical)
A procedure for adjusting the radial mass distribution of a rotor so that the mass centerline approaches the rotor geometric centerline, thereby reducing the vibratory forces generated by rotation.
Balancing Resonance Speed(s)
A rotative speed that corresponds to a natural resonance frequency.
Balanced Condition
For rotating machinery, a condition where the shaft geometric centerline coincides with the mass centerline.
Band-Pass Filter
A filter with a single transmission band extending from lower to upper cutoff frequencies. The width of the band-pass filter (band width) is determined by the separation of frequencies at which amplitude is attenuated by 3 dB (0.707).
Bandwidth (Measurement)
The frequency range (usually stated in hertz or Hz) within which a measuring system can accurately measure an amplitude quantity.
Bandwidth (Digital Filters)
The spacing between frequencies at which a band-pass filter attenuates the signal by 3 dB. In a digital signal analyzer, the measurement bandwidth is equal to [(frequency span)/(number of filters) x (window factor)]. Window factors are: 1 for uniform , 1.5 for Hanning, and 3.63 for flat top (Uniform, Hanning and Flat Top are digital filter shapes).
Baseline spectrum
A vibration spectrum taken when a machine is in good working condition (new or just overhauled), which is then used as reference for future monitoring or analysis.
Bearing (rolling Element)
A Rolling element bearing has four parts: an inner race, an outer race, balls or rollers, and a cage to maintain the proper separation of the rolling elements.
Bearing (sleeve)
A sleeve bearing is a cylinder of alloy metal surrounding the rotating shaft. Contact between the shaft and sleeve is prevented by a lubrication film.
Beat Frequency
Where two cyclic components are close together in frequency they combine in such a way that their sum will vary in amplitude at a rate equal to the difference in frequency between the two components. This phenomenon is known as beating, and its frequency is known as the beat frequency.
Bias
Refers to a more or less persistent tendency for the measurements, as a group, to be too large or too small.
Biased Hall Effect Sensor
Sensor that operates with either magnets or ferrous metal targets, Generates a square wave output.
Bins (lines)
In an FFT spectrum, the individual frequencies at which the amplitudes are calculated.
BIT
Short for binary digit. A number expressed in binary notation utilizes the digits 1 and 0, and these are called bits. Any number can be expressed with combinations of them.
Blade Passing Frequency
A potential vibration frequency on any bladed machine (turbine, axial compressor, fan, etc.). It is represented by the number of blades times shaft-rotating frequency.
Block Size
The number of samples used in a Digital Signal Aanalyser (DSA) to compute the Fast Fourier Transform (FFT). Also the number of samples in a DSA time display. Most DSAs use a block size of 1024. Smaller block size reduces frequency resolution.
Bode
Rectangular coordinate plot of 1x component amplitude and phase (relative to a keyphasor) vs. running speed / Magnitude and phase of vibration in a machine plotted against speed.
Broadband
Vibration (or other) signals which are unfiltered. Signals at all frequencies contribute to the measured value.
Bode plot
The magnitude of vibration at 1x shaft speed, also its phase relative to the key phaser, both plotted against running speed.
Bow
In Rotating Machinery. A shaft condition in which the geometric shaft centerline is not straight.Can be caused due to inadequate baring during cool down.
BPFO, BPFI.
Common abbreviations for ball pass frequency of defects on outer and inner bearing races, respectively.
Brinneling
Impressions made by bearing rolling elements on the bearing race; typically caused by external vibration when the shaft is stationary.
Broadband
Vibration (or other) signals which are unfiltered. Signals at all frequencies contribute to the measured value.
Buffer
A memory location in a computer or digital instrument that is set aside for temporarily storing digital information while it is waiting to be processed.
Buzz
A sound exemplified by loose power transformer laminations (dominated by 120 Hz where the power frequency is 60 Hz). | 924 | 4,482 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2019-26 | latest | en | 0.900206 |
http://forums.na.leagueoflegends.com/board/showthread.php?s=&p=32951756 | 1,405,071,529,000,000,000 | text/html | crawl-data/CC-MAIN-2014-23/segments/1404776426486.74/warc/CC-MAIN-20140707234026-00074-ip-10-180-212-248.ec2.internal.warc.gz | 54,831,306 | 9,093 | ### @ricklessabandon Why u math so good?
First Riot Post
Super Explosion
Senior Member
Quote:
Originally Posted by ricklessabandon
like every ad carry champion in season 2 building the same 5 items every game.
Why you no nerf IE-PD (or Crit) harder?
As long as the damage scaling is that extreme on only one item path, items like Hurricane will just about soundly be suboptimal in all cases.
ricklessabandon
qa analyst
Quote:
Originally Posted by ForrestLump
Great post! Thank you, but I do have one question.
Does +30 | +32% magic penetration mean +30 AND +32% or +30 OR +30% magic penetration?
| is an operator that means OR in some languages. I am not talking about C/C++ (even though it is similar ||).
oh, sorry—that's how it shows up in the character sheet in games. "x | y" translates as "x flat penetration and y % penetration" when i use it. so the scenario to which i was referring would be 7.8 | 8% before and 37.8 | 40% after, and both get applied in damage calculations with the % happening first.
example of the former case versus a target with 100 magic resist:
100 - 8% = 92
92 - 7.8 = 84.2
example of the latter case versus a target with 100 magic resist:
100 - 40% = 60
60 - 37.8 = 22.2
so you'd be penetrating 62 more magic resist with those items against a target with 100 magic resist.
Whisperx5
Senior Member
Do Riot have any plans for buffing armor?
Is Riot happy with the current state of the game?
RazyUltim8
Member
this red, utterly clueless, im done with this thread lol.
as shown by the release of that monstrosity known as black cleaver, it looks like riot really did their homework when it comes to s3
Skaarrjj
Senior Member
Quote:
Originally Posted by ricklessabandon
oh, sorry—that's how it shows up in the character sheet in games. "x | y" translates as "x flat penetration and y % penetration" when i use it. so the scenario to which i was referring would be 7.8 | 8% before and 37.8 | 40% after, and both get applied in damage calculations with the % happening first.
example of the former case versus a target with 100 magic resist:
100 - 8% = 92
92 - 7.8 = 84.2
example of the latter case versus a target with 100 magic resist:
100 - 40% = 60
60 - 37.8 = 22.2
so you'd be penetrating 62 more magic resist with those items against a target with 100 magic resist.
Xerath works pretty great with the new magic pen calculations. While the champion might be good, he's just not very fun to play.
ForrestLump
Senior Member
Quote:
Originally Posted by ricklessabandon
oh, sorry—that's how it shows up in the character sheet in games. "x | y" translates as "x flat penetration and y % penetration" when i use it. so the scenario to which i was referring would be 7.8 | 8% before and 37.8 | 40% after, and both get applied in damage calculations with the % happening first.
example of the former case versus a target with 100 magic resist:
100 - 8% = 92
92 - 7.8 = 84.2
example of the latter case versus a target with 100 magic resist:
100 - 40% = 60
60 - 37.8 = 22.2
so you'd be penetrating 62 more magic resist with those items against a target with 100 magic resist.
Thank you... yeah, that is really strong!
WesIey
Member
Quote:
Originally Posted by ricklessabandon
best core damage build (for 'skill gatling') is something like:
dfg, haunting guise, void staff, sorc shoes
7880g for +195 ability power, +30 | +32% magic penetration, +15% cdr, +200 health, +45 movement speed, and the dfg active.
thats my eve delete-champion core build D;
Super Explosion
Senior Member
Quote:
Originally Posted by Skaarrjj
Xerath works pretty great with the new magic pen calculations. While the champion might be good, he's just not very fun to play.
Xerath's trouble is requirement-based gamplay.
He requires the player to perform a hardlocked combo in order to be effective, rather than having a baseline effectiveness and rewarding the player for combos.
Zed is an excellent example of reward-based gameplay.
All of his abilities do something very effective-- but when combined through the skill of the player, even better effects are achieved.
ricklessabandon
qa analyst
Quote:
Originally Posted by Yndyr
Considering the person with the Caitlyn avatar is supposedly really good at math, it's amazing that Cait's passive tooltip still says it's "Every 8th shot" actives Headshot.
Or does 8 = 6 in Runeterra?
it's 8 for levels 1-6, 7 for levels 7-12, and 6 for levels 13-18.
the tooltip should be updating appropriately, but sometimes there's a slight delay so it might not happen the exact moment you hit 7 or 13.
Noblepeasant
Senior Member
Quote:
Originally Posted by ricklessabandon
summary of feelings:
best core damage build (for 'skill gatling') is something like:
dfg, haunting guise, void staff, sorc shoes
7880g for +195 ability power, +30 | +32% magic penetration, +15% cdr, +200 health, +45 movement speed, and the dfg active.
This is my eve build.
Haunting Guise, DFG, Sorc Shoes, Void Staff.
Instantly nuke everything. | 1,318 | 5,003 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2014-23 | longest | en | 0.94395 |
http://conversion.org/length/fathom/pica-postscript | 1,721,099,210,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514726.17/warc/CC-MAIN-20240716015512-20240716045512-00670.warc.gz | 5,483,821 | 7,216 | # fathom to pica (PostScript) conversion
Conversion number between fathom [ftm] and pica (PostScript) is 432. This means, that fathom is bigger unit than pica (PostScript).
### Contents [show][hide]
Switch to reverse conversion:
from pica (PostScript) to fathom conversion
### Enter the number in fathom:
Decimal Fraction Exponential Expression
[ftm]
eg.: 10.12345 or 1.123e5
Result in pica (PostScript)
?
precision 0 1 2 3 4 5 6 7 8 9 [info] Decimal: Exponential:
### Calculation process of conversion value
• 1 fathom = (exactly) (1.8288) / ((254/60000)) = 432 pica (PostScript)
• 1 pica (PostScript) = (exactly) ((254/60000)) / (1.8288) = 0.0023148148148148 fathom
• ? fathom × (1.8288 ("m"/"fathom")) / ((254/60000) ("m"/"pica (PostScript)")) = ? pica (PostScript)
### High precision conversion
If conversion between fathom to metre and metre to pica (PostScript) is exactly definied, high precision conversion from fathom to pica (PostScript) is enabled.
Decimal places: (0-800)
fathom
Result in pica (PostScript):
?
### fathom to pica (PostScript) conversion chart
Start value: [fathom] Step size [fathom] How many lines? (max 100)
visual:
fathompica (PostScript)
00
104320
208640
3012960
4017280
5021600
6025920
7030240
8034560
9038880
10043200
11047520
Copy to Excel
## Multiple conversion
Enter numbers in fathom and click convert button.
One number per line.
Converted numbers in pica (PostScript):
Click to select all
## Details about fathom and pica (PostScript) units:
Convert Fathom to other unit:
### fathom
Definition of fathom unit: ≡ 6 ft.
Convert Pica (PostScript) to other unit:
### pica (PostScript)
Definition of pica (PostScript) unit: ≡ 12 points .
← Back to Length units | 532 | 1,727 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2024-30 | latest | en | 0.560278 |
https://www.coursehero.com/file/9003914/Find-the-density-function-of-Density-function-of-with-Solution/ | 1,498,731,434,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128323895.99/warc/CC-MAIN-20170629084615-20170629104615-00347.warc.gz | 889,040,371 | 23,046 | # Find the density function of density function of with
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Unformatted text preview: g and Tang, 1975). , where , we have . or We know that, Following the above equation, density function of strain energy, U will be: for fU (u ) f U u u exp 2cu 2c 1 0 u This is a Chi-square type distribution with one degree of freedom. Graphically the distribution follows as shown in the figure. Problem 3. The height of earth dams must allow sufficient freeboard above the maximum reservoir level to prevent waves from washing over the top. The determination of this height would include the consideration of wind tide and wave height. where, = wind speed in miles; =fetch or length of water surface over which the wind blows, in feet; =average depth of lake along the fetch, in feet. If wind speed has an exponential distribution with mean speed, , Then determine the distribution of the tide, . (Ang and Tang, 1975) Solution. Denoting , we have . Thus and then we get, In this case, , Thus we have,...
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## This note was uploaded on 03/18/2014 for the course CE 5730 taught by Professor Dr.rajibmaity during the Spring '13 term at Indian Institute of Technology, Kharagpur.
Ask a homework question - tutors are online | 321 | 1,362 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2017-26 | latest | en | 0.908283 |
https://phibonacciblog.wordpress.com/2015/07/ | 1,531,840,081,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676589752.56/warc/CC-MAIN-20180717144908-20180717164908-00194.warc.gz | 729,332,468 | 16,413 | # ANCIENT GEOMETRY, 4. RENAISSANCE
Estándar
The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics as a relevant subject needed to understand nature and the arts. Two major reasons drove Renaissance artists towards the pursuit of mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas.
Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms. In light of these factors, Renaissance artists became some of the best applied mathematicians of their times.
The first printed illustration of a rhombicuboctahedron, byLeonardo da Vinci, published in De divina proportione.
Written by Luca Pacioli in Milan from 1496–98, published in Venice in 1509, De Divina Proportione was about mathematical and artistic proportionLeonardo da Vinci drew illustrations of regular solids in De divina proportione while he lived with and took mathematics lessons from Pacioli. Leonardo’s drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back. Skeletonic solids, such as the rhombicuboctahedron, were one of the first solids drawn to demonstrate perspective by being overlaid on top of each other. Additionally, the work also discusses the use of perspective by painters such as Piero della FrancescaMelozzo da Forlì, and Marco Palmezzano.
It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.
“The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)
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# ANCIENT GEOMETRY, 3. Great Mosque of Kairouan
Estándar
The oldest mosque in North Africa is the Great Mosque of Kairouan (Tunisia), built by Uqba ibn Nafi in 670 A.D. Boussora and Mazouz’s study of the mosque dimensions reveals a very consistent application of the golden ratio in its design.
Floor plan of the Great Mosque of Kairouan.
The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organization. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period.
Because of urban constraints, the mosque floor plan is not a perfect rectangle. Even so, for example, the division of the courtyard and prayer hall is almost a perfect golden ratio.
# ANCIENT GEOMETRY, 2. Parthenon
Estándar
The Parthenon is a temple dedicated the Greek goddess Athena, built in the 5th century BC on the Athenian Acropolis. It is contended that Phidias, the main Greek sculptor in charge of decorating the Parthenon, also knew about the golden ratio and its aesthetic properties. In fact, the Greek symbol for the Golden Ratio is named Phi (φ) because of Phidias The golden rectangle, a rectangle whose length to width ratio is the Golden Ratio and considered the most pleasing to the eye, is almost omnipresent in the façade and floor plans of the Parthenon. The entire façade may be enclosed within a golden rectangle.
The ratio of the length of a metope and triglyph to the height of the frieze, as well as the height of the columns and stylobate to the entire height of the temple is also the golden ratio.
Phidias himself constructed many Parthenon statues that meticulously embody the golden ratio. He is also notable for his contributions to the Athena Parthenos and the Statue of Zeus. As with the Pyramids however, more recent historians challenge the purposeful inclusion of the golden ratio in Greek temples, such as the Parthenon, contending that earlier studies have purposefully fitted in measurements of the temple until it conformed to a golden rectangle.
# ANCIENT GEOMETRY
Estándar
The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, 427- 347 bC
writing that “Plato said God geometrizes continually”. In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying “God arithmetizes”.
At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists.
According to Stephen Skinner, (Sydney, Australia, 1948) the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein.
Many forms observed in nature can be related to geometry, for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape
Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are sometimes interpreted in terms of sacred geometry and considered to be further proof of the natural significance of geometric forms
Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra.
Many of the sacred geometry principles of the human body and of ancient architecture have been compiled into the Vitruvian Man drawing by Leonardo da Vinci, itself based on the much older writings of the Roman architect Vitruvius. | 1,420 | 6,189 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2018-30 | latest | en | 0.892766 |
https://www.diynot.com/diy/threads/rope-puzzle.164539/#post-1113426 | 1,638,988,183,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363520.30/warc/CC-MAIN-20211208175210-20211208205210-00245.warc.gz | 811,254,682 | 13,856 | # Rope puzzle
Discussion in 'General Discussion' started by 2scoops0406, 18 Jan 2009.
1. ### 2scoops0406
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Here's another one but with far less safety implications (I may have lifted this from this forum so apologies if I have)
Assume the earth is a perfect sphere. You have a rope that stretches arounf the equator so the the ends of the rope just touch.
Now you want the same rope to be uniformly suspended 1 foot above the surface, how much extra rope do you need?
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3. ### blondini
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That ones as easy as pi
4. ### 2scoops0406
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If your saying it's pi feet, then I'm sorry, that's not correct.
5. ### 2scoops0406
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Well sorry about not being intuitive, the point is well made, in that I had to check the calculation a couple of times as I originally thought I'd made a mistake.
6. ### Softus
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The answer is (2 * pi) feet.
7. ### 2scoops0406
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Correct.
8. ### blondini
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I see my enlargement on my pie joke has been removed
9. ### 2scoops0406
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My fault, hijacked a post.
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sorry to get in a bit late on this one, are you saying the rope only has to be 6' longer?
12. ### tim west
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what if it's bungee rope?
13. ### blondini
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Close. Just two times the value of pi.
(1' extra radius = 2' extra circumference...)
14. ### Softus
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Huh?
15. ### blondini
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Uhuh.
16. ### Softus
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I beg to differ.
17. ### peteftw
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so what if it was round the circumferance of an elephants belly would it still need to be 6' longer to hover 1'?
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245 | 852 | 2,845 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2021-49 | latest | en | 0.907157 |
https://www.physicsforums.com/threads/small-cannon-displacement-problem.527499/ | 1,550,271,631,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247479627.17/warc/CC-MAIN-20190215224408-20190216010408-00586.warc.gz | 936,048,942 | 12,567 | # Small Cannon Displacement Problem
1. Sep 5, 2011
### mteykl
I had a small cannon on a desk where the projectile was shot out at 1.08 meters above the floor. The projectile's time taken to hit the floor was 0.452 seconds. The cannon is set to 0 degrees above horizontal. The horizontal distance traveled is 2.3 meters. Acceleration is gravity or -9.81 meters per seconds squared. we found velocity to be 5.0885 meters per second.
Here's where the real problem comes in. Our teacher gave us a new angle to shoot the ball and we must predict where the ball will hit on the ground. Our angle was 50 degrees above horizontal. So how far away from the cannon does the ball end up? Please tell me your work, equations used, and final answer. Thank you.
2. Sep 6, 2011
### Spinnor
You will need the following equations and know how to use them,
x = x_i + v_ix*t
y = y_i + v_iy*t + a*t^2/2
3. Sep 6, 2011 | 248 | 906 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2019-09 | latest | en | 0.930622 |
https://datasciencelk.com/probability-density-function/ | 1,723,721,812,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641291968.96/warc/CC-MAIN-20240815110654-20240815140654-00835.warc.gz | 147,401,908 | 29,457 | Probability Density Function
Earlier we used Probability Mass Function to describe how the total probability of 1 is distributed among the possible values of the Discrete Random Variable X. But we cannot define Probability Mass Function for a Continous Random Variable. We use the Probability Density Function to show the distribution of probabilities for a continuous random variable.
Definition
Let X be a continuous random variable. Then a probability distribution function (pdf) of X is a function f(x); such that for any two numbers a and with a ≤ b;
f(x) should satisfy the following 2 conditions:
1. f(x) > 0 for all x
2. Integral is equal to the area under the graph of f(x) which is equal to 1
Detailed Example
Let X be a continous random variable whose probability density function is;
f(x) = 2x2 for 0< x <1
But f(x) ≠ P(X = x)
ex : f(2) = 2(2)2 = 4 this is clearly not a probability.
f(x) is the height of the curve at X = x so that the area under the curve is 1.
In our future posts, we will be discussing about several probability density functions such as Uniform Distribution, Normal Distribution, Gamma Distribution etc. | 266 | 1,149 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2024-33 | latest | en | 0.868108 |
https://tianrunhe.wordpress.com/2012/08/26/string-to-integer-atoi/ | 1,638,787,252,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00516.warc.gz | 620,279,819 | 24,994 | (String to Integer (atoi))
Implement atoi to convert a string to an integer.
Thoughts:
Hint: Carefully consider all possible input cases. If you want a challenge, please do not see below and ask yourself what are the possible input cases.
Notes: It is intended for this problem to be specified vaguely (ie, no given input specs). You are responsible to gather all the input requirements up front.
Requirements for atoi:
The function first discards as many whitespace characters as necessary until the first non-whitespace character is found. Then, starting from this character, takes an optional initial plus or minus sign followed by as many numerical digits as possible, and interprets them as a numerical value.
The string can contain additional characters after those that form the integral number, which are ignored and have no effect on the behavior of this function.
If the first sequence of non-whitespace characters in str is not a valid integral number, or if no such sequence exists because either str is empty or it contains only whitespace characters, no conversion is performed.
If no valid conversion could be performed, a zero value is returned. If the correct value is out of the range of representable values, INT_MAX (2147483647) or INT_MIN (-2147483648) is returned.
Code (Java):
``` public int atoi(String number) {
if (number == null || number.trim().length() == 0) {
return 0;
}
int result = 0;
number = number.trim();
boolean negate = false;
char sign = number.charAt(0);
if (sign == '+' || sign == '-') {
if (sign == '-') {
negate = true;
}
number = number.substring(1);
}
int length = number.length();
int start = -1;
int index = 0;
for (; index < length; index++) {
char a = number.charAt(index);
if (a >= '0' && a <= '9') {
if (start == -1)
start = index;
} else {
break;
}
}
int end = index - 1;
if (start == -1)
return 0;
for (int i = start; i <= end; ++i) {
char a = number.charAt(i);
int digit = a - '0';
if (!negate) {
if (result + digit * (int) Math.pow(10, end - i) >= result)
result += digit * (int) Math.pow(10, end - i);
else
result = Integer.MAX_VALUE;
} else {
if (result - digit * (int) Math.pow(10, end - i) <= result)
result -= digit * (int) Math.pow(10, end - i);
else
result = Integer.MIN_VALUE;
}
}
return result;
}
Code (C++):
class Solution {
public:
int atoi(const char *str) {
if (str == NULL) {
return 0;
}
string number(str);
if(number.size() == 0)
return 0;
int i = 0;
for(; i < number.size(); ++i) {
if(!isspace(number[i])) {
break;
}
}
number = number.substr(i);
int result = 0;
bool negate = false;
char sign = number[0];
if (sign == '+' || sign == '-') {
if (sign == '-') {
negate = true;
}
number = number.substr(1);
}
int length = number.size();
int start = -1;
int index = 0;
for (; index < length; index++) {
char a = number[index];
if (a >= '0' && a <= '9') {
if(start == -1)
start = index;
} else {
break;
}
}
int end = index - 1;
if (start == -1)
return 0;
for (int i = start; i <= end; ++i) {
char a = number[i];
int digit = a - '0';
if (!negate) {
if (result + digit * (int) pow(10, end - i) >= result)
result += digit * (int) pow(10, end - i);
else
result = INT_MAX;
} else {
if (result - digit * (int) pow(10, end - i) <= result)
result -= digit * (int) pow(10, end - i);
else
result = INT_MIN;
}
}
return result;
}
};
__ATA.cmd.push(function() {
sectionId: '370373',
});
});
__ATA.cmd.push(function() {
__ATA.initDynamicSlot({
location: 120,
formFactor: '001',
label: {
},
creative: { | 953 | 3,474 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2021-49 | latest | en | 0.66029 |
https://pp.isofts.kiev.ua/ojs1/article/view/344/0 | 1,675,691,315,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00357.warc.gz | 487,185,803 | 9,170 | Propositional logics of partial predicates with composition of predicate complement | Nikitchenko | PROBLEMS IN PROGRAMMING
DOI: https://doi.org/10.15407/pp2019.01.003
### Propositional logics of partial predicates with composition of predicate complement
M.S. Nikitchenko, O.S. Shkilniak, S.S. Shkilniak, T.A. Mamedov
#### Abstract
The paper studies new software-oriented logical formalisms – the logics of partial predicates with predicate complement. Such logics are denoted LC. A characteristic feature of these logics is the presence of a special non-monotonic operation (composition) of the predicate complement. Such operations are used in various versions of the Floyd-Hoare logic with partial pre- and post-conditions. Properties of LC propositional compositions are similar to the properties of the traditional logical connectives. Properties of the new composition of the predicate complement are investigated. The class of P-predicates (partial single-valued) is closed under the composition of the predicate complement, but the class of T-predicates (total) is not closed. Therefore, it is possible to consider the general class LC – the logic of R-predicates (relational predicates) with the composition of the predicate complement, and its subclass LPC – the logic of P-predicates with such a composition. The focus of the work is the study of PLC – propositional LC. Propositional composition algebras and PLC languages are described. For LC of partial single-valued predicates, an irrefutability logical consequence relation |=IR^ is proposed and investigated under the conditions of undefinedness. The conditions for the validity of the |=IR^ and the properties of the decomposition of formulas are given. Based on the properties of the |=IR^, for PLC of P-predicates a calculus of sequential type is constructed. The basic sequential forms of this calculus and closure conditions of the sequents are given. For the constructed calculus, correctness and completeness theorems are hold. Proofs of these theorems will be given in the forthcoming articles.
Problems in programming 2019; 1: 03-13
#### Keywords
logic; partial predicate; logical consequence; sequent сalculus
PDF (Ukrainian)
#### References
Abramsky S., Gabbay D. and Maibaum T. (editors). (1993–2000). Handbook of Logic in Computer Science Oxford University Press, Vol. 1–5.
Avron A. and Zamansky A. (2011). Non-deterministic semantics for logical systems. In Handbook of Philosophical Logic, D.M. Gabbay, F. Guenthner (eds.), 2nd ed., Vol. 16, Springer Netherlands. P. 227–304. CrossRef
Gries D. and Schneider F. (1995). Avoiding the undefined by underspecification. Springer Berlin Heidelberg. CrossRef
Hähnle R. (2005). Many-valued logic, partiality, and abstraction in formal specification languages. In Logic Journal of the IGPL, 13. P. 415–433. CrossRef
Jones C. (2006). Reasoning about partial functions in the formal development of programs. In Proceedings of AVoCS'05. Vol. 145. Elsevier, Electronic Notes in Theoretical Computer Science. P. 3–25. CrossRef
Nikitchenko M. and Shkilniak S. (2015). Semantic Properties of Logics of Quasiary Predicates. In Workshop on Foundations of Informatics: Proceedings FOI-2015. Chisinau, Moldova. P. 180–197.
Nikitchenko M., Shkilniak O. and Shkilniak S. (2016). Pure first-order logics of quasiary predicates. In Problems in Progamming. N 2–3. P. 73–86 (in ukr).
Nikitchenko M. and Shkilniak S. (2017). Algebras and logics of partial quasiary predicates. In Algebra and Discrete Mathematics. Vol. 23. N 2. P. 263–278.
Ivanov I. and Nikitchenko M. (2018). On the sequence rule for the Floyd-Hoare logic with partial pre- and post-conditions. In Proceedings of the 14th International Conference on ICT. Vol. 2104 of CEUR Workshop Proc. P. 716–724.
Kleene S. (1973) Mathematical Logic. Moscow: Mir (in rus).
Kleene S. (1952) Introductions to Metamathematics. Van Nostrand, Princeton.
Nikitchenko M., Shkilniak O. and Shkilniak S. (2018). Logics of general non-deterministic predicates: semantic aspects. In Problems in Progamming. N 2–3. P. 31–45 (in ukr). CrossRef
DOI: https://doi.org/10.15407/pp2019.01.003
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• There are currently no refbacks. | 1,096 | 4,214 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2023-06 | longest | en | 0.896772 |
https://www.convert-measurement-units.com/convert+Exajoule+to+Terawatt+hour.php | 1,627,352,467,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046152168.38/warc/CC-MAIN-20210727010203-20210727040203-00264.warc.gz | 724,668,633 | 17,385 | Convert EJ to TWh (Exajoule to Terawatt hour)
## Exajoule into Terawatt hour
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Exajoule+to+Terawatt+hour.php
## How many Terawatt hour make 1 Exajoule?
1 Exajoule [EJ] = 277.777 777 777 78 Terawatt hour [TWh] - Measurement calculator that can be used to convert Exajoule to Terawatt hour, among others.
# Convert Exajoule to Terawatt hour (EJ to TWh):
1. Choose the right category from the selection list, in this case 'Energy'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Exajoule [EJ]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Terawatt hour [TWh]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '109 Exajoule'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Exajoule' or 'EJ'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Energy'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '37 EJ to TWh' or '33 EJ into TWh' or '92 Exajoule -> Terawatt hour' or '26 EJ = TWh' or '18 Exajoule to TWh' or '78 EJ to Terawatt hour' or '68 Exajoule into Terawatt hour'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(33 * 81) EJ'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '109 Exajoule + 327 Terawatt hour' or '97mm x 56cm x 80dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 2.487 792 693 410 5×1031. For this form of presentation, the number will be segmented into an exponent, here 31, and the actual number, here 2.487 792 693 410 5. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 2.487 792 693 410 5E+31. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 24 877 926 934 105 000 000 000 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | 946 | 3,799 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2021-31 | longest | en | 0.839819 |
https://www.teachoo.com/8141/2723/Triangle/category/Classifying-triangles/ | 1,685,316,925,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224644571.22/warc/CC-MAIN-20230528214404-20230529004404-00091.warc.gz | 1,119,139,722 | 32,775 | Classifying triangles
Chapter 5 Class 6 Understanding Elementary Shapes
Concept wise
Triangle is made by joining any 3 non-collinear points.
It is a 3 sided polygon.
Polygon
Polygon is a simple closed figure made of line segments
A triangle has
3 vertices − A, B, C
3 sides − AB, BC, AC
3 angles − ∠A, ∠B, ∠C
#### Interior and Exterior of a Triangle
Here,
• P, Q are in interior of ∆ABC
• R is on triangle ∆ABC
• X, Y are in exterior of ∆ABC
Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class | 152 | 534 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2023-23 | longest | en | 0.923473 |
http://www.aspmessageboard.com/showthread.php?140463-algorithm | 1,477,168,254,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719041.14/warc/CC-MAIN-20161020183839-00521-ip-10-171-6-4.ec2.internal.warc.gz | 323,791,523 | 15,715 | algorithm
1. Member
Join Date
Dec 1969
Posts
45
## algorithm
I know this is not ASP related but i'm still going to post my question... here it comes...<BR><BR>i'm looking for a formula or algorithm for determining the matches to be played in a league. Like for example in a soccer season every team plays every other team twice in one season. Once at home and once away. I don't feel like doing this manually (just try it for 16 teams, you'll be amazed to find how hard it is) so if anyone has any extra information or usefull links, all help is appreciated<BR><BR>greetz<BR>Jo
2. Senior Member
Join Date
Dec 1969
Posts
973
## looks like 16!
looks like a factorial solution...<BR><BR>Team 1 plays the remaining 15 teams at home.<BR>Team two has already played Team 1 away so...<BR>Team two plays the remaining 14 teams at home....<BR><BR>and so on....
3. oli
Senior Member
Join Date
Dec 1969
Posts
3,961
## RE: algorithm
I think this could be adapted, it gives every combination of an array of numbers, but this could be the team names. The only change would be home and away, but I guess you just duplicate the list for the two types of matches.<BR><BR>http://63.236.18.31/forum/asp.asp?M=450582&T=450030&F=20&P=1<BR><BR>
4. Member
Join Date
Dec 1969
Posts
45
## RE: looks like 16!
try and write it out, it's realy not that simple!
5. oli
Senior Member
Join Date
Dec 1969
Posts
3,961
.
6. Member
Join Date
Dec 1969
Posts
45
## RE: algorithm
i think adapting that would be as much work as building the formula from scratch
7. Member
Join Date
Dec 1969
Posts
45
## RE: looks like 16!
i what order...<BR>what makes it so difficult is that a team can only play one match in one match day
8. Senior Member
Join Date
Dec 1969
Posts
16,931
## RE: looks like 16!
team 1 plays 15 games at home and 15 games away.<BR>team 2 plays 15 games at home and 15 games away. However, two of those games (against team 1) are already counted. So that's 28 games.<BR>team 3 plays 15 games at home and 15 games away. However, four of those games (against team 1 and 2) are already counted. So that's 26 games.<BR><BR>If there are 16 teams, they play:<BR>30+28+26+....0<BR>Which is ((2n-2)!), I think. Can someone else confirm that - brain hurts.<BR><BR>Craig.
