url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3 values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93 values | snapshot_type stringclasses 2 values | language stringclasses 1 value | language_score float64 0.06 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
https://excellenttermpapers.com/crashing/ | 1,695,311,904,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506028.36/warc/CC-MAIN-20230921141907-20230921171907-00737.warc.gz | 290,254,626 | 13,108 | # Crashing
Problem 17-13
Here is a list of activity times for a project as well as crashing costs for its activities. Determine which activities should be crashed and the total cost of crashing if the goal is to shorten the project by three weeks as cheaply as possible. (Omit the “\$” sign in your response.)
Activity Duration (weeks) First Crash Second Crash 1-2 5 \$8 \$10 2-4 6 7 9 4-7 3 14 15 1-3 3 9 11 3-4 7 8 9 1-5 5 10 15 5-6 5 11 13 6-7 5 12 14
Activity Cost First crash \$ \$ \$ [removed] | 172 | 506 | {"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-2023-40 | latest | en | 0.912996 |
https://questioncove.com/updates/4d61382877f38b0b33d83fdb | 1,708,698,237,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474412.46/warc/CC-MAIN-20240223121413-20240223151413-00340.warc.gz | 475,757,786 | 5,367 | Mathematics 68 Online
OpenStudy (anonymous):
Hi there, i was wondering if you could help me find the range of this function: 3ln(y + 4) = 2ln(x + 2) -2ln(x + 9) + 3ln(x^2 -1)
OpenStudy (anonymous):
(range of values for x and y)
OpenStudy (anonymous):
hello
OpenStudy (anonymous):
ok we see that we have ln (y+4) on the left side. so y+4 > 0 , since ln 0 or ln (negative) is not defined. so y > -4 for the range. for the domain we have x + 2 > 0 , x + 9 > 0 and x^2 -1 > 0 you take the intersection of these conditions. so x > -2 , x > -9, x > 1 or x < -1 . so the domain is (-2, -1) union (-1, oo)
OpenStudy (anonymous):
OpenStudy (anonymous):
Thank you very much for your reply. i spent the whole day looking at it and i couldnt get my head around it. i was trying to gather the terms on the right hand side to simplify the function. guess i was looking at it the wrong way!
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http://scholar.cnki.net/result.aspx?q=%27Geometry+of+continued+fractions%27 | 1,586,280,302,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585371803248.90/warc/CC-MAIN-20200407152449-20200407182949-00382.warc.gz | 153,408,141 | 25,675 | 高级检索
作者：R. Daniel Mauldin , Mariusz Urbanski 来源：[J].tran(IF 1.019), 1999, Vol.351 (12), pp.4995-5025 摘要：... We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmeti...
作者：Oleg Karpenkov , Matty van Son 来源：[J].Journal of Number Theory(IF 0.466), 2020 摘要：... The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.
作者：R. Daniel Mauldin , Mariusz Urbański 来源：[J].Transactions of the American Mathematical Society(IF 1.019), 1999, Vol.351 (12), pp.4995-5025 摘要：... We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmeti...
来源：[J].The American Mathematical Monthly(IF 0.292), 1989, Vol.96 (8), pp.696-703
作者：Vladimir I. Arnold 来源：[J].Functional Analysis and Other Mathematics, 2009, Vol.2 (2-4), pp.129-138 摘要：Abstract(#br)The article describes the interrelations between the minimal integer number N ( a , b , c ) which belongs to the additive semigroup of integers generated by a , b , c together with all greater integers, on the one hand, and the geometrical theory of continued fractions...
作者：Oleg Karpenkov , Matty van-Son 来源：[J].Journal de théorie des nombres de Bordeaux, 2019, Vol.31 (1), pp.131-144 摘要：... We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.
作者：Patrick Popescu-Pampu 来源：[C].Singularities in Geometry and Topology 20042007 摘要：We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the...
作者：Oleg Karpenkov 来源：[B].Algorithms and Computation in Mathematics;;Springer Textbook2013 摘要：Abstract In the beginning of this book we discussed the geometric interpretation of regular continued fractions in terms of LLS sequences of sails. Is there a natural extension of this interpretation to the case of continued fractions with arbitrary elements? The aim of this cha...
作者：Oleg Karpenkov 来源：[B].Algorithms and Computation in Mathematics2013 摘要：Abstract Continued fractions play an important role in the geometry of numbers. In this chapter we describe a classical geometric interpretation of regular continued fractions in terms of integer lengths of edges and indices of angles for the boundaries of convex hulls of all int...
作者：Oleg Karpenkov 来源：[B].Algorithms and Computation in Mathematics;;Cambridge Studies in Advanced Mathematics;;Series on Knots and Everything2013 摘要：Abstract It turns out that the frequency of a positive integer k in a continued fraction for almost all real numbers is equal to $$\frac{1}{\ln2}\ln \biggl(1+\frac{1}{k(k+2)} \biggr),$$ i.e., for a general real x we have 42 % of 1, 17 % of 2, 9 % of 3, etc. This distribution i... | 887 | 3,378 | {"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": 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} | 2.578125 | 3 | CC-MAIN-2020-16 | latest | en | 0.628959 |
https://calculator.academy/leontief-production-equation-calculator/ | 1,685,998,493,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224652161.52/warc/CC-MAIN-20230605185809-20230605215809-00576.warc.gz | 183,883,733 | 53,887 | Enter the internal consumption and the external demand into the Calculator. The calculator will evaluate the Leontief Production Equation.
## Leontief Production Equation Formula
PTO = IC + ED
Variables:
• PTO is the Leontief Production Equation (total output)
• IC is the internal consumption
• ED is the external demand
To calculate Leontief Production Equation, sum the internal consumption and the external demand.
## How to Calculate Leontief Production Equation?
The following steps outline how to calculate the Leontief Production Equation.
1. First, determine the internal consumption.
2. Next, determine the external demand.
3. Next, gather the formula from above = PTO = IC + ED.
4. Finally, calculate the Leontief Production Equation.
5. After inserting the variables and calculating the result, check your answer with the calculator above.
Example Problem :
Use the following variables as an example problem to test your knowledge.
internal consumption = 23
external demand = 235 | 203 | 1,003 | {"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} | 2.625 | 3 | CC-MAIN-2023-23 | latest | en | 0.744387 |
https://pt.slideshare.net/ambasports/a-population-of-values-has-a-normal-distribution-with-1538-and-87pdf | 1,696,160,558,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510888.64/warc/CC-MAIN-20231001105617-20231001135617-00698.warc.gz | 515,256,585 | 45,994 | # A population of values has a normal distribution with =153.8 and =87..pdf
30 de Mar de 2023
1 de 1
### A population of values has a normal distribution with =153.8 and =87..pdf
• 1. A population of values has a normal distribution with =153.8 and =87.2. A random sample of size n=74 is drawn. Find the probability that a sample of size n=74 is randomly selected with a mean less than 154.8. Round your answer to four decimal places. P(M<154.8)= | 127 | 449 | {"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} | 3.171875 | 3 | CC-MAIN-2023-40 | latest | en | 0.88915 |
https://forum.43oh.com/topic/3441-2n-for-real-numbers-using-integer-math/ | 1,696,474,227,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511717.69/warc/CC-MAIN-20231005012006-20231005042006-00738.warc.gz | 268,307,200 | 18,466 | 43oh
# 2^N for real numbers using integer math
## Recommended Posts
When working with electronic music (MIDI synth and such) it is sometimes necessary to adjust pitch by units of octaves, semitones, or cents.
An octave is a 2:1 ratio. So the ratio between any number of octaves is 2^N. This is a simple bit shift. Easy.
A semitone is 1/12 of an octave (typically), so the ratio for that is 2^(N/12). A cent is 1/1200 of an octave, so the ratio for that is 2^(N/1200). Can't use bit shifting for that!
The C stdlib provides the pow() function that can be used for these calculations, but it uses rather slow floating point math.
This code will do it with relatively fast fixed point integer math. This code could easily be adapted to almost any base and exponent by changing the look-up tables. It uses the property of exponents that 2 ^ (1 + 2 + 4 + 8) == (2 ^ 1) * (2 ^ 2) * (2 ^ 4) * (2 ^ 8)
```//
// 2 ^ (n / (1200 * 8192))
//
// n = 1 / 8,192 cent
// = 1 / 819,200 semitone
// = 1 / 9,830,400 octave
//
// range of n: -134,217,727 to +78,643,199
// = -13.65 to +7.99 octaves
//
// returned value is 8.24 fixed point
//
// + exponents in 8.24 format
static const uint32_t etp[27] = {
0x01000001, // 00000001 1.0000000705
0x01000002, // 00000002 1.0000001410
0x01000005, // 00000004 1.0000002820
0x01000009, // 00000008 1.0000005641
0x01000013, // 00000010 1.0000011282
0x01000026, // 00000020 1.0000022563
0x0100004C, // 00000040 1.0000045127
0x01000097, // 00000080 1.0000090254
0x0100012F, // 00000100 1.0000180509
0x0100025E, // 00000200 1.0000361021
0x010004BB, // 00000400 1.0000722054
0x01000977, // 00000800 1.0001444161
0x010012EE, // 00001000 1.0002888530
0x010025DE, // 00002000 1.0005777895
0x01004BC1, // 00004000 1.0011559129
0x01009798, // 00008000 1.0023131618
0x01012F8B, // 00010000 1.0046316744
0x0102607D, // 00020000 1.0092848012
0x0104C6A1, // 00040000 1.0186558100
0x0109A410, // 00080000 1.0376596592
0x0113A513, // 00100000 1.0767375682
0x0128CC11, // 00200000 1.1593637909
0x01581889, // 00400000 1.3441243996
0x01CE81F4, // 00800000 1.8066704016
0x0343994D, // 01000000 3.2640579400
0x0AA77169, // 02000000 10.6540742354
0x71826157 // 04000000 113.5092978129
};
// - exponents in 8.24 format
static const uint32_t etn[27] = {
0x00FFFFFF, // 00000001 0.9999999295
0x00FFFFFE, // 00000002 0.9999998590
0x00FFFFFB, // 00000004 0.9999997180
0x00FFFFF7, // 00000008 0.9999994359
0x00FFFFED, // 00000010 0.9999988718
0x00FFFFDA, // 00000020 0.9999977437
0x00FFFFB4, // 00000040 0.9999954873
0x00FFFF69, // 00000080 0.9999909747
0x00FFFED1, // 00000100 0.9999819495
0x00FFFDA2, // 00000200 0.9999638992
0x00FFFB45, // 00000400 0.9999277998
0x00FFF689, // 00000800 0.9998556048
0x00FFED13, // 00001000 0.9997112304
0x00FFDA28, // 00002000 0.9994225441
0x00FFB455, // 00004000 0.9988454217
0x00FF68C1, // 00008000 0.9976921765
0x00FED1DC, // 00010000 0.9953896791
0x00FDA51C, // 00020000 0.9908006133
0x00FB4FC4, // 00040000 0.9816858552
0x00F6B582, // 00080000 0.9637071184
0x00EDC157, // 00100000 0.9287314100
0x00DCCF8E, // 00200000 0.8625420320
0x00BE7564, // 00400000 0.7439787570
0x008DB277, // 00800000 0.5535043908
0x004E6E13, // 01000000 0.3063671106
0x00180743, // 02000000 0.0938608065
0x0002415D // 04000000 0.0088098510
};
uint32_t tpow(int32_t n)
{
const uint32_t *et = etp; // Assume + exponent
if(n < 0) n = -n, et = etn; // Adjust for - exponent
//
uint32_t r = 1L << 24; // Init result to 1.0 in 8.24 format
//
do { //
if(n & 1) r = mul824(r, *et); // Multiply by exponent if lsb of n is 1
++et; // Next exponent
} while(n >>= 1); // Next bit, loop until no more 1 bits
//
return r; // Return result
}
```
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• Create New... | 1,896 | 5,329 | {"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-2023-40 | latest | en | 0.378279 |
https://getfem.readthedocs.io/en/latest/userdoc/model_time_integration.html | 1,653,770,228,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652663019783.90/warc/CC-MAIN-20220528185151-20220528215151-00242.warc.gz | 328,967,608 | 10,610 | # The model tools for the integration of transient problems¶
Although time integration scheme can be written directly using the model object by describing the problem to be solved at each iteration, the model object furnishes some basic tools to facilitate the writing of such schemes. These tools are based on the following basic principles:
• The original variables of the model represent the state of the system to be solved at the current time step (say step n). This is the case even for a middle point scheme, mainly because if one needs to apply different schemes to different variables of the system, all variable should describe the system at a unique time step.
• Some data are added to the model to represent the state of the system at previous time steps. For classical one-step schemes (for the moment, only one-step schemes are provided), only the previous time step is stored. For instance if u is a variable (thus represented at step n), Previous_u, Previous2_u, Previous3_u will be the data representing the state of the variable at the previous time step (step n-1, n-2 and n-3).
• Some intermediate variables are added to the model to represent the time derivative (and the second order time derivative for second order problem). For instance, if u is a variable, Dot_u will represent the first order time derivative of u and Dot2_u the second order one. One can refer to these variables in the model to add a brick on it or to use it in GWFL, the generic weak form language. However, these are not considered to be independent variables, they will be linked to their corresponding original variable (in an affine way) by the time integration scheme. Most of the schemes need also the time derivative at the previous time step and add the data Previous_Dot_u and possibly Previous_Dot2_u to the model.
• A different time integration scheme can be applied on each variable of the model. Note that most of the time, multiplier variable and more generally variables for which no time derivative is used do not need a time integration scheme.
• The data t represent the time parameter and can be used (either in GWFL or as parameter of some bricks). Before the assembly of the system, the data t is automatically updated to the time step n.
• The problem to be solved at each iteration correspond to the formulation of the transient problem in its natural (weak) formulation in which the velocity and the acceleration are expressed by the intermediate variables introduced. For instance, the translation into GWFL of the problem
$\dot{u}(t,x) - \Delta u(t,x) = \sin(t)$
can simply be:
(even though, of course, in this situation, the use of linear bricks is preferable for efficiency reasons)
• For all implemented one-step schemes, the time step can be changed from an iteration to another for both order one and order two in time problems (or even quasi-static problems).