9. Senior Member
Join Date
Dec 1969
Posts
973
## RE: looks like 16!
team 1 plays 15 games at home and 15 games away. <BR>team 2 plays 15 games at home and 14 games away<BR>team 3plays 15 games at home and 13 games away<BR><BR><BR>30,29,28.... ?
10. Senior Member
Join Date
Dec 1969
Posts
16,931
## Order?!
Order's not a problem.<BR><BR>Wait a second. Surely this is a statistics function? Isn't it that aPb and aCb thing, isn't it? The number you're after is the total number of UNIQUE combinations of two teams (if you care about home/away), or the total number of combinations (no duplicates), if you don't care about home/away.<BR><BR>Craig.
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• | 926 | 3,119 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2016-44 | latest | en | 0.952844 |
https://answerofmath.com/solved-how-to-know-whether-pearsons-or-spearmans-correlation-is-better-to-use/ | 1,679,615,588,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945218.30/warc/CC-MAIN-20230323225049-20230324015049-00312.warc.gz | 143,424,086 | 19,726 | # Solved – How to know whether Pearson’s or Spearman’s correlation is better to use
I have this question that is really confusing. I already know that Pearson is used for normal distribution and Spearman for the opposite, but how do we apply this to the question? As you can see, question "b" asks for that:
Contents
To me, Spearman's \$rho\$ is the first choice because
• it doesn't assume linearity
• it is resistant to outliers
• its statistical test is more powerful than the linear correlation test if you average over the distributions you're likely to see in practice
• it handles ordinal data that are not interval-scaled
Looking at the data to drive which statistic you use will change the operating characteristics including invalidating confidence interval coverage.
So to me the big question is what would shake me off the default position of using \$rho\$? The only answer I can think of is when you need to think about variation explained on the original data scale. \$r^{2} = R^{2}\$ in the \$p=1\$ case, and is the fraction of variance in \$Y\$ explained by \$X\$. We don't have a similar interpretation for \$r\$. But there are other natural interpretations of \$rho\$ related to concordance probabilities, and in some ways such probabilities are also natural.
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http://present5.com/liberty-tax-service-online-basic-income-tax-course-2/ | 1,542,814,913,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039749054.66/warc/CC-MAIN-20181121153320-20181121175320-00433.warc.gz | 269,945,572 | 18,906 | Скачать презентацию Liberty Tax Service Online Basic Income Tax Course
d3a47985d4b01e2f858a4c99043c7e84.ppt
• Количество слайдов: 70
Liberty Tax Service Online Basic Income Tax Course. Lesson 5 1
HOMEWORK CHAPTER 4 HOMEWORK 1: Answer the questions for each situation. 1. Frankie is employed by Sherman Bros. Insurance. In 2008, his salary was \$48, 500. Frankie participates in the 401(k) retirement plan his employer has set up. In 2008, Frankie contributed \$4, 000 to the plan. Frankie also received the following from his employer in 2008: § § A Florida vacation worth \$3, 264 as a prize for meeting his sales goals. Frankie was not able to take the vacation until January 2008. \$200 tickets to a pro basketball game as a Christmas present. Group health insurance premiums valued at \$3, 400. \$2, 500 as reimbursement for his travel costs. Frankie does not have to account to Sherman Bros. for the reimbursements or return any money he does not spend. a. Which of the above 4 items are taxable to Frankie in 2008? Florida vacation, pro basketball tickets, travel cost reimbursements b. What is the amount of employee compensation that will be shown in box 1 of Frankie’s Form W-2? \$50, 464 which consists of: \$48, 500 (wages) - \$4, 000 (401(k) contribution) + \$3, 264 (vacation) + \$200 (tickets) + \$2, 500 (travel) 2
HOMEWORK CHAPTER 4 HOMEWORK 1: Answer the questions for each situation. 2. Cameron Smith worked for a cab company for 6 months in 2008. He received the following Form W-2. 3
HOMEWORK CHAPTER 4 HOMEWORK 1 Cameron reported his tips as required to his employer. The monthly totals in his tip diary are as follows: January \$34 February \$20 March \$18 April \$25 May \$16 June \$28 a. What is the total amount of tips that Cameron reported to his employer? \$107 ( \$34 + \$20 +\$25 + \$28 = \$107) b. What is the total amount of Cameron’s tips that is subject to income tax? \$141 (\$107 + \$18 + \$16 = \$141) c. What amount does Cameron enter on Form 1040, line 7? \$ 10, 357 (\$10, 323 + \$18 + \$16 = \$10, 357) a. How much is his federal withholding? \$1, 032 4
HOMEWORK CHAPTER 4 HOMEWORK 2: Fill out page 1 of Form 1040 through line 7 and page 2, line 62, for the following using the information and forms provided. 1. Austin L. (SSN 032 -78 -6543, born 5/16/1975) and Felicity N. Geary (SSN 044 -65 -4321, born 7/18/1977) are married and lived together in 2008 at 14 Shady Way, Brooklyn, NY 11201. Their son Ronnie (SSN 511 -33 -9999, born 8/25/1996) and their daughter Darlene (SSN 511 -51 -1111, born 5/31/2000) are qualifying children for the child tax credit. 5
HOMEWORK CHAPTER 4 6
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HOMEWORK CHAPTER 4 2. Craig R. Gregory (SSN 333 -98 -7654, born 9/24/1973) is divorced. He keeps up a home for himself and his son Barry (SSN 233 -32 -3232, born 6/7/1999), who lives with him. Craig has stated in writing that his former wife can claim the exemption for Barry. Craig pays 70% of the total support for his widowed stepfather Lucian Alexander (SSN 277 -77 -8787, born 12/12/1935), who does not live with him. Lucian’s gross income in 2008 was \$2, 400. On some weekends, Craig works as a waiter at a banquet hall. In January his tips were \$17 and in March his tips were \$19 in cash plus two tickets to the Bulls game (value \$55). He did not report these tips to his employer. For every other month, Craig received \$20 or more in tips and he reported these to his employer. 11
HOMEWORK CHAPTER 4 12
HOMEWORK CHAPTER 4 13
HOMEWORK CHAPTER 4 14
Chapter 5: Interest, Dividends, and Other Income Chapter Content Taxable Interest U. S. Savings Bonds Tax-Exempt Interest Dividends Taxable State and Local Income Tax Refunds Alimony Received Unemployment Compensation Other Income Key Ideas Objectives Determine the Reporting of Interest and Dividends Know How to Distinguish Between Taxable and Tax-Exempt Interest and Dividends Understand How to Report Taxable Refunds, etc. , From State and Local Income Taxes Identify Taxable Alimony and How to Report It Learn About Unemployment Compensation and How to Report It Report Other Sources of Taxable Income Identify when Back up Withholding is Required 15
Key Terms and Definitions Earned Income – All amounts received from providing a service, including wages, tips, bonuses and self-employment income in the form of money, services or property. Investment income, such as dividends and interest, is not counted as earned income. Unearned Income – Money received for the investment of money or other property, such as interest, dividends and royalties. It also includes pensions, alimony, unemployment compensation and other income that is not earned for services performed. Taxable Interest – Includes interest you receive from bank accounts, loans you make to others, and other sources. Tax-Exempt Interest - Interest income that is not subject to income tax. Tax-exempt interest income is earned from bonds issued by states, cities, or counties and the District of Columbia. 16
Key Terms and Definitions Dividends – A stockholder’s share of the profit paid on an investment in a corporation reported on Form 1099 -DIV. “Dividends” from a savings and loan association or from a credit union are actually reported as interest. Capital Gain Distribution – Shareholder’s portion of gain from the sale of capital assets, such as mutual funds and real estate investment trusts. Capital gain distributions are taxed in the year received and are always considered to be held long term. Unemployment Compensation – Includes benefits to unemployed individuals that a state or the District of Columbia paid from the Federal Unemployment Trust Fund. 17
Reporting Interest Income and Dividends ü ü Interest income and dividends are common types of unearned income. They are considered unearned income because money, and not a person, is working to earn the income. ü Most types of interest and dividends are taxable. All interest and dividends must be reported on your tax return. Interest over \$10 is usually reported to you on a Form 1099 -INT. Dividends over \$10 are reported to you on a Form 1099 DIV. ü Substitute Forms 1099 -INT and 1099 -DIV may also be used ü Interest and dividends over \$1, 500 must be reported on Schedule B of Form 1040. ü Taxable interest is reported on line 8 a of Form 1040. ü Tax-exempt interest is reported on line 8 b. 18
Reporting Interest Income and Dividends Form 1099 -INT 19
Reporting Interest Income and Dividends Form 1099 -DIV 20
Taxable Interest Most interest income is taxable. In general, any interest that you receive or that is credited to your account and can be withdrawn is taxable income. Sara earned \$49 in interest on money in her bank savings account. She must report the \$49 as interest income even though she did not withdraw it from the bank. 21
Taxable Interest Taxable interest income includes interest from: ü bank accounts ü interest on loans you make to others ü interest from most other sources. Report the total taxable interest income on line 8 a of Form 1040. If your total interest income is more than \$1, 500, Part I and III of Schedule B must be completed. 22
Taxable Interest Zachary received \$1, 500 in interest income from BBT Bank this year. He will report the \$1, 500 directly on line 8 a of Form 1040. If his interest income from BBT Bank were \$1, 501, he would also need to report the \$1, 501 on Part I of Schedule B as in the second example below. Form 1040, Page 1 23
Taxable Interest Some of the other common sources of taxable interest come from interest on: ü Certificates of deposit (CDs) ü Deposits or share accounts from credit unions, mutual savings banks, cooperative banks, and federal and domestic savings and loan associations ü U. S. obligations such as U. S. Treasury bills, notes and bonds ü U. S. savings bonds ü Installment sale payments ü Life insurance proceeds remaining with the insurance company ü Tax refunds ü Gifts for opening accounts 24
U. S. SAVINGS BONDS Interest on U. S. Savings Bonds is reported in box 3 of Form 1099 -INT. If you use the cash method of accounting, as most individual taxpayers do, you generally report interest on U. S. savings bonds in the year that you receive it. There are three types of U. S. Savings Bonds: HH bonds EE bonds I bonds ü HH Bonds are purchased at face value. Interest is paid semiannually. You report the interest as income in the year it is received. ü EE Bonds are purchased at a discount and the interest on these bonds is taxed when the bond is redeemed. The taxable interest is the difference between the purchase amount and the redemption value. ü I Bonds are newer U. S. bonds. They are purchased at face value and interest is paid at maturity. You can report the interest on series EE, series E, and series I bonds in either of the following ways: Method 1. Report the interest at maturity of the bond or when you cash it. Method 2. Report the increase in redemption value as interest each year. 25
U. S. SAVINGS BONDS Sometimes U. S. Savings Bonds are owned by more than one person. Table 5 -1 clarifies who pays tax on U. S. Savings Bond interest. Interest on U. S. savings bonds is exempt from state and local taxes Table 5 -1. Who Pays the Tax on U. S. Savings Bond Interest 26
OTHER INTEREST CERTIFICATE OF DEPOSIT (CD) 1. If you buy CDs with maturity of more than one year, include part of the interest as income each year. 2. Early withdrawal penalty is reported in box 2 of Form 1099 -INT and reported on line 30 of Form 1040. 3. Early withdrawal penalty is for withdrawing money from CDs or other time -deposit savings accounts before the maturity date. It is a forfeit of some of the interest paid 27
OTHER INTEREST LIFE INSURANCE PROCEEDS Life insurance proceeds paid to a beneficiary are not usually taxable unless the benefits received are more than the amount that would have been payable at time of insured person’s death. (Interest can accrue before distribution) 28
OTHER INTEREST TAX REFUNDS Interest received on tax refunds is taxable income. GIFT/OPENING AN ACCOUNT The fair market value of a gift or service you receive for opening an account in a savings institution must be reported as interest in the year you receive it. 29
OTHER INTEREST Tom has a CD that matures in January 2009 and paid \$75 in interest in 2008. Does Tom include the \$75 interest as income on his 2008 return? Yes or No? 30
OTHER INTEREST Tom has a CD that matures in January 2009 and paid \$75 in interest in 2008. Does Tom include the \$75 interest as income on his 2008 return? Yes 31
TAX-EXEMPT INTEREST Some types of interest are exempt from federal income tax. If you are required to file a return you must show any tax-exempt interest you receive for informational purposes only. Tax-exempt interest is reported on line 8 b of Form 1040 and generally you will not receive a Form 1099 INT. 32
DIVIDENDS ü Distributions of money, stock, or other property paid to you by a corporation. ü Report dividends over \$1, 500 on Part II of Schedule B. ü Dividends may also come from a partnership, estate, trust, or an S corporation and be reported to you on a Form K 1. ü Major types of dividends are: ordinary dividends, capital gain distributions, nondividend distributions, and other distributions. 33
DIVIDENDS Ordinary dividends are taxable income and are paid out of the earnings and profits of a corporation. They are NOT capital gains. 34
DIVIDENDS Schedule B, Part II 35
DIVIDENDS Pauline Adams received a Form 1099 -DIV for \$1, 646 from NY Money Market Fund. 36
DIVIDENDS Qualified dividends are the ordinary dividends that are subject to the same 0% or 15% maximum tax rate that applies to net capital gain. Qualified dividends should be shown in box 1 b of Form 1099 -DIV. Capital gains will be covered in Chapter 11. 37
DIVIDENDS Capital Gain Distributions Mutual funds pass capital gains to investors as capital gain distributions. Capital gain distributions are reported in box 2 a of 1099 -DIV and are reported directly on line 13 of Form 1040 if a Schedule D is not required. 38
DIVIDENDS Corey does not have to file a Schedule D. He received a capital gain distribution of \$695 from JTH investments in 2008. Form 1040, Page 1 39
DIVIDENDS Nondividend distributions A return of capital is reported to you on box 3 of Form 1099 -DIV. Report the return of capital as a capital gain once your basis has been reduced to zero. 40
DIVIDENDS Jesse purchased stock in 1996 for \$4, 000. He received a return of capital of \$500 on the stock in 1999. Jesse reduced his basis in the stock to \$3, 500 (\$4, 000\$500). In 2008, he received a return of capital of \$4, 500. Since he only had a basis of \$3, 500, his basis was reduced to zero. What is the taxable amount for 2008? A. \$4, 500 B. \$3, 500 C. \$1, 000 41
DIVIDENDS § Jesse purchased stock in 1996 for \$4, 000. He received a return of capital of \$500 on the stock in 1999. Jesse reduced his basis in the stock to \$3, 500 (\$4, 000 -\$500). In 2008, he received a return of capital of \$4, 500. Since he only had a basis of \$3, 500, his basis was reduced to zero. What is the taxable amount for 2008? C. \$1, 000 42
OTHER DISTRIBUTIONS § Alaska Permanent Fund Dividends are not dividends. They are reported on line 21 of Form 1040 as other income. 43
BACKUP WITHHOLDING ON INTEREST AND DIVIDENDS § Interest and dividends are generally not subject to withholding. However, if you fail to give the payer your social security number or you give an incorrect number, the payments are subject to mandatory withholding (backup withholding). § Backup withholding will be shown in box 4 of Form 1040 -INT or Form 1040 -DIV. 44
TAXABLE STATE AND LOCAL INCOME TAX REFUNDS State or local income tax refunds, etc. Reported on Form 1099 -G. If you itemized deductions and claimed state and local income taxes as an itemized deduction and receive a state or local refund, generally reported on line 10 of Form 1040. If you either took standard deduction or claimed state and local general sales taxes as a deduction the state or local tax refund is not taxable. 45
TAXABLE STATE AND LOCAL INCOME TAX REFUNDS In 2008, Natalie, age 42, received a Form 1099 -G for her state refund of \$325. Her total itemized deductions in 2007 were \$7, 200. Her filing status was single. Using the worksheet on the next page, Natalie will report \$325 on line 10 of her Form 1040. 46
TAXABLE REFUNDS, CREDITS, OR OFFSETS OF STATE AND LOCAL INCOME TAXES Form 1040, Page 1 47
ALIMONY RECEIVED ü ü Report alimony received on line 11 of Form 1040. Child support payments are NOT alimony. Use Table 5 -2 (page 5 -13) to clarify alimony requirements Child support payments are NOT alimony. If you make child support payments, do not deduct them. If you are receiving child support payments, you do not have to include them as income. 48
ALIMONY RECEIVED The following table shows what is, and what is not considered alimony. Table 5 -2 Alimony Requirements (Instruments Executed After 1984) 49
ALIMONY RECEIVED Wanda receives \$250 per month of alimony. What amount is reported on Form 1040? A. \$250 B. \$3, 000 C. \$2, 500 50
ALIMONY RECEIVED Wanda receives \$250 per month of alimony. What amount is reported on Form 1040? B. \$3, 000 She reports the \$3, 000 (\$250 x 12) on line 11 of Form 1040. 51
UNEMPLOYMENT COMPENSATION Unemployment compensation is taxable. ü Reported to you on Form 1099 -G ü You report it on line 19 of Form 1040. 52
UNEMPLOYMENT COMPENSATION Anthony Sanders (073 -22 -8990) received \$463 in unemployment compensation in 2008 and it was reported on the following Form 1099 -G. He records the amount from box 1 on line 19 of Form 1040, Page 1 53
OTHER INCOME Report other income not covered on lines 7 -20 b on line 21 of Form 1040. 1. Includes prizes, awards, lottery winnings, and jury duty 2. Gambling income includes winnings from lotteries, raffles, horse and dog races and casinos, as well as the fair market value of prizes such as cars, houses, trips or other noncash prizes. Gambling winnings of \$600 or more are reported on Form W-2 G. Describe the type of income on dotted line next to line 21. 54
OTHER INCOME Donna won \$2, 500 in her state lottery. It was reported to her on a Form W-2 G. Line 21 of Form 1040 is shown below. Form 1040, Page 1 55
WITHHOLDING ON GAMBLING INCOME § Certain types of gambling winnings are subject to mandatory withholding. Generally, tax will be withheld from winnings of more than \$5, 000. All gambling winnings are taxable even if you do not receive a Form W-2 G. 56
VOLUNTARY WITHHOLDING § If you receive income from unemployment compensation, you can choose to have income tax withheld from the payments. To make this choice, you will have to fill out a Form W-4 V, Voluntary Withholding Request. § Box 4 of Form 1099 -G, Certain Government Payments, shows the taxes withheld from your unemployment compensation. For unemployment compensation, the payer is permitted to withhold 10% from each payment. No other percentage or amount is allowed. 57
Interest, Dividends, and Other Income KEY IDEAS ü ü ü ü Report interest and/or dividend income over \$1, 500 on Schedule B, Form 1040. Report early withdrawal penalties on line 30 of Form 1040. Report capital gain distributions on lines 13 of Form 1040 if you are not required to file Schedule D. State and local tax refunds are included in taxable income if you itemized deductions for the refund year and you received a tax benefit by including the state and local tax in itemized deductions. Alimony and separate maintenance payments are taxable income to the recipient of these payments and reported on line 11 of Form 1040. Other income, such as prizes, awards, gambling winnings, and jury duty pay, is reported on line 21 of Form 1040; include the amount and a description of the income. You can choose to have income tax withheld from other types of income such as unemployment compensation. 58
Interest, Dividends, and Other Income CLASSWORK 1: True or False 1. Interest income of less than \$10 is not required to be reported. 2. Total interest income of more than \$1, 500 must be reported on Schedule B, Part I. 3. Report a dividend from a credit union of less than \$1, 500 on line 8 a of Form 1040 if you have no other similar income. 4. Interest credited to a savings account is unearned income. 5. Interest on a Roth IRA is reported on line 8 b of Form 1040. 6. Interest on an EE or I bond can only be reported at the maturity date or when you cash it. 7. Interest on U. S. savings bonds is taxable on the state return. 8. If money is withdrawn from a CD before the maturity date and you forfeited some of the interest paid, report this amount on line 30 of Form 1040 as an early withdrawal penalty. 59
Interest, Dividends, and Other Income CLASSWORK 1: True or False 9. Interest received on tax refunds is not taxable income. 10. Life insurance proceeds you receive as a beneficiary are usually not taxable. 11. Jury duty pay is reported as income on line 21 of Form 1040. 12. Unemployment compensation is not taxable. 13. Alimony payments you receive are reported on Form 1040, line 11 and are taxable. 14. Child support payments you make can be deducted from your total income. 15. A state income tax refund is reported to you on Form 1099 -G. 16. All state tax refunds are nontaxable. 60
Interest, Dividends, and Other Income CLASSWORK 1: True or False 17. Income from the Alaska Permanent Fund is reported on line 9 a of Form 1040. 18. Interest and dividends are considered unearned income. 19. Gambling winnings are not taxable. 20. Child support payments are required to be reported on line 21. 61
Interest, Dividends, and Other Income CLASSWORK 1: True or False 1. Interest income of less than \$10 is not required to be reported. F 2. Total interest income of more than \$1, 500 must be reported on Schedule B, Part I. T 3. Report a dividend from a credit union of \$1, 500 or less on line 8 a of Form 1040 if you have no other similar income. T 4. Interest credited to a savings account is unearned income. T 5. Interest on a Roth IRA is reported on line 8 b of Form 1040. F 6. Interest on an EE or I bond can only be reported at the maturity date or when you cash it. F 7. Interest on U. S. savings bonds is taxable on the state return. F 8. If money is withdrawn from a CD before the maturity date and you forfeited some of the interest paid, report this amount on line 30 of Form 1040 as an early withdrawal penalty. T 62
Interest, Dividends, and Other Income CLASSWORK 1: True or False 9. Interest received on tax refunds is not taxable income. F 10. Life insurance proceeds you receive as a beneficiary are usually not taxable. T (only interest earned after death is taxable) 11. Jury duty pay is reported as income on line 21 of Form 1040. T 12. Unemployment compensation is not taxable. F 13. Alimony payments you receive are reported on Form 1040, line 11 and are taxable. T 14. Child support payments you make can be deducted from your total income. F 15. A state income tax refund is reported to you on Form 1099 -G. T 16. All state tax refunds are nontaxable. F 63
Interest, Dividends, and Other Income CLASSWORK 1: True or False 17. Income from the Alaska Permanent Fund is reported on line 9 a of form 1040. F 18. Interest and dividends are considered unearned income. T 19. Gambling winnings are not taxable. F 20. Child support payments are required to be reported on line 21. F 64
Interest, Dividends, and Other Income CLASSWORK 2: Tom and Sally are filing a joint return. They have the following interest and/or dividend income: Nations Savings and Loan (joint) \$1, 390 First Federal Bank (Sally) \$125 Citizens Credit Union (Tom) \$40 U. S. Series HH Savings Bonds (joint) \$45 Pullman Mutual Fund (joint) \$1, 590 Determine the total amount of interest and/or dividend income and where it is reported on Form 1040 or other tax form. 65
Interest, Dividends, and Other Income Classwork 2 Nations Saving and Loan \$1, 390 First Federal Bank 125 Citizens Credit Union 40 U. S. Series HH Savings Bonds 45 \$1, 600 would be reported on Schedule B, Part I and line 8 a of Form 1040. Pullman Mutual Fund \$1, 590 would be reported on Schedule B, Part II and line 9 a of Form 1040. 66
Interest, Dividends, and Other Income CLASSWORK 3: Assuming no other interest or dividend income, where are the following reported on the Federal tax return? (line of Form 1040 and the appropriate schedules, if any. ) Ordinary dividends over \$1, 500 Credit union dividends over \$1, 500 Savings account interest of \$8 Tax-exempt interest of \$325 U. S. Treasury notes interest of \$1, 150 Early withdrawal penalty on a CD of \$750 Capital gain distributions of \$2, 333 Alaska Permanent Fund dividends of \$68 67
Interest, Dividends, and Other Income CLASSWORK 3: Assuming no other interest or dividend income, where are the following reported on the Federal tax return? (line of Form 1040 and the appropriate schedules, if any. ) Ordinary dividends over \$1, 500 - reported on Schedule B, Part II and line 9 a of Form 1040 Credit union dividends over \$1, 500 – reported on Schedule B, Part I and on line 8 a of Form 1040 Savings account interest of \$8 – reported on line 8 a of Form 1040 Tax-exempt interest of \$325 – reported on line 8 b of Form 1040 68
Interest, Dividends, and Other Income CLASSWORK 3: U. S. Treasury notes interest of \$1, 150 – reported on line 8 a of Form 1040 Early withdrawal penalty on a CD of \$750 – reported on line 30 of Form 1040 Capital gain distributions of \$2, 333 – reported on line 13 of Form 1040 and on Schedule D, (if required to file a Schedule D) Alaska Permanent Fund dividends of \$68 - reported on line 21 of Form 1040 69 | 6,067 | 23,846 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2018-47 | longest | en | 0.939908 |
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# Elasticities and WelfareLesson 6
## Price Elasticity of Demand
1. Price elasticity of demand - the sensitivity of quantity demanded to a change in price. Price elasticity is
The symbol delta, D means change, percent change in price is:
and percent change in quantity demanded is:
We can use algebra to reduce the equation to:
Note - The DQ over DP is the inverse of a slope; Elasticity does not equal a line's slope
Note - Economists change frequently the denominator of the fractions for percent change. What number should you divide with? Should you divide by the initial point (P1, Q1) or the final (P2, Q2), or an average of these two points.
• Elasticity has no units!
• Can compare apples to oranges
• Also called elasticity coefficient
• The elasticity of demand has a minus sign; it shows the Law of Demand, where price and quantity have an inverse relationship.
• Some economists drop the minus sign
• Exam questions drop the minus sign
• Three categories
1. Inelastic demand elasticity - quantity demanded is not sensitive to changes in market price
2. Unitary elastic demand elasticity - if market price decreases by 1%, then quantity demanded increases by 1%
3. Elastic demand elasticity - quantity demanded is sensitive to changes in market price
• Example:
1. ED = -0.25 for coffee
• If the price of coffee decreases by 1%, then quantity demanded increases by 0.25%, vice-versa
2. ED = -1 for movies
• If the price of movies increases by 1%, then quantity demanded decreases by 1%, vice-versa.
3. ED = -4.0 for air travel
• If the price of air travel decreases by 1%, then quantity demanded increases by 4%, vice-versa.
• Note: Can multiply elasticities by a number
1. ED = -0.25 for coffee
• If the price of coffee decreases by 10%, then quantity demanded increases by 2.5% (vice-versa).
2. ED = -4.0 for air travel
• If the price of air travel decreases by 10%, then quantity demanded increases by 40% (vice-versa).
2. Calculations
• Example 1 - a college raises its tuition from \$20,000 to \$25,000 and students enrollments falls from 10,000 to 8,000.
• Compute the price elasticity of demand.
• Is it elastic or inelastic
• Average price = \$22,500
• Average quantity = 9,000
• Elasticity = 1
• Unitary elastic
3. Determinants of price elasticity of demand
1. Substitution Effect
• Elastic goods tend to have many substitutes
• Inelastic goods tend to have few substitutes
• Cigarettes, gasoline, and alcohol
2. Income Effect
• Elastic goods tend to take a large portion of income.
• Cars, computers, an houses.
• Inelastic goods tend to take a small portion of income.
• Matches, toothpicks, and salt.
3. Luxury versus necessity
• Luxury goods tend to be elastic
• Consumers may be sensitive to price in buying expensive clothes, jewelry, etc.
• Necessity goods tend to be inelastic
• Person needs heart medication
4. Time
• Second Law of Demand
• Goods are more elastic in the long run than the short run
• More time to adjust to price changes
• Example - During 1970's, OPEC cut back on production of oil (supply curve shifted left)
• Petroleum and gasoline prices increased
• Short run:
• Quantity demanded dropped very little (inelastic)
• Long run:
• Bought fuel efficient Japanese cars
• Moved closer to work
• Quantity demanded dropped significantly (more elastic)
• U.S. car manufacturers were hurt in the 1980s, because they could not make small cars
4. Demand function has two forms
Nonlinear demand functions - have a slight curvature to them
• Nonlinear - not a straight line
• Have constant elasticity at any point along function
• P = b Qa
• ED = a
Elastic Demand Curve Inelastic Demand Curve
Price Price
Quantity Quantity
• Relatively Elastic - any movement along this line has a constant elasticity
• Tend to be flat
• A small change in price leads to a large change in quantity demanded
-µ < ED < 1
• Relativity Inelastic - any movement along this curve has a constant elasticity:
• Tend to be steep
• A large change in price leads to a small change in quantity demanded
- 1< ED < 0
Unitary Elastic
Price
Quantity
Unitary Elasticity- any movement along this demand curve always has an elasticity of -1.
ED = -1
Linear demand functions- a straight line
• Have an elasticity that ranges from 0 to negative infinity
• P = b - a Q
• ED = 1 - b / (a*Q)
• Has two exceptions
• Perfectly inelastic - vertical demand function
• Quantity demanded does not respond to changes in price
• Perfectly elastic - horizontal demand function
• Quantity is perfectly sensitive to a change in market price
Linear Demand Function
Price
Quantity
Linear (Straight line) Demand Curve
-¥< Ed < 0
Perfectly Elastic Perfectly Inelastic
Price Price
Quantity Quantity
Slope: a = 0 and ED = -¥
Slope: a= ¥ and ED = 0
## Total Revenue and Price Elasticity
1. Total revenue (TR)- consumers pay revenue to a business for a product or service.
Total Expenditures = Total Revenue (TR) = Q P
Example: Consumers buy 1 million pizzas for \$10 each, so total expenditures = \$10 million. The firms collect this money so total revenue is \$10 million. Total revenue is an area under the demand function. Shown below:
• If the the market price decreases to \$5 per pizza, then pizza producers collect a new rectangle for total revenue, which is light blue and yellow rectangles.
• Because the light blue rectangle is common to both revenues for both prices, we can ignore it.
• The lower price causes a loss of the green rectangle and a gain in the yellow rectangle, causing revenue to decrease.
• We are assuming pizza is an inelastic good.
Linear Demand Function
Price
Linear Demand Function
P = b - a Q
Intercepts
If Q = 0, then P = b
If P = 0, then Q = b / a
Quantity
Total Revenue
Two Equations:
TR = P Q and P = b - a Q
Substitute demand function into
total revenue function
TR = (b - a Q) Q
TR = b Q - a Q2
If Q = 0, then TR = 0
If Q = b / a, then TR = 0
max. TR where Q = b / 2 a
Quantity
2. Conclusion:
• When price, P, increases, quantity demanded, Q, decreases. Change in total revenue is an interaction between P and Q. How TR changes depends on elasticity.
1. If demand is inelastic, an increase in price will cause total revenue to increase (and vice-versa).
2. If demand is elastic, a decrease in price will cause total revenue to increase (and vice-versa).
3. If demand is unitary elastic, an increase in price will cause no change in total revenue. (The price increase exactly offsets the decreases in quantity demanded).
• Example 1
• Demand for higher education is elastic. Oklahoma State University wants to maximize total revenue from the students.
• If OSU increases tuition, total revenues will decrease.
• If OSU decreases tuition, total revenues will increase.
• Price decreases a little, but quantity demanded increases a lot.
• Example 2
• Cigarettes are inelastic. Firms want to maximize total revenue from sells.
• If firms increase price, then total revenue increases.
• Price increases a lot, but quantity demanded decreases a little.
## Demand Elasticities
1. Income elasticity - indicates the responsiveness of the demand for a product to a change in income. Income elasticity is:
• Normal goods- goods with income elasticity of demand > 0 (i.e. positive).
• As income increases, the demand for normal goods will rise.
• Demand curve shifts right!
• Necessity 0 < EI < 1
• Food, EI = 0.51
• As income increases by 1%, then demand for food increases by 0.51 %.
• Luxury EI => 1
• New cars, EI = 2.45
• As income increases by 1%, then demand for cars increases by 2.45 %.
• Inferior goods- goods with a income elasticity < 0 (i.e. negative).
• As income increases, the demand for inferior goods will decrease.
• Demand curve shifts left!
• Margarine is -0.20
• As income increases by 1%, then demand for margarine decreases by 0.20%.
• Rice, bus travel, etc.
As a country becomes richer (higher income), then the production of normal goods will expand, while production for inferior goods will decrease!
2. Cross Price Elasticity - can determine if demands for two products are related. Products are defined as X and Y.
• If EXY> 0, then products X and Y are substitutes
• Example: EXY = 0.5 and the products are steak and chicken; if the price of chicken increases by1%, then demand for steak increases by 0.5 percent.
• If EXY= 0, then products are not related
• If EXY < 0, then products X and Y are complements
• Example: EXY = -0.9 and the products are DVDs and DVD players; if the price of DVD players increase by 1%, then demand for DVDs fall by 0.9%
Economists look at cross price elasticities to determine if products are in the same market or in different markets. Further, government officials can use these elasticities to determine if a monopolist has any substitutes for his product or service.
## Price Elasticity of Supply
1. Analogous to the price elasticity of demand. Price elasticity of supply is:
The price elasticity of supply will be positive because of the Law of Supply.
• Elasticity is similar to demand
• If ES < 1, then supply elasticity is inelastic
• If ES = 1, then supply is unitary elastic
• If ES > 1, then supply is elastic
• Elasticity is related how fast producers can expand production
• Short Run - firms do not have enough time to change plant size
• Supply tends to be inelastic
• Inelastic - is not sensitive to price changes
• Long Run - firms have enough time to change plant size
• Supply tends to be more elastic
• Elastic - is sensitive to price changes
• Example: The price of a Honda Civics increases
• Short-run, Honda can produce more cars by using more labor and resources
• Long-run, Honda can build additional factories
Elastic Supply Curve Inelastic Supply Curve
Price Price
Quantity Supplied Quantity Supplied
## Social Welfare
1. Consumer Surplus - the area below the demand curve but above the actual price paid.
• Measure of social welfare.
• An aggregate benefit to all consumers in the market.
• The market price of coffee is \$1.50 and consumers buy 15 (million) pounds of coffee.
• I place a \$2.50 value on this soda, but bought it for \$1.50
• I received a benefit of \$1.00
• If the market price of the soda decreases to \$0.50, consumers' surplus increases!
• Social welfare increases
Demand for Coffee Demand for Coffee
Price Price
Quantity (in thousands) Quantity (in thousands)
2. Producer Surplus - the area above the supply curve but below the actual sales price.
• Measure of social welfare
• An aggregate benefit to all producers in the market
• Producers' surplus is total fixed costs + profits
Supply of Coffee
Price
Quantity (in thousands)
3. Social Welfare is the sum of consumer plus producers' surpluses
Supply of Coffee
Price
Quantity (in thousands)
## The Impact of a Tax
1. Tax incidence - how the "economic" burden of tax is shared between buyers and sellers.
• Statutory incidence of tax - the legal assignment of who pays a tax; i.e. who sends the taxes to the government
• Tax incidence and statutory incidences differ.
2. Example: Gov. places a \$1 tax on each pizza sold on pizza producers.
• Statutory incidence falls on producers.
• Tax rate - the per-unit tax
• \$1 per pizza.
• Do not use percent tax!
• Percent tax changes the slope of the supply function
• Supply curve shifts left by exactly \$1.
• Market price was at P*, \$10 per pizza. New price, Pt, does not equal \$11.
• Price lies between \$11 and \$10.
• The tax changes consumer's behavior.
• New market price is higher (price + tax)
• Tax base - the total amount of goods, which are taxed.
• The higher the tax rates, the smaller the tax base.
• Tax rates change consumers' and producers' behavior and thus the size of the tax base
• Tax revenue from pizza = Qt X \$1.
• (area of a rectangle width X height)
• blue area + yellow area.
• "Yellow area" - actual tax burden on sellers.
• "Blue area"- actual tax burden on buyers.
• Deadweight loss of taxation - the red area.
• "Excess burden of taxation"
• A loss to society, because government interfered with the market
Pizza Market
Price
Quantity
• What if the \$1 pizza tax was placed on the buyer.
• (switching statutory incidence from sellers to buyers).
• The buyers send the tax to the government
• The demand curve will shift to the left by \$1, but the end result is exactly the same!
• Statutory incidence of tax changed, but the tax burden remained the same.
Pizza Market
Price
Quantity
• Inelastic demand (relative to supply) - quantity demanded is not sensitive to price changes.
• Tax burden falls more heavily on consumers
• Gov. loves to tax inelastic goods
• Gasoline (short-run)
• Beer / liquor
• Cigarettes
## Terminology
price elasticity of demand elastic demand inelastic demand unitary elastic demand nonlinear demand function linear demand function perfectly inelastic demand perfectly elastic demand total revenue test (TR) income elasticity of demand normal goods inferior goods cross-price elasticity of demand price elasticity of supply short run long run consumers' surplus producers' surplus social welfare tax incidence statutory incidence of tax tax rate tax base deadweight loss of taxation
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# △ABC and △DEF are similar such that 2AB=DE and BC=8 cm. Find EF.
Correct Answer
A
16 cm
Your Answer
B
12 cm
Your Answer
C
8 cm
Your Answer
D
4 cm
Solution
## The correct option is A 16 cmGiven that △ABC∼△DEF. If two traingles are similar then their corresponding sides are proportional. EFBC=DEAB EF8=2ABAB=2 EF=2×8 cm EF=16 cm
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Problem Sets Information Select Problem Runs Ranklist
ZOJ Problem Set - 2114
Transportation Network
Time Limit: 4 Seconds Memory Limit: 32768 KB
A transportation network consists of a central node and some other nodes directly or indirectly (through some intermediate nodes) connected to the central node, and also the transporting channels which connect them. It is known that for any node in the transportation network there is exactly one path leads to the central. Every channel has its own length, the distance between any two nodes is defined as the total length their shortest path.
For example, in the following transportation network (the numbers in the parentheses represents the length of the channels):
1
/\
(2) / \ (3)
2 3
(2)| |(2)
| |
4 5
The distance between 2 and 3 is 5, the distance between 1 and 4 is 4, and the distance between 3 and 5 is 2.
Two transportation network can be combined, combining network A to network B means building a channel that connects the central of A and the central of B, and take the central of B as the central of the combined network. That combined network contains all the nodes and channels that is in network A or in network B.
Country-A has N (3 <= N <= 20,000) nodes that needs transporting between. Initially there is no channel between them, and each node is a solo transportation network. For the transporting the substance more expediently, Country-A has decided to build some transporting channels.
To maintain the information of the networks, Country-A has bought a super computer, which can receive and execute two kinds of instructions:
1) Q i j (1 <= i, j <= N, i <> j)
Querying the distance between node i and node j in the network. If they are not in the same network, output "Not connected." (Without quotation), output the distance otherwise.
2) U i j l (1 <= i, j <= N, node i and node j are not in the same network)
Combine the network containing i to the network containing j, makes the channel between them be l. (l is an integer, and 0 < l <= 1,000)
To avoid hackers' invasion which may cause the leak of the information, you should input the instruction U i' j' l instead of U i j l (where i' = (i + last_result) mod n + 1, j' = (j + last_result) mod n + 1 here last_result is the result of last valid query (i.e. the query that i and j are in the same network). Initially last_result = 0.
Your task is to write the program for the super computer.
Input
The first line of the input is an integer X (0 < X <= 6) represents number test cases of this problem. Then X blocks each represents
a test case.
The first line of each block contains two numbers N and M (5 <= M <= 40,000) representing country-A has N transportation nodes and the super computer will receive M instructions. Then M lines each represents an instruction, has the format Q i j or U i' j' l, which has been described above.
There're NO breakline between two continuous test cases.
Output
For each querying output one line an integer represents the distance or "Not connected." (Without quotation)
There're NO breakline between two continuous test cases.
Sample Input
2
4 6
U 3 4 4
Q 2 3
U 3 2 2
Q 1 2
U 2 3 1
Q 2 4
4 6
U 3 4 4
Q 2 3
U 3 2 2
Q 1 2
U 2 3 1
Q 2 4
Sample Output
4
6
7
4
6
7
Author: XIN, Tao
Source: Online Contest of Christopher's Adventure
Submit Status | 866 | 3,369 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.328125 | 3 | CC-MAIN-2019-35 | latest | en | 0.939166 |
neuromorphic.eecs.utk.edu | 1,544,607,557,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376823817.62/warc/CC-MAIN-20181212091014-20181212112514-00585.warc.gz | 200,473,301 | 6,815 | # Demo: The pole balancer on DANNA
James S. Plank.
Summer 2016
This is a classic application from control theory. A pole is to be balanced on a cart that can move horizontally within a fixed area. The pole has a mass on its top. The pole starts in some imbalanced starting state, at some angle from vertical, rising or falling at some velocity. The goal of the system is to apply periodic forces to move the cart left or right, to keep the pole from falling, and to keep the cart from moving beyond its boundaries.
Our goal is for our neuromorphic models to "solve" instances of the pole balancing problem. Before we get to that, though, it's good to visualize the problem. In the video below, we start with a stationary cart, with the pole just a little to the right of vertical. (The angle from the pole to vertical is 0.001 radians). If you push the play button, you'll see that the pole starts falling to the right, slowly at first, and then faster, until it reaches an angle that we deem too big (0.209 radians). When that happens, we turn the pole red, and show the failing angle:
Our simulation is set up so that every 1/50s, one can apply a force to the cart to move it either to the left or to the right. The force is always the same. The goal is that whoever is applying the force can keep the pole balanced and can keep the cart between the borders of its track.
The following video shows another example of the pole balancer. Here, the starting point is at a greater angle from vertical (0.15 radians), so to keep it from falling too much, we apply force to the right. Unfortunately, we never apply force to the left, so even though the pole never falls too much, the car rams into the right wall!
Now, to have a neuromorphic implementation solve the pole balancing problem, we need to translate instances of the problem into charge events that are input to the neuromorphic implementation. Then, we need to interpret the output charge events from the neuromorphic implementation, and turn them into input for the pole balancing problem.
Our solution works as follows. Let's use the DANNA neuromorphic model as an example. Every 1/50s, we communicate the state of the pole balancing to the DANNA system, which has been programmed by evolutionary optimization to solve the problem. This state is composed of four parameters:
1. The x position of the cart.
2. The x velocity of the cart.
3. The angle θ of the pole from vertical.
4. The velocity of how the angle θ is changing.
Our DANNA network has 12 input neurons, three for each of these parameters. The values of each parameter are split into three bins. For example, the x position of the cart is split into "Left", "Middle" and "Right." When the state of the system is input into DANNA, a fixed charge is fired into the neurons corresponding to the four parameters.
The DANNA network then runs for 100 cycles, and there are two output synapses, whose firing events are counted. If one of them fires more than the other, then the result of the simulation is to apply force to the cart, to the left. If the other fires more, then the result of the simulation is to apply force to the cart, to the right. If the two fire equally, then no force is applied.
The picture below summarizes:
Example of turning an instance of the Inverted Pendulum on a Cart into input and output that our networks can understand.
And the picture below shows a 15 X 15 DANNA network that has been programmed to solve the problem. In the picture, the neurons are light blue, and the synapses are tan. The picture shows the 12 input neurons, and the two output synapses. Yes, some of the inputs are not connected to anything -- more on that later.
### Watch DANNA Balance the Pole for a Minute
The following video shows DANNA balancing the pole for a minute. This is just a clip -- on this input, the DANNA network keeps the pole balanced within the given boundaries for over a simulated week (we stopped running the simulation).
As you play the video, you'll see the positions and velocities of the cart and the pole go through various combinations of "high/middle/low." You'll also see the various states of the DANNA network that keep the pole balanced. In the video, we highlight neurons and synapses that fire in the 100-cycle intervals that translate input into output. These are highlighted by putting a dark border around the element.
The following video shows the first second of the above video, but slowed down so that each timestep takes a second, rather than a 50/th of a second. You can use this video to walk through the annotated examples that we give below.
### Example 1 - Showing input and output with DANNA.
In the first screen shot, we show the starting state of the pole balancing simulation, plus how DANNA has reacted to the first 100 cycles. First, you can see how the state of the simulation is transmitted to the input neurons of DANNA. For example, the fact that the x value is in the "medium" state is communicated by pulsing input to the second neuron in the leftmost column.
You can also see how various neurons and synapses fire in the interior of the DANNA array. Most importantly, you can see that the synapse on the right side that corresponds to "pulse the cart to the right" has fired. Therefore, at the next interval, the cart will be forced to the right:
Very little has changed from the previous screen shot to this one; however, you can see that the cart has been forced to the right, which results in the cart having velocity along the x direction (0.192686 units per second, if you can read the tiny print at the bottom of the green panel), and the pole having velocity as well (-0.241804 radians per second).
As before, the "pulse cart to the right" synapse is firing, which means that the cart will continue to be pushed to the right at the next time step.
By now, the observant reader will have noticed that several of the input neurons aren't connected to anything. In particular, the input neurons for low x, middle x, low x velocity, middle theta, and middle theta velocity have no synapses coming out of them. That means that these input values are ignored, yet this DANNA network still solves the pole-balancing problem!
We are going to skip a time-step, and go to time-step 0.06 next:
There are two things to note about this screen shot. First, the theta velocity parameter has changed from the "middle" state to the "low" state. As such, its input neuron has changed. Second, although charge has gone through various neurons and synapses, neither output synapses is firing, so on the next step, there will be no pulse to the cart:
At time 0.10, theta's velocity has moved from the "low" state back to the "middle" state.
This is the same state of the system as in timesteps 0.00 through 0.04. However, you'll notice that unlike those timesteps, the output synapses here are not firing. Go double-check that -- in the screen shot above for states 0.00 and 0.02, the four parameters are in the same state, yet in those timesteps, the "pulse to the right" synapse is firing, and here, it is not.
The reason is that the neurons and synapses compose a form of memory from state to state, and their internal states at timestep 0.10 are different from what they were at, for example, timestep 0.00, which has caused the output synapses not to fire at timestep 0.10.
This is one of the features that makes our spiking neuromorphic models different from conventional neural networks in general, and Deep Learning in particular. A Deep Learning system does not have a temporal component, which means that if Deep Learning to be applied to this problem, each timestep would have to be an independently solved problem. Deep Learning, in this case, would be equivalent to a Python program with 34 = 81 if statements.
At time 0.12, the input parameters remain the same, but the output synapse fires again, so the cart will be pulsed to the right at the next timestep.
### Example 2 - More subtle things that you might not see if we didn't point them out.
We'll start at timestep 0.14, where x velocity moves into the high state, and theta velocity moves into the low state:
You'll note that the input neuron corresponding to x velocity is not firing. The reason is that each input pulses in 10 units of charge. The threshold for the input neuron is 57, so 10 units of charge won't make it fire. That neuron needs five more input pulses for the charge to exceed its threshold so that it fire. The x velocity parameter stays the same for the next five timesteps, so indeed, at timestep 0.24, the neuron fires:
Our next screen shot is from timestep 0.36, where the cart and pole finally get into a position when the "Pulse Left" synapse fires:
You can see the pulse at the next timestep (0.38). This timestep is interesting as well, because you can see neuron/synapse outlines of both red and black. In this picture, the red outlines denote that the neuron/synapse fired once during the 100-cycle interval, adn the black outlines denote that the neuron/synapse fired twice.
Skipping foward to timestep 0.50, here you see that both output synapses have fired in the 100 cycles.
Since the "pulse left" synapse has fired twice (black outline), but the "pulse right" synapse has fired only once (red outline), on our next timestep, we will pulse the cart to the left. Here is that timestep (0.52):
You can see that the "pulse right" synapse has fired twice, and the "pulse left" synapse has fired just once, so in the next timestep (0.54), the cart will be pulsed to the right:
And finally, during timestep 0.54, the two output synapses each fired once, so there will be no pulse to the cart on the next timestep (which I don't picture).
### A few more videos from different starting positions
This first video shows a 10-second example where the cart is to the far left (x = -2.35), and its angle is theta = 0.18:
And the second shows 10-second example where the cart is to the far right (x = -1.80), its velocity is going to the right (x velocity = 0.20), and its angle velocity is falling to the right (theta velocity = 0.20): | 2,302 | 10,143 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.6875 | 4 | CC-MAIN-2018-51 | longest | en | 0.934215 |
https://www.varsitytutors.com/calculus_2-help/integrals?page=98 | 1,555,927,698,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578551739.43/warc/CC-MAIN-20190422095521-20190422121521-00163.warc.gz | 837,144,292 | 40,839 | # Calculus 2 : Integrals
## Example Questions
### Example Question #971 : Integrals
Evaluate the following integral:
Explanation:
To integrate, we must first make the following substitution:
Next, rewrite the integral and integrate:
The integration was performed using the following rule:
Finally, replace u with our original x term:
### Example Question #37 : Solving Integrals By Substitution
What is the integral of the following equation?
Explanation:
We can solve this integral with u substitution
let , so , or,
Making this substitution, and moving our constants gives us:
, solving the integral, we get , plugging our value for u back into the equation
### Example Question #38 : Solving Integrals By Substitution
Explanation:
To make this integral simpler, we will need to make a substitution. You want to pick a substitution where the derivative also exists in the integral. Here, we want to choose:
. Now, we want to rewrite the integral interms of the new variable.
.
The last step is just to substitute the original substitution back in.
.
Explanation:
### Example Question #41 : Solving Integrals By Substitution
Solve the indefinite integral using trigonemtric substitution
Explanation:
We substitute
to solve the integral. Solving for dx,
Substituting these values into the integral yields
Solving for from
gives us
And so the indefinite integral is
### Example Question #42 : Solving Integrals By Substitution
Evaluate the following indefinite integral:
Explanation:
The integrand is composed of a function as well as its derivative multiplied by a constant. Hence, we can find the antiderivative via u-substitution as follows:
Let . Then , and so . Thus,
### Example Question #43 : Solving Integrals By Substitution
Evaluate the following indefinite integral:
Explanation:
The integrand can be evaluated by means of the u-substitution method, as follows:
Let . Then , and so
### Example Question #44 : Solving Integrals By Substitution
Evaluate the following indefinite integral:
Explanation:
Here, an understanding of trigonometric identities, as well as the appropriate selection of a dummy variable for u-substitution, is required. To figure out which function to represent "u" (cosine or sine), simply re-write the integrand as
Remembering that ,
Now, we can substitute to yield
because if , then , which implies .
At this point, all that is left to do is expand the polynomial and evaluate the integrand:
### Example Question #45 : Solving Integrals By Substitution
Find the value of
.
Explanation:
To perform this integration, we use a substitution.
Since the derivative of is , we choose our substitution to be .
Differentiating gives us,
.
Now we can substitute this into our integral. We will have,
.
Along with this substitution, we must also change our limits of and . To do so, we take these values and plug them in for in the formula .
Doing so, we obtain and .
Now our integral will be transformed as follows,
.
This integral is now easy to integrate, for the function integrates to .
Thus we have,
.
Therefore, the answer to the integral is,
.
### Example Question #44 : Solving Integrals By Substitution
Evaluate the following integral:
Explanation:
To integrate, we must first make the following substitution:
Now, rewrite the integral in terms of u and integrate:
The integral was performed using the following rule:
Note that the rule contains a fraction in front of the inverse trig function. Do not confuse this fraction with the fraction coming from the u substitution!
Finally, replace u with our original term and multiply the constants: | 769 | 3,661 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2019-18 | latest | en | 0.775664 |
https://www.perlmonks.org/?node_id=694252 | 1,529,420,433,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267863043.35/warc/CC-MAIN-20180619134548-20180619154548-00199.warc.gz | 904,031,551 | 5,743 | Syntactic Confectionery Delight PerlMonks
### Re: find closest element of array without going over
by jds17 (Pilgrim)
on Jun 26, 2008 at 18:40 UTC ( #694252=note: print w/replies, xml ) Need Help??
Hi, I am afraid I am reinventing the wheel here, but I had fun writing it, so here it goes. Comments:
1. It's just a binary search, as suggested by everyone else!
2. The performance should be quite good, but I did not test it against big arrays (and I would obviously need to write something to test against...)
3. In case two numbers in the array are equally close, the bigger number will be chosen as the winner.
Update: Of course I have to assume the target array is sorted.
```use strict;
use warnings;
my @array = (1,4,5,6,7,9,10,23,34,44,55,56,57,59,70,80,90,100);
print "\nenter number\n";
chomp (my \$find = <STDIN>);
my \$nearest = @{nearest(\@array)}[0];
print "nearest to \$find in array is: \$nearest\n";
sub nearest {
my (\$a) = @_;
my \$size = @\$a;
return \$a if \$size == 1;
my \$mid = int((\$size-1) / 2);
my \$test = @\$a[\$mid];
return \$test <= \$find ?
(abs(\$test-\$find)<abs(@\$a[\$mid+1]-\$find) ? [\$test] :
\$find <= @\$a[\$mid+1] ? [@\$a[\$mid+1]] : nearest([@\$a[\$mid+1 .. \$
+size-1]]))
:
(abs(\$test-\$find)<abs(@\$a[\$mid-1]-\$find) ? [\$test] :
\$find >= @\$a[\$mid-1] ? [@\$a[\$mid-1]] : nearest([@\$a[0 .. \$mid]]
+));
}
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Other Users? | 509 | 1,525 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2018-26 | latest | en | 0.808774 |
https://trustedpaperwriters.com/a-paper-mill-has-installed-three-steam-generators-boilers-to-provide-process-steam-and-also-to-1-answer-below/ | 1,627,734,848,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154089.6/warc/CC-MAIN-20210731105716-20210731135716-00309.warc.gz | 579,345,577 | 9,934 | # A paper mill has installed three steam generators (boilers) to provide process steam and also to… 1 answer below »
A paper mill has installed three steam generators (boilers) to provide process steam and also to use some its waste products as an energy source. Since there is extra capacity, the mill has installed three 10-MW turbine generators to take advantage of the situation. Each generator is a 4160-V, 12.5 MVA, 60 Hz, 0.8- PF-lagging, two-pole, Y-connected synchronous generator with a synchronous reactance of 1.10 W and an armature resistance of 0.03 W. Generators 1 and 2 have a characteristic power-frequency slope of 5 MW/Hz, and generator 3 has a slope of 6 MW/Hz. sP
(a) If the no-load frequency of each of the three generators is adjusted to 61 Hz, how much power will the three machines be supplying when actual system frequency is 60 Hz?
(b) What is the maximum power the three generators can supply in this condition without the ratings of one of them being exceeded? At what frequency does this limit occur? How much power does each generator supply at that point?
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(c) What would have to be done to get all three generators to supply their rated real and reactive powers at an overall operating frequency of 60 Hz?
(d) What would the internal generated voltages of the three generators be under this condition? | 370 | 1,625 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2021-31 | latest | en | 0.951247 |
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# Vincenzo Bontempo
##### Last seen: 10 Monate ago
31 total contributions since 2019
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Return unique values without sorting
If the input vector A is [42 1 1], the output value B must be the unique values [42 1] The *values of B are in the s...
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Replicate elements in vectors
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Find the largest value in the 3D matrix
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Check if number exists in vector
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Remove any row in which a NaN appears
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Determine whether a vector is monotonically increasing
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Arrange Vector in descending order
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Column Removal
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etwa ein Jahr ago | 1,467 | 4,939 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2020-50 | longest | en | 0.428878 |
https://www.open.edu/openlearncreate/mod/oucontent/view.php?id=563§ion=2.7 | 1,624,085,004,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487643703.56/warc/CC-MAIN-20210619051239-20210619081239-00421.warc.gz | 852,758,432 | 22,435 | # 5.1.7 Length in the Metric System
## Activity: Units of Length
(a) Write 327 centimeters as meters.
### Discussion
How many centimeters are in 1 meter? It’s okay to go back to the previous screen and look at the diagram at the bottom.
(a) There are 100 cm in 1 m and we are converting from a small unit to a large unit, so we divide by 100 to change cm into m.
Hence, .
(b) Write 6.78 meters as centimeters. | 115 | 416 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2021-25 | longest | en | 0.934652 |
https://www.traditionaloven.com/metal/precious-metals/platinum/convert-ounce-oz-of-platinum-to-grain-gr-of-platinum.html | 1,542,512,309,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039743963.32/warc/CC-MAIN-20181118031826-20181118053826-00208.warc.gz | 1,034,559,975 | 13,945 | Platinum ounce (avoirdupois) to grains of platinum converter
# platinum conversion
## Amount: ounce (avoirdupois) (oz) of platinum mass Equals: 437.50 grains (gr) in platinum mass
Calculate grains of platinum per ounce (avoirdupois) unit. The platinum converter.
TOGGLE : from grains into avoirdupois ounces in the other way around.
### Enter a New ounce (avoirdupois) Amount of platinum to Convert From
* Enter whole numbers, decimals or fractions (ie: 6, 5.33, 17 3/8)
## platinum from ounce (avoirdupois) to grain Conversion Results :
Amount : ounce (avoirdupois) (oz) of platinum
Equals: 437.50 grains (gr) in platinum
Fractions: 437 1/2 grains (gr) in platinum
CONVERT : between other platinum measuring units - complete list.
## Platinum Amounts (solid platinum)
Here the calculator is for platinum amounts (solid platinum volume; dense, precious, gray to white metal rare in abundance on the planet earth. Its annual production is only a very few hundred tons. It is a very highly valuable metal. Platinum performs real well in resisting corrosion. Not only beautiful jewellery is made out of platinum, this metal enjoys quite a wide variety of uses. For instance in electronics, chemical industries and also in chemotherapy applications against certain cancers. Traders invest money in platinum on commodity markets, in commodity future trading as this material is also one of the major precious commodity metals. Thinking of going into investing in stocks? It would be a wise idea to start learning at least basics at a commodity trading school first, to get used to the markets, then start with small investments. Only after sell and buy platinum.)
Is it possible to manage numerous units calculations, in relation to how heavy other volumes of platinum are, all on one page? The all in one Pt multiunit calculation tool makes it possible to manage just that.
Convert platinum measuring units between ounce (avoirdupois) (oz) and grains (gr) of platinum but in the other direction from grains into avoirdupois ounces.
conversion result for platinum: From Symbol Equals Result To Symbol 1 ounce (avoirdupois) oz = 437.50 grains gr
# Precious metals: platinum conversion
This online platinum from oz into gr (precious metal) converter is a handy tool not just for certified or experienced professionals. It can help when selling scrap metals for recycling.
## Other applications of this platinum calculator are ...
With the above mentioned units calculating service it provides, this platinum converter proved to be useful also as a teaching tool:
1. in practicing avoirdupois ounces and grains ( oz vs. gr ) exchange.
2. for conversion factors training exercises with converting mass/weights units vs. liquid/fluid volume units measures.
3. work with platinum's density values including other physical properties this metal has.
International unit symbols for these two platinum measurements are:
Abbreviation or prefix ( abbr. short brevis ), unit symbol, for ounce (avoirdupois) is: oz
Abbreviation or prefix ( abbr. ) brevis - short unit symbol for grain is: gr
### One ounce (avoirdupois) of platinum converted to grain equals to 437.50 gr
How many grains of platinum are in 1 ounce (avoirdupois)? The answer is: The change of 1 oz ( ounce (avoirdupois) ) unit of a platinum amount equals = to 437.50 gr ( grain ) as the equivalent measure for the same platinum type.
In principle with any measuring task, switched on professional people always ensure, and their success depends on, they get the most precise conversion results everywhere and every-time. Not only whenever possible, it's always so. Often having only a good idea ( or more ideas ) might not be perfect nor good enough solutions. Subjects of high economic value such as stocks, foreign exchange market and various units in precious metals trading, money, financing ( to list just several of all kinds of investments ), are way too important. Different matters seek an accurate financial advice first, with a plan. Especially precise prices-versus-sizes of platinum can have a crucial/pivotal role in investments. If there is an exact known measure in oz - avoirdupois ounces for platinum amount, the rule is that the ounce (avoirdupois) number gets converted into gr - grains or any other unit of platinum absolutely exactly. It's like an insurance for a trader or investor who is buying. And a saving calculator for having a peace of mind by knowing more about the quantity of e.g. how much industrial commodities is being bought well before it is payed for. It is also a part of savings to my superannuation funds. "Super funds" as we call them in this country.
Conversion for how many grains ( gr ) of platinum are contained in a ounce (avoirdupois) ( 1 oz ). Or, how much in grains of platinum is in 1 ounce (avoirdupois)? To link to this platinum - ounce (avoirdupois) to grains online precious metal converter for the answer, simply cut and paste the following.
The link to this tool will appear as: platinum from ounce (avoirdupois) (oz) to grains (gr) metal conversion.
I've done my best to build this site for you- Please send feedback to let me know how you enjoyed visiting. | 1,145 | 5,195 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2018-47 | longest | en | 0.826125 |
https://www.coursehero.com/file/5979099/homework-1/ | 1,516,470,508,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084889677.76/warc/CC-MAIN-20180120162254-20180120182254-00195.warc.gz | 853,991,239 | 112,016 | homework 1 - Chapter 1 Section 1 1 Which of the following...
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Chapter 1, Section 1 1. Which of the following are statements? (I also discuss the truth value, where applicable) a) The moon is made of green cheese. YES (false) b) He is certainly a tall man. YES (it depends on who the man is) c) Two is a prime number YES (true) d) Will the game ever be seen? NO e) Next year interest rates will rise. YES (who knows about next year?) f) Next year interest rates will fall. YES (who knows about next year?) g) x 2 – 4 = 0 YES (true) 2. Given the truth values A true, B false, and C true, what is the truth value of each of the following wff's? a) A ( B C ) [T (F T)] [T T] T b) ( A B ) C [(T F) T] [F T] T c) ( A B )' C [(T F)' T] [F’ T] [T T] T d) A ' ( B ' C )' [T' (F' T)'] [F (T T)'] [F T'] [F F] F 3. What is the truth value of each of the following statements? a) 8 is even or 6 is odd T F true b) 8 is even and 6 is odd T F false c) 8 is odd or 6 is odd F F false d) 8 is odd and 6 is odd F F false e) If 8 is odd, then 6 is odd F Æ F true f) If 8 is even, then 6 is odd T Æ F false g) If 8 is odd, then 6 is even F Æ T true h) If 8 is odd and 6 is even, then 8 < 6 (F T) Æ F true 4. Find the antecedent and consequent in each of the following statements. a) Healthy plant growth follows from sufficient water. antecedent: sufficient water consequent: healthy plant growth b) Increased availability of information is a necessary condition for further technological advances. antecedent: further technological advances consequent: increased availability of information c) Errors will be introduced only if there is a modification of the program. antecedent: errors will be introduced consequent: there is a modification of the program d) Fuel savings implies good insulation or storm windows throughout. antecedent: fuel savings consequent: good insulation or storm windows throughout 5. Negate the following statements: a) The answer is two or three 1. Neither two nor three is the answer. 3. The answer is not 2 and it is not 3. b) Cucumbers are green and seedy. 2. Cucumbers are not green or not seedy. c) 2 < 7 and 3 is odd. 4. 2 7 or 3 is even.
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6. Write the negation of each statement a) If the food is good, then the service is excellent. F Æ S is F' S. Negated: (F' S)' is F S' The food is good, but the service is poor. b) Either the food is good or the service is excellent. F S. Negated: F' S' The food is poor and so is the service. c) Either the food is good and the service is excellent, or else the price is high (F S) P Negated: (F S)' P'. Either the food is poor or the service is poor, but the price is low. d) Neither the food is good nor the service is excellent. F' S' Negated: F S Either the food is good or the service is excellent.
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homework 1 - Chapter 1 Section 1 1 Which of the following...
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Ask a homework question - tutors are online | 942 | 3,240 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2018-05 | latest | en | 0.890044 |
https://www.coursehero.com/file/6655802/211A-1-final2009sol/ | 1,498,637,641,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128322873.10/warc/CC-MAIN-20170628065139-20170628085139-00122.warc.gz | 845,066,663 | 219,446 | 211A_1_final2009sol
# 211A_1_final2009sol - EE211A Fall Quarter 2009 Digital...
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EE211A Digital Image Processing I Fall Quarter, 2009 1 Final Exam Solutions 1. 2D DFT (20 points) The above 256x256 image contains a periodic pattern of 64x64 squares of intensity 255 on a background of intensity 0. The center (not the upper left corner) of the above image is the (0, 0) location. The square centered at (0, 0) goes from . All the other squares are the same size and are centered at locations offset from the central square (both vertically and horizontally) by integer multiples of 128. Identify all the locations where , the 2D DFT of will be nonzero. Answer: The image in the space domain can be understood as a 2D rect of dimension 64x64 convolved with a 2D array of delta functions with spacing (128, 128) in each dimension. Convolution in the space domain corresponds to multiplication in the frequency domain. The nonzero locations of the 2D DFT of the image above can be answered by determining the nonzero locations of the 2D DFT of the array of delta functions multiplied by the 2D DFT of the rect. An array of delta functions with space domain spacing 128 transforms into an array of delta functions with a spacing of 256/128 = 2 pixels in each dimension in the frequency domain. A 64x64 rect is transformed into a 2D sinc function with nulls spaced at all and multiples of 256/64 = 4 (except for ).