• A scheme for second order in time problem (resp. first order in time) can be applied to a second or first order in time or even to a quasi-static problem (resp. to a first order or quasi-static problem) without any problem except that the initial data corresponding to the velocity/displacement have to be initialized with respect ot the order of the scheme. Conversely, of course, a scheme for first order problem cannot be applied to a second order in time problem.
## The implicit theta-method for first-order problems¶
For a problem which reads
$M\dot{U} = F(U)$
where $$F(U)$$ might be nonlinear (and may depend on some other variables for coupled problems), for $$dt$$ a time step, $$V = \dot{U}$$ and $$U^n, V^n$$ the approximation of $$U, V$$ at time $$ndt$$, theta-method reads
$\begin{split}\left\{ \begin{array}{l} U^n = U^{n-1} + dt(\theta V^n + (1-\theta) V^{n-1}), \\ MV^n = F(U^n), \end{array}\right.\end{split}$
for $$\theta \in (0, 1]$$ the parameter of the theta-method (for $$\theta = 0$$, the method corresponds to the forward Euler method and is not an implicit scheme) and for $$U^{n-1}, V^{n-1}$$ given.
Before the first time step, $$U^0$$ should be initialized, however, $$V^0$$ is also needed (except for $$\theta = 1$$). In this example, it should correspond to $$M^{-1}F(U^0)$$. For a general coupled problem where $$M$$ might be singular, a generic precomputation of $$V^0$$ is difficult to obtain. Thus $$V^0$$ have to be furnisded also. Alternatively (see below) the model object (and the standard solve) furnishes a mean to evaluate them thanks to the application of a Backward Euler scheme on a (very) small time step.
The following formula can be deduced for the time derivative:
$V^n = \frac{U^n - U^{n-1}}{\theta dt} - \frac{1-\theta}{\theta}V^{n-1}$
When applying this scheme to a variable “u” of the model, the following affine dependent variable is added to the model:
"Dot_u"
which represent the time derivative of the variable and can be used in some brick definition.
The following data are also added:
"Previous_u", "Previous_Dot_u"
which correspond to the values of “u” and “Dot_u” at the previous time step.
Before the solve, the data “Previous_u” (corresponding to $$U^0$$ in the example) has to be initialized (except for $$\theta = 1$$). Again, “Previous_Dot_u” has to be either initialized or pre-computed as described in the next section. The affine dependence of “Dot_u” is thus given by:
Dot_u = (u - Previous_u)/(theta*dt) - Previous_Dot_u*(1-theta)/theta
Which means that “Dot_u” will be replaced at assembly time by its expression in term of “u” (multipied by $$1/(\theta*dt)$$) and in term of a constant remaining part depending on the previous time step. The addition of this scheme to a variable is to be done thanks to:
add_theta_method_for_first_order(model &md, const std::string &varname, scalar_type theta);
## Precomputation of velocity/acceleration¶
Most of the time integration schemes (except, for instance, the backward Euler scheme) needs the pre-computation of the first or second order time derivative before the initial time step (for instance $$V^0$$ for the theta-method for first order problems, $$A^0$$ for second order problems …).
The choice is let to the user to either initialize these derivative or to ask to the model to automatically approximate them.
The method used (for the moment) to approximate the supplementary derivatives may be explained in the example of the solve of
$M\dot{U} = F(U)$
with a theta-method (see the previous section). In order to approximate $$V_0$$, the theta-method is applied for $$\theta = 1$$ (i.e. a backward Euler scheme) on a very small time step. This is possible since the backward Euler do not need an initial time derivative. Then the time derivative computed thanks to the backward Euler at the end of the very small time step is simply used as an approximation of the initial time derivative.
For a model md, the following instructions:
model.perform_init_time_derivative(ddt);
standard_solve(model, iter);
allows to perform automatically the approximation of the initial time derivative. The parameter ddt corresponds to the small time step used to perform the aproximation. Typically, ddt = dt/20 could be used where dt is the time step used to approximate the transient problem (see the example below).
## The implicit theta-method for second-order problems¶
For a problem which reads
$M\ddot{U} = F(U)$
where $$F(U)$$ might be nonlinear (and may depend on some othere variables for coupled problems), for $$dt$$ a time step, $$V = \dot{U}$$, $$A = \ddot{U}$$ and $$U^n, V^n, A^n$$ the approximation of $$U, V, A$$ at time $$ndt$$, the first oder theta-method reads
$\begin{split}\left\{ \begin{array}{l} U^n = U^{n-1} + dt(\theta V^n + (1-\theta) V^{n-1}), \\ V^n = V^{n-1} + dt(\theta A^n + (1-\theta) A^{n-1}), \\ MA^n = F(U^n), \end{array}\right.\end{split}$
for $$\theta \in (0, 1]$$ the parameter of the theta-method (for $$\theta = 0$$, the method correspond to the forward Euler method and is not an implicit scheme) and for $$U^{n-1}, V^{n-1}, A^{n-1}$$ given.
At the first time step, $$U^0, V^0$$ should be given and $$A^0$$ is to be given or pre-computed (except for $$\theta = 1$$).
The following formula can be deduced for the time derivative:
\begin{align}\begin{aligned}V^n = \frac{U^n - U^{n-1}}{\theta dt} - \frac{1-\theta}{\theta}V^{n-1}\\A^n = \frac{U^n - U^{n-1}}{\theta^2 dt^2} - \frac{1}{\theta^2dt}V^{n-1} - \frac{1-\theta}{\theta}A^{n-1}\end{aligned}\end{align}
When aplying this scheme to a variable “u” of the model, the following affine dependent variables are added to the model:
"Dot_u", "Dot2_u"
which represent the first and second order time derivative of the variable and can be used in some brick definition.
The following data are also added:
"Previous_u", "Previous_Dot_u", "Previous_Dot2_u"
which correspond to the values of “u”, “Dot_u” and “Dot2_u” at the previous time step.
Before the solve, the data “Previous_u” and “Previous_Dot_u” (corresponding to $$U^0$$ in the example) have to be initialized and “Previous_Dot2_u” should be either initialized or precomputed (see the previous section, and except for $$\theta = 1$$). The affine dependences are thus given by:
Dot_u = (u - Previous_u)/(theta*dt) - Previous_Dot_u*(1-theta)/theta
Dot2_u = (u - Previous_u)/(theta*theta*dt*dt) - Previous_Dot_u/(theta*theta*dt) - Previous_Dot2_u*(1-theta)/theta
The addition of this scheme to a variable is to be done thanks to:
add_theta_method_for_second_order(model &md, const std::string &varname,
scalar_type theta);
## The implicit Newmark scheme for second order problems¶
For a problem which reads
$M\ddot{U} = F(U)$
where $$F(U)$$ might be nonlinear (and may depend on some othere variables for coupled problems), for $$dt$$ a time step, $$V = \dot{U}$$, $$A = \ddot{U}$$ and $$U^n, V^n, A^n$$ the approximation of $$U, V, A$$ at time $$ndt$$, the first oder theta-method reads
$\begin{split}\left\{ \begin{array}{l} U^n = U^{n-1} + dtV^n + \frac{dt^2}{2}(2\beta V^n + (1-2\beta) V^{n-1}), \\ V^n = V^{n-1} + dt(\gamma A^n + (1-\gamma) A^{n-1}), \\ MA^n = F(U^n), \end{array}\right.\end{split}$
for $$\beta \in (0, 1]$$ and $$\gamma \in [1/2, 1]$$ are the parameters of the Newmark scheme and for $$U^{n-1}, V^{n-1}, A^{n-1}$$ given.
At the first time step, $$U^0, V^0$$ should be given and $$A^0$$ is to be given or pre-computed (except for $$\beta = 1/2, \gamma = 1$$).
The following formula can be deduced for the time derivative:
\begin{align}\begin{aligned}V^n = \frac{\gamma}{\beta dt}(U^n - U^{n-1}) + \frac{\beta-\gamma}{\beta}V^{n-1} + dt(1-\frac{\gamma}{2\beta})A^{n-1}\\A^n = \frac{U^n - U^{n-1}}{\beta dt^2} - \frac{1}{\beta dt}V^{n-1} - (1/2-\beta)A^{n-1}\end{aligned}\end{align}
When aplying this scheme to a variable “u” of the model, the following affine dependent variables are added to the model:
"Dot_u", "Dot2_u"
which represent the first and second order time derivative of the variable and can be used in some brick definition.
The following data are also added:
"Previous_u", "Previous_Dot_u", "Previous_Dot2_u"
which correspond to the values of “u”, “Dot_u” and “Dot2_u” at the previous time step.
Before the first solve, the data “Previous_u” and “Previous_Dot_u” (corresponding to $$U^0$$ in the example) have to be initialized. The data “Previous_Dot2_u” is to be given or precomputed (see Precomputation of velocity/acceleration and except for $$\beta = 1/2, \gamma = 1$$).
The addition of this scheme to a variable is to be done thanks to:
add_Newmark_scheme(model &md, const std::string &varname,
scalar_type beta, scalar_type gamma);
## The implicit Houbolt scheme¶
For a problem which reads
$(K+\frac{11}{6 dt}C+\frac{2}{dt^2}M) u_{n} = F_{n} + (\frac{5}{dt^2} M + \frac{3}{ dt} C) u_{n-1} - (\frac{4}{dt^2} M + \frac{3}{2 dt} C) u_{n-2} + (\frac{1}{dt^2} M + \frac{1}{3 dt} C) u_{n-3}$
where $$dt$$ means a time step, $$M$$ the matrix in term of “Dot2_u”, $$C$$ the matrix in term of “Dot_u” and $$K$$ the matrix in term of “u”. The affine dependences are thus given by:
Dot_u = 1/(6*dt)*(11*u-18*Previous_u+9*Previous2_u-2*Previous3_u)
Dot2_u = 1/(dt**2)*(2*u-5*Previous_u+4*Previous2_u-Previous3_u)
When aplying this scheme to a variable “u” of the model, the following affine dependent variables are added to the model:
"Dot_u", "Dot2_u"
which represent the first and second order time derivative of the variable and can be used in some brick definition.
The following data are also added:
"Previous_u", "Previous2_u", "Previous3_u"
which correspond to the values of “u” at the time step n-1, n-2 n-3.
Before the solve, the data “Previous_u”, “Previous2_u” and “Previous3_u” (corresponding to $$U^0$$ in the example) have to be initialized.
The addition of this scheme to a variable is to be done thanks to:
add_Houbolt_scheme(model &md, const std::string &varname);
## Transient terms¶
As it has been explained in previous sections, some intermediate variables are added to the model in order to represent the time derivative of the variables on which the scheme is applied. Once again, if “u” is such a variable, “Dot_u” will represent the time derivative of “u” approximated by the used scheme.
This also mean that “Dot_u” (and “Dot2_u” in order two in time problems) can be used to express the transient terms. In GWFL, the term:
$\int_{\Omega} \dot{u} v dx$
can be simply expressed by:
Dot_u*Test_u
Similarly, every existing model brick of GetFEM can be applied to “Dot_u”. This is the case for instance with:
which adds the same transient term.
VERY IMPORTANT: When adding an existing model brick applied to an affine dependent variable such as “Dot_u”, it is always assumed that the corresponding test function is the one of the corresponding original variable (i.e. “Test_u” here). In other words, “Test_Dot_u”, the test variable corresponding to the velocity, is not used. This corresponds to the choice made to solve the problem in term of the original variable, so that the test function corresponds to the original variable.
Another example of model brick which can be used to account for a Kelvin-Voigt linearized viscosity term is the linearized elasticity brick:
getfem::add_isotropic_linearized_elasticity_brick(model, mim, "Dot_u", "lambda_viscosity", "mu_viscosity");
when applied to an order two transient elasticity problem.
## Computation on the sequence of time steps¶
Typically, the solve on the different time steps will take the following form:
for (scalar_type t = 0.; t < T; t += dt) { // time loop
// Eventually compute here some time dependent terms
iter.init();
getfem::standard_solve(model, iter);
// + Do something with the solution (plot or store it)
model.shift_variables_for_time_integration();
}
Note that the call of the method:
model.shift_variables_for_time_integration();
is needed between two time step since it will copy the current value of the variables (u and Dot_u for instance) to the previous ones (Previous_u and Previous_Dot_u).
## Boundary conditions¶
Standard boundary conditions can of course be applied normally to the different variables of the unknown. By default, applying Dirichlet, Neumann or contact boundary conditions to the unknown simply means that the conditions are prescribed on the variable at the current time step n.
## Small example: heat equation¶
The complete compilable program corresponds to the test program tests/heat_equation.cc of GetFEM distribution. See also /interface/tests/matlab/demo_wave_equation.m for an example of order two in time problem with the Matlab interface.
Assuming that mf_u and mim are valid finite element and integration methods defined on a valid mesh, the following code will perform the approximation of the evolution of the temperature on the mesh assuming a unitary diffusion coefficient:
getfem::model model;
model.add_fem_variable("u", mf_u, 2); // Main unknown of the problem
getfem::add_generic_elliptic_brick(model, mim, "u"); // Laplace term
// Volumic source term.
// Dirichlet condition.
(model, mim, "u", mf_u, DIRICHLET_BOUNDARY_NUM);
// transient part.
gmm::iteration iter(residual, 0, 40000);
model.set_time(0.); // Init time is 0 (not mandatory)
model.set_time_step(dt); // Init of the time step.