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EE211A Digital Image Processing I Fall Quarter, 2009 2 When a function that is nonzero only at = multiple of 2 is multiplied by a function that is zero at all multiples of 4 except 0, the result is a function that is nonzero in the following places: All combinations where is one of 0, ± 2, ± 6, ± 10, ± 14, ± 18 . . . . ± 126 etc. AND is one of 0, ± 2, ± 6, ± 10, ± 14, ± 18 . . . . ± 126. Note: This answer can also be equivalently expressed using the range from 0 to 255 by taking the above values modulo 256.
EE211A Digital Image Processing I Fall Quarter, 2009 3 2. Transforms ( 20 points) Consider a 1D transform defined as follows: a) (5 points) In matrix notation, the transform can be described as
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## This note was uploaded on 12/27/2011 for the course EE211A 211A taught by Professor Villasenor during the Spring '11 term at UCLA.
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211A_1_final2009sol - EE211A Fall Quarter 2009 Digital...
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Ask a homework question - tutors are online | 731 | 2,816 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2017-26 | longest | en | 0.849766 |
https://ultimosedusupport.com/2020/12/27/mathematics-primary-five-third-term-questions/ | 1,675,188,130,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499888.62/warc/CC-MAIN-20230131154832-20230131184832-00693.warc.gz | 613,464,453 | 27,209 | MATHEMATICS PRIMARY FIVE THIRD TERM QUESTIONS
MATHEMATICS PRIMARY FIVE THIRD TERM QUESTIONS
1 of 3
MATHEMATICS PRIMARY FIVE THIRD TERM QUESTIONS
1. Express 24040046 in words.
A. Two hundred and forty thousand and four hundred and forty-six
B. Twenty-four thousand four hundred and forty-six
C. Twenty-four million, forty thousand and forty-six
D. Twenty-four million four hundred thousand and forty-six
E. Two million forty million four hundred thousand and forty-six
2. Write in figures: Two million, five thousand and forty-two.
A. 2005042
B. 205042
C. 104836
D. 20050042
E. 250042
3. Subtract the H.C.F of 9, 18, and 27 from the L.C.M of 16 and 2
A. 9
B. 27
C. 7
D. 39
E. 48
4. Find the quotient of 46.8 and 10
A. 46.8
B. 4.68
C. 0.468
D. 4.86
E. 0.0468
5. Simplify 3x + x – 2x + 4x
A. 5x
B. 7x
C. 6x
D. 4x
E. 3x
6. The product of two numbers is 24. If one of the numbers is -3, find the second number.
A. -8
B. -6
C. 4
D. 6
E. 8
7. Convert 0.25 to a simple fraction.
A. ½
B. 2/3
C. 3/5
D. 1/4
E. 4/5
8. A trader bought an article for N500 and sold it for N650. Find her profit percent.
A. 5%
B. 10%
C. 20%
D. 30%
E. 331/3%
9. What is the value of 12.6 x 0.32 to the nearest whole number?
A. 4
B. 6
C. 8
D. 12
E. 16
10. Sum together 13.2, 1.63, and 0.729.
A. 36.79
B. 15.559
C. 15.379
D. 14.794
E. 14.559
2 of 3
11. By what is 1.327 multiplied to get one thousand, three hundred and twenty seven?
A. 10000
B. 1000
C. 100
D. 10
E. 0.1
12. I have 40 mangoes; I gave 12 out to John and 3 to Musa. What fraction reduced to its term did I give out?
A. 7/10
B. 5/5
C. 3/8
D. 3/10
E. 3/40
13. Subtract the smallest of these fractions from the largest: ½, 2/5, ¾, and 2/3
A. ½
B. 2/5
C. 1/8
D. 2/3
E. ¼
14. Express 270 as a product of its prime factors.
A. 2x3x3x3x5
B. 2x3x5x9
C. 2x3x5x59
D. 2x3x15
E. 3x3x5x6
15. If 10 is multiplied by a number, x and the result is added to 5, this statement may be written in algebra as:
A. 15
B. 5 + 10x
C. 20
D. 15x
E. 25x
16. Simplify 1½ of ¼ + 5½ ÷ ¾
A. 413/32
B. 615/24
C. 717/24
D. 75/6
E. 11½
17. Simplify 28axy2 ÷ 7xy2
A. 4a
B. 4ay
C. 4axy
D. 7ay
E. 4ay2
19. 178.4673 rounded to 2 decimal places is
A. 180
B. 1780.00
C. 178.47
D. 178.46
E. 170.47
The marks scored in primary science by ten
pupils are as follows:
16, 14, 13, 16, 18, 19, 20, 15, 18, 18
Use the above given data to answer questions 20 to 23
20. What is the modal mark?
A. 14
B. 15
C. 16
D. 18
E. 20
3 of 3
21. What is the average mark?
A. 21
B. 19
C. 17
D. 15
E. 13
22. What is the value of the median mark?
A. 16.0
B. 16.7
C. 17.0
D. 19.7
E. 20.0
23. Arrange the following in ascending order 15, 5, -10, -18, -35
A. 15, 5, -10, -18, -35
B. -35, -18, 15, -10, 5
C. 5, -10, 15, -18, -35
D. -35, -18, -10, 5, 15
E. -35, -18, 5, -10, 15
25. The area of the floor of a room is 169cm2. Calculate its length given that it is a square floor.
A. 13cm
B. 25cm
C. 52cm
D. 26cm
E. 169cm
26. Calculate the circumference of a circle whose radius is 14cm. (π = 22/7)
A. 44cm
B. 88cm
C. 154cm
D. 616cm
E. 1232cm
27. A football fields is 120m long and 65m wide. Find its perimeter.
A. 110m
B. 185m
C. 370m
D. 480m
E. 7800m
28. A box has a volume of 140cm3. If its breadth is 5cm and its length is 7cm, find its height.
A. 6cm
B. 5cm
C. 4cm
D. 3½cm
E. 2½cm
32. In the diagram below, find the angles marked x.
A. 40o
B. 60o
C. 70o
D. 120o
E. 140o
33. When angle is greater than 180o but less than 360o, what is it called?
A. Reflex angle
B. Obtuse angle
C. Right angle
D. Isosceles angle
E. Angle on a straight line
34. Find the value of x in the equation:
2x – 4 = 16
A. 7
B. 5
C. 8
D. 9
E. 10
35. Simplify – 4 + (-7) – 4
A. -9
B. -15
C. +15
D. -11
E. +10
36. Select the correct answer to:
9 x 2 – 12 ÷ 2 + 2
A. 243
B. 218
C. 5
D. 14
E. 15 | 1,759 | 3,761 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.859375 | 4 | CC-MAIN-2023-06 | latest | en | 0.819439 |
https://www.physicsforums.com/threads/is-1-2z-the-correct-result-for-differentiating-ln-3y-2z-with-respect-to-z.145976/ | 1,719,097,223,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862420.91/warc/CC-MAIN-20240622210521-20240623000521-00381.warc.gz | 814,365,991 | 16,040 | # Is -1/2z the correct result for differentiating ln(3y-2z) with respect to z?
• cabellos
In summary, the conversation involved differentiating ln(3y-2z) with respect to z, and the resulting answer being -2/(3y-2z). The use of the chain rule was suggested and applied to get this answer.
cabellos
I should know this, but i just wanted to check...differentiating ln(3y-2z) with respect to z...does this = -1/2z ?
You must differentiate the whole thing first, then differentiate what is on the inside.
To check your answer take the integral of -1/2z, you will see it is not equal ln(3y-2z)
As KoGs suggested, an appropriate use of the chain rule should do just fine.
ok so is it -2/3y + z
cabellos said:
ok so is it -2/3y + z
No, it is not. As mentioned before, try to apply the chain rule:http://mathworld.wolfram.com/ChainRule.html"
Last edited by a moderator:
I did apply it...this is how i calculated that result:
d/dz In(3y-2z)
y=In u therefore dy/du = 1/u
u=3y-2z therefore du/dz = -2
dy/du x du/dz = -2/(3y-2z)
where am i going wrong?
cabellos said:
I did apply it...this is how i calculated that result:
d/dz In(3y-2z)
y=In u therefore dy/du = 1/u
u=3y-2z therefore du/dz = -2
dy/du x du/dz = -2/(3y-2z)
where am i going wrong?
Now you're not going wrong, since the result is correct. Good work!
## 1. What is differentiation check?
Differentiation check is a process used in mathematics to find the rate of change of a function at a specific point. It involves calculating the derivative of the function and evaluating it at the given point.
## 2. Why is differentiation check important?
Differentiation check is important because it allows us to analyze the behavior of a function and make predictions about its values. It is also a fundamental tool in calculus and is used to solve various real-world problems involving rates of change.
## 3. How do you perform a differentiation check?
To perform a differentiation check, you first need to find the derivative of the function. This can be done by using differentiation rules or formulas. Once you have the derivative, you can then plug in the given point to find the rate of change or slope at that point.
## 4. What is the difference between differentiation and integration?
Differentiation and integration are two fundamental operations in calculus. Differentiation involves finding the rate of change of a function, while integration involves finding the area under a curve. In other words, differentiation is used to find the slope of a curve, while integration is used to find the area between the curve and the x-axis.
## 5. What are some common applications of differentiation check?
Differentiation check has many applications in various fields such as physics, engineering, and economics. Some common applications include finding maximum and minimum values of a function, analyzing motion and acceleration, and optimizing production and profit functions in business.
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1K | 895 | 3,517 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2024-26 | latest | en | 0.935014 |
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# Error-Correcting Codes : a Mathematical Introduction
Author: John Baylis Boca Raton, FL : Chapman & Hall/CRC, 2018. Chapman & Hall mathematics. eBook : Document : EnglishView all editions and formats (not yet rated) 0 with reviews - Be the first.
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Content-negotiable representations | 1,206 | 4,489 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2020-10 | latest | en | 0.598982 |
http://ftp.cs.rochester.edu/u/brown/172/lectures/9_avl/9avl2.html | 1,511,609,176,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934809778.95/warc/CC-MAIN-20171125105437-20171125125437-00720.warc.gz | 124,936,039 | 2,783 | ## AVL Details
Weiss Ch. 4.4.1 - 4.4.2
Pawlicki's PPT (56 oheads, diagrams, movies, code...)
### Questions
1. What's a double rotation?
2. What are examples of rotations with real nodes and arcs, not triangles?
3. What's wrong with Figs 4.35, 4.36 (double rotation with triangles) anyway?
4. How know which of the four rotation cases we're in?
5. How keep track of heights?
6. Wouldn't it be easier to look at the code directly? Maybe easier than English? Or pictures?
### What's a double rotation?
Literally it is two single rotations performed on three consecutive nodes k3, k1, k2 (headed down a path from the root) to address imbalance in the top node k3. k2 winds up a new subtree root with k1 and k3 the roots of k2's left and right subtrees. Weiss's labeling of these k's may not be consistent....
Generally, we recall that imbalance from insertion into subtrees A or D can be dealt with by a single rotation. We saw why that won't fix that problem arising from insertion into B or C.
The figure below shows cases 1, 2, 3, 4: balance condition ruined by insertion into A, B, C, D respectively.
What's wrong with Figs 4.35 and 4.36?
It's meant to show double rotation caused by insertion into B that causes an imbalance condition at k3. It's badly drawn. The k1-k3 circles aren't lined up with the arrows, nor are the tops of the triangles for subtrees A-D. There aren't enough levels, AND finally subtree B, which insertion has made too deep, should be shown with a level more altitude, not the same height as others.
### Double rotations with nodes and arcs
Weiss goes thru a long sequence of insertions, with figures and commentary, in the book. Here, we see two double rotations and a single.
Another single shown, another couple suppressed, we get to last insertion of 9, causing a double rotation because of an imbalance condition at 13.
Above k3 is 10, k1 is 8, k2 (sketchy) is the 9 we insert as 8's R. child. Below is the result of the double rot.
### How know which case we're in?
That is, which of the four cases (just inserted into A, B, C, or D) below are we in?
This decision, along with balancing itself, is done by Weiss's `balance` method (Fig. 4.39, p. 134).
Given you've found a height imbalance at node t, use subtree heights to figure which side is bigger, (determines cases 1 or 2 vs 3 or 4). then look at sub-subtree heights (e.g. height(t.left.left) vs height(t.left.right) to pick between the remaining two cases (single or double rot.).
### How keep track of heights?
Only need to update tree heights on a path from the insertion point back up to root. The depths of other trees may change but not their heights.
Weiss manages the operations like this in Fig. 4.39:
`insert` recursively calls itself on the left or right subtree of the current node it's considering until it finds the null pointer. The base case is to return a newly-created node with the inserted value in it.
In the recursive case, after every call to insert and before returning, the tree whose root is the node being considered at that level is balanced. So the rebalancing proceeds up the path followed by the search to the root, rebalancing all the way.
### Code vs. English
I'm pretty convinced here that the clearest and neatest way to see what's going on is to head straight to the code (which I could flash up but will not).
So for this I'm a Pawlickian! I assume you have the book. Read it. As you see, I mark mine up a bit... it's not a romance novel (hmmm...I mark those up, too...).
"Her breathless excitement surged to a crescendo *wrong word!* as his strong sure fingers..."
Last update: 7/17/13 | 914 | 3,636 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2017-47 | longest | en | 0.928603 |
https://www.topperlearning.com/answer/find-value/98r5ww77 | 1,679,317,691,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296943483.86/warc/CC-MAIN-20230320114206-20230320144206-00562.warc.gz | 1,129,036,357 | 59,207 | Request a call back
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# CBSE Class 8 Answered
find value
Asked by pushpadevipushpadevi19 | 13 Jan, 2023, 08:07: PM
Expert Answer
Hint
To find angle 6, either find any of the angle 3, 2 or 4.
as angle 1 = 70 degrees.
and angle 1 + angle 2 = 180 degrees (linear pair)
so, angle 2 = 110 degrees
Angle 2 = angle 6 (corresponding pair)
So, (b) is the correct option.
Answered by | 15 Jan, 2023, 12:25: AM
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ANSWERED BY EXPERT | 545 | 1,334 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2023-14 | longest | en | 0.813056 |
http://ujihisa.blogspot.com/2009/05/about-array-aka-spacecraft-operator.html | 1,386,575,866,000,000,000 | text/html | crawl-data/CC-MAIN-2013-48/segments/1386163936569/warc/CC-MAIN-20131204133216-00014-ip-10-33-133-15.ec2.internal.warc.gz | 194,013,409 | 14,220 | # Standards
Blogged by Ujihisa. Standard methods of programming and thoughts including Clojure, Vim, LLVM, Haskell, Ruby and Mathematics written by a Japanese programmer. github/ujihisa
## Thursday, May 7, 2009
### About Array#<=> (a.k.a. Spacecraft Operator)
The documentations about `Array#<=>` are:
For example:
``````[1, 2, 3, 4] <=> [1, 2, 3, 3] #=> 1
[1, 2, 3, 4] <=> [1, 2, 3, 4] #=> 0
[1, 2, 3, 4] <=> [1, 2, 3, 5] #=> -1
[1, 2, 3, 4] <=> [1, 2, 3] #=> 1
``````
The implementation in MRI 1.9 is:
``````VALUE
rb_ary_cmp(VALUE ary1, VALUE ary2)
{
long len;
VALUE v;
ary2 = to_ary(ary2);
if (ary1 == ary2) return INT2FIX(0);
v = rb_exec_recursive(recursive_cmp, ary1, ary2);
if (v != Qundef) return v;
len = RARRAY_LEN(ary1) - RARRAY_LEN(ary2);
if (len == 0) return INT2FIX(0);
if (len > 0) return INT2FIX(1);
return INT2FIX(-1);
}
static VALUE
recursive_cmp(VALUE ary1, VALUE ary2, int recur)
{
long i, len;
if (recur) return Qnil;
len = RARRAY_LEN(ary1);
if (len > RARRAY_LEN(ary2)) {
len = RARRAY_LEN(ary2);
}
for (i=0; i<len; i++) {
VALUE v = rb_funcall(rb_ary_elt(ary1, i), id_cmp, 1, rb_ary_elt(ary2, i));
if (v != INT2FIX(0)) {
return v;
}
}
return Qundef;
}
``````
I translated it from C to Ruby literally:
``````class Array
def yet_another_cmp(you)
you = you.to_ary
return 0 if self == you
v = nil
(0...[self.length, you.length].min).each do |i|
v = self[i] <=> you[i]
return v if v != 0
end
len = self.length - you.length
return 0 if len == 0
return 1 if len > 0
-1
end
end
``````
It works exactly the same.
You may think it is easy to write a simpler equivalent implement with `Array#zip`, but unfortunately I found that it was not so simple. The following is the simplest code I can write. Of course it works exactly the same as original `<=>`.
``````class Array
def simple_cmp(you)
self.zip(you) {|x, y|
break if (x && y).nil?
(v = x <=> y) == 0 or return v
}
self.length <=> you.length
end
end
``````
In conclusion, `Array#<=>` itself is complicated one. Enjoy your space travel!
1. Indeed, it's not trivial to compare arrays.
Since you have such a good eye at finding my errors in rubyspecs, I hope you won't mind if note a couple of differences in your "translation".
First, the test for equality should be using equal?, not == (otherwise you'd loop forever!). More importantly, the Ruby code handles recursive arrays, so the ruby translation is actually much more complicated!
Finally, your "zip" version won't work for arrays containing nil values, like [nil, :foo].yet_another_cmp [nil, :bar] # ==> 0, should be 1
Checking the ruby code led me to find a cute quirk of Ruby 1.8.7/1.9. Here's the quiz for you: without writing any method/block/lambda, can you find ways to have:
x == y # ==> true
y == x # ==> false
There are many ways!
I'll post a bug report tomorrow (although I'm not sure they will consider it a bug!)
2. I saw merely the tip of the iceberg! I never imagine about [nil, ...].
Although I had been thinking about the asymmetricity quiz until now, I couldn't find it. At first I suspected the following is the answer, but not.
a, b = [], []
a << a
b << a | 996 | 3,119 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2013-48 | longest | en | 0.677508 |
http://nrich.maths.org/7188/solution | 1,505,958,176,000,000,000 | text/html | crawl-data/CC-MAIN-2017-39/segments/1505818687592.20/warc/CC-MAIN-20170921011035-20170921031035-00379.warc.gz | 248,550,279 | 5,046 | Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Calendar Capers
Choose any three by three square of dates on a calendar page...
Days and Dates
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Stage: 3 Short Challenge Level:
The number is less than $100$, but five times the number is greater than $100$, so it must be between $20$ and $100$.
Reversing the digits makes a prime number, so the first digit must be $1$, $3$, $7$ or $9$, as all two-digit prime numbers are odd and not divisible by $5$. The number must be at least 20, so this rules out $1$.
Since the digits add to a prime number, the possibilities are:
First Digit Second Digit
$3$ $2,4,8$
$7$ $4,6$
$9$ $2,4,8$
The number must be one more than a multiple of $3$, so the digits must sum to give one more than a multiple of $3$, as multiples of $3$ have digit sums that are multiples of $3$. This leaves $34$, $76$ and $94$.
The number must have exactly one prime digit, which rules out $94$.
The number must have exactly four factors. $34$ has $1$, $2$, $17$ and $34$, but $76$ has $1$, $2$, $4$, $19$, $38$ and $76$.
Therefore the number is $34$.
This problem is taken from the UKMT Mathematical Challenges. | 362 | 1,335 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2017-39 | latest | en | 0.923702 |
centurycarpetcleaners.biz | 1,675,416,279,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500044.66/warc/CC-MAIN-20230203091020-20230203121020-00793.warc.gz | 179,302,160 | 12,631 | NP (complexity) Wikipedia In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from …
## Karp's 21 NP-complete problems Wikipedia
Class NP-complete and NP-hard problems. Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential, An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP..
An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp NP-complete problems off all shapes and colors. •These are universal NP-problems...if you can solve them efficiently, you can solve ANY problem in NP efficiently. •L is NP-complete if: –L is in NP –ANY other problem in NP reduces to L. •If you come up with an efficient algorithm to 3-color a map, then P=NP.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from … 1 NP-complete problems In the previous lecture notes, we defined the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are
• NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second
Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … Although I think "List of NP-complete problems" is an excellent article (especially as a source of new articles to write), I'm concerned that we may also be vulnerable to claims of copyright violation because we essentially steal the NP Guide's presentation.
Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem
Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … A Some NP-Complete Problems To ask the hard question is simple. But what does it mean? What are we going to do? W. H. Auden In this appendix we present a brief list of NP-complete problems; we restrict
NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision The Partition-Knapsack Problem This problem is what we originally referred to as “knapsack.” Given a list of integers L, can we partition it into two disjoint sets whose sums are equal? Example: L={3,4,5,6,14,18}, Solution: 3+4+18=5+6+16 Partition-Knapsack is NP-complete; reduction from Knapsack.
Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey and Johnson's 1979 book Computers and Intractability: A Guide to the Theory of NP-Completeness. NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem
Computer Algorithms Design and Analysis Known NP-Complete Problem Garey & Johnson: Computer and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979 About 300 problems i.e. SAT, Clique, Hamiltonian, Partition, Knapsack … Note: 0-1 knapsack problem is NPC problem, but it can be solved by using dynamic programming in polynomial time, think about why and Although I think "List of NP-complete problems" is an excellent article (especially as a source of new articles to write), I'm concerned that we may also be vulnerable to claims of copyright violation because we essentially steal the NP Guide's presentation.
Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in
Tutorial 8 NP-Complete Problems Nanjing University. • NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard)., NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or.
### NP-Completeness
Annotated List of Selected NP-complete Problems. On dit qu’un problème de décision est « NP-complet » lorsque le langage correspondant est NP-complet. C’est une notion informelle car il existe plusieurs moyens de coder les instances d’un problème, mais cela ne pose pas de difficultés dans la mesure où on emploie un codage raisonnable du problème vers le langage considéré (voir la section NP-complétude faible)., 1 NP-complete problems In the previous lecture notes, we defined the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are.
1 NP-complete problems The University of Edinburgh. Connecting problems together. NP-Completeness What are the hardest problems in NP? The Cook-Levin Theorem A concrete NP-complete problem. Recap from Last Time. The Limits of Computability RE A HALT TM L D co-RE R ADD 0*1* A HALT TM L D EQ TM EQ TM. The Limits of Efficient Computation P NP R. P and NP Refresher The class P consists of all problems solvable in deterministic polynomial time. …, NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or.
### Tutorial 8 NP-Complete Problems Nanjing University
Annotated List of Selected NP-complete Problems. NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in A Useful List of NP-Complete Problems Graphs. Vertex Cover Decision Problem(VC): Given a graph G=(V,E) and a positive integer k, is there a subset V' of V of vertices which form a Vertex Cover for G with the size of V' no more than k. .(A Vertex Cover for G is a set of vertices with the property that every edge has at least one vertex from that set as an endpoint.).
• NP-complete problems
• NP-complete problems
• More NP-Complete and NP-hard Problems
• LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certificate.” At worst, all solutions w must be checked, giving exponential running time. Or trying giving `NP-complete' or `NP and complete' as a search term to http://liinwww.ira.uka.de/searchbib/index> (This is basically a bibliography database, but, you can click on the `on-line papers' button to list electronically readable full texts).
NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems … THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a
Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- 4 NP –HARD AND NP –COMPLETE PROBLEMS BASIC CONCEPTS •The computing times of algorithms fall into two groups. •Group1–consists of problems whose solutions are bounded by …
An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp • NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard).
NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a …
Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in
Most Tensor Problems Are NP-Hard CHRISTOPHER J. HILLAR, Mathematical Sciences Research Institute LEK-HENG LIM, University of Chicago We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list includes: determining the feasibility of a system of bilinear equations, de- Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete.
NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems … A Useful List of NP-Complete Problems Graphs. Vertex Cover Decision Problem(VC): Given a graph G=(V,E) and a positive integer k, is there a subset V' of V of vertices which form a Vertex Cover for G with the size of V' no more than k. .(A Vertex Cover for G is a set of vertices with the property that every edge has at least one vertex from that set as an endpoint.)
P, NP, and NP-Completeness The Basics of Computational Complexity The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and compu-tational models. The P versus NP Question asks whether finding solutions is harder than checking NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,
NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, NP-complete problems off all shapes and colors. •These are universal NP-problems...if you can solve them efficiently, you can solve ANY problem in NP efficiently. •L is NP-complete if: –L is in NP –ANY other problem in NP reduces to L. •If you come up with an efficient algorithm to 3-color a map, then P=NP.
• NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). P, NP, and NP-Completeness The Basics of Computational Complexity The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and compu-tational models. The P versus NP Question asks whether finding solutions is harder than checking
## NP-complete problem mathematics Britannica
Combinatorial Optimization with Graph Convolutional. NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,, 16 NP-Hard Problems (December 3 and 5) 16.1 ‘E cient’ Problems A long time ago1, theoretical computer scientists like Steve Cook and Dick Karp decided that a minimum requirement of any e cient algorithm is that it runs in polynomial time: O(nc) for some constant c. People recognized early on that not all problems can be solved this quickly.
### NP-Complete Problems Virginia Tech
Karp's 21 NP-complete problems Wikipedia. LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certificate.” At worst, all solutions w must be checked, giving exponential running time., Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential.
Optimization Problems NP-complete problems are always yes/no questions. In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a … NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second
Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey and Johnson's 1979 book Computers and Intractability: A Guide to the Theory of NP-Completeness. NP-complete problems are closely related to each other and all can be reduced to each other in polynomial time. (Of course, not all such reductions are efficient.) In this work we focus on four canonical NP-hard problems [23]. Maximal Independent Set (MIS). Given an undirected graph, find the largest subset of vertices in
Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a
algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,
NP-hard problem. Deciding if a graph have a MIS of size k, deciding if a subset is a MIS, are NP-complete problems (this is typically proven via a reduction to SAT). This problem was one of Richard Karp’s original 21 problems shown NP-complete in his 1972 seminal article [34]. NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second
2.Select a problem Z known to be NP-Complete. 3.Consider an arbitrary instance s Z of problem Z. Show how to construct, in polynomial time, an instance s X of problem X such that (a)If s Z 2 Z, then s X 2 X and (b)If s X 2 X, then sz 2 z. T. M. Murali December 2, 2009 CS 4104: NP-complete problems 1 NP-complete problems In the previous lecture notes, we defined the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are
16 NP-Hard Problems (December 3 and 5) 16.1 ‘E cient’ Problems A long time ago1, theoretical computer scientists like Steve Cook and Dick Karp decided that a minimum requirement of any e cient algorithm is that it runs in polynomial time: O(nc) for some constant c. People recognized early on that not all problems can be solved this quickly NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia-batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation,
Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential 1 NP-complete problems In the previous lecture notes, we defined the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction from … NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems. So-called easy, or
• NP-complete problems are always yes/no questions. • In practice, we tend to want to solve optimization problems, where our task is to minimize (or maximize) a function, f(x), of the input, x. • Optimization problems, strictly speaking, can’t be NP-complete (only NP-hard). LECTURE NOTES: NP-COMPLETE PROBLEMS 3 checked in polynomial time. W is known as the ”witness” or ”certificate.” At worst, all solutions w must be checked, giving exponential running time.
More NP-Complete Problems Stanford University. 1 NP-complete problems In the previous lecture notes, we defined the notion of completeness for a com-plexity class. A priori, it is not clear that complete problems even exist for natural complexity classes such as NP or PSPACE. Fascinatingly, completeness turns out to be a pervasive phenomenon - most natural problems in NP are, Adiabatic quantum optimization fails for random instances of NP-complete problems Boris Altshuler,1,2, Hari Krovi, 2,yand Jeremie Roland z 1Columbia University 2NEC Laboratories America Inc.
### Problème NP-complet — Wikipédia
NP-complete problems web.cse.msstate.edu. NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second, NP-hard problem. Deciding if a graph have a MIS of size k, deciding if a subset is a MIS, are NP-complete problems (this is typically proven via a reduction to SAT). This problem was one of Richard Karp’s original 21 problems shown NP-complete in his 1972 seminal article [34]..
CMSC 451 Reductions & NP-completeness. NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision, NP-Hard and NP-Complete Problems 2 – The problems in class NPcan be verified in polynomial time If we are given a certificate of a solution, we can verify that ….
### NP-Completeness Stanford University
More NP-Complete Problems. Computer Algorithms Design and Analysis Known NP-Complete Problem Garey & Johnson: Computer and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979 About 300 problems i.e. SAT, Clique, Hamiltonian, Partition, Knapsack … Note: 0-1 knapsack problem is NPC problem, but it can be solved by using dynamic programming in polynomial time, think about why and NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision.
• Liste de problГЁmes NP-complets — WikipГ©dia
• NP hard and NP Complete problems Basic Concepts
• NP hard and NP Complete problems Basic Concepts
• NP-Hard and NP-Complete Problems
• NP-Complete: can be solved in Polynomial time only using a Non-deterministic method. NP-Complete may not last. Oh, one more thing, it is believed that if anyone could *ever* solve an "NP-Complete" problem in "P" time, then *all* "NP-complete" problems could also be solved that way by using the same method, and the whole class of "NP-Complete NP-complete problems are closely related to each other and all can be reduced to each other in polynomial time. (Of course, not all such reductions are efficient.) In this work we focus on four canonical NP-hard problems [23]. Maximal Independent Set (MIS). Given an undirected graph, find the largest subset of vertices in
NP Complete (abbreviated as NPC) problems, standing at the crux of deciding whether P=NP, are among hardest problems in computer science and other related areas. Through decades, NPC problems are treated as one class. Observing that NPC problems have different natures, it is unlikely that they will have the same complexity. Our intensive study shows that NPC problems are not all equivalent in algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP
NP Certification algorithm intuition. ・Certifier views things from "managerial" viewpoint. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. Def. Algorithm C(s, t) is a certifier for problem X if for every string s, s ∈ X iff there exists a string t such that C(s, t) = yes. Def. NP is the set of problems for which there exists a An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP.
NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. Clearly, all problems in P are also in NP, so P NP. NP-Complete problems are in a formal sense the hardest problems in NP|if any one of them can be solved in poly time, then they can all be solved in poly time; this would result in the set equality P = NP. Similarly, if any one of the NP-Complete problem can be shown to require exponential
Class NP, NP-complete, and NP-hard problems W. H¨am¨al¨ainen November 6, 2006 1 Class NP Class NP contains all computational problems such that the corre- sponding decision problem can be solved in a polynomial time by a nondeterministic Turing machine. algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP
Some First NP-complete problem We need to nd some rst NP-complete problem. Finding the rst NP-complete problem was the result of the Cook-Levin theorem. We’ll deal with this later. For now, trust me that: Independent Set is a packing problem and is NP-complete. Vertex Cover is a covering problem and is NP-complete. An Annotated List of Selected NP-complete Problems. Université de Liverpool, Département d'informatique, COMP202. Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger. A compendium of NP optimization problems. KTH NADA. Stockholm. Voir aussi. 21 problèmes NP-complets de Karp
On dit qu’un problème de décision est « NP-complet » lorsque le langage correspondant est NP-complet. C’est une notion informelle car il existe plusieurs moyens de coder les instances d’un problème, mais cela ne pose pas de difficultés dans la mesure où on emploie un codage raisonnable du problème vers le langage considéré (voir la section NP-complétude faible). NP-Hard and NP-Complete Problems For many of the problems we know and study, the best algorithms for their solution have computing times can be clustered into two groups- 1. Solutions are bounded by the polynomial 2. Solutions are bounded by a nonpolynomial . No one has been able to device an algorithm which is bounded by the polynomial of small degree for the problems belonging to the second
NP Certification algorithm intuition. ・Certifier views things from "managerial" viewpoint. ・Certifier doesn't determine whether s ∈ X on its own; rather, it checks a proposed proof t that s ∈ X. Def. Algorithm C(s, t) is a certifier for problem X if for every string s, s ∈ X iff there exists a string t such that C(s, t) = yes. Def. NP is the set of problems for which there exists a NP-Completeness How would you define NP-Complete? They are the “hardest” problems in NP Definition of NP-Complete Q is an NP-Complete problem if: 1) Q is in NP 2) every other NP problem polynomial time reducible to Q Getting Started How do you show that EVERY NP problem reduces to Q? One way would be to already have an NP-Complete problem
2.Select a problem Z known to be NP-Complete. 3.Consider an arbitrary instance s Z of problem Z. Show how to construct, in polynomial time, an instance s X of problem X such that (a)If s Z 2 Z, then s X 2 X and (b)If s X 2 X, then sz 2 z. T. M. Murali December 2, 2009 CS 4104: NP-complete problems NP problems have their own significance in programming, but the discussion becomes quite hot when we deal with differences between NP, P , NP-Complete and NP-hard. P and NP- Many of us know the difference between them. P- Polynomial time solving. Problems …
algorithm, then P = NP. If any problem in NP cannot be solved by a polynomial-time deterministic algorithm, then NP-complete problems are not in P. • This theorem makes NP-complete problems the focus of the P=NP question. • Most theoretical computer scientists believe that P ≠ NP. But no one has proved this yet. NP P NP P NP-complete NP NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are p-time solvable How to formally compare easiness/hardness of problems? Reductions Reduce language L 1 to L 2 via function f: 1. Convert input x of L 1 to instance f(x) of L 2 2. Apply decision algorithm for L 2 to f(x) Running time = time to compute f + time to apply decision
However, David Zuckerman showed in 1996 that every one of these 21 problems has a constrained optimization version that is impossible to approximate within any constant factor unless P = NP, by showing that Karp's approach to reduction generalizes to a specific type of approximability reduction. THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a | 7,896 | 35,084 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2023-06 | latest | en | 0.880283 |
https://www.encyclopediaofmath.org/index.php/Dunford%E2%80%93Pettis_operator | 1,539,706,578,000,000,000 | text/html | crawl-data/CC-MAIN-2018-43/segments/1539583510853.25/warc/CC-MAIN-20181016155643-20181016181143-00418.warc.gz | 933,881,894 | 7,237 | # Dunford-Pettis operator
(Redirected from Dunford–Pettis operator)
In their classic 1940 paper [a5], N. Dunford and B.J. Pettis (with a bit of help from R.S. Phillips, [a8]) showed that if is a weakly compact operator (cf. Dunford–Pettis property; Grothendieck space) acting on a space of Lebesgue-integrable functions, then is completely continuous (cf. also Completely-continuous operator); hence, if and are weakly compact, then is compact. Here, an operator is (weakly) compact if it takes bounded sets into (weakly) compact sets and completely continuous if it takes weakly compact sets into norm-compact sets. See also Dunford–Pettis property.
The Dunford–Pettis result was recognized by A. Grothendieck for what it was and, in his seminal 1953 paper [a6], he isolated several isomorphic invariants inspired by the work of Dunford and Pettis. In particular, he said that a Banach space has the Dunford–Pettis property if for any Banach space , any weakly compact operator is completely continuous, while has the reciprocal Dunford–Pettis property if regardless of the Banach space , the weak compactness of a linear operator is ensured by its complete continuity. These definitions were the first formulations in terms of how classes of operators on a space relate to each other and a clear indication of the impact homological thinking was having on Grothendieck and, through him, on functional analysis.
Grothendieck did more than define the properties; he showed that for any compact Hausdorff space , the space of continuous scalar-valued functions on enjoys both the Dunford–Pettis property and the reciprocal Dunford–Pettis property. Soon after, Grothendieck used ideas related to the Dunford–Pettis property to show that for a finite measure , any linear subspace of that is closed in () is finite-dimensional. After Grothendieck, efforts at adding new, significant examples of spaces with the Dunford–Pettis property met with little success; in the late 1970s, J. Elton and E. Odell discovered that any infinite-dimensional Banach space contains either a copy of , or a subspace without the Dunford–Pettis property. Interest in the serious study of the Dunford–Pettis property was renewed, although new and different examples of spaces with the Dunford–Pettis property were still elusive.
The logjam was broken in 1983, when J. Bourgain [a1] showed that the poly-disc algebras, poly-ball algebras and the spaces of continuously differentiable functions all enjoy the Dunford–Pettis property; Bourgain showed much more: the aforementioned spaces and all their duals enjoy the Dunford–Pettis property. Bourgain's work was to lead to a rash of new, interesting examples and techniques, Bourgain algebras were borne (cf. [a11]) and the already tight relations between Banach space theory and harmonic analysis were further solidified.
#### References
[a1] J. Bourgain, "The Dunford–Pettis property for the ball-algebras, the polydisc algebras and the Soboler spaces" Studia Math. , 77 (1984) pp. 245–253 [a2] J.A. Cima, R.M. Timoney, "The Dunford–Pettis property for certain planar uniform algebras" Michigan Math. J. , 34 (1987) pp. 99–104 [a3] J. Diestel, "A survey of results related to the Dunford–Pettis property" , Integration, Topology, and Geometry in Linear Spaces. Proc. Conf. Chapel Hill 1979 , Contemp. Math. , Amer. Math. Soc. (1980) pp. 15–60 [a4] J. Diestel, J.J. Uhl Jr., "Vector Measures" , Surveys , 15 , Amer. Math. Soc. (1977) [a5] N. Dunford, B.J. Pettis, "Linear operations on summable functions" Trans. Amer. Math. Soc. , 47 (1940) pp. 323–390 [a6] A. Grothendieck, "Sur les Applications linéaires faiblement compactes d'espaces du type " Canad. J. Math. , 5 (1953) pp. 129–173 [a7] E. Odell, "Applications of Ramsey theorems in Banach spaces" H.E. Lacey (ed.) , Notes in Banach Spaces , Austin Univ. Texas Press (1981) [a8] R.S. Phillips, "On linear transformations" Trans. Amer. Math. Soc. , 48 (1940) pp. 516–541 [a9] S.F. Saccone, "Banach space properties of strongly tight uniform algebras" Studia Math. , 114 (1985) pp. 159–180 [a10] P. Wojtaszczyk, "Banach spaces for analysts" , Studies Adv. Math. , 25 , Cambridge Univ. Press (1991) [a11] K. Yale, "Bourgain algebras" , Function spaces (Edwardsville, IL, 1990) , Lecture Notes Pure Appl. Math. , 136 , M. Dekker (1992) pp. 413–422
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Dunford–Pettis operator. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dunford%E2%80%93Pettis_operator&oldid=22366 | 1,255 | 4,508 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2018-43 | latest | en | 0.948133 |
https://www.physicsforums.com/threads/linear-operators-eigenvalues.140520/ | 1,544,893,343,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376826892.78/warc/CC-MAIN-20181215152912-20181215174912-00211.warc.gz | 996,352,842 | 12,671 | # Linear Operators, Eigenvalues
1. Oct 29, 2006
### frederick
If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!
Last edited: Oct 29, 2006
2. Oct 29, 2006
### StatusX
e^A e^B = e^(A+B) is true when A and B commute (AB=BA). Or are you asking when e^A e^B Y=e^(A+B) Y, where Y is an eigenvector of A and B? That is always true.
Last edited: Oct 29, 2006
3. Oct 29, 2006
### mathman
A and B commute is sufficient - I don't know if it's necessary.
4. Oct 29, 2006
### Office_Shredder
Staff Emeritus
I'm not quite sure what the question is asking here....
I definitely did pick up that A and B must be the same size for A+B to exist (and must be square to have eigenvectors and values). Also, you need AB=BA (I think?). Someone else should verify this | 294 | 865 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2018-51 | latest | en | 0.884479 |
https://tumericalive.com/how-much-money-do-you-start-with-in-monopoly-europe/ | 1,713,215,932,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817033.56/warc/CC-MAIN-20240415205332-20240415235332-00657.warc.gz | 538,447,263 | 11,229 | Each player is given \$1500 divided as follows: 2 \$500’s, 2 \$100’s, 2 \$50’s, 6 \$20’s, 5 \$10’s, 5 \$5’s, and 5 \$1’s. All remaining money and other equipment go to the Bank.
## What currency is Monopoly money?
Monopoly Dollar
Monopoly Dollar, (represented by the currency sign ‘M'[1]) shortened to “Dollar”, is the main currency used in Monopoly. At the beginning of a game, each player gets 1,500 dollars, but his or her total changes almost every turn.
How much does each player get in Monopoly?
\$1,500
the Chance and Community Chest cards facedown on their allotted spaces on the board. Each player chooses one token to represent himther while traveling around the board. Each player is given \$1,500 divided as follows: P each of \$500s, \$ 1 0 0 ~ and \$50~; 6 \$40~; 5 each of \$105, \$ 5 ~ and \$Is.
How much money do you get in the old Monopoly?
In Monopoly, each player starts the game with 1,500 dollars. They’re broken down into two \$500, four \$100, one \$50, one \$20, two \$10, one \$5, and five \$1.
How much money do you start with in Monopoly? Players begin with \$1,500 in Monopoly money, according to Hasbro game instructions. Here is the breakdown of how much money each player gets: Two \$500s.
### How much money do you start with in Monopoly Here and Now?
The \$15 million each player starts with is divided as follows: 2 each of \$5,000,000s, \$1,000,000s, and \$500,000s; 6 \$200,000s; and 5 each of \$100,000s, \$50,000s, and \$10,000s. Properties have been changed to famous locations around the USA, such as Times Square, the White House, Las Vegas, and the Gateway Arch.
Is Monopoly money illegal?
It is different from most currencies, including the American currency or British currency upon which it is based, in that it is smaller, one-sided, and does not have different imagery for each denomination. It is not legal tender and has no monetary value in any jurisdictions.
How much is a Monopoly Dollar?
The base value of Monopoly money is defined because replacement packs of Monopoly cash cost \$3.50 USD, there are \$15,140 worth of Monopoly currency in the pack, so it comes out to \$ 0.000231176 USD.
## Can you bail yourself out of jail in Monopoly?
To Get Out of Jail
A player gets out of Jail “early” by: Rolling Doubles on any of that player’s next three turns in Jail. If a player succeeds in doing this, he or she immediately moves forward the number of spaces shown by the throw.
## What is the best Monopoly strategy?
Develop property as aggressively as you can.
• Buy orange and red properties, as they are the most landed-on.
• Don’t bother with utilities.
• Develop three houses or hotels as quickly as possible.
• Later in the game, don’t try to get out of jail right away.
• How does Monopoly end?
Game over – quick end
Officially MONOPOLY ends only when one player has achieved ownership of everything, crushing opponents one by one.
Can you pay to get out of jail in Monopoly?
After you have been sent to jail and everyone else has had a turn, you may choose to pay \$50 to get out. After paying, you roll the dice and move your token as normal.
### Is copying money for fun illegal?
Federal Crimes
Under federal law, the use or attempted use of counterfeit currency is illegal if the person has the intent to defraud the recipient. A conviction for the offense carries up to 20 years in prison and a fine.
### Is Monopoly money printed more than real money?
More monopoly money is printed than real money
Last year, the U.S. Bureau of Engraving and Printing printed \$974 million in real money. About 95 percent of that money printed in the U.S. is to replace old, worn-out bills. You can bet your little, old shoe that money fact is true!
Is Monopoly money worth more than ruble?
Monopoly money, which is widely regarded as worthless bits of paper used in a popular board game, is now substantially more valuable than the oil-backed Russian currency.
Can you pay \$50 to get out of jail Monopoly?
Immediately after throwing the dice for his/her third turn, if the player does not roll Doubles, he or she must pay the \$50 fine. He then comes out and moves forward from Jail the number of spaces shown by his/her roll, as normal.
## Do you collect 200 if you land on Go To jail?
You collect \$200 every time your token lands on or passes Go, whether by throwing the dice or by drawing a card. The only time that you don’t collect \$200 is when you’re sent to jail by a Chance or Community Chest card. This card clearly states that you ‘Go directly to Jail, do not pass Go’.
## Is Monopoly a skill or a luck?
Monopoly is a game of both luck and skills, as it involves a combination of people skills, some luck, as well as strategy. One cannot win Monopoly purely based on luck as the player has to make wise decisions on how to handle their money and investments after the roll of the dice has made a few decisions for them.
What is the secret to winning Monopoly?
Get three houses as quickly as possible.
As soon as you get a monopoly, start building, and don’t stop building until you’ve got three houses on each property. You will make far more money after you get up to three houses per property. This extra income will increase your chances of winning the game.
Has anybody ever finished a game of Monopoly?
Monopoly is one of the most well-known and popular board games in the world. It’s also one of the longest. In fact, since the game’s inception in 1936, nobody has actually finished a game.
### Do you get 200 for landing on Go?
According to the official rules of Monopoly, you collect \$200 for landing on Go. This is the same salary that you get every time you pass Go, you don’t earn double money for landing on the Go space itself.
### Do you collect rent in jail?
Buying Property and Collecting Rent While in Jail
Your play does not come to a complete halt while you are in jail in Monopoly. You can still buy, sell, and trade properties and collect rent.
Is it illegal to rip a dollar bill?
It’s also illegal to tear a dollar bill and even flatten a penny under the weight of a locomotive on the railroad tracks. The laws making defacing and debasing currency a crime have their roots in the federal government’s use of precious metals to mint coins.
Can an ATM detect counterfeit money?
Banks typically don’t have a way of knowing if cash came from their branch or ATM, even if you have a receipt, so a claim that it did is handled on a case-by-case basis. Whether your bank will swap out a bogus bill for a genuine one is up to its discretion.
## Is it legal to print Monopoly money?
It’s perfectly legal to print your own Monopoly money, which is a great option if you’ve lost some notes, the dog’s interrupted a game by snacking on your funds, or if you play bigger games and want to expand the bank. | 1,617 | 6,827 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2024-18 | longest | en | 0.958338 |
https://www.jeremyong.com/klein/case_studies/ga_skeletal_animation/ | 1,642,678,983,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00552.warc.gz | 874,828,783 | 16,741 | # Basic Skeletal Animation with Geometric Algebra
## Intro
One of the most direct applications of Geometric Algebra is to model a hierarchy of kinematic chains, also referred to as a skeleton. Most tutorials and material you'll find in books and online for this subject matter will be expressed in terms of the quaternion and dual-quaternion algebra. So here, let's break down how we might model a skeleton and its animation data with Geometric Algebra with simple code snippets using Klein.
## Data Modeling
First, let's model the data in a joint structure. A joint is an individually controllable set of degrees of freedom in our skeleton (your elbow or shoulder is a good example), and we'll assume for the moment that all joints in our skeleton can rotate the attached bone but not extend it or twist it in place (these are known as "prismatic" and "cylindrical joints" respectively).
struct joint
{
kln::motor inv_bind_pose;
uint8_t parent_offset;
uint8_t group_size;
};
Joints! Not Bones!
Often in the industry, you may hear people say "bones" but this is honestly a misnomer. The transforms applied during animation act on joints, and the bones are just occupying the space between the joints as it were.
Typically, when an animator rigs a character, it's done when the character is positioned in what is known as a "bind pose" or "T-pose." After all, associating vertices of a skinned mesh with nearby joints isn't very practical if all the joints are collapsed to the origin. As a result, it's common to cache on the joint itself, a means to transform the joint out of the bind pose. In our case, we'll use a motor called the inv_bind_pose.
A familiar face rocking the T-pose
What are motors again?
If you're used to working primarily with matrices, quaternions, and dual quaternions don't be too put off by the terminology. A motor is isomorphic to the dual quaternions but embedded in a "fuller" algebra with a richer structure. In practice, anywhere you would have needed a rotation plus a translation, or a dual quaternion, a motor is often a suitable choice.
If you have rotation data and translation data from an external data source, you can easily convert it to a motor by constructing a rotor and a translator and taking the product to produce a motor.
As we mentioned, joints are part of a skeletal hierarchy, so we need a way to reference the parent joint. The representation I prefer is to store a negative offset to the parent joint, so given a joint j, its parent would be *(&j - parent_offset). We can easily identify the root bone as having an offset of zero when encoded this way. Another trick used here is to use the additional padding we have left in the structure to store the size of the joint's "group" which includes itself and all of its children. A joint that is a leaf of the skeletal hierarchy will have a group_size of 1.
The joints themselves reside in a skeleton, so let's just use the simplest arrangement we can think of.
struct skeleton
{
joint* joints;
char const** joint_names;
uint16_t size;
};
Here, joints are stored on the skeleton as a single allocation with size elements. Its common to refer to joints by name for both debugging and authorship, but since the joint names aren't needed at runtime, we'll store them in a separately allocated array.
Now, all we've done is established a nice representation of the skeletal hierarchy, but we haven't done any animation yet! To do this, we're going to need to store a sequence of poses (also known as an animation clip). Each pose will encode a transform for every joint in our skeleton. Then, by interpolating from pose to pose, we'll have our first rudimentary animation system. Here's what our pose structure could be modeled as:
struct pose
{
kln::motor* joint_poses; // Array of poses for each joint
// NOTE: some engines allow animators to include scale data with each joint
// pose, but we'll ignore that for the time being.
};
struct clip
{
pose* poses; // Array of poses
uint16_t size; // Number of poses in the animation clip
uint16_t timestamps; // Array of timestamps for each skeletal pose
uint32_t timestamp_us; // Conversion from timestamp to microseconds
};
Again, we use the kln::motor to model the position of a joint. Typically, the rotation/translation of each joint in a pose is encoded relative to the parent joint. Why? Because the relative encoding allows us to mix and match animations on different parts of the skeleton, or perturb various joint transforms depending on gameplay. For example, suppose we want to allow a character to play its reload animation, while adding an additional twist at the hip as the player turns the camera. This type of emergent pose is much easier to tackle when we can use the joint poses of the reload animation clip directly, and simply apply the additional transform of the hip rotation.
## Forward Kinematics
Now that we have our clip containing an array of pose objects, we can now animate an instance of our skeleton! First, let's do this without any interpolation.
At a given pose (keyframe), we start at the root joint, apply its transform to itself, then to all its children, and then we repeat this process for all the other joints.
struct skeleton_instance
{
// All positions here are in world coordinate space
kln::point* joint_positions;
kln::point world_location;
};
void animate_keyframe(skeleton const& parent,
skeleton_instance& instance,
pose const& target)
{
// We need to write out the final transforms to the instance of the parent
// skeleton. The clip is the set of joint poses we need to apply.
// First, initialize the position of every joint to the world location
for (uint16_t i = 0; i != parent.size; ++i)
{
instance.joint_positions[i] = instance.world_location;
}
// Locomoting the world location of the instance according to the animation
// clip is known as "root motion" animation and is a relatively common
// technique, although it does have some tradeoffs outside the scope of this
// tutorial.
// For each joint, apply its corresponding joint pose motor to every
// position in its group.
for (uint16_t i = 0; i != parent.size; ++i)
{
// To apply the joint pose motor, we use the call operator. Here, we
// use the overload that is efficient when applying the same motor to
// a set of different positions.
target.joint_poses[i](
&instances.joint_positions[i], // Position input
&instances.joint_positions[i], // Position output
parent.joints[i].group_size); // Count
}
}
And in just a few lines of code, we have a "stepping" version of our animation system. Of course, there's a big problem with what we have so far. To get smooth animations, we'd need more keyframes than is reasonable. Before getting to that though, let's consider why we opted to apply a joint pose motor $$N$$ times across $$N$$ joint positions in a single call as opposed to $$N$$ separate calls. To see why, we'll need to look at the expanded motor conjugation operation $$mP\widetilde{m}$$ for some motor $$m$$ and point $$P$$.
Conjugation?
Often, you may hear the term "conjugate" used as a noun. For example, the complex conjugate of $$a + bi\mkern1mu$$ is $$a - bi\mkern1mu$$. However, the term is also used frequently to mean a "sandwich multiplication" such as $$pq\widetilde{p}$$. Those familiar with quaternions will recognize this as the application of a quaternion $$q$$ to a point $$p$$. Conjugation is used through Geometric Algebra because the fundamental action is reflection through a plane (produced by a conjugation with a vector quantity). Rotations and translations are modeled as two reflections, and so their action manifests itself as a conjugation by a bivector quantity.
First, let's give variable names to all the coordinates of a point $$P$$:
$P \equiv a_0\ee_{123} + a_1\ee_{021} + a_2\ee_{013} + a_3\ee_{023}$
and the motor $$m$$:
\begin{aligned} m \equiv b_0 &+ b_1 \ee_{12} + b_2 \ee_{31} + b_3 \ee_{23} \\ &+ c_1 \ee_{01} + c_2 \ee_{02} + c_3 \ee_{03} + c_0 \ee_{0123} \end{aligned}
Here, we've labeled the coefficients with prefixes $$b$$ and $$c$$ to distinguish between elements that contain the degenerate $$\ee_0$$ from those that don't. With these definitions, the group action of the motor is performed via conjugation as follows:
\begin{aligned} mP\widetilde{m} = &a_0 (b_0^2 + b_1^2 + b_2^2 + b_3^2) \ee_{123} + \\ \\ + (2&a_0(b_3 c_2 - b_0 c_3 - b_2 c_1 - b_1 c_0) + \\ 2&a_2(b_1 b_2 - b_0 b_3) + \\ 2&a_3(b_0 b_2 + b_1 b_3) + \\ &a_1(b_0^2 + b_1^2 - b_2^2 - b_3^2)) \ee_{021} + \\ \\ + (2&a_0(b_1 c_1 - b_0 c_2 - b_3 c_3 - b_2 c_0) + \\ 2&a_3(b_2 b_3 - b_0 b_1) + \\ 2&a_1(b_0 b_3 + b_1 b_2) + \\ &a_2(b_0^2 + b_2^2 - b_1^2 - b_3^2)) \ee_{013} + \\ \\ + (2&a_0(b_2 c_3 - b_0 c_1 - b_1 c_2 - b_3 c_0) + \\ 2&a_1(b_1 b_3 - b_0 b_2) + \\ 2&a_2(b_0 b_1 + b_2 b_3) + \\ &a_3(b_0^2 + b_3^2 - b_1^2 - b_2^2)) \ee_{032} \end{aligned}
Now, this is admittedly a mouthful, but if you stare at it long enough, some patterns should emerge. First, note that the result could be factorized in such a way that $$a_0$$, $$a_1$$, $$a_2$$, and $$a_3$$ don't need to participate in the computation until the very end. This is an optimization opportunity! All the arithmetic for the terms involving factors of $$b$$ and $$c$$ can be computed once and reused for each point $$P$$.
Internally, Klein uses a template variable to determine if it should loop over an array of entities when applying the motor and this optimization is done automatically, provided the code is written as above. If the motor is applied to a single entity (as in m(p)), that will be equivalent to the application of a dual-quaternion, so still not slow by any means, but still, the optimization mentioned above is often too good to pass up. To see the exact SSE code where this is optimization is made, feel free to refer to the Klein code here (search for the function sw312 which means "sandwich partition 3 with partitions 1 and 2").
Why did you label the motor coefficients with $$b$$s and $$c$$s
The answer to this is that the $$c$$ coefficients which were attached to basis elements with an ideal component $$\ee_0$$ produce a translational effect. If you look at the expanded motor conjugation above and set $$c_0 = c_1 = c_2 = c_3 = 0$$, a number of terms drop out and you'll be left with a purely rotational action. A rotor (aka a quaternion)! In fact, the code internally shares a bunch of code this way and the extraneous code when no translation is desired is optimized out at compile time.
OK, now we have code that will apply compute the joint pose positions in a global coordinate space given a specified pose. In practice though, an arbitrary time sample in our animation clip could be requested. In particular, we may need to render a pose between two keyframes. How should we go about doing this?
## Normalized Interpolation
What we need is a mechanism for interpolating between two motors, say, $$m_1$$ and $$m_2$$. There are at least two ways of performing this interpolation, a fast and moderately accurate way, and a slower but truly accurate way. By "accurate" here, what we mean is that given a parameter $$t \in [0, 1]$$ that maps $$m$$ to $$m_1$$ when $$t = 0$$ and maps $$m$$ to $$m_2$$ when $$t = 1$$, the speed of a particle moving along the path taken by $$m$$ is constant.
The reason why a simple linear interpolation such as $$m = m_1 (1 - t) + m_2 t$$ doesn't work is because the norm of $$m$$ must be $$1$$ to represent a rigid-body transform. It's easy to prove that the norm of an $$m$$ produced this way from two normalized motors isn't normalized in general. Let's compute the norm directly:
\begin{aligned} m &\equiv m_1 (1 - t) + m_2 t \\ \\ m\widetilde{m} &= \left(m_1 (1 - t) + m_2 t\right) \left(\widetilde{m}_1 (1 - t) + \widetilde{m}_2 t\right) \\ &= m_1\widetilde{m}_1 (1-t)^2 + (m_2\widetilde{m}_1 + m_1\widetilde{m}_2)t(1-t) + m_2\widetilde{m}_2t \\ &= 1 - t + t^2 + (m_2\widetilde{m}_1 + m_1\widetilde{m}_2)t(1-t) \end{aligned}
If $$t = 0$$ or $$t = 1$$, then no interpolation happened at all, and we can see that the expression above works out to $$1$$ as we'd expect. Otherwise, we can see that the norm of a linearly interpolated motor is $$1$$ if and only if $$m_2\widetilde{m}_1 + m_1\widetilde{m}_2 = 1$$ which is not true in general.
To correct for this, a "fast and dirty" approach is to just linearly interpolate anyways, but then normalize the result so that we at least we're guaranteed to end up with a rigid body transform. This is commonly referred to as nlerp, and a function that does this might look like the following:
// NOTE: t is expected to be between 0 and 1
kln::motor nlerp(kln::motor const& m1, kln::motor const& m2, float t)
{
return ((1 - t) * m1 + t * m2).normalize();
}
Not much to it! The main benefit of something like nlerp is that it is fast and requires no transcendental functions except a single fast Newton-Raphson square root to normalize the result.
## The Exp and Log Map
The contents of this section are slightly more math heavy and less programming heavy. It's useful knowledge to know, but if its a bit much and you just want a smooth constant speed interpolation, you can safely skip to the next section where we simply apply the techniques learned here in a provided API. Don't be intimidated though! I, the author, sincerely wish that the material here is presented in a way that can be grasped even if unfamiliar given a little bit of patience.
What if we wanted to maintain constant velocity around the curve? We can accomplish this by linearizing the transition motor. Let's step back for a second. Motors are the result of an exponential map, but to see why this might be plausible, let's look at complex numbers first as they are likely more familiar. Recall Euler's formula:
$e^{i\mkern1mu \theta} = \cos \theta + i\mkern1mu \sin \theta$
The reason this works is because if we Taylor expand the LHS:
$e^{i\mkern1mu \theta} = 1 + i\mkern1mu\theta - \frac{\theta^2}{2} - \frac{i\mkern1mu \theta^3}{6} + \dots = \left(1 - \frac{\theta^2}{2} + \dots\right) + i\mkern1mu\left(\theta - \frac{\theta^3}{6} + \dots\right)$
we seemingly miraculously end up with a well defined rotation, recognized on the RHS of Euler's formula. The $$i\mkern1mu$$ is the key ingredient. Because the square of $$i\mkern1mu$$ is $$-1$$, repeated multiplication of $$i\mkern1mu$$ doesn't grow to infinity. Instead, it "rotates" with a well-defined periodicity. Suppose I had two rotations $$r_1$$ and $$r_2$$ as below:
\begin{aligned} r_1 &\equiv \cos{\theta_1} + i\mkern1mu \sin{\theta_1} \\ r_2 &\equiv \cos{\theta_2} + i\mkern1mu \sin{\theta_2} \end{aligned}
and suppose I want a rotation that takes me halfway between $$r_1$$ to $$r_2$$. How would we produce such a rotation? The answer is obvious. We simply produce a rotation that's the average of $$\theta_2$$ and $$\theta_1$$ (assuming that we bisect the shorter arc between them). Then, the desired rotation is simply given as
$\cos{\frac{\theta_1 + \theta_2}{2}} + i\mkern1mu\sin{\frac{\theta_1 + \theta_2}{2}}$
This was easy because of the form I expressed $$r_1$$ and $$r_2$$ to you. The angles we needed to blend between was in plain sight! What if the rotation was given as $$\alpha + i\mkern1mu \beta$$ instead? Well, in this case, we can retrieve the angle by taking the logarithm of the rotation. Let's do this precisely. Suppose now that the rotations are given as follows:
\begin{aligned} r_1 &\equiv \alpha_1 + i\mkern1mu \beta_1 \\ r_2 &\equiv \alpha_2 + i\mkern1mu \beta_2 \end{aligned}
and again we are asked to find the rotation $$r$$ halfway between $$r_1$$ and $$r_2$$. The first thing we can realize is identify a new quantity I'm just going to call $$r_\delta$$. Suppose that $$r_\delta$$ represents the rotation halfway between $$r_1$$ and $$r_2$$. That is:
$r_\delta^2 r_1 = r_2$
Then, we can multiply both sides by $$r_1^\dagger$$ (the complex conjugate of $$r_1$$) to isolate $$r_\delta$$ on the LHS. Solving for $$r_\delta$$ can proceed as follows:
\begin{aligned} r_\delta^2 r_1 &= r_2 \\ r_\delta^2 r_1 r_1^\dagger &= r_2 r_1^\dagger \\ 2 \ln\left|r_\delta\right| &= \ln\left|r_2 r_1^\dagger\right| \tag{1} \\ r_\delta &= \exp{\left(\frac{1}{2}\ln\left|r_2 r_1^\dagger\right|\right)} \end{aligned}
$r_\delta r_1 = \exp{\left(\frac{1}{2}\ln\left|r_2 r_1^\dagger\right|\right)} r_1 \tag{2}$
In the last step, we right multiplied by $$r_1$$ (our starting rotation) since $$r_\delta$$ was the halfway rotation between $$r_1$$ and $$r_2$$. Hopefully, taking the natural logarithm of a complex number isn't two scary. After all, we know that $$r_1$$ and $$r_2$$ have corresonding angles $$\theta_1$$ and $$\theta_2$$ (defined as the arctangents of their $$\alpha$$ and $$\beta$$ components) along with polar forms that make simplifying the RHS above easy.
\begin{aligned} r_\delta r_1 &= \exp{\left(\frac{1}{2}\ln\left|r_2 r_1^\dagger\right|\right)} r_1 \\ &= \exp{\left(\frac{1}{2}\ln\left|e^{i\mkern1mu \theta_2}\right|\ln\left|e^{-i\mkern1mu \theta_1}\right|\right)} r_1 \\ &= \exp{\left(\frac{i\mkern1mu (\theta_2 - \theta_1)}{2}\right)} \exp{(i\mkern1mu \theta_1)} \\ &= \cos{\frac{\theta_1 + \theta_2}{2}} + i\mkern1mu\sin{\frac{\theta_1 + \theta_2}{2}} \end{aligned}
Different path but same result! Now, this might seem needlessly complicated to achieve what could have been done more easily by reading off the angles, but this is only because converting complex numbers to their polar representations is relatively easy. The angle can be determined by taking the arctangent of the ratio of the imaginary and real component. The most important step to appreciate in the second method we used above, is the part where we divide both sides by $$2$$ (see the equation marked $$(1)$$ above). The exponent there was our desired subdivision (we wanted to split the arc in two, so the incremental rotation ended up being squared to take us from $$r_1$$ to $$r_2$$). If we wanted to subdivide the arc into $$n$$ segments, we would have needed a power of $$n$$. By taking the logarithm of both sides, we linearized the rotation so that we could simply divide our arc in the correct number of subdivisions.
For a motor in 3D projective geometric algebra, there is a closed-form solution for the logarithm which means we can apply the same trick as above! In fact, we technically also don't yet know how to exponentiate the logarithm of a motor, but Klein provides implementations of both the exp and log functions taking bivectors to motors and vice versa that we can use. The derivation for both is a bit involved to flesh out here, but code demonstrating how this is done can be referred to here (a fuller derivation will be the subject of a future post). Taking the journey above as inspiration, given two motors $$m_1$$ and $$m_2$$, we have a recipe for blending between them smoothly. First, we multiply $$m_2$$ by $$\widetilde{m}_1$$ (the reversion operator is the Geometric Algebra analogue of the complex conjugate). This gives us $$m_2\widetilde{m}_1$$ which is the motor that takes $$m_1$$ to $$m_2$$. Next we take the logarithm of $$m_2\widetilde{m}_1$$, divide the logarithm by the number of segments in our interpolation, re-exponentiate, and finally multiply by $$m_1$$ to produce the interpolated result. If this was difficult to follow, feel free to refer again to the process we went through for complex numbers above. The RHS of equation $$(2)$$ is precisely what we want after substituting $$r$$s for $$m$$s.
Huh? This doesn't look like the slerp I'm familiar with
Chances are, you're used to seeing slerp in the following form (credit: wikipedia):
$\frac{[\sin{(1 - t)\phi]}}{\sin\phi}p_1 + \frac{\sin{[t\phi]}}{\sin\phi}p_2$
The derivation used with exponentials and logarithms is completely equivalent but it might take some staring (or pencil and paper) to work out why that is so. The key lies in realizing that the formula given here uses $$\phi$$ which is angle of the arc subtended by the two points of the arc (computed by the inner product $$p_1 \cdot p_2$$). This angle already captures the information provided by the logarithm and the sine ratios after reconstitute the non-linearized map as opposed to exponentiation.
The issue with this formula is that it doesn't generalize well to dual-quaternions or motors because the angle of the subtended arc isn't quite as easy to compute.
## Spherical Interpolation
We can now implement our motor blend function as follows:
// Blend between two motors with a parameter t in the range [0, 1]
kln::motor slerp(kln::motor const& a, kln::motor const& b, float t)
{
// Starting from a, the motor needed to get to b is b * ~a.
// To perform this motion continuously, we can take the principal
// branch of the logarithm of b * ~a, and subdivide it before
// re-exponentiating it to produce a motor again.
// In practice, this should be cached whenever possible.
line motor_step = log(b * ~a);
// exp(log(m)) = exp(t*log(m) + (1 - t)*log(m))
// = exp(t*(log(m))) * exp((1 - t)*log(m))
motor_step *= t;
// The exponential of the step here can be cached if the blend occurs
// with fixed steps toward the final motor. Compose the interpolated
// result with the start motor to produce the intermediate blended
// motor.
return exp(motor_step) * a;
}
VoilĂ . A motor slerp, also known as a "dual quaternion slerp." Now, you may be thinking, isn't this slower? The answer is yes, log and exp both require transcendentals after all. However, the choice between slerp and nlerp isn't necessarily as cut and dry as you may think. First, higher quality interpolation can mean that fewer keyframes are needed to produce the desired result. Second, as is evident in the code snippet above, the logarithm (called motor_step) can be cached if the motors do not change from frame to frame. This effectively cuts the cost of the slerp in half at the cost of some memory.
With this blend function, we can now sample our animation clip at any time.
// Given a skeleton, an instance of the skeleton, a clip, and a timestamp,
// transform the instance to the correct pose sampled from the clip.
void animate_sample(skeleton const& parent,
skeleton_instance& instance,
clip const& active_clip,
skeleton_instance const& instance,
int32_t timestamp_ms,
// scratch is a mutable pose with sufficient memory
// to hold our interpolated joint poses.
pose& scratch)
{
pose* previous;
pose* next;
float* t;
// This function isn't provided, but it takes a clip and timestamp
// and produces the poses that straddle the requested time and the
// interpolation parameter.
query_pose_endpoints(clip, timestamp, &previous, &next, &t);
for (uint16_t i = 0; i != parent.size; ++i)
{
// This could use slerp or nlerp if we wanted. A possible
// implementation of this slerp function was given above.
scratch.joint_poses[i] = slerp(
previous->joint_poses[i],
next->joint_poses[i],
*t
);
}
// Reuse our keyframe forward kinematic routine from above
animate_keyframe(parent, instance, scratch);
}
Of course, there are myriad optimizations that should jump out to us from the implementation given here, but as a starting point and considering how few lines of code we used, it's not bad in my opinion! Example optimizations include caching the logarithms from the previous frame, or reworking the code above so that all the temporary interpolated results do not need to reside in memory at once. The code provided here was written thusly in the interest of remaining terse.
What about inv_bind_pose??
We defined this kln::motor on our joint and never used it. "What gives?" you might ask. Well, we didn't use it because we didn't need to transform to the joint's local coordinate space. This will be needed for skinning which will be the subject of a future tutorial. I'm impressed you noticed this (if you did)!
## Conclusion
We have developed from the ground up the barebones making of an animation library. To be anything close to resembly a production library, it would need animation blending, vertex skinning/morphing, animation retargeting, and a whole host of other features, but at the very least, it should have been illustrative in the basic underpinnings of modeling kinematic motion with Geometric Algebra and Klein. Of course, there's much more to geometry than rigid motion, so stay tuned for future write-ups on collision detection and a whole host of other topics!
Feedback? Questions? Comments? Suggestions on what you'd like to see next? Feel free to drop by our discord and say hi! | 6,535 | 24,582 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2022-05 | latest | en | 0.940183 |
https://blog.sheetgo.com/google-sheets-formulas/and-formula-google-sheets/ | 1,679,818,282,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00208.warc.gz | 180,314,340 | 33,311 | # How to use the Google Sheets AND formula
The utility of the entire range of formulas wouldn’t be complete without the Google Sheets AND formula. It accepts logical expressions as input arguments, evaluates them to either a logical TRUE or FALSE. If all of the input arguments evaluate to TRUE, then the formula returns TRUE as the end result. However, if any of the input arguments evaluate to FALSE, the formula returns FALSE as an output.
### Syntax
AND(logical_expression1, [logical_expression2, …])
logical_expression1 – is an expression that evaluates to a logical TRUE or FALSE. This can be a direct expression or a reference to the cell that represents a logical expression.
[logical_expression2, …] – these are optional and additional expressions that return a logical TRUE or FALSE.
### Usage: Google Sheets AND Formula
Let us try various combinations of input arguments within the AND formula and examine the results. You may have already guessed that the first two variations of the formula with a single input argument, aren’t that useful to us.
It is interesting to note that the AND formula works even when we provide numbers as the input arguments. What happens when the Google Sheets application encounters a number when it is expecting a logical TRUE or FALSE? It simply converts them to a logical TRUE or FALSE. A zero is converted to FALSE and a non-zero to TRUE. Doesn’t really matter if the numbers contain decimals. Please see the examples in the snapshot below.
The AND formula also takes a range of cells as an input argument, as illustrated in the image below.
While it is hard to deny the utility of the AND formula in the real world, probably it has a limited application as a standalone formula. However, when it is used in conjunction with other formulas like IF, we can see its magic coming to life. In the snapshot below, consider the first and third formulas. The second and fourth formulas are their alternatives. Notice how simple it gets when we use AND instead of multi-level IF formula nesting.
### AND formula
And there you go! Use the AND formula in Google Sheets to evaluate logical expressions.
If you’d like to learn more about the various formulas of Google Sheets, why not take a look at our blog post on the
Alternatively, check out related blog posts below!
## How to use the LOG formula in Google Sheets
Needing to calculate the logarithm of your data inside spreadsheets? No need for a calculator;...
## How to use the YIELD formula in Google Sheets
Google Sheets is a powerful tool that offers a wide range of benefits for its users. In addition... | 539 | 2,617 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2023-14 | longest | en | 0.834568 |
sezweh.feuerwehr-hueffenhardt.de | 1,701,456,138,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100304.52/warc/CC-MAIN-20231201183432-20231201213432-00212.warc.gz | 579,890,693 | 149,689 | geom_smooth () and stat_smooth () are effectively aliases: they both use the same arguments.
# Ggplot logistic regression
liberator file 3dThe argument method of function with the value “glm” plots the logistic regression curve on top of a ggplot2 plot. bettendorf iowa baseball tournament 2023 schedule
One option would be to use geom_polygon with stat="density" where we could invert the density using after_stat (1 - density). . If you are using the same x and y values that you supplied in the ggplot() call and need to plot the linear regression line then you don't need to use the formula inside. .
12 Survival Analysis; 8.
Continue exploring.
.
You can use the R visualization library ggplot2 to plot a fitted linear regression model using the following basic syntax: ggplot (data,aes (x, y)) +.
.
1 Answer. If you are using the same x and y values that you supplied in the ggplot() call and need to plot the linear regression line then you don't need to use the formula inside. cookbook-r. .
. Either a double histogram, a double boxplot or a double dotplot, which could be modified or integrated with other graphical elements of ggplot2. com/Statistical_analysis/Logistic_regression/#SnippetTab" h="ID=SERP,5663.
COVID-19 has put a bit of a damper on this, but a question we can all relate to is whether to go out tonight, or not.
args ) Parameter:.
Oct 29, 2020 · One easy way to visualize these two metrics is by creating a ROC curve, which is a plot that displays the sensitivity and specificity of a logistic regression model. data).
Feb 16, 2017 · 1 Answer. .
.
Plotting the results of your logistic regression Part 2: Continuous by continuous interaction. 1 input and 3 output.
I tried to plot the results of an ordered logistic regression analysis by calculating the probabilities of endorsing every answer category of the dependent variable (6-point Likert scale, ranging from "1" to "6").
.
Output.
e. 5 Linear Regression; 8. . One easy way to visualize these two metrics is by creating a ROC curve, which is a plot that displays the sensitivity and specificity of a logistic regression model.
com/Statistical_analysis/Logistic_regression/#SnippetTab" h="ID=SERP,5663. ggplot (data = mtcars, aes (x = mpg, y = vs, color = as. it generates predictions by a model by holding the non-focal variables constant and varying the focal variable(s). 7 Another linear regression example; 8.
data).
License. Additionally I added a geom_path for the black colored outline ( geom_polygon will connect the endpoints too): library (ggplot2) ggplot (ex, aes (x = x1, y = y1)) + geom_point (alpha = 0. .
blue bistro tampa
8.
. . 12 Survival Analysis; 8.
monica yates mitchell
To assess how well a logistic regression model fits a dataset, we can look at the.
Logistic regression assumptions. . . Or, you can do it in ggplot2! library(ggplot2); library(tidyr) # first you have to get the information into a long dataframe, which is what ggplot likes :) plot. | 719 | 2,986 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2023-50 | latest | en | 0.848556 |
https://edurev.in/course/quiz/attempt/-1_JEE-Main-Physics-Mock-8/ab8cc6b0-54d3-4050-8199-7ee861494358 | 1,670,417,772,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446711162.52/warc/CC-MAIN-20221207121241-20221207151241-00285.warc.gz | 231,498,481 | 44,113 | JEE > JEE Main Physics Mock - 8
# JEE Main Physics Mock - 8
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## 25 Questions MCQ Test JEE Main & Advanced Mock Test Series | JEE Main Physics Mock - 8
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JEE Main Physics Mock - 8 - Question 1
### Quantity that remains unchanged in a transformer is
JEE Main Physics Mock - 8 - Question 2
### The working of dynamo is based on principle of
Detailed Solution for JEE Main Physics Mock - 8 - Question 2 The dynamo operates on the principle of the production of dynamically induced emf. Hence when ever flux is cut by the conductor, emf is poduced in it according to the law of electromagnetic induction.
JEE Main Physics Mock - 8 - Question 3
### A constant force acts on a body of mass 0.9 kg at rest for 10 s. If the body moves a distance of 250 m, the magnitude of the force is
Detailed Solution for JEE Main Physics Mock - 8 - Question 3
Here d= 250m
t= 10 seconds
initial velocity(u)= 0
using equation d= ut +1/2 at2, we will get
250 = 50 xa
a=5m/s2
force= mass x acceleration = 0.9ks x 5m/s2 = 4.5 newton
JEE Main Physics Mock - 8 - Question 4
Two bodies of mass 10 kg and 5 kg moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal acceleration is
Detailed Solution for JEE Main Physics Mock - 8 - Question 4
JEE Main Physics Mock - 8 - Question 5
The directions of electric and magnetic fields in J.J. Thomson's experiment for the determination of e/m for an electron are
Detailed Solution for JEE Main Physics Mock - 8 - Question 5 The specific charge of an electron can be determined when electron moves in both magnetic field and electric field which are mutually perpendicular to each other so that the net force on the electron is made zero. IN this situation the direction of motion of electron remains perpendicular to both electric and magnetic field.
JEE Main Physics Mock - 8 - Question 6
An electric dipole is placed along the X-axis at the origin O. A point P is at a distance of 20 cm from this origin such that OP makes an angle π/3 with the X axis. If electric field at P makes an angle θ with X-axis, the value of θ is
Detailed Solution for JEE Main Physics Mock - 8 - Question 6
JEE Main Physics Mock - 8 - Question 7
A glof ball of mass 0.05 kg placed on a tee, is struck by a golf club. The speed of the golf ball as it leaves the tee is 100 m/s, the time of contact between them is 0.02 s. If the force decreases to zero linearly with time, then the force at the beginning of the contact is
JEE Main Physics Mock - 8 - Question 8
A semi circle arc of radius 'a' is charged uniformly and the charge per unit length is λ. The electric field at its centre is
JEE Main Physics Mock - 8 - Question 9
The figure shows the symbol of a
JEE Main Physics Mock - 8 - Question 10
A liquid flows through a pipe of non-uniform cross-section. If A₁ and A₂ are the cross-sectional areas of the pipe at two points, the ratio of velocities of the liquid at these points will be
JEE Main Physics Mock - 8 - Question 11
A particle is moving in a uniform magnetic field then
JEE Main Physics Mock - 8 - Question 12
Which of the following equations is definitely wrong?
JEE Main Physics Mock - 8 - Question 13
In neutron discovery experiment, Berillium element is bombarded by
JEE Main Physics Mock - 8 - Question 14
In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer-
Assertion(A): A particle move from A (X1,Y1,Z1) to B (X2,Y2,Z2), displacement is given as (X2-X1)i, (Y2-Y1)j, (Z2-Z1)k.
Reason (R):Displacement can be positive, negative or zero.
JEE Main Physics Mock - 8 - Question 15
In the following question, a Statement of Assertion (A) is given followed by a corresponding Reason (R) just below it. Read the Statements carefully and mark the correct answer-
Assertion(A): The density of air at the top of the troposphere is about 10 times the density near the earth's surface.
Reason(R): The atmosphere between the heights of 12 km and 50 km is called troposphere.
JEE Main Physics Mock - 8 - Question 16
For a gas, if the ratio of specific heats at constant pressure and volume is γ, then the value of degree of freedom is
Detailed Solution for JEE Main Physics Mock - 8 - Question 16
Cp=(f/2+1)RT
Cv= (f/2)RT
Taking ratio we get
Cp/Cv = y = (f/2 + 1)/(f/2)
y=(f + 2)/f
y=1 + 2/f
y -1=2/f
f = 2/(y-1)
Hope it helps
JEE Main Physics Mock - 8 - Question 17
A 2 kg mass is rotating on a circular path of radius 0.8 m with angular velocity of 44 rad-s-1. If radius of the path becomes 1 m, then value of angular velocity will be
Detailed Solution for JEE Main Physics Mock - 8 - Question 17
JEE Main Physics Mock - 8 - Question 18
Which is not a unit of electric field
JEE Main Physics Mock - 8 - Question 19
Ultrasonic waves are those waves
JEE Main Physics Mock - 8 - Question 20
In a mixture of H−He+ gas (He+ is singly ionized He atom) , H atoms and He+ ions are excited to their respective first excited states. Subesquently, H atoms transfer their total excitation energy to He+ ions (by collisions). Assume that the Bohr model of atom is exactly vaild.
Q. The quantum number n of the state finally populated in He+ ions is
*Answer can only contain numeric values
JEE Main Physics Mock - 8 - Question 21
A biconvex thin lens is prepared from glass of refractive index µ2 = 3/2. The two converging surface have equal radii of 20cm each. One of the surface is silvered from outside to make it reflecting. It is placed in a medium of refractive index µ1 = 5/3. This system will behave as concave mirror of focal length f, find value of |f| in cm.
Detailed Solution for JEE Main Physics Mock - 8 - Question 21
*Answer can only contain numeric values
JEE Main Physics Mock - 8 - Question 22
Find the maximum kinetic energy (in eV) of the photoelectron liberated from the surface of lithium (work function φ = 2.15eV) by electromagnetic radiation whose electric component varies with time as E = a (1 + cos ωt)cosω0t,
where a is a constant, ω = 12 × 1014 rads–1 and ω0 = 3.6 × 1015 rads–1 (h = 6.6 × 10–34 in SI units)
Detailed Solution for JEE Main Physics Mock - 8 - Question 22
*Answer can only contain numeric values
JEE Main Physics Mock - 8 - Question 23
A photosensitive surface is irradiated with light of wavelength λ, the stopping potential is V. When the same surface is irradiated with the light of wavelength 2λ, stopping potential is V/3. Then the ratio of threshold wavelength (λmax) and the λ is
Detailed Solution for JEE Main Physics Mock - 8 - Question 23
*Answer can only contain numeric values
JEE Main Physics Mock - 8 - Question 24
One of the circuits for the measurement of resistance by potentiometer is shown. The galvanometer is connected at point A and zero deflection is observed at length PJ = 30 cm. In second case the secondary cell is changed. Take ES = 10 V and r = 1Ω in 1st reading and ES = 5V and r = 2Ω in 2nd reading. In second case, the zero deflection is observed at length PJ = 10 cm. What is the resistance R (in ohm)?
Detailed Solution for JEE Main Physics Mock - 8 - Question 24
*Answer can only contain numeric values
JEE Main Physics Mock - 8 - Question 25
A uniform triangular plate of triangular area 1m2, base length 60 cm and thickness 10 mm (prism like shape) is lying vertically on a smooth ground as shown in figure. Find maximum value of cotθ for which it does not topple.
Detailed Solution for JEE Main Physics Mock - 8 - Question 25
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http://www.physicsforums.com/showthread.php?p=2088038 | 1,409,720,780,000,000,000 | text/html | crawl-data/CC-MAIN-2014-35/segments/1409535924501.17/warc/CC-MAIN-20140909014259-00427-ip-10-180-136-8.ec2.internal.warc.gz | 1,265,461,104 | 12,681 | # Irrational Numbers
by timjones007
Tags: irrational, numbers
P: 10 why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?
Emeritus
PF Gold
P: 16,091
Quote by timjones007 why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?
Your question doesn't really make sense. If you know the proof, then what's your problem?
Are you using the word "why" in some unusual way? If so, you really should have said that up front....
P: 122 Well, your question does seem odd, but my guess is that you want to ask a philosophical question. Let me ask you a question. Do rational numbers exist? how do you know this?
Sci Advisor HW Helper P: 3,684 Irrational Numbers For me, the intuitive answer is "because there aren't nearly enough rationals to 'fill in' all the gaps".
P: 122 Well, I don't understand why people think rational numbers exist and some numbers don't. It's just easier to think that all numbers are mathematical constructs and real numbers are simply, yes, way to fill in gaps between rational numbers. I wonder what the op thinks of complex numbers.
P: 287 Oh jeez, I'll try to tread softly in this thread. It's actually a very interesting question the OP is getting at, and one I've often thought about myself. How much information do we need to have about a number before we can consider the number to be well-defined? Are all numbers which provably exist well-defined under our definitions of well-definedness of a number? Is there a definition of the well-definedness of numbers? One can also talk about whether numbers are computable or not. It's interesting that the real numbers are most incomputable... what does this mean? What can even be meant by incomputable number? I think it's an interesting discussion. To the OP: do you think that sqrt(2) exists, and in what sense do you think this? I mean, we both know that there is a proof that it is not rational. It's a relatively tame irrational number. Why do you feel the way you do? I could enjoy this conversation.
P: 10 no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely. So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself. It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined. In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
HW Helper
P: 3,684
Quote by timjones007 no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.
So you don't accept 1/9 = 0.111111... or 1/10 = 0.10000... either?
What about this: I define "foo" as an ordered pair (a, b) where (a, b) = (c, d) iff (a - c)(b - d) = 0 and the operations plus and times are defined by (a, b) + (c, d) = (a + c, b + d) and (a, b) * (c, d) = (ac + 2bd, bc + ad). Do "foo"s exist?
How about "bar"s, where "bar" is an ordered pair (a, b) where (a, b) = (c, d) iff ad = bc and the operations plus and times are defined by (a, b) + (c, d) = (ad + bc, bd) and (a, b) * (c, d) = (ac, bd)? Do "bar"s exist?\
Maybe "baz", where a "baz" is (a) where (a) = (b) iff a - b = 0 and the operations plus and times are defined by (a) + (b) = (a + b) and (a) * (b) = (ab). Do "baz"s exist?
P: 122
Quote by timjones007 no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely. So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself. It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined. In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
So tell me, what does "exist" mean? Do you think 2 exists? How so? Is there a realm of numbers where 2 exists but sqrt(2) doesn't?
what about sqrt(2) km? Does that exist?
Math Emeritus Sci Advisor Thanks PF Gold P: 39,569 It might be good to point out that while asking if a number is "well-defined" it makes no sense to focus entirely on the decimal representation of the number, as timjones007 does in #7. The decimal representation of a number is just that- a representation- and has little to do with the properties of the number itself.
Emeritus
PF Gold
P: 16,091
Quote by csprof2000 It's actually a very interesting question the OP is getting at ...
I would be very surprised if he was actually asking interesting questions about formal language, computability theory, or anything like that. I think he simply doesn't have a clear understanding of what others (and he) means by 'number', and lacking such clarity, is flailing about with his intuition.
P: 287 "no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely. " So you take the definition of a number as its decimal representation? This would take a little elaboration to take into account the (very valid) objection raised by CRGreathouse. For instance, you could say that a number is well defined if its decimal representation repeats with a string of digits of finite length L for all places N at least N_0 to the right of the decimal. This covers repeating decimals (1/9 = 0.1111... letting N_0 = 1 and the string of digits being "1", and 1/10 = 0.1000... is covered letting N_0 = 2 and the string of digits being "0", etc.) Obviously, the choice of 10 as the base doesn't make any difference... you could allow this to vary as well. But the real problem with that is that you're taking the properties of rational numbers and saying that's what makes a number well-defined. Does that make sense? I mean, if we are trying to show that irrational numbers are not well defined, it's a little self-serving to equate a property of rational numbers with well-definedness. Savvy? "So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself. " Well, the problem with this is that, as CRG said, 1/10 = 0.100... and this technically also goes on forever... perhaps a better way of saying what you're thinking is that you have a finite set of rules with which you can always generate the next digit in the decimal (or some other sensible) representation. For instance, 1/10 is well-defined because I can say "tenths' place 1, all other places 0" and you can use the two rules to write out the number to any desired number of digits. Does this sound alright, tim? The only snag with that, of course, is that sqrt(2) is also well defined by this definition of well-definedness. Consider this: sqrt(2) can be found as follows: sqrt(n):: x := 0 p := n // could be made more efficient, but who cares? for p = n to p_min begin while x <= n begin x = x + 10^p end x = x - 10^p end Let's see this operating on n = 2. x = 0. x = 100, p = 2. x = 0, p = 2. x = 10, p = 1. x = 0, p = 1. x = 1, p = 0. x = 2, p = 0. x = 1, p = 0. x = 1.1, p = -1 x = 1.2, p = -1 x = 1.3, p = -1 x = 1.4, p = -1 x = 1.5, p = -1 x = 1.4, p = -1 x = 1.41, p = -2 etc. As you can see, this will always allow you to find the nth decimal digit in a finite number of steps... so you would need a stricter definition than the one I provided to exclude sqrt(2).
P: 287 What does everybody else think about what it takes to define a number? Do numbers have to have a value? If so, and you know a number exists for which you cannot possibly find its value... does this mean anything?
Emeritus
PF Gold
P: 16,091
What does everybody else think about what it takes to define a number?
A type of number system is defined by a list of properties. If a particular set* has those properties, it's a model of that number system, and we would call its elements numbers (of the appropriate type).
(The properties don't have to be complete -- though the definitions for common number systems like the integers or the reals are complete in the appropriate sense)
*: Or type or class or language or whatever foundational gadget you want to use.
Once you actually have an actual, concrete list of properties to work with, you can usually answer simple questions relatively easily. e.g. it's fairly straightforward to show that
* in the rational numbers, 2 doesn't have a square root. (what would its factorization be?)
* in the real numbers, 2 does have a square root. (construct it as the least upper bound)
* for fields the question is undecidable -- some fields do and some fields don't have a square root of 2. (as shown by the previous two examples)
P: 41
Quote by timjones007 no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely. So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself. It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined. In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
You are right! sqrt(2) does not exist.
And in fact also 1 does not exist.
1 is the multiplication of 13/7 and 7/13 now
13/7 and 7/13 are just symbols for their decimal representations which are
13/7 = 1,85714285......
7/13 = 0,53846153......
and the decimal places continue on infinitely.
So, if we will never reach the last digit in the decimal places for these two numbers, how can we multiply them together???
In other words, 1 is a number that you get multiplying 13/7 by 7/13.
However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
PF Gold P: 2,022 This is an interesting question and I think it is one that has been discussed since ancient times.According to David Wells (the penguin dictionary of curious and interesting numbers) pi is the only irrational and transcedental number that occurs naturally.People here have been using root 2 as an example and I have been trying to think of an example where this number can be given a unit.Suppose we were told that a square had an area of root 2 metres squared.Does this mean anything when such a square cannot be consructed or have I picked on a dopey example?
P: 287 Dadface: That could be a dopey example. What about the distance between opposite corners of a square of area 1? Hurkyl: I see what you're saying, but I think the problem we're all having is in communicating. I agree that you're absolutely right about numbers... a very clear and thoughtful exposition. However, I think that the OP means to talk about the value of numbers, not their properties... to know what the number is, not whether it is there or not. I mean, 2 *is* an integer, but how big is 2? We can get to 2 using a finite number of logical steps. Is sqrt(2) a real number? The OP didn't think so, but perhaps after my last post he will agree that sqrt(2) must exist as well... since we can get as close as we like to it on a whim. But in what sense do the numbers which we cannot find values for have these values - even if we know the number must exist? I apologize that the discussion is a little vague. I'd love to give you an example of such a number, but obviously I can't... I don't know, maybe the reason this topic isn't more mainstream is that it's a rabbit hole, makes no sense, and has no good answer.
PF Gold P: 2,022 Yes csprof2000,yours is a good example,so numbers like root 2 come up but we can't measure them.It is a rabbit hole.
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x exp(x) musing tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 07/03/2009, 07:34 PM (07/03/2009, 03:40 PM)BenStandeven Wrote: Actually, I think you get exp(G(x)) ^^ 2^n = exp(G(x + n C)) = G(x + n C)/G(x + n C - C), where the operation is "symmetric" tetration (referring to the association method). I don't see any obvious way to get the usual "top down" tetration from this. Quote:and maybe f(x) can be found by current attempts for tetration. and G(x) might be found by inversing the ( modified ) carleman matrix of F(x) It would probably be simpler to find G directly. Quote:notice also that x exp(x) has a unique real fixed point !! Moreover, it is at zero, so it doesn't even need shifting. 1) interating exp ( 'anything' ) ^^ (2^n) seems serious overkill and overclocked speed !! so i dont think that can be correct. 2) simply finding G directly ? do you know a fixed point for G perhaps ? why do you think so ? 3) no we dont need shifting , 0 * exp(0) = 0 , i know that of course , in fact , thats partially why i mentioned it. regards tommy1729 « Next Oldest | Next Newest »
Messages In This Thread x exp(x) musing - by tommy1729 - 07/01/2009, 03:19 PM RE: x exp(x) musing - by BenStandeven - 07/03/2009, 03:40 PM RE: x exp(x) musing - by tommy1729 - 07/03/2009, 07:34 PM RE: x exp(x) musing - by BenStandeven - 07/04/2009, 11:25 PM RE: x exp(x) musing - by bo198214 - 07/05/2009, 07:12 PM RE: x exp(x) musing - by tommy1729 - 07/05/2009, 07:33 PM
Possibly Related Threads... Thread Author Replies Views Last Post almost periodic musing tommy1729 1 4,060 06/17/2014, 12:24 PM Last Post: tommy1729 just another uniqueness musing Gottfried 5 10,176 02/12/2009, 12:10 AM Last Post: bo198214
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http://www.matematicasvisuales.com/english/html/geometry/space/octahedron.html | 1,719,175,290,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198864850.31/warc/CC-MAIN-20240623194302-20240623224302-00854.warc.gz | 46,697,535 | 6,117 | The Octahedron
It is a very interesting experience to build and touch a model of an octahedron.
We can use cardboard (the octahedron consists of eight equilateral triangles):
Very basic origami (the six vertices are in three squares in three orthogonal planes):
Or you can use twelve plastic tubes:
An octahedron is composed by two pyramids of square base.
We can see the height of these two pyramides as the diagonal of a square.
The diagonal of a square of edge length 1 is:
Therefore, the volume of an octahedron of edge length 1 is (remember that the volume of a pyramid is one third of the base area times the perpendicular height):
And the volume of an octahedron of edge length a is:
Using that we can calculate the volume of a tetrahedron. We can consider a tetrahedron of edge length 2:
We can write a relation:
A tetrahedron of edge length 2 is made of one octahedron and four tetrahedra of edge length 1:
Then, the volume of an octahedron is four times the volume of a tetrahedron and we can recalculate the volume of a tetrahedron.
Origami: octahedron and tetrahedron.
REFERENCES
Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Luca Pacioli - De divina proportione - (La divina proporción) Ediciones Akal, 4ª edición, 2004. Spanish translation by Juan Calatrava.
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Demonstration of Pythagoras Theorem inspired in Euclid.
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
Using cardboard you can draw plane nets and build polyhedra.
A very simple technique to build complex and colorful polyhedra.
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Examples of polyhedra built using tubes.
Modular Origami is a nice technique to build polyhedra.
Examples of polyhedra built using tensegrity.
Examples of polyhedra built using Zome.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated octahedron.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the stellated octahedron (stella octangula).
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron. | 1,186 | 4,755 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2024-26 | latest | en | 0.845587 |
https://www.comsol.ch/blogs/overview-integration-methods-space-time/?setlang=1 | 1,571,279,121,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986672548.33/warc/CC-MAIN-20191017022259-20191017045759-00072.warc.gz | 853,445,197 | 21,158 | # Overview of Integration Methods in Space and Time
January 29, 2014
Integration is one of the most important mathematical tools, especially for numerical simulations. Partial Differential Equations (PDEs) are usually derived from integral balance equations, for example. Once a PDE needs to be solved numerically, integration most often plays an important role, too. This blog post gives an overview of the integration methods available in the COMSOL software and shows you how you can use them.
### The Importance of Integrals
COMSOL uses the finite element method, which transforms the governing PDE into an integral equation — the weak form, in other words. Having a closer look at the COMSOL simulation software, you may realize that many boundary conditions are formulated in terms of integrals. A couple of examples of these are Total heat flux or floating potential. Integration also plays a key role in postprocessing, as COMSOL provides many derived values based on integration, like electric energy, flow rate, or total heat flux. Of course, our users can also use integration in COMSOL for their own means, and here you will learn how.
### Integration by Means of Derived Values
A general integral has the form
\int_{t_0}^{t_1}\int_{\Omega}F(u)\ \mathrm{d A} \mathrm{d} t
where [t_0,t_1] is a time interval, \Omega is a spatial domain, and F(u) is an arbitrary expression in the dependent variable u. The expression can include derivatives with respect to space and time or any other derived value.
The most convenient way to obtain integrals is to use the “Derived Values” in the Results section of the new ribbon (or the Model Builder if you’re not running Windows®).
How to add volume, surface, or line integrals as Derived Values.
You can refer to any available solution by choosing the corresponding data set. The Expression field is the integrand and allows for dependent or derived variables. For transient simulations, the spatial integral is evaluated at each time step. Alternatively, the settings window offers Data Series Operations, where Integration can be selected for the time domain. This results in space-time integration.
Example of Surface Integration Settings with additional time integration via the Data Series Operation.