// Null initial value for the temperature.
gmm::clear(model.set_real_variable("Previous_u"));
// Automatic computation of Previous_Dot_u
model.perform_init_time_derivative(dt/20.);
iter.init();
standard_solve(model, iter);
// Iterations in time
for (scalar_type t = 0.; t < T; t += dt) {
iter.init();
getfem::standard_solve(model, iter);
// + Do something with the solution (plot or store it)
// Copy the current variables "u" and "Dot_u" into "Previous_u"
// and "Previous_Dot_u".
model.shift_variables_for_time_integration();
} | 4,411 | 16,777 | {"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": 2, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2022-21 | longest | en | 0.910585 |
http://scanftree.com/programs/python/python-program-to-convert-kilometers-to-miles/ | 1,550,595,072,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247490806.45/warc/CC-MAIN-20190219162843-20190219184843-00492.warc.gz | 229,821,819 | 11,026 | Python Program to Convert Kilometers to Miles
Levels of difficulty: / perform operation:
Source Code:
```# Program to convert kilometers into miles
# Input is provided by the user in kilometer
# take input from the user
kilometers = float(input('How many kilometers?: '))
# conversion factor
conv_fac = 0.621371
# calculate miles
miles = kilometers * conv_fac
print('%0.3f kilometers is equal to %0.3f miles' %(kilometers,miles))```
Output:
```How many kilometers?: 5.5
5.500 kilometers is equal to 3.418 miles```
Explanation
In this program, we use the ask the user for kilometers and convert it to miles by multiplying it with the conversion factor. With a slight modification, we can convert miles to kilometers. We ask for miles and use the following formula to convert it into kilometers.
`kilometers = miles / conv_fac`
`About Me` | 202 | 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} | 2.78125 | 3 | CC-MAIN-2019-09 | latest | en | 0.83664 |
http://anil7pute.blogspot.com/2013/02/funny-addition.html | 1,585,980,450,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585370520039.50/warc/CC-MAIN-20200404042338-20200404072338-00166.warc.gz | 11,104,930 | 17,262 | ## Thursday, February 7, 2013
It is a very interesting topic for my eBook "Miraculous World of Numbers". Funny Addition. See the following Addition.
I added three numbers 1) 74613, 2) 340, 3) 340 and I got the addition as 437. The details are as follows:
Similarly, two more additions are given below.
1) 74613, 274613, 3) 340, 4) 340 and their addition will be 43374613 .
As shown in the following diagram.
1) 437, 274613, 3) 340 and their addition will be 43373414.
As shown in the following diagram.
Do you like my addition? If you have anything to say, please feel free to email me at my email id anil@7pute.com along with your details:
Wish you all the best to correct me and give your correct answer.
Anil Satpute | 211 | 728 | {"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-2020-16 | latest | en | 0.891995 |
https://www.physicsforums.com/threads/calculate-the-wavelength-for-red-light.342269/ | 1,532,121,342,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676591831.57/warc/CC-MAIN-20180720193850-20180720213850-00312.warc.gz | 961,511,452 | 12,681 | # Homework Help: Calculate the wavelength for red light
1. Oct 2, 2009
### max2040uk
1. The problem statement, all variables and given/known data
The frequency of red light is 430THz calculate its wave length. Is the answer below correct?
2. Relevant equations
(3*10^8) / (430*10^12) = wave length
3. The attempt at a solution
(3*10^8) / (430*10^12) = 6.98 to 1 Decimal Place
2. Oct 2, 2009
### Staff: Mentor
You are off by some power of ten. What's 108/1012?
3. Oct 2, 2009
### max2040uk
Hi thanks for the reply at the end of the calculator is said e-7 (10^-7) stupid mistake. Thank you for your time :) | 199 | 618 | {"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} | 3.125 | 3 | CC-MAIN-2018-30 | latest | en | 0.826938 |
https://mrelementarymath.com/ | 1,653,414,347,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662573189.78/warc/CC-MAIN-20220524173011-20220524203011-00179.warc.gz | 464,544,810 | 27,388 | Take the stress out of implementing math centers in your classroom...
## Latest from the Blog
### 3rd Grade Math: How to Build Multiplication and Division Understanding
In third grade, one of the most important math goals is teaching multiplication and division understanding. Don’t make the mistake of jumping right into solving equations without first building a solid understanding of the concepts of multiplication and division The
### Reflect and Reset: Tips for Becoming a Better Math Teacher
Have you ever felt like you were stuck in a teaching rut? Or pictured yourself spinning on that endless teaching hamster wheel? Then, you need time for some teacher reflection. No matter how good your lesson is, there is always
### 3rd Grade Math: How to Teach Addition and Subtraction within 1,000
Teaching 3rd grade addition and subtraction can be challenging. Students often make mistakes when adding and subtracting larger multi-digit numbers. This is usually because they need a stronger understanding of place value. When the standard algorithm is used correctly, it
### 3rd Grade Math: How to Build Multiplication and Division Understanding
In third grade, one of the most important math goals is teaching multiplication and division understanding. Don’t make the mistake of jumping right into solving equations without first building a solid understanding of the concepts of multiplication and division The
### Reflect and Reset: Tips for Becoming a Better Math Teacher
Have you ever felt like you were stuck in a teaching rut? Or pictured yourself spinning on that endless teaching hamster wheel? Then, you need time for some teacher reflection. No matter how good your lesson is, there is always
### 3rd Grade Math: How to Teach Addition and Subtraction within 1,000
Teaching 3rd grade addition and subtraction can be challenging. Students often make mistakes when adding and subtracting larger multi-digit numbers. This is usually because they need a stronger understanding of place value. When the standard algorithm is used correctly, it | 404 | 2,066 | {"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} | 3.78125 | 4 | CC-MAIN-2022-21 | longest | en | 0.917385 |
https://notesbylex.com/eigenvector | 1,721,913,994,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763858305.84/warc/CC-MAIN-20240725114544-20240725144544-00698.warc.gz | 366,374,087 | 3,933 | ## Eigenvector
An Eigenvectors of a Matrix Transformation is any non-zero vector that remains on its Vector Span after being transformed.
That means that performing the transformation is equivalent to scaling the vector by some amount. The amount it scales the Eigenvector is called the Eigenvalue.
For example, if we transform the basis vectors with matrix $\begin{bmatrix}2 && 1 \\ 0 && 2\end{bmatrix}$, we can see that $\hat{j}$ is knocked off its span, where $\hat{i}$ is simply scaled by 2.
One other particular case of vector that remains on its span is the zero vector: $\vec{v}=\begin{bmatrix}0\\0\end{bmatrix}$, but that's not an Eigenvector.
When performing 3d rotations, the Eigenvectors are particularly useful as they describe the axis of rotation.
The notation for describing the relationship between the matrix transformation of the vector and same scaling quality equivalent:
$A\vec{v} = \lambda\vec{v}$ | 221 | 926 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "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} | 3.296875 | 3 | CC-MAIN-2024-30 | latest | en | 0.898863 |
http://itfeature.com/statistics/range-absolute-measure-of-dispersion | 1,484,657,922,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560279915.8/warc/CC-MAIN-20170116095119-00217-ip-10-171-10-70.ec2.internal.warc.gz | 146,733,746 | 31,181 | # Range: An Absolute Measure of Dispersion
Measure of Central Tendency provides typical value about the data set, but it does not tell the actual story about data i.e. mean, median and mode are enough to get summary information, though we know about the center of the data. In other words, we can measure the center of the data by looking at averages (mean, median, mode). These measure tell nothing about the spread of data. So for more information about data we need some other measure, such as measure of dispersion or spread.
Spread of data can be measured by calculating the range of data; range tell us over how many numbers of data extends. Range (an absolute measure of dispersion) can be found by subtracting highest value (called upper bound) in data from smallest value (called lower bound) in data. i.e.
Range = Upper Bound – Lowest Bound
OR
Range = Largest Value – Smallest Value
This absolute measure of dispersion have disadvantages as range only describes the width of the data set (i.e. only spread out) measure in same unit as data, but it does not gives the real picture of how data is distributed. If data has outliers, using range to describe the spread of that can be very misleading as range is sensitive to outliers. So we need to be careful in using range as it does not give the full picture of what’s going between the highest and lowest value. It might give misleading picture of the spread of the data because it is based only on the two extreme values. It is therefore an unsatisfactory measure of dispersion.
However range is widely used in statistical process control such as control charts of manufactured products, daily temperature, stock prices etc., applications as it is very easy to calculate. It is an absolute measure of dispersion, its relatives measure known as the coefficient of dispersion defined the the relation
$Coefficient\,\, of\,\, Dispersion = \frac{x_m-x_0}{x_m-x_0}$
Coefficient of dispersion is a pure dimensionless and is used for comparison purpose.
### Download pdf file: Range:Measure of Dispersion 64.40 KB Range:Measure of Dispersion
Updated: Feb 22, 2014 — 3:48 pm
#### Muhammad Imdadullah
Student and Instructor of Statistics and business mathematics. Currently Ph.D. Scholar (Statistics), Bahauddin Zakariya University Multan. Like Applied Statistics and Mathematics and Statistical Computing. Statistical and Mathematical software used are: SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis etc. | 546 | 2,537 | {"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} | 3.71875 | 4 | CC-MAIN-2017-04 | longest | en | 0.944794 |
http://www.cardtrick.ca/trick/As-Many-As-You.html | 1,542,766,031,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039746926.93/warc/CC-MAIN-20181121011923-20181121033923-00120.warc.gz | 384,528,766 | 6,276 | Informational Site Network
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Home Card Tricks Coin Tricks Magic Tricks Rope Tricks Other Tricks Sleights Search
## Card Tricks
Chocolate Sandwich Variation
Effect: This is a variation on The Sandwiched Aces trick w...
2 Card Monte
Effect: Performer gives spectator a heart faced card (say ...
magic Card Jump - magic card trick
The magician lets the spectator pick up a card. Th...
In and Out Deck
A deck of cards is fanned for a choice. But the magician ...
Effect: A spectator picks a card. You dial a number on the...
Pair 'em Up
Effect: By picking 2 cards from the deck, you will be able...
Royal Confidante
Sleight of hand required: Glimpsing the bottom card of the...
Card Trick 1
Effect: The magician has three rows of cards. An audie...
Magic In A Glass
Effect: The magician takes a long-stemmed glass and announ...
The Changing Ace
Effect: You hold the Ace of Clubs, Diamonds, and Spades, s...
## As Many As You
This mathematical trick can be performed with any pack
Effect: The magician can reveal the exact number of cards taken by a spectator.
Method: The magician asks a spectator to take a few cards from the top of the pack but to conceal them from the magician to the extent that the neither the performer nor the spectator can tell how many were taken.
The magician then also takes a bunch of cards secretly making sure he takes more than the spectator.
Next, the magician asks the spectator to turn his back and count his cards silently so as to give no clue to the performer.
The magician counts his own at the same time then in a novel albeit roundabout way reveals the exact number of cards held by the spectator.
How the Trick is Done
When the magician counts his cards - say for example he has 17 cards. He takes 3 from the total making a new total of 14 then says to the spectator: "I've got as many as you....3 more .....and enough to make your number up to 14". He then asks the spectator to reveal how many cards he is holding and the prediction is proven true as the magician first deals out the number of cards to match the spectators total (in this case 10) - then three more - then enough to make the spectators total up to 14. (magician deals his remaining 4 cards).
This will work whatever number the spectator or magician is holding.
Another example: if the magician's total 20 then the he would say: "I've got as many as you....3 more .....and enough to make your number up to 17". He would then ask the spectator how many cards he is holding and on hearing "8" the magician would first deal out the number of cards to match the spectators total (in this case 8 - then three more - then enough to make the spectators total up to 17. (magician deals his remaining 9 cards).
This is a self working mathematical trick.
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Privacy | 643 | 2,842 | {"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} | 3.203125 | 3 | CC-MAIN-2018-47 | longest | en | 0.948381 |
http://www.tinspireapps.com/?a=EQSOL | 1,495,982,453,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463609837.56/warc/CC-MAIN-20170528141209-20170528161209-00101.warc.gz | 814,355,220 | 8,087 | # Step by Step Equation Solver - Step by Step with the TI-Nspire CX (CAS)
## Solve Equation Solver problems stepwise using the Ti-Nspire Calculator
Step by Step Equation Solver
Runs on TI-Nspire CX CAS only.
It does not run on computers!
TI-Nspire CX CAS
App Purchase
Enter the last 8 digits of your 27-digit TI-Nspire's Product ID.
Located under 5:Settings → 4:Status → About
ID may look like: 1008000007206E210B0BD92F455. HELP.
If this was your ID you would only type in BD92F455.
At the end of the PayPal checkout, you will be sent an email containing your key and download instructions.
Description
Solve any Equations or Systems of Equations Step by Step. Works for any function. Also, performs Step by Step the Quadratic Formula, Complete the Square, Solve Equations via Grouping & Substitution and much more.
A Math with Steps.
FUNCTIONALITY & MENU ITEMS OF APP :
• Equations
• Step by Step Equation Solver
• Solve any Equation or Inequality
• Solve Equations via Grouping & Substitution
• Complete the Square
• Solve 2x2 system
• Read: PEMDAS - Order of Operation
• Identity Checker
• Newton Method
• Algebra
• Simplify Expression
• Factor
• Expand
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• Exponents & Logarithms | 407 | 1,536 | {"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-2017-22 | latest | en | 0.697806 |
https://stats.stackexchange.com/questions/139989/what-is-the-difference-between-variable-and-random-variable?noredirect=1 | 1,723,251,880,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640782288.54/warc/CC-MAIN-20240809235615-20240810025615-00404.warc.gz | 425,713,939 | 45,649 | # What is the difference between variable and random variable?
I know that "variable" means "values which vary." In a simple linear regression model :
$$Y=\beta_0+\beta_1X+\epsilon$$
$X$ is variable that is the values of $X$ vary. Why is $X$ not a random variable? What is the difference between a variable and a random variable?