The Average is another Derived Value related to integration. It equals an integral, which is divided by the volume, area, or length of the considered domain. The Average Data Series Operation additionally divides by the time horizon. Derived Values are very useful, but because they are only available for postprocessing, they cannot handle every type of integration. That is why COMSOL provides more powerful and flexible integration tools. We demonstrate these methods with an example model below.
### Spatial and Temporal Integration for a Heat Transfer Example Model
We introduce a simple heat transfer model, a 2D aluminum unit square in the (x,y)-plane. The upper and right sides are fixed at room temperature (293.15 K) and on the left and lower boundary, a General inward heat flux of 5000W/m^2 is prescribed. A stationary solution and a time-dependent solution after 100 seconds are shown in the following figures.
Stationary solution, click image to enlarge.
Transient solution after 100 s, click image to enlarge.
### Spatial Integration by Means of Component Coupling Operators
Component Coupling Operators are, for example, needed when several integrals are combined in one expression, when integrals are requested during calculation, or in cases where a set of path integrals are required. Component Coupling Operators are defined in the Definitions section of the respective component. At that stage, the operator is not evaluated yet. Only its name and domain selection are fixed.
How to add Component Coupling Operators for later use.
For our example, we first want to calculate the spatial integral over the stationary temperature, which is given by
\int_{\Omega}T(x,y)\ \mathrm{d}x\mathrm{d}y = 301.65
In the COMSOL software, we use an integration operator, which is named intop1 by default.
Settings window of the integration operator.
How to evaluate the Integration operator.
In the next step, we demonstrate how an Integration operator can also be used within the model. We could, for example, ask what heating power we need to apply to obtain an average temperature of 303.15 K, which equals an average temperature increase of 10 K compared to room temperature. First, we need to compute the difference between the desired and the actual average temperature. The average is calculated by the integral over T, divided by the integral over the constant function 1, which gives the area of the domain. Fortunately, this type of calculation can easily be done with an Average operator in COMSOL. By default, such an operator is named aveop1. (Note that the average over the domain is the same as the integral for our example. That is because the domain has unit area.) The corresponding difference is given by
303.15-\int_{\Omega}T(x,y)\mathrm{d} x\mathrm{d} y = 1.50
Next, we need to find the General heat flux on the left and lower boundary, so that the desired average temperature is satisfied. To this end, we introduce an additional degree of freedom named q_hot and an additional constraint as a global equation. The General inward heat flux is replaced by q_hot.
How to add an additional degree of freedom and a global equation, which forces the average temperature to 303.15 K.
Solving this coupled system with a stationary study results in q_{hot}=5881.30 W/m^2. This value has to be prescribed as a General inward heat flux boundary condition to achieve an average temperature of 303.15 K in the whole domain.
### Computing the Antiderivative by Means of Integration Coupling
A frequently asked question we receive in Support is: How can one obtain the spatial antiderivative? The following application of integration coupling answers this question. The antiderivative is the counterpart of the derivative, and geometrically, it enables the calculation of arbitrary areas bounded by function graphs. One important application is the calculation of probabilities in statistical analyses. To demonstrate this, we fix y=0 in our example and denote the antiderivative of T(x,0) by u(x). This means that \frac{\partial u}{\partial x}=T(x,0). A representation of the antiderivative is the following integral
u(\bar x) = \int_0^{\bar x}T(x,0)\mathrm{d} x
where we use \bar x in order to distinguish the integration and the output variable. In contrast to the integrals above, we here have a function as a result, rather than a scalar quantity. We need to include the information that for each \bar x\in[0,1] the corresponding value of u(\bar x) requires an integral to be solved. Fortunately, this is easy to set up in the COMSOL environment and requires only three ingredients, so to speak. First, a logical expression can be used to reformulate the integral as
u(\bar x) = \int_0^1T(x,0)\cdot(x\leq\bar x)\ \mathrm{d} x
Second, we need an integration operator that acts on the lower boundary of our example domain. Let’s denote it by intop2. Third, we need to include the distinction of integration and output variable. The notation for this situation is source and destination for x and \bar x, respectively. When using an integration coupling operator, the built-in operator dest is available, which indicates that the corresponding expression does not belong to the integration variable. More precisely, it means \bar x=dest(x) in COMSOL. Putting the logical expression and the dest operator together, results in the expression T*(x<=dest(x)), which is exactly the input expression that we need for intop2. Altogether, we can calculate the antiderivative by intop2(T*(x<=dest(x))), resulting in the following plot in our example:
How to plot the antiderivative by Integration coupling, the dest operator, and a logical expression.
COMSOL provides two other integration coupling operators, namely general projection and linear projection. These can be used to obtain a set of path integrals in any direction of the domain. In other words, integration is performed only with respect to one dimension. The result is a function of one dimension less than the domain. For a 2D example the result is a 1D function, which can be evaluated on any boundary. Some more details on how to use these operators are subject to a forthcoming blog post on component couplings.
### Spatial Integration by Means of an Additional Physics Interface
The most flexible way of spatial integration is to add an additional PDE interface. Let’s remember the example of the antiderivative and assume that we want to calculate the antiderivative not only for y=0. The task can be formulated in terms of the PDE
\frac{\partial u}{\partial x}=T(x,y)
with Dirichlet boundary condition u=0 on the left boundary. The easiest interface to implement this equation is the Coefficient Form PDE interface, which only needs the following few settings:
How to use an additional physics interface for spatial integration.
The dependent variable u represents the antiderivative with respect to x and is available during calculation and postprocessing. Besides flexibility, a further advantage of this method is accuracy, because the integral is not obtained as a derived value, but is part of the calculation and internal error estimation.
### Temporal Integration by Means of Built-In Operators
We have already mentioned the Data Series Operations, which can be used for time integration. Another very useful method for time integration is provided by the built-in operators timeint and timeavg for time integration or time average, respectively. They are readily available in postprocessing and are used to integrate any time-dependent expression over a specified time interval. In our example we may be interested in the temperature average between 90 seconds and 100 seconds, i.e.:
\frac{1}{10}\int_{90}^{100}T(x,y,t)\ \mathrm{d} t
The following surface plot shows the resulting integral, which is a spatial function in (x,y):
How to use the built-in time integration operator timeavg.
Similar operators are available for integration on spherical objects, namely ballint, circint, diskint, and sphint.
### Temporal Integration by Means of Additional Physics Interfaces
If temporal integrals have to be available in the model, you need to define them as additional dependent variables. Similar to the Coefficient Form PDE example shown above, this can be done by adding an ODE interface of the Mathematics branch. Suppose, for example, that at each time step, the model requests the time integral from start until now over the total heat flux magnitude, which measures the accumulated energy. The variable for the total heat flux is automatically calculated by COMSOL and is named ht.tfluxMag. The integral can be calculated as an additional dependent variable with a Distributed ODE, which is a subnode of the Domain ODEs and DAEs interface. The source term of this domain ODE is the integrand, as shown in the following figure.
How to use an additional physics interface for temporal integration.
What is the benefit of such a calculation? The integral can be reused in another physics interface, which may be influenced by the accumulated energy in the system. Moreover, it is now available for all kinds of postprocessing, which is more convenient and faster than built-in operators. For an example, check out the Carbon Deposition in Hetereogeneous Catalysis model, where a domain ODE is used to calculate the porosity of a catalyst as a time-dependent field variable in the presence of chemical reactions.
### Integration of Analytic Functions and Expressions
So far, we have shown how to integrate solution variables during calculation or in postprocessing. We have not yet covered integrals of analytic functions or expressions. To this end, COMSOL provides the built-in operator integrate(expression, integration variable, lower bound, upper bound).
The expression might be any 1D function, such as sin(x). It is also possible to include additional variables, such as sin(x*y). The second argument specifies over which variable the integral is calculated. For example integrate(sin(x*y),y,0,1) yields a function in x, because integration only eliminates the integration variable y. Note that the operator can also handle analytic functions, which need to be defined in the Definitions node of the current component.
How to add an analytic function.
How to integrate over an analytic function.
#### Categories
##### Jing Huang
February 21, 2014
Interesting, but I am wondering how to extend the “Spatial Integration by Means of an Additional Physics Interface” to 2 dimensional?
December 1, 2014
It is a very interesting topic and well presented too. I gain much from it and I believe many other COMSOL users will benefit from it if the author could make a webinar based on this blog. (I mean webinars are advertised better and have more attention).
Many thanks.
##### neeraj mishra
May 13, 2016
sir my topic is simulation of dielectric elastomer actuator i m using comsol multyphysics 5.0
i m not getting result plz help me .
PFA: this is my base paper
http://www.sciencedirect.com/science/article/pii/S0924424707004335 | 2,825 | 13,340 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2019-43 | latest | en | 0.908117 |
https://www.unitconverters.net/electric-field-strength/newton-coulomb-to-statvolt-centimeter.htm | 1,709,448,024,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476205.65/warc/CC-MAIN-20240303043351-20240303073351-00226.warc.gz | 1,013,074,259 | 2,873 | Home / Electric Field Strength Conversion / Convert Newton/coulomb to Statvolt/centimeter
# Convert Newton/coulomb to Statvolt/centimeter
Please provide values below to convert newton/coulomb [N/C] to statvolt/centimeter [stV/cm], or vice versa.
From: newton/coulomb To: statvolt/centimeter
### Newton/coulomb to Statvolt/centimeter Conversion Table
Newton/coulomb [N/C]Statvolt/centimeter [stV/cm]
0.01 N/C3.335646048E-7 stV/cm
0.1 N/C3.335646048E-6 stV/cm
1 N/C3.33565E-5 stV/cm
2 N/C6.67129E-5 stV/cm
3 N/C0.0001000694 stV/cm
5 N/C0.0001667823 stV/cm
10 N/C0.0003335646 stV/cm
20 N/C0.0006671292 stV/cm
50 N/C0.001667823 stV/cm
100 N/C0.003335646 stV/cm
1000 N/C0.0333564605 stV/cm
### How to Convert Newton/coulomb to Statvolt/centimeter
1 N/C = 3.33565E-5 stV/cm
1 stV/cm = 29979.19999934 N/C
Example: convert 15 N/C to stV/cm:
15 N/C = 15 × 3.33565E-5 stV/cm = 0.0005003469 stV/cm | 349 | 895 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2024-10 | latest | en | 0.52186 |
http://gamedev.stackexchange.com/questions/tagged/2d?sort=faq&pagesize=30 | 1,469,406,225,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824201.28/warc/CC-MAIN-20160723071024-00149-ip-10-185-27-174.ec2.internal.warc.gz | 93,528,545 | 27,738 | # Tagged Questions
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https://holmarc.com/ultra_acousto_detector.php | 1,610,912,102,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703513144.48/warc/CC-MAIN-20210117174558-20210117204558-00382.warc.gz | 383,442,823 | 9,842 | Detector Based Apparatus for
Ultrasonic Diffraction -
Acousto optic effect
Model: HO-ED-A-01
The Ultrasonic diffraction apparatus is used to study diffraction of light by ultrasonic waves. Ultrasonic sound refers to sound with a frequency greater than the human audible range (20Hz to 20 KHz). When an ultrasonic wave propagates through a medium, the molecules in that medium vibrate over very short distance in a direction parallel to the longitudinal wave. The apparatus consists of a graduated long rail and rail carriages appropriately fitted with laser head, an RF oscillator and a detector with translation stage. The ultrasonic diffraction setup uses laser as light source. As the laser beam is intense and monochromatic, we get clear higher order diffraction pattern.
The ultrasonic waves generated by the transducer travels down the medium (liquid) and gets reflected at the bottom (flat glass plate) of the cell. The incident and reflected waves interfere and a stationary / standing wave pattern is formed. The laser head is mounted using a kinematic holder. This helps to direct the laser beam through the liquid and then to the detector conveniently. The diffraction pattern is scanned using a translation stage with freedom in X axis. The velocity of ultrasonic waves in liquids can be calculated from this experiment. This instrument is designed to give accurate and best results.
Experiment Examples
To find the velocity of ultrasonic wave in liquids
The velocity of ultrasonic wave in a liquid,
V = ʋ ʌ
Where υ is the frequency of the crystal oscillator and Λ is the wavelength of ultrasonic wave.
We have,
Λ = n λ / Sin θ
Where n is the order of diffraction, λ is the wavelength of the laser used and θ is the angle of diffraction.
θ = tan-1 ( D / L )
D is the order length and L is the distance measured from the crystal oscillator to the detector.
To find the bulk modulus of the given liquid
The bulk modulus of the liquid,
β = ρV2
Where ρ is the density of the liquid and V is the velocity of the ultrasonic wave.
To find the compressibility of the liquid
The compressibility of a liquid is the reciprocal of bulk modulus,
K = 1 / ρV2
Where ρ is the density of the liquid and is V the velocity of the ultrasonic wave.
Specification of Piezo Electric Crystals
Dimension : 20 mm diameter x 0.7 mm thickness
Resonant frequency fr : 3 MHz ± 50 KHz
Resonant impedance Zm : ≤ 6 Ω.
Static capacitance Cs : 570 0pF ± 15% @ 1 kHz
Dimension : 20 mm diameter x 0.4 mm thickness
Resonant frequency fr : 5 MHz ± 100 KHz
Resonant impedance Zm : ≤ 0.48 Ω.
Static capacitance Cs : 3800 pF ± 20% @ 60 Hz/1 V
Fig: Optical system for observation of diffraction by ultrasonic waves
Fig: Graph shows distance Vs. detector current
Features
Precision design
Diode laser is used as light source
5 MHz,3 MHz crystals
Corrosion free
High quality photo detector
Related Topics
Interference
Standing waves
Huygens' Principle
Bulk modulus
Compressibility
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Optical Rail
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Length
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1500mm
Material
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Black anodized Aluminum alloy
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Kinematic Laser Mount
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Material
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Black anodized Aluminum alloy
:
:
+/-4 degrees
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1 no.
Glass tank Mount with Crystal Holder
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Material
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Black anodized Aluminum alloy
:
Quantity
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Detector Mount with X- Translation
Model No: ED-A-01-DMXT
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Black anodized Aluminum alloy
Travel
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Micrometer controlled
Resolution
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0.01 mm
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3, 5 MHz
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650 nm
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Photo Transistor
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7 segment, 3 ½ digit
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\$ 46.00
| | 1,151 | 4,276 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2021-04 | longest | en | 0.887247 |
https://de.mathworks.com/matlabcentral/answers/763981-how-do-you-save-values-from-a-for-loop-in-a-vector?s_tid=prof_contriblnk | 1,632,750,681,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00351.warc.gz | 240,664,448 | 24,030 | # How do you save values from a for-loop in a vector?
1 view (last 30 days)
Buttercup12 on 5 Mar 2021
Commented: Buttercup12 on 5 Mar 2021
I have the following:
for ml=100:50:1000
z=fzero(@(x) y(x,ml), [1 10]);
disp(num2str(z))
end
But I need each value of z saved as a vector to use it in a plot later. How do you do that?
Stephen on 5 Mar 2021
Edited: Stephen on 5 Mar 2021
With MATLAB it is almost always better to loop over an index than to loop over data:
ml = 100:50:1000;
n = numel(ml);
z = nan(1,n); % preallocate
for k = 1:n % loop over indices
z(k) = fzero(@(x) y(x,ml(k)), [1,10]);
end %^^^ indexing ^^^
Buttercup12 on 5 Mar 2021
Thank you! :) | 232 | 656 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2021-39 | latest | en | 0.792114 |
https://physics.stackexchange.com/questions/221642/tachyons-and-lorentz-velocity-transformation | 1,701,754,491,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100545.7/warc/CC-MAIN-20231205041842-20231205071842-00702.warc.gz | 512,654,538 | 41,399 | # Tachyons and Lorentz velocity transformation
In general, is it possible to apply the Lorentz velocity transformation to a tachyon?
I have tried to do so but the results seemed very illogical. Here's my attempt: suppose an evil spaceship named XYZ is moving away from Earth at speed $0.6c$ in the +x direction. We send a futuristic tachyonic spaceship named TY to attack XYZ at speed $4.0c$, also in the +x direction.
Now consider a person on the tachyonic spaceship. He attempts to find the speed which the XYZ is approaching himself (therefore, the speed must be negative). Using the Lorentz velocity transformation: $$\begin{eqnarray*} v'&=&\frac{v-u}{1-uv/c^2}\\ &=&\frac{0.6c-4.0c}{1-(0.6)(4.0)}\\ &=&2.4c \end{eqnarray*}$$
This implies that the XYZ is moving forward, away from TY at a speed even faster! The tachyonic spaceship will not be able to catch up to XYZ. What is the problem here?
• Dec 1, 2015 at 16:51
• There is no transformation between frames with a relative velocity greater than $c$. They are causally disconnected. Applying any formula derived from the Lorentz transformation will just give meaningless results. Dec 1, 2015 at 16:53 | 320 | 1,163 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.34375 | 3 | CC-MAIN-2023-50 | latest | en | 0.920863 |
https://gmatclub.com/forum/if-2-x-2-x-2-exponent-x-2-3-2-13-what-is-value-of-x-26940.html?fl=similar | 1,508,219,188,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187820927.48/warc/CC-MAIN-20171017052945-20171017072945-00062.warc.gz | 885,138,572 | 38,560 | It is currently 16 Oct 2017, 22:46
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# If 2^x - 2^x-2(exponent x-2) = 3(2^13), what is value of x?
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If 2^x - 2^x-2(exponent x-2) = 3(2^13), what is value of x? [#permalink]
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24 Feb 2006, 20:02
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
If 2^x - 2^x-2(exponent x-2) = 3(2^13), what is value of x?
a) 9
b) 11
c) 13
d) 15
e) 17
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24 Feb 2006, 20:23
X = 5
2^x (1-1/4) = 3*(2^13)
=> (2^x)*3/4 = 3*(2^13)
=> 2^x = 2^15
=> x = 15
Kudos [?]: 17 [0], given: 0
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Joined: 14 Dec 2004
Posts: 1685
Kudos [?]: 168 [0], given: 0
### Show Tags
24 Feb 2006, 21:50
D) x= 15.
2^x-2 * (4-1) = 3 * (2^13)
= 2^x-2 = 2^13
=> x = 15
Kudos [?]: 168 [0], given: 0
24 Feb 2006, 21:50
Display posts from previous: Sort by | 589 | 1,540 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2017-43 | latest | en | 0.817291 |
https://www.excelforum.com/excel-formulas-and-functions/912570-aging-a-date-in-column-a-unless-there-is-a-value-in-column-b.html | 1,653,464,296,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662580803.75/warc/CC-MAIN-20220525054507-20220525084507-00090.warc.gz | 835,433,764 | 14,806 | # aging a date in column A unless there is a value in column B
1. ## aging a date in column A unless there is a value in column B
Hello,
I need help coming up with a formula. In column A I have various dates that a customer began doing business with us. In column B is the date that that customer began generating revenue. I need to put a formula in column C that calculates the number of days between now and when the customer begain doing business with us (column A) unless there is a date in column B.
Book1.xlsx
In this example I would need a value returned in column C on rows 3, 4, 5, and 7 but not on 2 and 6.
Any help would be greatly appreciated!
2. ## Re: aging a date in column A unless there is a value in column B
Maybe this...
Entered in C2 and copied down:
=IF(B2="","",TODAY()-A2)
3. ## Re: aging a date in column A unless there is a value in column B
That is a good start, but the column still displays a value even if there is something in column B.
4. ## Re: aging a date in column A unless there is a value in column B
I figured it out. The formula I needed was =IF(and(A2>0, B2=""), TODAY()-A2, "")
Thanks for the help!
5. ## Re: aging a date in column A unless there is a value in column B
Good deal. Thanks for the feedback!
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https://www.mathworks.com/matlabcentral/cody/problems/1958-add-a-row-of-zeros-on-top-of-a-matrix/solutions/1204819 | 1,576,143,487,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540542644.69/warc/CC-MAIN-20191212074623-20191212102623-00220.warc.gz | 789,212,832 | 15,561 | Cody
# Problem 1958. Add a row of zeros on top of a matrix
Solution 1204819
Submitted on 5 Jun 2017 by Abid Hasan
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
x = rand(4); y_correct = [zeros(1,size(x,2));x]; assert(isequal(addrow(x),y_correct))
2 Pass
x = []; y_correct = zeros(1,0); assert(isequal(addrow(x),y_correct))
3 Pass
x = rand(8,1); y_correct = [zeros(1,size(x,2));x]; assert(isequal(addrow(x),y_correct))
4 Pass
x = zeros(0,1); y_correct = 0; assert(isequal(addrow(x),y_correct)) | 202 | 625 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2019-51 | latest | en | 0.620761 |
https://exceljet.net/glossary/lifting | 1,726,868,496,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725701423570.98/warc/CC-MAIN-20240920190822-20240920220822-00476.warc.gz | 213,533,737 | 9,539 | The term "lifting" refers to an array calculation behavior in Excel formulas. When you give a range or array to a function not programmed to accept arrays natively, Excel will "lift" the function and call it multiple times, one time for each value in the array. The result is an array with the same dimensions as the input array.
Lifting is a built-in behavior that happens automatically. Lifting can occur when an array is fed into a function argument that doesn't accept an array (but does accept single values), or when a range is fed into a function argument that doesn't accept a range or an array.
Note: in Excel 2021 or later, you will see lifting happen in real-time as multiple results spill onto the worksheet. In earlier versions of Excel, lifting still occurs, but only one result is displayed in the cell that contains the formula.
### Example
The example shown illustrates what happens if you call the LEN function on the range C5:C7, which contains three values. LEN isn't programmed to handle arrays natively, so LEN is run three times, once for each value in an operation like this:
``````=LEN(C5:C7)
=LEN({"dog";"kitten";"fish"})
={3;6;4}
``````
Notice the result is a vertical array with three values, just like the source range.
### Dealing with multiple results
When lifting occurs in a formula, there will be multiple results, and these need to be catered for.
In the example above, because LEN returns three separate values in an array, we need handle the output with a function that can work with arrays. One option is to calculate a total character count in the range C5:C7 using SUMPRODUCT:
``````=SUMPRODUCT(LEN(C5:C7))
``````
SUMPRODUCT will handle arrays natively, so this formula does not require control + shift + enter.
The SUM function could be used as well, but would need to be entered with CSE:
``````{=SUM(LEN(C5:C7))}
``````
Note: A special case of lifting is called "pairwise lifting", which is combining arrays in a pairwise fashion. | 463 | 1,987 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2024-38 | latest | en | 0.853408 |
https://www.reddit.com/r/AskReddit/comments/p8o3z/does_everyone_who_plays_monopoly_pay_the_200_and/ | 1,490,222,405,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218186353.38/warc/CC-MAIN-20170322212946-00610-ip-10-233-31-227.ec2.internal.warc.gz | 967,261,407 | 18,429 | This is an archived post. You won't be able to vote or comment.
[–] 7 points8 points (0 children)
only if the 200 is less than the 10%, its not that hard to figure out
[–] 0 points1 point (0 children)
I haven't played that version in a long time actually. I've been playing the new Tycoon version that deals in millions of dollars and credit cards.
The board has a new set amount for the 10%, and it's 2 million.
But to answer your question, after a certain point (once I've generated over \$2000 of wealth in the old game), I just pay the \$200 because it's cheaper.
[–] 0 points1 point (1 child)
I never figure out what the 10%... It's too much work.
[–] 0 points1 point (0 children)
i think it's the sum of all the cash you have + the mortgage values of all your properties.
[–] 0 points1 point (0 children)
We always played where you could pick. 10% is a great deal when you're poor.
[–] 0 points1 point (0 children)
no, i always calculate the 10% and it's almost always less than \$200. | 284 | 1,011 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2017-13 | latest | en | 0.965193 |
https://www.proprofs.com/quiz-school/story.php?title=earthquakes-assessment-mcfadden-period-2-2015 | 1,713,986,085,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296819847.83/warc/CC-MAIN-20240424174709-20240424204709-00310.warc.gz | 872,349,374 | 107,373 | # Earthquakes Assessment Mcfadden Period 2 2015
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• 1.
### When stress causes rocks to break: vibrations called _____________________ are produced.
• A.
Earthquakes
• B.
Tsunami
• C.
Faults
• D.
Elastic Limit
A. Earthquakes
Explanation
When stress causes rocks to break, the release of energy creates vibrations known as earthquakes. These vibrations travel through the Earth's crust, causing the ground to shake. Earthquakes can occur along fault lines, which are areas where rocks have broken and slipped past each other. The elastic limit refers to the maximum amount of stress a material can handle before permanently deforming, but it is not directly related to the production of vibrations. Tsunamis, on the other hand, are large ocean waves typically caused by underwater earthquakes, but they are not the vibrations themselves.
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• 2.
### When tension forces pull rock apart a ________________ occurs.
• A.
Reverse Fault
• B.
Normal Fault
• C.
Strike-slip Fault
• D.
Transform Fault
B. Normal Fault
Explanation
When tension forces pull rock apart, a normal fault occurs. In a normal fault, the hanging wall moves downward relative to the footwall, resulting in the extension and stretching of the rock mass. This type of faulting is commonly associated with divergent plate boundaries, where two tectonic plates move away from each other, creating tensional forces that cause the rock to fracture and slide. The movement along the fault plane is primarily vertical, with the hanging wall moving down and the footwall moving up. This can lead to the formation of fault scarps and the displacement of rock layers.
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• 3.
### _______________________ waves cause particles in rocks to move at right angles to the direction of the waves.
• A.
Surface Waves
• B.
Primary Waves
• C.
Secondary Waves
• D.
Tertiary Waves
C. Secondary Waves
Explanation
Secondary waves, also known as shear waves, are a type of seismic wave that causes particles in rocks to move at right angles to the direction of the wave. These waves are slower than primary waves and can only travel through solid materials. They are responsible for the side-to-side shaking motion during an earthquake and are the second wave to arrive at a seismograph station, hence the name "secondary waves."
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• 4.
### The ___________ of an earthquake is the point of Earth's surface above the focus.
• A.
Focus
• B.
Epicenter
• C.
Foci
• D.
Fault
B. Epicenter
Explanation
The epicenter of an earthquake refers to the point on the Earth's surface directly above the focus, which is the underground origin or starting point of the earthquake.
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• 5.
### The measure of energy released by an earthquake is the earthquake's ________________________
• A.
Magnitude
• B.
Destructive Force
• C.
Focus
• D.
Epicenter
A. Magnitude
Explanation
The measure of energy released by an earthquake is known as its magnitude. Magnitude is a quantitative measurement of the seismic energy released during an earthquake, which is calculated using various seismological techniques. It is an important factor in understanding the strength and impact of an earthquake, as it provides a standardized scale to compare different seismic events. Magnitude is typically reported using the Richter scale or the moment magnitude scale (Mw), both of which take into account the amplitude of seismic waves recorded by seismographs.
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• 6.
### At a _____________________, the rocks above the fault surface are forced up and over the rocks below the fault surface.
• A.
Reverse Fault
• B.
Normal Fault
• C.
Strike-Slip Fault
• D.
Transform Fault
A. Reverse Fault
Explanation
A reverse fault occurs when the rocks above the fault surface are pushed up and over the rocks below the fault surface. This type of fault is characterized by a steeply inclined fault plane and is caused by compressional forces in the Earth's crust. The movement along a reverse fault is vertical and the hanging wall moves up relative to the footwall. Reverse faults are commonly associated with convergent plate boundaries, where two tectonic plates collide and cause compression.
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• 7.
### By studying seismic wave information, a scientist discovered that boundary between Earth's crust and its upper mantle, which is called the ______________.
• A.
Moho
• B.
Lithosphere
• C.
Asthenosphere
• D.
Crust
A. Moho
Explanation
The correct answer is Moho. The Moho is the boundary between the Earth's crust and its upper mantle. It was discovered by studying seismic wave information. Seismic waves travel at different speeds through different layers of the Earth, and the Moho is characterized by a significant increase in seismic wave velocity. This discovery has provided valuable insights into the structure and composition of the Earth's interior.
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• 8.
### A ___________________ is a seawave that is caused by seismic activity and can cause great devastation.
• A.
P-waves
• B.
S-waves
• C.
Surface Waves
• D.
Tsunami
D. Tsunami
Explanation
A tsunami is a seawave that is caused by seismic activity and can cause great devastation. Tsunamis are typically triggered by underwater earthquakes, volcanic eruptions, or landslides. These events generate powerful waves that can travel across the ocean at high speeds, and when they reach shallow waters near the coastline, they can rapidly increase in height and cause widespread destruction. Tsunamis are known for their destructive power and have the potential to cause loss of life and property damage on a massive scale.
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• 9.
### Most destruction is caused by __________________ waves
• A.
S-waves
• B.
Surface Waves
• C.
P-waves
• D.
Sound Waves
B. Surface Waves
Explanation
Surface waves are the correct answer because they are responsible for causing the most destruction during an earthquake. These waves travel along the Earth's surface and have a horizontal motion that can cause buildings and infrastructure to collapse. They are slower than P-waves and S-waves but have a larger amplitude, making them more destructive. Surface waves are also responsible for the shaking and rolling motion felt during an earthquake, which can lead to landslides and tsunamis in coastal areas.
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• 10.
### An instrument called a _____________________ is used to record seismic waves
• A.
Seismograph
• B.
Seismogram
• C.
Seismometer
A. Seismograph
Explanation
A seismograph is an instrument used to record seismic waves. It consists of a heavy weight attached to a stationary base, and a pen or stylus attached to the weight. When an earthquake occurs, the base remains stationary while the weight and pen move with the shaking ground. This movement is recorded on a rotating drum or a digital display, creating a seismogram. A seismometer, on the other hand, is an instrument that measures the motion of the ground during an earthquake but does not record it. Therefore, the correct instrument used to record seismic waves is a seismograph.
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• 11.
### A fault between two plates that are moving sideways past each other is called a _______________ fault
• A.
Normal
• B.
Reverse
• C.
Transform
• D.
Strike-slip
D. Strike-slip
Explanation
A fault between two plates that are moving sideways past each other is called a strike-slip fault. In this type of fault, the movement occurs horizontally along the fault line, with one block of rock sliding past the other in a sideways motion. This can result in shearing and displacement of the rocks on either side of the fault. Strike-slip faults are commonly associated with transform plate boundaries, such as the San Andreas Fault in California.
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• 12.
### ________________ waves cause particles in the Earth to move back and forth in the same direction (similar to a slinky) as the wave travels
• A.
Secondary
• B.
Light
• C.
Surface
• D.
Primary
D. Primary
Explanation
Primary waves, also known as P-waves, are a type of seismic wave that travel through the Earth's interior. These waves cause particles in the Earth to move back and forth in the same direction as the wave travels. This means that as the P-wave travels, the particles in the Earth move in the same direction as the wave, similar to how a slinky moves back and forth when stretched and released. Therefore, the correct answer is Primary.
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• 13.
### Earthquakes generate energy waves called ____________________.
• A.
Seismic Waves
• B.
Ocean Waves
• C.
Sound Waves
• D.
Light Waves
A. Seismic Waves
Explanation
Earthquakes generate energy waves called seismic waves. These waves are produced by the release of energy during an earthquake and travel through the Earth's layers. Seismic waves are responsible for the shaking and vibrations felt during an earthquake. They can be categorized into two main types: body waves and surface waves. Body waves include primary (P) waves and secondary (S) waves, which travel through the Earth's interior. Surface waves, on the other hand, travel along the Earth's surface and are responsible for the most destructive effects of an earthquake.
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• 14.
### When the force on rocks is great enough, they break, producing vibrations called _____________.
• A.
Faults
• B.
Earthquakes
• C.
Strains
• D.
Stresses
B. Earthquakes
Explanation
When rocks experience a significant amount of force, they can fracture and create vibrations known as earthquakes. Faults, strains, and stresses are all related to the movement and deformation of rocks, but earthquakes specifically refer to the seismic events caused by the breaking of rocks under intense force.
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• 15.
### Once the elastic limit of rocks is passed, they break and move along surfaces called _______________.
• A.
Faults
• B.
Earthquakes
• C.
Strains
• D.
Stresses
A. Faults
Explanation
When rocks are subjected to stress beyond their elastic limit, they undergo deformation and eventually break. The broken pieces of rocks then move along surfaces known as faults. Faults are the result of the movement and displacement of rocks, often caused by tectonic forces. Earthquakes are closely associated with faults, as the sudden release of energy during an earthquake is often triggered by the movement along these fault surfaces. Therefore, faults are the correct answer in this context.
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• 16.
### Most earthquakes happen ____________.
• A.
Without warning
• B.
In areas where earthquakes have occurred in the past
• C.
Along plate boundaries
• D.
All the answers provided are correct
D. All the answers provided are correct
Explanation
All the answers provided are correct because most earthquakes can happen without warning, in areas where earthquakes have occurred in the past, and along plate boundaries. Earthquakes can occur suddenly and unpredictably, causing significant damage and loss of life. They are more likely to happen in regions that have a history of seismic activity, as well as along the boundaries of tectonic plates where the movement and interaction of plates can generate seismic activity. Therefore, all the given options accurately describe the occurrence of most earthquakes.
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• 17.
### A person twice as far from the epicenter of an earthquake as another person will notice that the time between the arrival of the primary and secondary waves will be ________________.
• A.
The same
• B.
Larger
• C.
Reduced
• D.
Unnoticeable
B. Larger
Explanation
The person who is twice as far from the epicenter of an earthquake as another person will notice that the time between the arrival of the primary and secondary waves will be larger. This is because the primary waves, also known as P-waves, travel faster than the secondary waves, also known as S-waves. As the distance from the epicenter increases, the time gap between the arrival of these waves also increases. Therefore, the person who is farther away will experience a longer time interval between the arrival of the primary and secondary waves.
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• 18.
### Scientists discovered the different layers in Earth's interior by studying___________.
• A.
Tsunamis
• B.
Tides
• C.
Changes in seismic waves
• D.
All of the answers provided are correct
C. Changes in seismic waves
Explanation
Scientists discovered the different layers in Earth's interior by studying changes in seismic waves. Seismic waves are waves of energy that are generated by earthquakes and travel through the Earth. As these waves pass through different layers of the Earth, they can be refracted, reflected, or absorbed, providing valuable information about the composition and structure of the Earth's interior. By analyzing the behavior of seismic waves, scientists have been able to identify and study the various layers of the Earth, such as the crust, mantle, and core.
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• 19.
### ____________ is the force that squeezes rocks together.
• A.
Tension
• B.
Shear
• C.
Elastic Limit
• D.
Compression
D. Compression
Explanation
Compression is the force that squeezes rocks together. When rocks are subjected to compression, they are pushed together, causing them to be compressed and potentially resulting in deformation or fracturing. This force is commonly associated with convergent plate boundaries, where tectonic plates collide, causing rocks to be compressed and folded. Compression is also responsible for the formation of mountains and the creation of geological features such as fault lines and folds.
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• 20.
### ____________ is the force that pulls rocks apart.
• A.
Tension
• B.
Shear
• C.
Elastic Limit
• D.
Compression
A. Tension
Explanation
Tension is the force that pulls rocks apart. When a force is applied in opposite directions to an object, tension occurs, causing the object to stretch or elongate. In the case of rocks, tension can lead to the formation of cracks and fractures, as the rocks are pulled apart along their weakest points. This force is commonly observed in geological processes such as faulting and the formation of rift valleys.
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• 21.
### ____________ is the force that causes plates to move sideways past each other.
• A.
Tension
• B.
Shear
• C.
Elastic Limit
• D.
Compression
B. Shear
Explanation
Shear is the force that causes plates to move sideways past each other. When two plates are under stress and the forces acting on them are parallel but in opposite directions, shear forces occur. These forces cause the plates to slide past each other, resulting in lateral movement. Shear is a common force in areas where tectonic plates meet, such as fault lines, and is responsible for earthquakes and the formation of features like transform boundaries.
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• 22.
### ______________ faults are caused by compressional forces.
• A.
Normal
• B.
Strike-Slip
• C.
Reverse
• D.
Elastic
C. Reverse
Explanation
Reverse faults are caused by compressional forces. In a reverse fault, the hanging wall moves upward relative to the footwall, resulting in a steeply inclined fault plane. This type of fault is commonly found in areas where two tectonic plates are colliding, such as convergent plate boundaries. The compressional forces push the rocks together, causing them to buckle and fracture, leading to the formation of a reverse fault.
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• 23.
### ______________ faults are caused by shear forces.
• A.
Normal
• B.
Strike-Slip
• C.
Reverse
• D.
Elastic
B. Strike-Slip
Explanation
Strike-slip faults are caused by shear forces. In these faults, the rocks on either side of the fault move horizontally past each other, with no significant vertical displacement. This type of fault occurs when there is horizontal compression or tension in the Earth's crust, causing the rocks to slide past each other in a sideways motion. Examples of strike-slip faults include the San Andreas Fault in California.
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• 24.
### Along a(n) ________ fault, rock above the fault surface moves downward in relation to rock below the fault surface
• A.
Normal
• B.
Strike-Slip
• C.
Reverse
• D.
Elastic
A. Normal
Explanation
Along a normal fault, the rock above the fault surface moves downward in relation to the rock below the fault surface. This occurs due to tensional forces pulling the rocks apart, causing the overlying rock to slide down and create a gap. Normal faults are commonly found in areas of extension, such as divergent plate boundaries, where the Earth's crust is being pulled apart.
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• 25.
### The most destructive seismic waves are ________________.
• A.
Primary
• B.
Surface
• C.
Secondary
• D.
Tsunami
B. Surface
Explanation
Surface seismic waves are the most destructive because they travel along the Earth's surface and cause the most damage to buildings and infrastructure. These waves have a larger amplitude and longer period compared to other seismic waves, which allows them to transfer more energy and cause more destruction. Surface waves are responsible for the majority of the damage and casualties during an earthquake.
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• 26.
### The __________ waves are the first to reach a seismograph after an earthquake
• A.
Surface
• B.
Primary
• C.
Secondary
• D.
Tsunami
B. Primary
Explanation
Primary waves, also known as P-waves, are the first seismic waves to reach a seismograph after an earthquake. These waves are compressional waves that travel through the Earth's interior, causing particles to move in the same direction as the wave. They are the fastest seismic waves and can travel through solids, liquids, and gases. Due to their speed, they are the first to be detected by seismographs, providing valuable information about the earthquake's location and magnitude.
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• 27.
### At least ____________ seismographs are needed to accurately locate an earthquake's epicenter.
• A.
Two
• B.
Five
• C.
Four
• D.
Three
D. Three
Explanation
To accurately locate an earthquake's epicenter, at least three seismographs are needed. This is because seismographs record the arrival times of seismic waves at different locations. By comparing the arrival times of these waves at three different seismographs, scientists can triangulate the epicenter of the earthquake. If only two seismographs were used, it would be difficult to determine the exact location of the epicenter. Using more than three seismographs can provide additional data and improve accuracy, but three is the minimum number required for a reliable estimation.
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• 28.
### The point in the Earth's interior where the energy release of an earthquake occurs is called the _____________
• A.
Focus
• B.
Epicenter
• C.
Fault
• D.
Inner Core
A. Focus
Explanation
The point in the Earth's interior where the energy release of an earthquake occurs is called the focus. This is the exact location within the Earth where the seismic waves originate and the rupture of the fault begins. It is important to distinguish the focus from the epicenter, which is the point on the Earth's surface directly above the focus. The focus is usually located deep within the Earth, while the epicenter is the point that is felt and measured at the surface. The fault refers to the fracture or break in the Earth's crust where the movement occurs. The inner core, on the other hand, is the solid innermost part of the Earth.
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• 29.
### When seismic waves reach the Moho Discontinuity at the bottom of Earth's crust they will __________.
• A.
Slow Down
• B.
Stay the same
• C.
Stop
• D.
Speed up
D. Speed up
Explanation
Seismic waves, which are generated by earthquakes or other sources of energy, travel through different layers of the Earth's interior. When these waves reach the Moho Discontinuity, which is the boundary between the Earth's crust and the underlying mantle, they undergo a change in velocity. The seismic waves speed up as they transition from the crust to the denser mantle. This increase in speed is due to the difference in physical properties of the two layers, such as density and composition. Therefore, the correct answer is "Speed up."
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• 30.
### The magnitude of an earthquake is measured by the _______________
• A.
Richter Scale
• B.
Moho
• C.
Modified Mercalli Scale
• D.
Elastic limit
A. Richter Scale
Explanation
The magnitude of an earthquake is measured by the Richter Scale. This scale was developed by Charles F. Richter in 1935 and is used to quantify the energy released during an earthquake. It measures the amplitude of seismic waves recorded by seismographs, with each whole number increase on the scale representing a tenfold increase in amplitude and approximately 31.6 times more energy release. The Richter Scale is widely recognized and used by scientists and engineers to compare and classify earthquakes based on their magnitude.
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• 31.
### Primary waves ___________ when they hit the liquid outer core.
• A.
Slow down and bend
• B.
Stay the same
• C.
Stop
• D.
Speed up
A. Slow down and bend
Explanation
Primary waves, also known as P-waves, are seismic waves that can travel through both solid and liquid materials. When these waves reach the liquid outer core, they slow down and bend due to the change in density and composition of the material. This bending phenomenon is known as refraction. Therefore, the correct answer is "Slow down and bend."
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• 32.
### Secondary waves ___________ when they hit the liquid outer core
• A.
Slow down
• B.
Stay the same
• C.
Speed up
• D.
Stop
D. Stop
• 33.
### Liquefaction is when wet soil acts like a(n) ____________________.
• A.
Solid
• B.
Liquid
• C.
Gas
• D.
Semi-solid
B. Liquid
Explanation
Liquefaction is a phenomenon in which wet soil loses its strength and behaves like a liquid. This occurs when the soil is saturated with water and subjected to stress, such as during an earthquake. The excess water in the soil reduces the friction between particles, causing them to lose contact and flow like a liquid. This can lead to ground instability, sinking, and damage to structures built on the affected soil.
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• 34.
### The difference between a magnitude 1 and a magnitude 2 on the Ritcher Scale is ________________ times more powerful.
• A.
23
• B.
32
• C.
33
• D.
22
B. 32
Explanation
The difference between a magnitude 1 and a magnitude 2 on the Richter Scale is 32 times more powerful. The Richter Scale is a logarithmic scale used to measure the intensity of earthquakes. Each whole number increase on the scale represents a tenfold increase in the amplitude of the seismic waves and approximately 32 times more energy released. Therefore, a magnitude 2 earthquake would be 32 times more powerful than a magnitude 1 earthquake.
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• 35.
### This is an example of a ______________fault.
• A.
Normal
• B.
Reverse
• C.
Strike-slip
A. Normal
Explanation
This is an example of a normal fault. In a normal fault, the hanging wall moves downward relative to the footwall. This type of faulting occurs in areas undergoing tensional stress, where the crust is being pulled apart. The movement along the fault plane is typically vertical or near-vertical.
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• 36.
### This is an example of a ______________________fault.
• A.
Normal
• B.
Reverse
• C.
Strike-slip
B. Reverse
Explanation
This question is asking for the type of fault that is being described. The term "reverse" refers to a type of fault where the rock layers are pushed together, causing one side to move upwards relative to the other side. This is opposite to a normal fault where the rock layers are pulled apart. A strike-slip fault, on the other hand, involves horizontal movement along the fault line. Therefore, the correct answer for this question is "reverse" fault.
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• 37.
### This is an example of a ______________________fault.
• A.
Normal
• B.
Reverse
• C.
Strike-slip
C. Strike-slip
Explanation
This question is asking for the type of fault that is being described. The term "strike-slip" refers to a type of fault where the rocks on either side of the fault move horizontally past each other. In this case, the fault is not described as normal or reverse, but rather as a strike-slip fault.
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• 38.
### This fault is caused by ____________________ forces.
• A.
Tensional
• B.
Compressional
• C.
Shear
A. Tensional
Explanation
This fault is caused by tensional forces. Tensional forces occur when rocks are being pulled apart, causing the fault to form and the rocks to move away from each other. This type of fault is commonly found in areas where the Earth's crust is being stretched, such as divergent plate boundaries or rift zones.
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• 39.
### This fault is caused by ____________________ forces.
• A.
Tensional
• B.
Compressional
• C.
Shear
B. Compressional
Explanation
This fault is caused by compressional forces. Compressional forces occur when two tectonic plates collide or push against each other, causing rocks to be pushed together and compressed. This compression can cause the rocks to buckle or break, resulting in a fault. Compressional forces are common in areas where two continental plates collide, such as the Himalayas.
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• 40.
### This fault is caused by ____________________ forces.
• A.
Tensional
• B.
Compressional
• C.
Shear
C. Shear
Explanation
This fault is caused by shear forces. Shear forces occur when two tectonic plates slide past each other horizontally. This type of fault is characterized by a lateral movement along the fault line, causing rocks on either side to move in opposite directions. Shear forces can result in the formation of transform boundaries, where plates slide past each other, or strike-slip faults, where rocks on either side of the fault line move horizontally.
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• 41.
### Using the graphic provided, answer the following question. Which wave travels faster?
• A.
S-waves
• B.
P-waves
• C.
Surface Waves
B. P-waves
Explanation
P-waves travel faster than S-waves and surface waves. P-waves, also known as primary waves, are the fastest seismic waves and can travel through solids, liquids, and gases. They compress and expand the material they pass through, causing particles to move in the same direction as the wave. S-waves, or secondary waves, are slower and can only travel through solids. They move particles perpendicular to the direction of the wave. Surface waves, as the name suggests, only travel along the surface of the Earth and are slower than both P-waves and S-waves.
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• 42.
### Using the graphic provided, answer the following question. What is the time difference at 3,000km?
• A.
4 Minutes
• B.
8 Seconds
• C.
16 Minutes
• D.
8 Minutes
D. 8 Minutes
Explanation
Based on the graphic provided, it can be inferred that there is a time difference of 8 minutes at 3,000km. This can be determined by observing that each line on the graphic represents a time difference of 2 minutes, and there are 4 lines between 0km and 3,000km. Therefore, multiplying 2 minutes by 4 lines equals a time difference of 8 minutes.
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• 43.
### Using the graphic provided, answer the following question. Which wave has a more constant speed?
• A.
S-waves
• B.
P-waves
• C.
Surface waves
A. S-waves
Explanation
S-waves have a more constant speed compared to P-waves and surface waves. S-waves, also known as shear waves, travel through solids and have a slower speed compared to P-waves. P-waves, also known as primary waves, can travel through solids, liquids, and gases, but their speed can vary depending on the medium. Surface waves, on the other hand, travel along the surface of the Earth and their speed can also vary. Therefore, S-waves have a more constant speed than P-waves and surface waves.
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• 44.
### Using the graphic provided, answer the following question. At what distance was the time difference 6 minutes?
• A.
2,000km
• B.
2,500km
• C.
3,000km 3,000km
• D.
3,500km
B. 2,500km
Explanation
Based on the graphic provided, it can be observed that the time difference is increasing as the distance increases. The time difference at 2,000km is less than 6 minutes, while the time difference at 3,000km is more than 6 minutes. Therefore, the distance at which the time difference is 6 minutes is in between these two values, which is 2,500km.
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• 45.
### What is the largest layer of the Earth?
• A.
Mantle
• B.
Outer Core
• C.
Inner Core
• D.
Crust
A. Mantle
Explanation
The mantle is the largest layer of the Earth. It is located between the crust and the outer core. The mantle is composed of hot, solid rock and extends about 2,900 kilometers below the Earth's surface. It accounts for approximately 84% of the Earth's volume and is responsible for the movement of tectonic plates, volcanic activity, and the convection currents that drive the Earth's geological processes.
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https://web7.staging02.com/ckiqo0/298111-expectation-of-exponential-function | 1,620,739,659,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243989614.9/warc/CC-MAIN-20210511122905-20210511152905-00124.warc.gz | 633,268,830 | 4,732 | This rule is true because you can raise a positive number to any power. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. Proof The probability density function of the exponential distribution is . Conditional expectation of bivariate normal. If X is discrete, then the expectation of g(X) is defined as, then E[g(X)] = X x∈X g(x)f(x), where f is the probability mass function of X and X is the support of X. Thus µ(θ) is an invertible function, therefore given µ(θ), we can uniquely determine θ. 1. The expectation value of the exponential distribution Last updated: Sep. 7, 2019 The probability density function of the exponential distribution is . Now all we need to do is consider taking the expectation of the exponential of the random variable, i.e. This the time of the first arrival in the Poisson process with parameter l. Recall that we computed its pdf to be f(t) = le lt, and its cdf to be F(t) = 1 e lt. 3. By definition, the expectation value is The key benefit of the MGF is that you can Taylor expand it as 1. If X is continuous, then the expectation … Related. We will now mathematically define the exponential distribution, and derive its mean and expected value. The expectation value for this distribution is . what is ? The domain of any exponential function is . The exponential distribution is often concerned with the amount of time until some specific event occurs. Memoryless conditional expectation of shifted function exponential. Lecture 19: Variance and Expectation of the Expo-nential Distribution, and the Normal Distribution Anup Rao May 15, 2019 Last time we defined the exponential random variable. Median for Exponential Distribution . The definition of expectation follows our intuition. Moment Generating Function of a nonlinear transformation of an exponential random variable. Definition 1 Let X be a random variable and g be any function. 2. Well, this is very similar to the moment generating function (MGF) of , which is defined as. Exponential Distribution can be defined as the continuous probability distribution that is generally used to record the expected time between occurring events. Finding the conditional expectation of independent exponential random variables 6 Evaluating integrals involving products of exponential and Bessel functions over the interval $(0,\infty)$ It is also known as the negative exponential distribution, because of its relationship to the Poisson process. 0. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. It is often used to model the time elapsed between events. Being the expectation of a strictly positive quantity, the expectation here must always be strictly positive, so the logarithm is well-de ned. The parent exponential function f(x) = b x always has a horizontal asymptote at y = 0, except when b = 1. This observation will prove useful later when obtaining the mle estimators of θ. The exponential distribution is one of the widely used continuous distributions. The function also contains the mathematical constant e, approximately equal to … The function cis called the cumulant function of the family. Conditional expectation of random vector given low-rank linear transform. 1.8 Regular Exponential Families κ (θ)) is an increasing function in θ. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … You can’t raise a positive number to any power and get 0 or a negative number. 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http://math.stackexchange.com/questions/221107/spectrum-of-mathbbzx | 1,467,425,246,000,000,000 | text/html | crawl-data/CC-MAIN-2016-26/segments/1466783404405.88/warc/CC-MAIN-20160624155004-00194-ip-10-164-35-72.ec2.internal.warc.gz | 200,563,823 | 20,901 | # Spectrum of $\mathbb{Z}[x]$
Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be proved simply using localizations, but can't quite remember how to do it! I definitely don't want a link to a long involved argument about polynomials, I can find quite enough of those!
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Are you counting 0 as a prime in $\mathbb{Z}$? Because isn't (x) a prime ideal? – Noah Snyder Oct 25 '12 at 20:53
@NoahSnyder: I've corrected the question now - I wrote it in a rush, so was a bit imprecise! – Edward Hughes Oct 25 '12 at 22:30
@Edward: IMO the most difficult part of the argument is that if $I$ is a non-zero prime ideal in $\mathbb{Q}[x]$, then it contains a generator $g$ such that $(g)$ is a prime ideal of $\mathbb{Z}[x]$. This requires some non-trivial facts about polynomial rings: e.g. the fact that $\mathbb{Z}[x]$ is a unique factorization domain, and what the primes elements are. – Hurkyl Oct 25 '12 at 22:56
@Hurkyl: It is Gauss' Lemma. – Martin Brandenburg Oct 26 '12 at 7:41
There's a wonderful picture by Mumford on this: neverendingbooks.org/index.php/mumfords-treasure-map.html – only Oct 27 '12 at 14:23
The prime ideals of $\mathbb Z[x]$ are of three kinds depending on their heights
1. (height $0$): $\{ 0\}$;
2. (height $1$): $F(x)\mathbb Z[x]$ with $F(x)$ an irreducible element in $\mathbb Z[x]$. Equivalently: $F(x)$ is a prime number $p$ or is primitive and irreducible in $\mathbb Q[x]$.
3. (height $2$, maximal ideals): the $(p, f(x))$ as you describe.
-
Here is a geometric (schemes!) way to think about it.
The inclusion $\Bbb Z\to\Bbb Z[x]$ defines a morphism $\operatorname{Spec}(\Bbb Z[x])\to\operatorname{Spec}(\Bbb Z).$ Thus to figure out the primes of $\operatorname{Spec}(\Bbb Z[x]),$ we can simply determine all the fibres of this map. How do we compute the fibres of this map?
For $\langle p\rangle\subseteq\Bbb Z$ a prime ideal, we pull back the morphism given above over the map $\operatorname{Spec}(\kappa(p))\to\operatorname{Spec}(\Bbb Z)$ induced by $\Bbb Z\to\Bbb Z_p/\frak{m_p}$ $=\kappa(p)$, where $\kappa(p)$ is the residue field of $p.$ The residue field $\kappa(0)=\Bbb Q$ and for all other primes $p$ we have $\kappa(p)=\Bbb F_p.$
The fibre over $\langle 0\rangle$ is thus $\operatorname{Spec}(\Bbb Q\otimes_{\Bbb Z}\Bbb Z[x])=\operatorname{Spec}(\Bbb Q[x])$ which is all irreducible polynomials over $\Bbb Q$ and the zero ideal. Similarly, the fibre over $\langle p\rangle$ is $\operatorname{Spec}(\Bbb F_p[x]),$ which is just the irreducible polynomials over $\Bbb F_p$ along with its zero ideal. (The zero ideals correspond to those in $\Bbb Z.$)
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The intersection of a prime ideal with $\mathbb Z$ is again prime, thus we obtain a prime $p$ (or 0). By localizing at $p$, we make all non-multiples of $p$ invertible and are left with an ideal in the principle ideal ring $\mathbb Z_p[X]$, i.e. $(f)$ with $f\in\mathbb Z_p[X]$. If we had a nontrivial factorization $f\equiv gh\pmod p$, this could be lifted to a factorization in $\mathbb Z_p[X]$, which is impossible. Hence $f$ is irreducible $\bmod p$. This also holds if we replace $f$ with an approximation in $\mathbb Z[X]$. We also see that any $g$ in the ideal becomes a multiple of $f$ in $\mathbb Z_p[X]$, hence can be written as a multiple of $p$ plus a multiple of $f$ in $\mathbb Z[X]$.
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I can't quite see the argument for when the prime ideal is (0). Could you possibly clarify it? Many thanks! – Edward Hughes Oct 25 '12 at 22:40
How is $\mathbb Z_p[X]$ a principal ideal domain? How about the ideal $(p,X)$? I don't think this ideal is principal, since if $f(X)$ divides both $p$ and $X$, dividing $p$ implies having degree $0$ and dividing $X$ while being of degree $0$ means being a unit. – Patrick Da Silva Jan 9 '14 at 0:36
Any polynomial ring F[x] over a field is a PID – rschwieb Jan 9 '14 at 3:41
There is a sketch here but I didn't proofread it. I am gambling that it is useful, so I apologize (and will delete this) if it turns out to be useless.
-
It was useful for me. Thanks! – Patrick Da Silva Jan 8 '14 at 23:45
Although after thinking about it I think it is wrong ; Z[x] is not an Euclidean domain, the Euclidean algorithm doesn't work. How could you compute the gcd of $x^2 - 1$ and $2x$? – Patrick Da Silva Jan 9 '14 at 0:00
Did you post this under the wrong thread or something? Z[x] is definitely not Euclidean because it's not a PID – rschwieb Jan 9 '14 at 3:39
It mentions the Euclidean algorithm in your link. I think I'm confused as to how they do it. What they say seems to work though (with my example of $x^2 - 1$ and $2x$, you can extract the constant term $2 = 2(x^2 - 1) - x(2x)$). No, not the wrong thread. – Patrick Da Silva Jan 10 '14 at 9:58
@PatrickDaSilva You can perform the extended Euclidean algorithm first in $\Bbb Q[x]$, and then multiply through by the gcd of any fractions that appear. If the two polynomials were coprime in $\Bbb Q[x]$, then they'll have linear combination in $\Bbb Z[x]$ that's an integer. – rschwieb Jan 10 '14 at 14:01 | 1,577 | 5,150 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2016-26 | latest | en | 0.848718 |
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# If412 02
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### If412 02
1. 1. IF412 – Konsep Bahasa Pemrograman Pertemuan : 02 PROGRAMMING WITH ASSIGNMENT Perhatikan potongan program berikut : while x < > A[i] do i := i - 1; end potongan program di atas mengandung 3 konsep utama dalam pemrograman, yaitu : • Assignment • Assignable data structures • Control flow statement Bahasa pemrograman Modula-2 dan C adalah contoh bahasa pemrograman Imperative, yang prinsip rancangannya berdasarkan pada : • Machine model • Structured programming Evolusi Bahasa Imperative Fortran Lisp Algol 60 ISWIM CPL* Prolog Algol 68 Concurrent BCPL Pascal Pascal Scheme Simula 67 ML C Distributed CSP Processes Mesa Smalltalk-80 Modula-2 C++ Ada Oberon Standard ML BINA NUSANTARA Edisi : 1 Revisi : 1 Sept - 1998
2. 2. IF412 – Konsep Bahasa Pemrograman Pertemuan : 02 Publication Format for Program • Contoh : publication format Pascal Modula-2 C While x < > A[i] do i := i - 1; while x < > A[i] do i := i - 1; while (x != A[i]) end i := i - 1; The Effect of an Assignment Assigment mengubah nilai dari lokasi tertentu di dalam mesin.Random access machine terdiri dari • Memori • Program (Assignment, I/O, Control Flow) • Input / Output File 1: read M[1] Input 2: read M[2] 3: M[1] := M[1] - M[2] Control 4: if M[1] >= 0 then goto 3 5: M[1] := M[1] + M[2] 6: write M[1] 7: halt Output 0 1 2 3 Memory 27 10 ... Fig. 3.1. A random-access machine (RAM). • Bentuk assignment pada gambar di atas : <expression>1 := <expression>2 <expression>1 = menunjuk ke suatu lokasi memori (disebut dengan L-value). <expression>2 = menunjukkan sebuah nilai (R-value). • Contoh Control Flow if M[j] > = 0 then goto I =================================== M[l] := n M[l] := M[j] + M[k] Assignments M[l] := M[j] - M[k] M[l] := M[M[j]] M[M[j]] := M[k] ________________________________________ read M[l] Input/Output write M[j] ________________________________________ Control Flow if M[j] >= 0 then goto I halt Fig. 3.2. Instruction set for a random-access machine. BINA NUSANTARA Edisi : 1 Revisi : 1 Sept - 1998
3. 3. IF412 – Konsep Bahasa Pemrograman Pertemuan : 02 • Assignment melalui indirect address : l j k M[l] := M[M[j]] l j k k n n k n (a) l j k M[M[j]] := M[k] l j k l n n l n (b) Fig. 3.3. Assignments through indirect addresses Dynamic Thread THREAD STATE Instruction Remaining Location Output Input 1 2 Produced 27, 10 1: read M[1] 10 27 2: read M[2] 27 10 3: M[1] := M[1] - M[2] 17 10 4: if M[1] >= 0 goto 3 17 10 3: M[1] := M[1] - M[2] 7 10 4: if M[1] >= 0 goto 3 7 10 3: M[1] := M[1] - M[2] -3 10 4: if M[1] >= 0 goto 3 -3 10 5: M[1] := M[1] + M[2] 7 10 6: write M[1] 7 10 7 7: halt 7 10 7 Fig. 3.4. A thread through the program in Fig. 3.1. STRUCTURED PROGRAMMING • Definisi • Latar Belakang Structured Control Flow • Composition Jika S1, S2, ... Sk adalah statement-statement maka komposisi urutan statement ditulis: S1; S2 ; ... Sk; Control flow : S1⇒ S2 ⇒ ... Sk; Jika k = 0 ⇒ disebut empty statement • Conditional BINA NUSANTARA Edisi : 1 Revisi : 1 Sept - 1998
4. 4. IF412 – Konsep Bahasa Pemrograman Pertemuan : 02 Jika E adalah sebuah ekspresi dan SL1 dan SL2 adalah statement list, kemudian Conditional Statement dibentuk sbb: if E then SL1 else SL2 end if E then SL1 end Control flow : SL1 bila E benar SL2 bila E salah • Loop forever Jika SL adalah statement list, maka bentuk iterasi / loop sbb : loop SL end Ada beberapa statement yang digunakan untuk keluar dari loop seperti : exit, break. • While - Loop Jika E adalah ekspresi, dan SL adalah statement list : while E do SL end Control flow : E & SL dikerjakan secara bergantian selama E benar, bila E salah kerjakan statement berikutnya setelah while-loop. if E then SL1 else SL2 end while E do SL end • • true E true E SL1 SL2 • SL • Fig. 3.5. Conditionals and while-loops. Invariant sebagai jembatan antara static & dynamic computation while x ≥ y do { apabila sampai di sini → x ≥ y } ⇐ invariant x := x - y; end • Tempat-tempat invariant : 1. while {loop invariant} E do SL end 2. loop {loop invariant} while E do SL end 3. Precondition → tepat sebelum statement 4. Postcondition → tepat sesudah statement • Contoh : Linear Search 0 1 n limit ... used free Fig. 3.6. Table organization for linear search with a sentinel. BINA NUSANTARA Edisi : 1 Revisi : 1 Sept - 1998 | 1,535 | 4,600 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2016-44 | latest | en | 0.230845 |
https://answercult.com/question/how-does-recursion-really-works-in-computer-science/ | 1,657,138,804,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00629.warc.gz | 147,315,477 | 18,403 | # How does recursion really works in computer science?
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How does recursion really works in computer science?
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Most implementations of recursion use an actual subroutine jump to save current state, invoke the function, and continue after it returns. There are optimizations that some compilers can make to exploit that the function is calling itself, but not all compilers do that.
Imagine you have a man that is neither smart nor dumb, he will simply follow instructions to the letter, this is a computer.
You hand him a stick and tell him: “Break this stick in half, now give me the sticks” he takes the stick, breaks it in half, and gives it back to you. This is an example of a non-recursive function since you give him precise instructions on what to do and a end result.
You hand him a stick and tell him: “Whenever you have a stick, break it in half” he takes the stick, breaks it in half, looks down and sees he now has 2 sticks, which he each breaks in half, then he looks down and sees he now has 4 sticks… This will go on forever until he has a stick so small he can no longer break it in half. Then he comes to you to tell you he couldn’t do his job. This is an example of a recursive function.
Recursion can be an amazing tool to use but is impossible to fully predict in more complex instructions, it is any instruction that will repeat itself until certain conditions are met (or not) you can use it to calculate absurdly large numbers (A good example is Pi) but there is never really a guarantee that it will finish what it was told to do without causing problems (errors)
Recursion just means something that calls itself.
Here’s an example: List all the files in all subfolders of a directory.
Your code outline might look something like this:
Function: parseFolder
Logic: List all files in the folder. Then, for each subfolder of this folder, call parseFolder.
Practically, what this is going to do is list all the files in each folder, go into the subfolders of that folder and list those files, go into the subfolders of *that* folder and list all the folders, etc, until there are no more subfolders to explore.
Depends on the language you are using but most operate the same way.
Functions take up space in the ram to operate so each function has it’s own dedicated space. When you call a function it creates that space for the function to inhabit. When a function calls itself, it creates another space in the ram for the this function to inhabit. This continues until (hopefully) a base case returns (where the space is unreserved and can be used for more stuff). Otherwise it’ll keep on creating new space on the ram for functions until you quite literally run out, although modern cpus will just kill the program before that happens.
This explanation is of course wildly oversimplified but the concept is like that.
Recursion works for computer science classes, as a way of misleading dev students about what actually goes on in software development.
In industry I’m not sure if I’ve ever seen a recursive function written by anyone other than a total beginner.
Recursion makes debugging extremely difficult and problems which are suitable for recursion can often be solved by having two functions and a couple of variables about state. One function manages the state variable and the other function actually does processing and the managing function has a loop that updates the state variables and keeps calling the processing function until the state variables represent a success state.
There is another use for recursion in industry which is when a tech company wants to know how good you are at writing unreadable, untestable, confusing “genius code”, they’ll give you some recursion problem to try it with. | 779 | 3,771 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2022-27 | latest | en | 0.951705 |
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