• X is a random variable in this example. You might instead be thinking of the distinction between fixed and random effects? These are terms used to describe the parameters being estimated. For example, $\beta_1$ could be thought of as a single value that is the same across observations, or as a random variable that can differ across observations. Commented Mar 2, 2015 at 15:04
• @DLDahly, what about controlled experiments or dummy variables? Is $X$ a random variable even then? Commented Mar 2, 2015 at 15:05
• Very roughly, random variable is a variable equipped with probability. In mathematics, you say $X$ is a variable if it is not fixed and can take multiple values. But does every possible value have the same chance (probability) of being selected? For example, if $X$ takes either 0 or 1 as its value, what can you say about the chance $X=1$ if $X$ is just a "variable"? If $X$ is a Bernoulli(p) "random variable", then you know $X=0$ with chance 1-p, and $X=1$ with chance p. Commented Mar 2, 2015 at 16:25
• In regression, you can regard $X$ as a random variable if you know/assume its distribution, but it doesn't help because regression cares for only conditional distribution/expectation of $Y$ given $X$. That is, $X$ is fixed as a constant for the moment. Commented Mar 2, 2015 at 16:27
• What a random variable is has been discussed extensively at stats.stackexchange.com/questions/50. $X$ might or might not be a random variable in this model, depending on how you view its values as arising.
– whuber
Commented Mar 2, 2015 at 18:05
A variable is a symbol that represents some quantity. A variable is useful in mathematics because you can prove something without assuming the value of a variable and hence make a general statement over a range of values for that variable.
A random variable is a value that follows some probability distribution. In other words, it's a value that is subject to some randomness or chance.
In linear regression, $X$ may be viewed either as a random variable that is observed or it can be considered as a predetermined fixed value which, as LEP already discussed, the investigator chooses. As you've pointed out, we usually assume the later (whether or not this assumption is correct is another story). However, the OLS estimator is unbiased whether or not you treat $X$ as random and the estimate of the variance of the OLS estimator is unbiased for the variance of $\hat{\beta}_{OLS}$ whether or not you treat $X$ as random. These are a couple reasons people don't get too caught up in whether or not to assume $X$ is random in regression.
If you treat $X$ as random, I will show that the OLS estimator is still unbiased below.
Let $X$ be a random variable and let $\hat{\beta}_{OLS} = (X^{T}X)^{-1} X^{T} Y$.
$E(\hat{\beta}_{OLS})=E[E[\hat{\beta}_{OLS}|X]]=E[E[(X^{T}X)^{-1} X^{T} Y|X]]=E[(X^{T}X)^{-1} X^{T}E[ Y|X]] =E[(X^{T}X)^{-1} X^{T}X\beta] =E[\beta]=\beta$
If you treat $X$ as random, I will show that the estimate of the variance of $\hat{\beta}_{OLS}$ is unbiased for the unconditional variance below.
$Var(\hat{\beta}_{OLS})=Var(E(\hat{\beta}_{OLS}|X)) + E(Var(\hat{\beta}_{OLS}| X))=Var(\beta)+ E(Var(\hat{\beta}_{OLS}|X))=E(Var(\hat{\beta}_{OLS}|X))=E(\sigma^{2}( X^{T} X)^{-1})$
When you wrote down your equation, you did not list the assumptions: $$Y=\beta_0+\beta_1X+\epsilon$$
Why is X not a random variable?
Yes, it is often assumed (for simplicity of exposition in the intro statistics textbooks) that $$X$$ is fixed, or as you put it non-random.
It is fixed (non-random) in controlled experiments, i.e. mostly in natural sciences such as physics and biology. You can set the parameter $$X$$ at the level you're interested, and measure the response $$Y$$. In this case you make a set of assumptions such as Gauss-Markov theorem. For instance, feed the mice 1 mg of ascorbic acid and measure their hair loss. You control how much of the substance to administer.
However, it can be random, and it usually $$is$$ random in observational studies, i.e. 99% of all economics and social sciences alike. I can't set Dow-Jones Index (DJIA) at the arbitrary level, and measure the response in GDP (gross domestic product). I can only observe both, and whatever it is DJIA the day of my observation, that's my $$X$$. That's why the $$X$$ is random. In this case I have to use a different set of assumptions than the controlled experiment above. Imagine, now how difficult it is to establish causality between DJIA and DGP, unlike the case with mice when I decided how much of what to feed.
• The Gauss-Markov Theorem and Random Regressors. Author(s): Juliet Popper Shaffer. The American Statistician, Vol. 45, No. 4 (Nov., 1991), pp. 269-273
• "Gauss–Markov Assumptions for Observational Research (Arbitrary x)" in Encyclopedia of Research Design, p.532:
A parallel but stricter set of Gauss–Markov assumptions is typically applied in practice in the case of observational data, where the researcher cannot assume that x is fixed in repeated samples.
Theoretically speaking the outcomes of an experiment (experiment = a random procedure) can be numerical (i.e rolling a die) or can be mapped to numbers by the designer (i.e flipping a coin, with outcomes 1=head and 0=tail).
This numerical representation of the outcomes defines the random variables.
In these examples, we can tell that there is some chance involved, in other words we don't control the experiment 100%. So a random variable is linked to observations in the real world, where uncertainty is involved, and that's where the "randomness" comes from.
Most importantly, as others have already pointed out, a random variable x (which is either discrete or continuous) is quantified by a probability density function (pdf). So we say that, random variables have distributions.
Now, in the case of linear regression, you ALREADY know the values of X and from that you try to figure out the values of Y, in other words, Y is the random variable (as you still don't have its values and they will depend on Xs' values).
Resources:
• Actually, if we're referring specifically to the OLS estimator, one of the assumptions is that the independent variables should be deterministic and not stochastic. Commented Feb 10, 2020 at 11:23 | 1,660 | 6,604 | {"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": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2024-33 | latest | en | 0.906555 |
http://www.computing.net/answers/office/auto-insert-and-number-row-based-on-an-value-input/16609.html | 1,484,786,183,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280410.21/warc/CC-MAIN-20170116095120-00506-ip-10-171-10-70.ec2.internal.warc.gz | 397,229,346 | 9,253 | Auto Insert and Number Row based on an value input
March 23, 2012 at 22:33:11
Specs: Windows XP
Hi all,I need help generate certain numbers of rows and number them. let say A1 =10, then I would want B1=1B2=2B3=3B4=4B5=5B6=6B7=7B8=8B9=9B10=10If A1 value is 5, then B1=1,B2=2.B3=3,B4=4,B5=5 I hope it is clear, let me know and really thank you in advance.
See More: Auto Insert and Number Row based on an value input
#1
March 24, 2012 at 12:01:50
Put this in B1 and drag it down as many rows as you need to cover the maximum number that A1 will ever be.Worst case, fill the entire Column B with it.=IF(ROW()>\$A\$1,"",ROW())Click Here Before Posting Data or VBA Code ---> How To Post Data or Code.
Report •
Related Solutions | 248 | 728 | {"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-2017-04 | latest | en | 0.865166 |
https://relate.cs.illinois.edu/course/cs357-f15/file-version/68fa8dd5f1a58eb5d5e81218d37a59f892ca861b/media/interp/Monomial%20interpolation.html | 1,501,231,978,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549448146.46/warc/CC-MAIN-20170728083322-20170728103322-00377.warc.gz | 695,377,589 | 66,046 | # Monomial interpolation¶
In [3]:
#keep
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
%matplotlib inline
In [4]:
#keep
x = np.linspace(0, 1, 200)
Now plot the monomial basis on the interval [0,1] up to $x^9$.
In [5]:
#keep
n = 10
for i in range(n):
pt.plot(x, x**i)
pt.vlines(np.linspace(0, 1, n), 0, 1, alpha=0.5, linestyle="--")
Out[5]:
<matplotlib.collections.LineCollection at 0x10564e160>
• How do the entries of the Vandermonde matrix relate to this plot?
• Guess the condition number of the Vandermonde matrix for $n=5,10,20$:
In [6]:
#keep
n = 5
V = np.array([np.linspace(0, 1, n)**i for i in range(n)]).T
la.cond(V)
Out[6]:
686.43494181859796
In [ ]: | 249 | 701 | {"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} | 2.84375 | 3 | CC-MAIN-2017-30 | latest | en | 0.507555 |
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# Built-up up shape in compression
• To: seaint(--nospam--at)seaint.org
• Subject: Built-up up shape in compression
• From: "Benjamin Cornelius" <bcorneli(--nospam--at)hotmail.com>
• Date: Fri, 18 Jun 2004 17:13:44 -0400
I am evaluating an existing steel truss with a built-up top chord composed of a two angles and a plate between. The plate is sandwiched between the two angles to form a tee-shape. The plate and angles are fastened together with a single line of rivets. The stem of the tee (the plate) points downward and I have concluded that it is slender.
```
```
I have axial compression and bending (due to the direct application of roof loads). I am evaluating the combined stresses according to the 3rd edition LRFD, and have calculated a Qs reduction factor per Appendix B5.3a to account for the slenderness of the stem plate. The Qs factor is dependent on the b/t value of the plate projecting from the built-up compression member.
```
```
B5.1(c) of the Spec says that the "b" of the plate is the distance from the free edge to the first row of fasteners or line of welds. For me, that would mean b = 10.5".
```
```
I am guessing that this provision was written so that it would be valid regardless of whether the unstiffened element was connected to the built-up member on one side only (single shear configuration) or on both sides (double shear configuration).
```
```
In my case, the downward legs of the angle confine the stem plate, such that one might convince himself that "b" could be taken as the depth of the stem plate beyond the toes of the angles. For me, that would mean b = 9".
```
Does anyone have any opinions?
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http://stickycompany.com/castrol/elizabethan/92036268c43518 | 1,675,503,277,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500095.4/warc/CC-MAIN-20230204075436-20230204105436-00222.warc.gz | 43,358,916 | 11,769 | ### multinomial coefficient examples
it has only two possible outcomes (e.g. 0 or 1).Some popular examples of its use include predicting if an e-mail is spam or not spam or if a tumor is malignant or not i=1 k For a 5% Conversely, the multinomial distribution makes use of the multinomial coefficient which comes from the multinomial theorem. Multinomial Coefficients and More Counting (PDF) 3 Sample Spaces and Set Theory (PDF) 4 Axioms of Probability (PDF) 5 Probability and Equal Likelihood (PDF) 6 Conditional Probabilities One is the dependent variable (that is nominal). Multinomial Coefficient example. So the probability of selecting exactly 3 red balls, 1 white ball and 1 black ball equals to 0.15. The goal is to take away some of the mystery by providing clean code examples that are easy to run and compare with other tools. Let $$X$$ be a set of $$n$$ elements. 1 Examples7. For a fixed n n n and k k k, what is Using multinomial theorem, we have. There are 3,360 unique partitions of the word ARKANSAS. Sum or product of two or more multinomials is also a multinomial, but their subtraction or division may not result in a multinomial.
log likelihood = What is the multinomial coefficient used for?
A group of six students consists of 3 seniors, 2 juniors, and 1 sophomore. In general we can use A Note on Interpreting Multinomial Logit Coefficients. Source File: test_multinomial.py. Example 1. c + d) 10 using multinomial theorem and by using coefficient property we can obtain the required result. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Examples. We plug these inputs into our multinomial distribution calculator and easily get the result = 0.15. The goal is to take away some of the mystery by providing clean code examples that are easy to run and compare with other tools. One group will have 5 students and the other three For example, we could use logistic regression to model the relationship between various measurements of a manufactured specimen (such as dimensions and chemical composition) to predict if a crack greater than 10 mils will occur (a binary variable: either yes or no). Example Model building. logistic coefficient is the expected amount of change in the logit for each one unit change in the predictor. Before we perform these algorithm in R, lets ensure that we have gained a concrete understanding using the cases below: Case 1 (Multinomial Regression) The modeling of program choices made by high school students can be done using Multinomial logit. Search: Glm Multinomial. One or more independent variable(s) (that is interval or ratio or dichotomous). example 2 Find the coefficient of x 2 y 4 z in the expansion of ( x + y + z) 7.
From the stars and bars method, the number of distinct terms in the multinomial expansion is C ( n + k 1, n) . 5x + 9, 6y 2 + 2y - 5 etc are the On any particular trial, the probability of drawing a red, white, or black ball is 0.5, 0.3, and 0.2, respectively. Draw samples from a multinomial distribution. In short, this counts for the number of possible combinations, with importance to the order of players. Multinomial Coefficients and More Counting (PDF) 3 Sample Spaces and Set Theory (PDF) 4 Axioms of Probability (PDF) 5 Probability and Equal Likelihood (PDF) 6 Conditional Probabilities (PDF) 7 Bayes' Formula and Independent Events (PDF) 8 Discrete Random Variables (PDF) 9 Expectations of Discrete Random Variables (PDF) 10 Variance (PDF) 11 Peoples occupational choices might be influenced by their parents occupations and their own education level. Search: Glm Multinomial. Basics and examples; Incompleteness of axiomatic systems; Exercises; III Set Theory; 9 Sets.
Section 2.7 Multinomial Coefficients. Multinomial Coefficient = 8! d2. I Answer: 8!/(3!2!3!) For example, number of terms in the expansion of (x + y + z) 3 is 3 + 3 -1 C 3 1 = 5 C 2 = 10. Note that this example is different from Example 5b because now the order of the two teams is irrelevant. The first important definition is the multinomial coefficient: For non-negative integers b 1, b 2, , b k b_1, b_2, \ldots, b_k b 1 , b 2 , , b k such that i = 1 k b i = n, \displaystyle \sum_{i=1}^{k} b_i = n, If 'Interaction' is 'off' , then B is a k 1 + p vector. Usage multichoose(n, bigz = FALSE) Arguments. r n! The difference ${Q_3 - Q_1}$ is called the inter quartile range. It depends on the lower quartile ${Q_1}$ and the upper quartile ${Q_3}$. Examples of Multinomial. 1.1 Example; 1.2 Alternate expression; 1.3 Proof; 2 Multinomial coefficients. It provides a selection of efficient tools for machine learning and statistical modeling \ (5 x^ {3}-2 x y+7 y^ {2}\) is a multinomial with three terms. In short, this counts for the number of possible combinations, with importance to the order of players.
3. Hence, is often read as " choose " and is called the choose function of and . The multinomial coefficient is also the number of distinct ways to permute a multiset of n elements, and ki are the multiplicities of each of the distinct elements. Project: sympy. We want to get coefficient of a 3 b 2 c 4 d this implies that r 1 = 3, r 2 = 2, r 3 = 4, r 4 = 1, Section 23.2 Multinomial Coefficients Theorem 23.2.1. As we mentioned previously, Cover_Type is the response and we use all other columns as predictors If the testing set is labeled, testing will be done and some statistics will be computed to measure the quality Glm Stamp Models Quite the same Wikipedia The GLM operator is used to predict the Future customer attribute of the Deals sample data set The GLM WikiMatrix The For dmultinom, it defaults to sum(x).. prob: numeric non-negative vector of length K, specifying the probability for the K classes; is internally normalized to sum 1. In other words, the number of distinct See Wikipedia article Gaussian_binomial_coefficient.. The greatest coefficient in the expansion of (a 1 + a 2 + a 3 +.. + a m ) n is (q!) m r ((q + 1)!) COUNTING SUBSETS OF SIZE K; MULTINOMIAL COEFFICIENTS 413 Formally, the binomial theorem states that (a+b)r = k=0 r k arkbk,r N or |b/a| < 1. In statistics, the corresponding multinomial series appears in the multinomial Math 461 Introduction to Probability A.J. Logistic regression models a relationship between predictor variables and a categorical response variable. On any given trial, the probability that a particular outcome will occur is constant. We can study the Statistics - Multinomial Distribution.
That is, there is no A and B team, but just a division consisting of 2 groups of The MCC is in essence a correlation coefficient value between -1 and +1. How many unique partitions of this group of students are there by grade? * 1! There are three types of logistic regression models, which are defined based on categorical response. INPUT: n, k the values $$n$$ and $$k$$ defined above q (default: None) the variable $$q$$; if None, then use a default variable in $$\ZZ[q]$$. a number appearing as a coefficient in the expansion of $$(x_1 + x_2 + \dotsb + x_m)^n$$ $$\binom{n}{i_1,i_2,\dotsc,i_m}$$ the coefficient on the term $$x_1^{i_1} x_2^{i_2} How many ways to do that? Get used to seeing log-likelihood functions in this form,! A property of multinomial data is that there is a dependency among the counts of the 6 faces. For example, suppose we conduct an experiment by rolling two dice 100 times. Lecture 1.4: Binomial and multinomial coe cients Matthew Macauley Department of Mathematical Sciences Clemson University We will motivate the following theorem with an example: (x + y)6 Also allows efficiently computing entire sets of binomial and multinomial coefficients in one go. sum of the squared coefficients to the loss function. Now try simple regression with a 3-category outcome. B = mnrfit (X,Y) returns a matrix, B, of coefficient estimates for a multinomial logistic regression of the nominal responses in Y on the predictors in X. example. You want to choose three for breakfast, two for lunch, and three for dinner. > # I think I have to make an mlogit data frame with just the vars I want. Multinomial logistic regression is an extension of logistic regression that adds native support for multi-class classification problems.. Logistic regression, by default, is limited to two-class classification problems. Odds males are admitted: odds(M) = P/(1-P) = .7/.3 = 2.33 Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference.. Integer mathematical function, suitable for both symbolic and numerical manipulation. The multinomial coefficient is used to denote the number of possible partitions of objects into groups having numerosity . Extended Keyboard Examples Upload Random multinomial coefficient calculator - Wolfram|Alpha Compute answers using Wolfram's breakthrough technology & knowledgebase, In combinatorics, is interpreted as the number of -element subsets (the -combinations) of an -element set, that is the number of ways that things can be "chosen" from a set of things. A multinomial coefficient describes the number of possible partitions of n objects into k groups of size n 1, n 2, , n k. The formula to calculate a multinomial coefficient is: I'll build two multinomial models, one with glmnet::glmnet(family = "multinomial"), and one with nnet::multinom(), predicting Species by Sepal.Length and Sepal.Width from everyone's favorite dataset. * 2!) Basics; Defining sets; Subsets and equality of sets; Complement, union, and intersection; The reason I like Stan is that it allows you extend beyond the standard multinomial logit model to hierarchical models, dynamic models and all sorts of fun stuff. (Here n = 1,2, and r = 0,1,,n. Worked Example 23.2.5. Partition problems I You have eight distinct pieces of food. . For example: a 2 b 0 c 1 {\displaystyle a^{2}b^{0}c^{1}} has the coefficient ( 3 Binary logistic regression: In this approach, the response or dependent variable is dichotomous in naturei.e. ylog() . One group will have 5 students and the other three groups will Determining a specific coefficient in a multinomial expansion. If \(q$$ is unspecified, then the variable is the generator $$q$$ for a univariate polynomial ring over the integers.. coefficient is equal to zero (i.e. size: integer, say N, specifying the total number of objects that are put into K boxes in the typical multinomial experiment. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample = the difference between the x-variable rank and the y-variable rank for each pair of data.
Multinomial logistic regression and logistic regression are generalized linear models. Intepretation for the sign of the coefficients: a positive coefficient represents a negative effect on $$y_i$$. Example 1: A new drug was tested for the treatment of certain types of cancer patients. The Greatest Coefficient in a multinomial expansion. 1 Answer. No longer just selecting balls (yes Multinomial. Admissions Example Calculating the Odds Ratio Example: admissions to a graduate program Assume 70% of the males and 30% of the females are admitted in a given year Let P equal the probability a male is admitted. Model Summary. Determine the coefficient on $$x^2 y z^6$$ in the expansion of $$(3 x + 2 y + z^2 + 6)^8\text{. Trinomial Theorem. However a type vector is itself a special kind of multi-index, one dened on the strictly positive natural numbers. def test_multinomial_coefficients(): assert multinomial_coefficients(1, 1) == {(1,): 1} assert For example, operating system preference of a universitys students could be classified as Windows, Mac, or Linux. Infinite and missing values are not allowed. log ( multi ( 7 , 4 , 2 ) ) // Prints: 25740 Observe that when r is not a natural number, the right-hand side is an innite sum and the condition |b/a| < 1 insures that the series converges. Example 118 Multinomial coefficients Suppose we are given k boxes labeled 1 from CPSC 320 at University of British Columbia In the example, just above, the DATA areyp33 and PROBABILITY is , thus ylo33g(/) .p The typical log-likelihood function is the sum of such terms (plus, sometimes, the binomial or multinomial coefficient, which does not involve the parameters). (n choose r). Multinomial Logistic Regression models how a multinomial response variable Y depends on a set of k explanatory variables, x = ( x 1, x 2, , x k). Using (2) and (4), we need to have , , , and . The example below demonstrates how to predict a multinomial probability distribution for a new example using the multinomial logistic regression model. With the above coefficient, the expansion will be read as follows: For n th power. The first formula is a general definition for the Like any other regression model, the multinomial output can be predicted using one or more independent variabl You are currently logged in from 5 GeneralizedLinearModels DavidRosenberg New York University April12,2015 David Rosenberg (New York University) DS-GA 1003 April 12, 2015 1 / 20 (squared error), "laplace" (absolute loss), The multinomial coefficients (n_1,n_2,,n_k)!=((n_1+n_2++n_k)!)/(n_1!n_2!n_k!) This function computes the multinomial coefficient by computing the factorial of each number on a log scale, differencing log(n!) Enter the number of times out of n that you One group will have 5 students and the other three groups will have 4 students. So, = 0.5, = 0.3, and = 0.2. There are different ways to form a set of ( r 1) non-redundant logits, and these will lead to different polytomous (multinomial) logistic regression models. Let's assume the first 13 cards are dealt to player 1, cards 14-26 to player 2, 27-39 to player 3 and the last 13 cards to player 4. Let Q equal the probability a female is admitted. Peoples occupational choices might be influenced by their parents occupations and their own education level. A teacher will divide her class of 17 students into four groups to work on projects. Probability Theory: Multinomial Coefficient Example: Have 10 balls. (5, 2, 1, 1, 2 1 1 ) = 8 3, 1 6 0. Scikit-Learn ii About the Tutorial Scikit-learn (Sklearn) is the most useful and robust library for machine learning in Python. For discrete examples, just replace integrals with summations. The logit is what is being predicted; it is the odds of membership in the category of the function). Example 1 Suppose that the joint density function of and is given by where , and . 4.2. 0. 5) are extensions of logistic and probit regressions for categorical data with more than two options, for example survey responses such as Strongly Agree, Agree, Indierent, Disagree, Strongly Disagree Adaptive LASSO in R The adaptive lasso was introduced by Zou (2006, JASA) for linear regression and by Zhang and Lu (2007, Biometrika) for proportional It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. Recall that the multinomial logit model estimates k-1 models, where the k th equation is relative to the referent group. }{\prod n_j!}. The coefficient takes its name from the following multinomial One can always make i. Feit, 2020. r!(nr)! Examples: To find the multinomial coefficient of 7, 16, and 1000, we binary expand each of them: Since no column has more than one 1, the multinomial coefficient is odd, and hence we should output something truthy. In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. License: View license. Cubic Units: Definition, Facts & Examples. Multinomial Probability = 4 4 3 3 2 2 1 ( | ,) 1 y y y y i i p p p p y n f y n pi = Enter the total number in the population (trials) in cell G4. n = sample This multinomial coefficient gives the number of ways of depositing 4 distinct objects into 3 distinct groups, with i objects in the first group, j objects in the second group and k objects in the / (3! An expression with one or more terms (the exponents of variables can be either positive or negative) \(4x^{-1} +2y+3z$$ You can find more information under "Definition of Variable, Constant, Term and Coefficient" section of this page. Search: Glm Multinomial. The multinomial theorem provides a formula for expanding an expression such as (x1 + x2 ++ xk)n for integer values of The formula is: In formal terms, the multinomial coefficient formula gives the expansion of (k 1 + k 2 + k n) where k are non-negative integers. View MULTINOMIAL COEFF.pdf from EECS 55 at University of California, Irvine. One easy and flexible way to estimate these models is in Stan. Coefficient estimates for a multinomial logistic regression of the responses in Y, returned as a vector or a matrix. It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n, and is given by the formula =!! Solution. We check if the proportional odds assumption is reasonable by comparing with a more flexible model allowing different $$\beta$$ for different category $$k$$. Suppose that we have two colors of paint, say red and blue, and we are going to choose a subset of $$k$$ elements to be Multinomial logistic regression Number of obs c = 200 LR chi2 (6) d = 33.10 Prob > chi2 e = 0.0000 Log likelihood = -194.03485 b Pseudo R2 f = 0.0786. b. Log Likelihood This where q is the quotient and r is the remainder when n is divided by m. Multinomial Coefficient example; Question A teacher will divide her class of 17 students into four groups to work on projects. The examples given here are continuous joint distributions.
Pick 3 of them to be colored red, pick 3 to be colored blue, and pick 4 to be colored purple. A weighting of the coefficients can be used that reduces the strength of the penalty from full penalty to a very slight penalty. > # First try The coefficient for x3 is significant at 10% (<0.10).
The Binomial Theorem gives us as an expansion of (x+y) n. The Multinomial Theorem gives us an
\binom{11}{5,2,1,1,2} = 83{,}160. How many ways can students be assigned to group if: A multinomial experiment is a statistical experiment and it consists of n repeated trials. Here we introduce the Binomial and Multinomial Theorems and see how they are used. \ (7 x y-9 y z+6 z x-7\) is a multinomial with four terms. Hildebrand Binomial coecients Denition: n r = n! Examples of multinomial logistic regression. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.Commonly, a binomial coefficient is indexed by a pair of integers n k 0 and is written (). Search: Glm Multinomial. random.multinomial(n, pvals, size=None) #. Fraction less than 1: Definition, Facts & Examples. The RRR of a coefficient indicates how the risk of the outcome falling in the comparison group compared to the risk of the outcome falling in the referent group changes with the variable in question. In general we can use multinomial models for multi-category target variables, or more generally, multi-count data. Some examples of constants are $$3, 6, \dfrac{-1}{2}, \sqrt{5}$$ etc. 2.1 Sum of all multinomial coefficients; 2.2 Number of multinomial coefficients; 2.3 Valuation of multinomial coefficients; 3 Interpretations. Let's assume the first 13 cards are dealt to player 1, cards 14-26 to player 2, 27 , although this coefficient is not significant. = 3,360. The above four Example: \ (5 x^ {2}+3 x\) is a multinomial with two terms. The usual value is 0.05, by this measure none of the coefficients have a significant effect on the log-odds ratio of the dependent variable. 3 Generalized Multinomial Theorem 3.1 Binomial Theorem Theorem 3.1.1 If x1,x2 are real numbers and n is a positive integer, then x1+x2 n = r=0 n nrC x1 n-rx 2 r (1.1) Binomial Coefficients - sum(log(x! multinomial coefficient. 1 5 Coefficients Multinomial uhm How many ways n ti in h hi thzt Thr why EY f 4 hi n no mi Nr cn n n hi nz h n Ln with more than two possible discrete outcomes. The Multinomial Coefficients The multinomial coefficient is widely used in Statistics, for example when computing probabilities with the hypergeometric distribution . Calculate the covariance and the correlation coefficient. In the case of the multinomial one has no intrinsic ordering; in contrast in the case of ordinal regression there is an association between the levels.For example if you examine the variable V1 that has green , yellow and red as independent levels then V1 encodes a multinomial variable. In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. i. Some extensions like one-vs-rest can allow logistic regression to be used for multi-class classification problems, although they require that the classification problem Details. * 1!
Informally, you can think of it The difference ${Q_3 - Q_1}$ divided by 2 is called semi-inter quartile range or the quartile deviation. Example var multi = require ( "multinomial" ) console . The program choices are general program, vocational program and academic program. Multinomial logistic regression with Python: a comparison of Sci-Kit Learn and the statsmodels package including an explanation of how to fit models and interpret coefficients with both. Each trial has a discrete number of possible outcomes. The factorial , double factorial , Pochhammer symbol , binomial coefficient , and multinomial coefficient are defined by the following formulas. = sum of the squared differences between x- and y-variable ranks. Example 2: Students by Grade. Multinomial logit models are a workhorse tool in marketing, economics, political science, etc. The multinomial coefficient Multinomial [ n 1, n 2, ], denoted , gives the number of ways of * 1! For example: In the coefficient of term x 1 y 1 z 2 uses i = 1, j = 1, and k = 2, which will be equal to. > # Excellent. By definition, the ()!.For example, the fourth power of 1 + x is n: a B = mnrfit (X,Y,Name,Value) returns a matrix, B, of coefficient estimates for a multinomial model fit with additional options specified by one or more Name,Value pair arguments. Generalized Linear Models and Extensions, Second Edition provides a comprehensive overview of the nature and scope of generalized linear models (GLMs) and of the major changes to the basic GLM algorithm that allow modeling of data that violate GLM distributional assumptions History and Etymology for This is a minimal reproducible
multinomial theorem, in algebra, a generalization of the binomial theorem to more than two variables. | 5,511 | 22,622 | {"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} | 3.984375 | 4 | CC-MAIN-2023-06 | latest | en | 0.875476 |
https://westudyhere.com/symbol-for-energy-in-physics/ | 1,701,897,362,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100603.33/warc/CC-MAIN-20231206194439-20231206224439-00275.warc.gz | 693,845,069 | 21,689 | # Symbol For Energy In Physics
## What are Physic Term – Symbol For Energy In Physics
The symbol for energy in physics is typically represented by the letter “E”. Energy is a fundamental concept in physics that refers to the ability of a system to do work or cause a change. It is a scalar quantity, meaning it has magnitude but no direction.
The symbol “E” is used in various physics concepts and equations to represent different forms of energy. For example, in the equation E = mcΒ², “E” represents the energy of an object, “m” represents its mass, and “c” represents the speed of light. This equation, known as Einstein’s mass-energy equivalence, shows the relationship between mass and energy.
In other equations, “E” can represent specific types of energy, such as kinetic energy (Β½mvΒ²) or potential energy (mgh). These equations allow physicists to calculate and analyze the energy of objects in motion or in different positions.
The symbol “E” is also used in the principle of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed. This principle is crucial in understanding the behavior of systems and is applied in various fields of physics, such as mechanics, thermodynamics, and electromagnetism.
Overall, the symbol “E” for energy in physics is significant as it represents a fundamental quantity that is essential for understanding and describing the behavior of physical systems.
## Explanation of Key Terms – Symbol For Energy In Physics
1. Kinetic Energy (KE):
– Definition: The energy possessed by an object due to its motion.
– Formula: KE = 1/2 * mass * velocity^2
– Example: A moving car possesses kinetic energy. The faster the car is moving and the heavier it is, the more kinetic energy it has. For instance, a car traveling at 60 mph will have more kinetic energy than the same car traveling at 30 mph.
2. Potential Energy (PE):
– Definition: The energy possessed by an object due to its position or state.
– Formula: PE = mass * gravity * height
– Example: A book placed on a shelf possesses potential energy. The higher the book is placed, the more potential energy it has. For instance, a book placed on a higher shelf will have more potential energy than the same book placed on a lower shelf.
3. Thermal Energy:
– Definition: The energy associated with the motion of particles within a substance.
– Example: When a pot of water is heated on a stove, the thermal energy increases, causing the water molecules to move faster and eventually boil.
4. Chemical Energy:
– Definition: The energy stored in the bonds between atoms and molecules.
– Example: The energy stored in food is chemical energy. When we eat food, our bodies convert the chemical energy into other forms, such as kinetic energy for movement.
5. Electrical Energy:
– Definition: The energy associated with the flow of electric charges.
– Example: When we turn on a light bulb, electrical energy is converted into light energy, allowing us to see.
6. Nuclear Energy:
– Definition: The energy stored in the nucleus of an atom.
– Example: Nuclear power plants use nuclear energy to generate electricity by harnessing the energy released during nuclear reactions.
7. Gravitational Potential Energy:
– Definition: The energy possessed by an object due to its position in a gravitational field.
– Formula: PE = mass * gravity * height
– Example: A roller coaster at the top of a hill possesses gravitational potential energy. As it descends, the potential energy is converted into kinetic energy, providing the thrill of the ride.
8. Elastic Potential Energy:
– Definition: The energy stored in a stretched or compressed elastic object.
– Formula: PE = 1/2 * spring constant * displacement^2
– Example: A stretched rubber band possesses elastic potential energy. When released, the potential energy is converted into kinetic energy, prop
## Applications in The real World – Symbol For Energy In Physics
1. The symbol for energy in physics, E, is applied in real-world scenarios such as electricity generation. Understanding this symbol helps engineers design efficient power plants and electrical grids, leading to practical applications like reliable electricity supply for homes and industries.
2. Another symbol for energy, W, is used in industries like manufacturing and transportation. Understanding this symbol helps optimize processes and reduce energy consumption, leading to practical applications such as improved production efficiency and reduced carbon emissions.
3. The symbol for kinetic energy, KE, is applied in technologies like transportation and sports. Understanding this symbol helps engineers design faster and more efficient vehicles, leading to practical applications like high-speed trains and record-breaking athletic performances.
4. The symbol for potential energy, PE, is used in scenarios like building construction and renewable energy. Understanding this symbol helps architects design stable structures and engineers harness energy from sources like wind and water, leading to practical applications like skyscrapers and hydroelectric power plants.
5. The symbol for thermal energy, Q, is applied in industries like heating and cooling. Understanding this symbol helps engineers design efficient HVAC systems, leading to practical applications like comfortable indoor environments and reduced energy consumption.
Overall, understanding and applying these symbols for energy in physics have numerous practical applications across various industries and technologies, leading to improved efficiency, sustainability, and technological advancements.
## Related Terms
1. Energy symbol
2. Energy equation
3. Energy transfer
4. Energy conservation
5. Potential energy symbol
6. Kinetic energy symbol
7. Work-energy theorem
8. Law of conservation of energy
9. Energy transformation
10. Energy unit
### Conclusion
If you’re interested in learning more about the fascinating world of physics and its applications, be sure to explore our website for more content. We have a wide range of articles, explanations, and examples that delve into various physics concepts, including energy and its symbols. Whether you’re a student, a professional, or simply curious about the workings of the physical world, our website offers valuable resources to enhance your understanding. So, dive in and expand your knowledge of physics today! | 1,244 | 6,424 | {"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-2023-50 | latest | en | 0.939943 |
https://www.perlmonks.org/index.pl/?displaytype=print;node_id=54957 | 1,718,205,982,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861173.16/warc/CC-MAIN-20240612140424-20240612170424-00053.warc.gz | 848,384,748 | 2,356 | http://www.perlmonks.org?node_id=54957
At first I had ignored this, then decided to do it. It was a more fun challenge than I thought. There are, not counting the order of the moves, actually 4 solutions in 15 moves for a 5x5 board. What follows is the throw-away script I wrote to find this. By default it solves a 5x5 board. Pass it an argument and it will solve an nxn board. (I tried it in the 1..10 range and found that there is 1 solution for 1, 2, 3, 6, 7, 8 and 10. As I mentioned, there are 4 for 5, plus 16 for 4 and 256 for 9. Don't ask me why, I merely report what I found...)
It would not be hard to extend this to handle arbitrary rectangular boards. I also didn't need the globals but this is throw-away code and it was easier that way. I make no apologies for the huge numbers of anonymous functions. The fact that I can feasibly find all 64 solutions for an 11x11 board by brute-force search on my old laptop speaks loudly enough for the efficiency of the method...
```use strict;
use Carp;
use vars qw(\$min \$max @board @soln @toggles);
\$min = 1;
\$max = shift(@ARGV) || 5;
@board = map [map 0, \$min..\$max], \$min..\$max;
foreach my \$x (\$min..\$max) {
foreach my \$y (\$min..\$max) {
push @toggles, ["\$x-\$y", ret_toggle_square(\$x, \$y)];
}
}
find_soln();
sub find_soln {
if (! @toggles) {
# Solved!
print join " ", "Solution:", map \$_->[0], @soln;
print "\n";
}
else {
my \$toggle = shift(@toggles);
# Try with, then without
if (\$toggle->[1]->()) {
push @soln, \$toggle;
find_soln();
pop @soln;
}
if (\$toggle->[1]->()) {
find_soln();
}
unshift @toggles, \$toggle;
}
}
# Returns a function that switches one square and returns
# true iff the new color is black
sub ret_swap_square {
my (\$x, \$y) = @_;
#print "Generated with \$x, \$y\n";
my \$s_ref = \(\$board[\$x-1][\$y-1]);
return sub {\$\$s_ref = (\$\$s_ref + 1) %2;};
}
# Returns a function that toggles one square and its
# neighbours, and returns whether or not any neighbour
# swapping again with \$x lower or \$x the same and \$y lower.
sub ret_toggle_square {
my (\$x, \$y) = @_;
my @fin_swaps;
my @other_swaps;
unless (\$x == \$min) {
push @fin_swaps, ret_swap_square(\$x - 1, \$y);
}
if (\$x == \$max) {
unless (\$y == \$min) {
push @fin_swaps, ret_swap_square(\$x, \$y - 1);
}
if (\$y == \$max) {
push @fin_swaps, ret_swap_square(\$x, \$y);
}
else {
push @other_swaps, ret_swap_square(\$x, \$y);
unless (\$y == \$max) {
push @other_swaps, ret_swap_square(\$x, \$y+1);
}
}
}
else {
unless (\$y == \$min) {
push @other_swaps, ret_swap_square(\$x, \$y - 1);
}
push @other_swaps, ret_swap_square(\$x, \$y);
push @other_swaps, ret_swap_square(\$x + 1, \$y);
unless (\$y == \$max) {
push @other_swaps, ret_swap_square(\$x, \$y + 1);
}
}
return sub {
\$_->() foreach @other_swaps;
my \$ret = 1;
\$ret *= \$_->() foreach @fin_swaps;
return \$ret;
}
} | 905 | 2,842 | {"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} | 3.234375 | 3 | CC-MAIN-2024-26 | latest | en | 0.83058 |
https://mathematica.stackexchange.com/questions/265989/changing-colors-of-contourplot | 1,722,980,415,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640508059.30/warc/CC-MAIN-20240806192936-20240806222936-00319.warc.gz | 326,921,671 | 41,177 | # Changing colors of ContourPlot
I have a problem with ContourPlot colors. In fact, i have the following table
A={x^2 + y^2 == 203.378, x^2 + y^2 == 59.1222, x^2 + y^2 == 35.2762, x^2 + y^2 == 25.4656, x^2 + y^2 == 20.1073, x^2 + y^2 == 16.7197, x^2 + y^2 == 14.37, x^2 + y^2 == 12.6233, x^2 + y^2 == 11.2367, x^2 + y^2 == 10.0089}
That i want to plot using the colors of "ThermometerColors". To do so, i used the following
ContourPlot[Evaluate@A, {x, -15, 15}, {y, -15, 15}, ColorFunction -> "ThermometerColors"]
However, all the plots have the same color. I think that ContourPlot associates only the first color to the functions of table A. Any idea how to associate one color from "ThermometerColors" to each function?
Try this:
ContourPlot[Evaluate[A], {x, -15, 15}, {y, -15, 15}, ContourStyle -> MapIndexed[{ColorData["ThermometerColors"][#2[[1]]/Length[A]]} &, A], PlotLegends -> A[[All, -1]], FrameStyle -> Black]
If we reverse the color order:
ContourPlot[Evaluate[A], {x, -15, 15}, {y, -15, 15}, ContourStyle -> Reverse@MapIndexed[{ColorData["ThermometerColors"][#2[[1]]/Length[A]]} &, A], PlotLegends -> A[[All, -1]], FrameStyle -> Black]
Assuming you want contour colors to reflect contour values:
ContourPlot[Evaluate @ A[[1, 1]], {x, -15, 15}, {y, -15, 15},
ColorData["ThermometerColors"] /@ Rescale[A[[All, -1]]]}],
PlotLegends -> Sort @ A[[All, -1]]]
Alternatively,
ContourPlot[Evaluate @ A[[1, 1]], {x, -15, 15}, {y, -15, 15},
ColorData[{"ThermometerColors", MinMax@A[[All, -1]]}] /@ A[[All, -1]]}],
PlotLegends -> Sort @ A[[All, -1]]]
same picture
\$Version
(* "13.0.1 for Mac OS X x86 (64-bit) (January 28, 2022)" *)
Clear["Global*"]
A = {x^2 + y^2 == 203.378, x^2 + y^2 == 59.1222, x^2 + y^2 == 35.2762,
x^2 + y^2 == 25.4656, x^2 + y^2 == 20.1073, x^2 + y^2 == 16.7197,
x^2 + y^2 == 14.37, x^2 + y^2 == 12.6233, x^2 + y^2 == 11.2367,
x^2 + y^2 == 10.0089};
ContourPlot[Evaluate@A[[1, 1]],
{x, -15, 15}, {y, -15, 15},
Contours -> A[[All, 2]],
` | 799 | 1,991 | {"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} | 2.921875 | 3 | CC-MAIN-2024-33 | latest | en | 0.638777 |
niswest.rghotels.ru | 1,532,249,472,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593142.83/warc/CC-MAIN-20180722080925-20180722100925-00170.warc.gz | 261,610,723 | 5,535 | # Carbon dating reference
15-Aug-2017 00:52
Since carbon is fundamental to life, occurring along with hydrogen in all organic compounds, the detection of such an isotope might form the basis for a method to establish the age of ancient materials. Libby, a Professor of Chemistry at the University of Chicago, predicted that a radioactive isotope of carbon, known as carbon-14, would be found to occur in nature. Radiocarbon decays slowly in a living organism, and the amount lost is continually replenished as long as the organism takes in air or food.Once the organism dies, however, it ceases to absorb carbon-14, so that the amount of the radiocarbon in its tissues steadily decreases.The currently accepted value for the half-life of will remain; a quarter will remain after 11,460 years; an eighth after 17,190 years; and so on.The equation relating rate constant to half-life for first order kinetics is $k = \dfrac \label$ so the rate constant is then $k = \dfrac = 1.21 \times 10^ \text^ \label$ and Equation $$\ref$$ can be rewritten as $N_t= N_o e^ \label$ or $t = \left(\dfrac \right) t_ = 8267 \ln \dfrac = 19035 \log_ \dfrac \;\;\; (\text) \label$ The sample is assumed to have originally had the same (rate of decay) of d/min.g (where d = disintegration).
Relative dating stems from the idea that something is younger or older relative to something else.
The abbreviation "BP", with the same meaning, has also been interpreted both of which requested that publications should use the unit "a" for year and reserve the term "BP" for radiocarbon estimations. A large quantity of contemporary oxalic acid dihydrate was prepared as NBS Standard Reference Material (SRM) 4990B. This value is defined as "modern carbon" referenced to AD 1950.
Some archaeologists use the lowercase letters bp, bc and ad as terminology for uncalibrated dates for these eras. Beginning in 1954, metrologists established 1950 as the origin year for the BP scale for use with radiocarbon dating, using a 1950-based reference sample of oxalic acid. Currie Lloyd: The problem was tackled by the international radiocarbon community in the late 1950s, in cooperation with the U. Radiocarbon measurements are compared to this modern carbon value, and expressed as "fraction of modern" (f M).
Once an organism is decoupled from these cycles (i.e., death), then the carbon-14 decays until essentially gone.
The half-life of a radioactive isotope (usually denoted by $$t_$$) is a more familiar concept than $$k$$ for radioactivity, so although Equation $$\ref$$ is expressed in terms of $$k$$, it is more usual to quote the value of $$t_$$."Radiocarbon ages" are calculated from f M using the exponential decay relation and the "Libby half-life" 5568 a.
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https://sde.whiletrue.live/dsa/graph-data-structure/ | 1,708,512,098,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473472.21/warc/CC-MAIN-20240221102433-20240221132433-00745.warc.gz | 520,116,789 | 10,044 | # Graph Data Structure
Graphs are powerful and versatile data structures that represent connections and relationships between entities. From social networks to transportation systems, graphs provide a flexible framework for modeling real-world scenarios.
## What is a Graph? #
• In simple terms, a graph is a collection of nodes, also known as vertices, connected by edges.
• The nodes represent entities, while the edges represent the relationships between those entities.
• Graphs are used to model a wide range of scenarios, where entities and their connections play a significant role.
## Properties of Graphs #
Let’s explore some fundamental properties of graphs:
Vertices (Nodes)
• Vertices are the fundamental building blocks of a graph.
• Each vertex represents an entity or an element within the system being modeled.
• Vertices can have attributes, such as labels or values, associated with them.
Edges
• Edges connect pairs of vertices and represent relationships or connections between them.
• They can be directed or undirected, indicating the directionality of the relationship.
Weighted Edges
• Some graphs have weighted edges, where each edge carries a numerical value or weight.
• These weights can represent factors like distance, cost, or strength of the relationship.
Degree
• The degree of a vertex in a graph is the number of edges incident to it.
• In a directed graph, we have separate in-degree and out-degree values.
Paths
• A path in a graph is a sequence of vertices connected by edges.
• It represents a route or a sequence of steps to reach from one vertex to another.
## Types of Graphs #
Graphs can be classified into different types based on their properties:
Undirected Graph
• In an undirected graph, the edges have no direction.
• The relationship between any two vertices is symmetric, and traversing the edges is bidirectional.
## Directed Graph (Digraph) #
• In a directed graph, also known as a digraph, the edges have a direction.
• The relationship between vertices is asymmetric, and traversing the edges follows the specified direction.
Weighted Graph
• A weighted graph is a graph where each edge has an associated weight or value.
• These weights can represent various factors, such as distances, costs, or probabilities.
Acyclic Graph
• An acyclic graph is a graph that contains no cycles.
• A cycle is a path that starts and ends at the same vertex, passing through distinct vertices.
## Applications of Graphs #
Graphs find extensive applications in numerous domains, including:
Social Networks
• Social networking platforms leverage graphs to model connections between users, allowing for friend recommendations, community detection, and analyzing the spread of information.
Routing and Network Analysis
• Graphs are widely used in network routing algorithms, where nodes represent network devices, and edges represent connections between them.
• Graph algorithms help optimize routing decisions and analyze network performance.
Web Page Ranking
• Search engines like Google employ graph-based algorithms, such as PageRank, to determine the relevance and ranking of web pages based on their link structure.
Recommendation Systems
• Graphs are used in recommendation systems to model user-item interactions and suggest relevant items based on connections and similarities.
Shortest Path Algorithms
• Graph algorithms like Dijkstra’s algorithm and Bellman-Ford algorithm help find the shortest paths between vertices, making them valuable in navigation systems and logistics planning.
## Conclusion #
• Graphs are powerful data structures that enable the modeling and analysis of complex relationships and connections.
• Their versatility allows us to represent and solve a wide range of real-world problems efficiently.
• By understanding the properties and types of graphs, as well as their applications, developers can harness the power of graphs to build intelligent systems and design innovative algorithms.
• So, whether you’re exploring social networks, optimizing routes, or building recommendation engines, a solid grasp of graphs will undoubtedly enhance your ability to tackle intricate challenges in the world of computer science. | 787 | 4,245 | {"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-2024-10 | latest | en | 0.950998 |
http://brainly.com/question/288331 | 1,477,670,058,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988722951.82/warc/CC-MAIN-20161020183842-00108-ip-10-171-6-4.ec2.internal.warc.gz | 36,285,892 | 10,047 | # Free help with homework
## Why join Brainly?
• find similar questions
# What is the y intercept of this line y+4=-4(x-2)
Rewrite this equation in standard form
What is the x intercept of this line
What is the equation in standard form of a perpendicular line that passes through (5,1)
What is the x intercept of the perpendicular line
What is the equation in standard form of a parallel line that passes through(0,-2)
On the parallel line find the ordered pair where x=-2
Any help is greatly appreciated
1
by Aliciacervantes | 127 | 530 | {"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} | 3.15625 | 3 | CC-MAIN-2016-44 | longest | en | 0.936593 |
https://python-forum.io/thread-32305.html | 1,627,905,916,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154320.56/warc/CC-MAIN-20210802110046-20210802140046-00008.warc.gz | 472,286,680 | 22,411 | ##### Generate Random operator, take user input and validate the user
Generate Random operator, take user input and validate the user mapypy Programmer named Tim Posts: 9 Threads: 6 Joined: Feb 2021 Reputation: Feb-02-2021, 02:56 PM Hi, I have the following code: ```import random number_one = random.randint(0, 100) number_two = random.randint(0, 100) rand_ops = ['+', '-', '/', '*'] while(True): correct_answer = eval (str(number_one) + random.choice(rand_ops) + str(number_two)) trial = input('What is the correct answer for:'+ str(number_one) +random.choice(rand_ops)+ str(number_two)+'=' ) if int(trial) != int(correct_answer): print('That is incorrect. Try again. :(') continue else: print('That is correct. Great job!'':)') break```This code does not recognize the correct answer. I would like to keep the structure of the code and keep it at a minimum to accomplish the following: Generate two random numbers between 0 and 100Generate a random operation ( +, -, /, * , ** ) Print the statement [based on random values]: e.g. random numbers are 10, 5 e.g. random operation is '-' it should print 10 - 5 = ?Get the input from the userValidate the answer Reply Posts: 7,567 Threads: 133 Joined: Sep 2016 Reputation: Feb-02-2021, 03:00 PM you choose operator twice. there is no guarantee the question and the "correct" answer are the same operation. As a side note - look at `operator` module, instead of using `eval` If you can't explain it to a six year old, you don't understand it yourself, Albert Einstein How to Ask Questions The Smart Way: link and another link Create MCV example Debug small programs Reply mapypy Programmer named Tim Posts: 9 Threads: 6 Joined: Feb 2021 Reputation: Feb-02-2021, 03:35 PM (Feb-02-2021, 03:00 PM)buran Wrote: you choose operator twice. there is no guarantee the question and the "correct" answer are the same operation. As a side note - look at `operator` module, instead of using `eval` What is your recommendation to standardize the operation in this case? Reply Posts: 7,567 Threads: 133 Joined: Sep 2016 Reputation: Feb-02-2021, 03:41 PM (Feb-02-2021, 03:35 PM)mapypy Wrote: What is your recommendation to standardize the operation in this case?select operator only once and assign it to variable. use that variable to ask the user and calculate the answer. Note, in this case using `eval` will not create risk, but if you have input from untrusted source - it could be a problem If you can't explain it to a six year old, you don't understand it yourself, Albert Einstein How to Ask Questions The Smart Way: link and another link Create MCV example Debug small programs Reply nilamo Last Thursdayist Posts: 3,443 Threads: 99 Joined: Sep 2016 Reputation: Feb-03-2021, 08:41 PM Instead of using `eval()`, using the operator module would look a little like this:```>>> import operator as op >>> import random >>> rand_ops = {"+": op.add, "-": op.sub, "/": op.truediv, "*": op.mul} >>> num_one = random.randint(0, 100) >>> num_two = random.randint(0, 100) >>> op_key = random.choice(list(rand_ops.keys())) >>> op_key '+' >>> print(f"what's {num_one} {op_key} {num_two} = ?") what's 37 + 1 = ? >>> answer = rand_ops[op_key](num_one, num_two) >>> answer 38``` mapypy likes this post Reply
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Announcement #3 8/6/2020 | 1,260 | 4,544 | {"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-2021-31 | latest | en | 0.84932 |
http://www.yourelectrichome.com/2011/05/transformer-inrush-current.html | 1,571,496,550,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986696339.42/warc/CC-MAIN-20191019141654-20191019165154-00558.warc.gz | 326,794,742 | 33,036 | ### Transformer Inrush Current
1. Definition
When transformers are energised (or when a transformer primary is switched on to the supply source), a transient inrush current up to 10-50 times the rated transformer current can flow and may last for several cycles. The worst condition exists when the transformer is switched on at or around the zero crossing of the voltage wave form. This inrush current can cause malfunctioning or unwanted tripping of the primary protective devices. The magnitude and duration of the inrush current depends on the following factors :
1) Size of the transformer bank
2) Size of the power system as well as the magnitude, and ratio of inductance to resistance of the system from source to the transformer bank
3) Type of iron used for the core
4) Previous history of the core (residual flux), initial recovery of the sympathetic flux wave,etc.
Hence it is very essential that a careful a careful analysis of the characteristic is made in selective the primary protective device for transformer. Table 1 and 2 give the relationship between the capacity of the transformers with the time constant of the inrush current and the ratio of inrush current to be the normal current for oil filled and cast resin type transformers respectively.
The relation between the primary current threshold setting and the time delay setting of the protective device is shown in the Fig. 1, where the y-axis represents the ratio of time delay setting to the inrush time constant and the x-axis represents the ratio of setting threshold of primary current to the inrush current.
Table 1 Inrush time constants for oil filled transformer
τinrush Ki =IPinrush / I1tr Transformer rating KVA Str 0.1 15 50 0.15 14 100 0.2 12 160 0.22 12 250 0.25 12 400-500 0.3 11 630 0.35 10 1000 0.4 9 1600 0.45 8 2000
Table 2 Inrush time constants for cast resin transformers
τinrush Ki =IPinrush / I1tr Transformer rating KVA Str 0.15 10.5 200 0.18 10.5 250 0.2 10 315 0.25 10 400-500 0.26 10 630 0.3 10 800-1000 0.35 10 1250 0.4 10 1600 0.4 9.5 2000
Where Str = rated capacity of the transformer - KVA
IPinrush = inrush current in primary winding - A
I1tr = primary rated current - A
Consider for example a 500 KVA oil filled transformer with a voltage ratio of 11 KV/433 V ans suitable for 50 Hz three phase operation.
From table 1
The ratio of inrush to rated primary current Ki = 12
Time constant of the inrush current = 0.25 sec
Full load rated current on the primary = 26.2 A
The inrush current = 12 x 26.2
= 314.4 A
Assuming a setting threshold for primary current I't of 35 A
I't/IPinrush = 35/314.4 = 0.111
From Fig 1, corresponding to I't/IPinrush = 0.111, the ratio tr /τinrush =1.8
Therefor,time delay setting = 1.8 x 0.25 = 0.45 seconds.
Fig. 1 Relation between threshold setting and time delay setting
Table 3 Fuse rating in the primary side of transformers (11 KV/433 V)
This is the minimum time delay required by the primary protective device to avoid unwanted tripping or malfunctioning of the protective device on the primary side, the fuse manufacturers provide information regarding the minimum and maximum size of the high voltage fuse ratings. Table 3 provides a typical data of the fuse ratings on the primary side of transformers.
2. Current inrush phenomenon ( Switching transient )
In the steady state operation, both V1 and Φ are sinusoidal and Φ lags V1 by 90° as shown in the Fig.2.
Fig. 2
When the primary voltage V1 is switched on to the transformer, the core flux and the exciting current undergo a transient before achieving the steady state. They pass through a transient period. The effect of transient is severe when voltage wave pass through origin.
In the inductive circuit flux can start with zero value. But the steady state value of flux at start is -Φm , as shown in the Fig. 2, at t = 0. Thus during transients a transient flux called off-set flux, Φt = Φm originates such that at t = 0, Φt ss is zero at the instant of switching. This transient flux Φt then decays according to circuit constants i.e. ratio L/R. This ratio is generally very small for transformers.
Thus during transients, the total flux goes through a maximum value of 2Φm. Such effect is called doubling effect. This is shown in the Fig.3.
Due to the doubling effect, core flux achieves a value of 2Φm due to which transformer draws a large exciting current. This is due to the fact that core goes into deep saturation region of magnetisation. Such a large exiting current can be as large as 100 times the normal exiting current. To withstand electromagnetic forces developed due to large current, the windings of transformer must be strongly braced. This large current drawn during transient is called inrush phenomenon.
Fig. 3 Doubling effect
Practically initial core flux can not be zero due to the residual flux Φ'r present, due to retentively property of core. The transient resultant flux goes through Φr = Φ'r + 2Φm and there is heavy inrush current in practice. The effect of transient is even severe in practice.
Such high transient current gradually decreases and finally acquires a steady state. It can last for several seconds. The transient flux Φt and exciting current both are unidirectional during transients. In steady state, exciting current becomes sinusoidal. The Fig. 4 shows the oscillogram of the inrush current.
Fig. 4 Waveform of inrush current
Related articles :
1. The bottom line is to bring electricity to the consumer safely. That requires the use of distribution and power transformers, along with necessary components. The major factors to be considered are power requirements, voltage specifications, topography, climate, and the sound levels.
Cast resin transformer
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4. hey your blog is seriously interesting and good i am impress.Cast Resin Transformer
5. Really i like your post it give me many ideas related to machinery.I love to read that.
Cast Resin Transformer
6. Nice blog! Its really an informative blog regarding electrical equipment as we are also manufacturer and distributor of electrical components.
Oil Immersed Transformer | Cast Resin Transformer
7. A transformer draws inrush current that can exceed saturation current at power up. The Inrush Current affects the magnetic property of the core. This happens even if the transformer has no load with its secondary open.
Power transformers in India | Transformer manufacturer in India
8. What is doubling effect in transformer core | 1,601 | 6,740 | {"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-2019-43 | latest | en | 0.838224 |
https://www.coursehero.com/file/11306627/84-Random-Block-ANOVA/ | 1,542,634,075,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039745762.76/warc/CC-MAIN-20181119130208-20181119152208-00085.warc.gz | 836,194,274 | 36,355 | 8.4 Random Block ANOVA
# 8.4 Random Block ANOVA - 6/1 lQlQ em Pvai 2125!”...
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Jill Tulane University ‘16, Course Hero Intern | 1,048 | 2,925 | {"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} | 2.53125 | 3 | CC-MAIN-2018-47 | latest | en | 0.529889 |
https://m.everything2.com/title/game+theory | 1,723,174,024,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640751424.48/warc/CC-MAIN-20240809013306-20240809043306-00457.warc.gz | 303,417,338 | 14,345 | Game Theory is based largely on the idea of John von Neumann. von Neumann wrote a book explaining his ideas in a publication entitled "Theory of Games and Economic Behaviour" which was co-authored with Oskar Morgernsten in 1934, shortly after he re-located to the United States. For more information on him I recommend reading Teiresias's excellent node, linked above.
Game Theory is an interesting branch of mathematics that attempts to study the decision making process of two or more players in a game. In this case a game is a conflict situation where the players are either attemping to fulfil different, mutually exclusive, objectives on the same system, or they are playing for control of a finite resource. Game Theory attempts to provide a mathematical structure for selecting an Optimum Strategy in the face of an opponent who will also be trying to play to an Optimum Strategy.
In Game Theory there are certain assumptions that must be made:
1. Each player has two or more choices at each stage of the game.
2. Every possible combination of plays leads to a defined end-state, either win, lose or draw.
3. Every end-state results in a pre-defined payoff for each player (such as gaining control over a certain amount of a resource). A Zero-Sum game is when, at a given end-state, the sum total of the payoffs to all the players is zero (i.e. the winnings are equal to the loses).
4. Each player has full knowledge of the game and all its possible end-states. This means that they know the payoffs both to themselves and their opponents at any given end-state.
5. All decisions are rational. A player will always choose the option that would result in a greater payoff to himself.
The last two points limit (to an extent) the usefullness of Game Theory in terms of its application to real-life situations. They have, however, been used to do research in economics and psycology.
There are several different types of game, but the most commonly seen type of game is an NxM game. This is any game where the outcomes can be plotted onto a Matrix of dimentions N by M, where N and M represtent the number of options that each player has available to him at that point.
Two common examples of NxM games are 'The Prisoners Dilemma' and 'Matching Pennies'
The "Prisoners Dilemma" is a NxM game of dimentions 2x2. In its simplest form it works thus: There are two prisoners who have been arrested by the police. If neither of them admit to the crime then they will both walk free. If they both admit to the crime then they will each get a sentence of five years. If one of them admits to the crime and the other denys it then the one who admitted the crime gets three years and the one who denied it gets nine years. What is the optimal strategy that they should follow?
This is best represented as a table:
``` B
| D | A |
---------- Where D = Deny, A = Admit
D|0/0|9/5| Outcomes are given as (A/B)
A ----------
A|5/9|3/3|
----------
```
From this you can easily see that while it is temping to deny the crime as this will give the greatest payoff, the strategy with the best payoff-to-risk ratio is for both sides to admit the crime. (Personal story - I played a game like this for a group of 10yr old kids. Even after playing five times niether side was willing to take the optimal strategy. Both sides kept playing the outcome where they both lost, rather than the less risky one where both sides would have won a little. This is why it doesn't work so well in real life)
The "Matching Pennies" game is apparently played by children. It is also a 2x2 games. Each player secretly turns a coin heads up or tails up. They then show the coins to each other. If they match then A has to pay B a penny. If they differ then B pays A a penny. This games continues untill they get bored.
As a table:
``` B
| H | T |
------------ Where H = Heads, A = Tails
H|-1/1|1/-1| Outcomes are given as (A/B)
A ------------
T|1/-1|-1/1|
------------
```
From this it is easy to see that there is no optimal strategy. In the long run it is best to play completely randomly as this means that both sides should come out even. Given that long term strategy this becomes a zero sum game.
A few more thoughts on game theory, intended to form an addition to elem_125's excellent write-up above.
It is important to understand that game theory applies beyond the bounds of what one would usually consider "games". Certainly game theory has something to say on games like Chess and Go, but it applies equally well to any system where parties are in potential competition. I say potential because game theory also handles ideas like the emergence of co-operation! You will perhaps now be getting a feeling for how broadly game theory is actually spread.
An example of a 'game theory' emerging in a situation which is certainly not considered a game by most affected would be in the recent firemen's strike here in the UK. This is certainly a game according to our earlier definition, having competing parties - firemen and the government - and a number of end results. Yet it is of the utmost seriousness! Note also that it needn't be a zero-sum 'game'; had it been resolved earlier, both parties might have come out of it better off. As it is, both parties will probably lose public favour.
It is these non-zero-sum games that most interested the great John Nash, subject of the film A Beautiful Mind. There's a fair amount about that great hero of game theory under his node, but within the topic of this one I'd like to point out just how far-reaching his ideas were. In nature, non-zero-sum games are just as common as zero-sum ones - unlike humanly constructed games, where we would usually consider it quite strange for both sides to lose! His work on Nash equilibrium had huge implications for an array of topics. To borrow a quote from that node, Nash equilibrium was described as "the most important idea in noncooperative game theory... whether analyzing election strategies, or causes of war."
That was the first point I wanted to make. Game theory is everywhere. It makes enough sweeping assumptions to satisfy any physicist, and yet it does have applications in the real world and provides us with much interesting mathematics to study.
The second thought that I think is important in relation to game theory is the idea of "solving" games. In this context, I use "game" to mean what is more traditionally considered a game - board games, card games et al. But to solve a game often has two meanings, and I will discuss both since they are both rather interesting.
Often we are satisfied that a game has been solved when a computer can be programmed to play it to a level comparable to the best humans. This is tricky for two reasons. Firstly, because people often find it difficult to articulate exactly what processes they use in choosing a next move, and this explicite explanation is obviously necessary when designing an algorithm. Also, a number of sub-concious checks are performed by humans that discard many theoretically possible moves without much consideration. These checks are more complex to translate to declarative heuristics.
A large number of games have been solved in this way. A simple example is chess. With Deeper Blue, a computer has finally beaten a Grand Master and the chess algorithm has been advanced to such a stage that it surpasses any human player. Quite how it got to that stage is a story in itself, and indeed Kasparov has claimed that the computer was given 'human assistance' at key points in the match. That's for another node, though.
On the other hand, there are games which have not been solved even in this fairly weak sense. Perhaps the classic example of this is Go. The best algorithms for Go can, I understand, still be soundly thrashed by a reasonably highly-ranked Go professional. There are a number of reasons for this weakness in the Go playing standards. As mentioned before, Go is often played on 'instinct' and players with much experience get a feel for good moves quite unconciously. Clearly, there is no such analogue when creating computer programs; further, such a player will find it hard to describe as a procedure the processes involved in selecting a good move. A furthur problem with Go is the fairly ill-defined goal. While the game is in motion, "Territory" is a rather fuzzy notion and, as any programmer will tell you, computers do not get on well with fuzziness. Chess, by constrast, has a very well defined goal - capture the king. Certainly, positional strength in chess is comparable in complexity to that of Go, but at least the end target in chess is somewhat better defined. Therefore chess lends itself, at least in this respect, to being made into a program - Go, however, does not. In addition, the space of possible outcomes is unimaginably large - a 19 x 19 board, with potentially hundreds of stones in play, cannot be over-run by brute processing power. Go does not succumb well to pure 'crunching power', which inevitably increases with each generation of computer - it needs a rather more subtle approach. If a Go program is to ever rival a professional human, many advances in our programming approach will have to come first.
Let us now look at the other sense in which a game is 'solved'. This sense is a much more rigorously defined one, and is therefore favoured by mathematicians, and it is also strongly related to Nash equilibrium. In this use, to solve a game means to define an algorithm that will play absolutely the best strategy all the time. This is often phrased as "an algorithm that cannot be beaten", and in a game devoid of probabilistic elements (as in chess) this is quite true. However, a perfect player of, to take an extreme example, Snakes and Ladders can certainly be beaten. That is not a reflection of the quality of the algorithm; it is an inherent flaw in trying to solve any game with random elements.
For most games, this is quite a huge undertaking. In the simple example of Tic Tac Toe, the space of possible outcomes is relatively small - but it is still a fairly difficult and certainly time-consuming task to devise a perfect strategy. Of course, Tic Tac Toe has long sinced been solved in this strictest of senses, but many other games - even those of perfect knowledge - have resisted attempts to do so.
Returning to the popular topic of chess, we are certainly nowhere near a perfect strategy. The space of possible outcomes is simply too large, and despite the smaller board, may be comparable to that of Go. There is still much argument among chess players as to the best first move, let alone the several hundred that frequently follow. Even simplified versions of chess - discounting complicating technicalities such as en passant and promotion of pieces - are far beyond the computative power available today. Such deep-rooted uncertainty as to 'best strategies' is undoubtedly part of the appeal of chess, and explains how professionals of similar skill can play in vastly different ways.
That said, other games have been solved. To give a recent example, scientists published just this year a solution to the African game Bantumi. This was done in what I consider to be a rather inelegant fashion - in short, the researchers compiled a 17GB database of all possible game scenarios and the best move to follow them. I don't doubt that creating a program to discover these 'best moves' required much skill and effort, but it is perhaps symptomatic of the future of mathematics - likewise the solution to the four colour map problem. Nevertheless, Bantumi has been solved and we know it is possible to create an unbeatable algorithm - genuinly unbeatable in this case, Bantumi being a game of perfect knowledge.
There is one final aspect to this game-solving business. As was recently published in the New Scientist, it has been mathematically proven that Tetris is a 'hard' game. That is to say, even knowing the pieces to be given in advance, there is no algorithm to play the game perfectly. Brings some respite to those of us whose Tetris skills leave much to be desired.
So there we have it. Games that can be solved - Bantumi. Games that can't be solved - Tetris. And finally, games which are as yet unallocated to either category - Chess.
I hope this has broadened your mind of the fascinating topic of game theory. That is all.
Well almost. A quick thanks to JerboaKolinowski, who raised some excellent points about my slightly unfair comparison between chess and Go. That probably arose since I am much less familiar with Go than chess, but I've tried to embrace his insight without having to completely re-write this whole node.
Log in or register to write something here or to contact authors. | 2,740 | 12,783 | {"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} | 3.265625 | 3 | CC-MAIN-2024-33 | latest | en | 0.967531 |
https://whatisconvert.com/322-kilometers-hour-in-knots | 1,586,300,921,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585371806302.78/warc/CC-MAIN-20200407214925-20200408005425-00187.warc.gz | 777,440,505 | 7,072 | # What is 322 Kilometers/Hour in Knots?
## Convert 322 Kilometers/Hour to Knots
To calculate 322 Kilometers/Hour to the corresponding value in Knots, multiply the quantity in Kilometers/Hour by 0.53995680345662 (conversion factor). In this case we should multiply 322 Kilometers/Hour by 0.53995680345662 to get the equivalent result in Knots:
322 Kilometers/Hour x 0.53995680345662 = 173.86609071303 Knots
322 Kilometers/Hour is equivalent to 173.86609071303 Knots.
## How to convert from Kilometers/Hour to Knots
The conversion factor from Kilometers/Hour to Knots is 0.53995680345662. To find out how many Kilometers/Hour in Knots, multiply by the conversion factor or use the Velocity converter above. Three hundred twenty-two Kilometers/Hour is equivalent to one hundred seventy-three point eight six six Knots.
## Definition of Kilometer/Hour
The kilometre per hour (American English: kilometer per hour) is a unit of speed, expressing the number of kilometres travelled in one hour. The unit symbol is km/h. Worldwide, it is the most commonly used unit of speed on road signs and car speedometers. Although the metre was formally defined in 1799, the term "kilometres per hour" did not come into immediate use – the myriametre (10,000 metres) and myriametre per hour were preferred to kilometres and kilometres per hour.
## Definition of Knot
The knot is a unit of speed equal to one nautical mile (1.852 km) per hour, approximately 1.151 mph. The ISO Standard symbol for the knot is kn. The same symbol is preferred by the IEEE; kt is also common. The knot is a non-SI unit that is "accepted for use with the SI". Worldwide, the knot is used in meteorology, and in maritime and air navigation—for example, a vessel travelling at 1 knot along a meridian travels approximately one minute of geographic latitude in one hour. Etymologically, the term derives from counting the number of knots in the line that unspooled from the reel of a chip log in a specific time.
### Using the Kilometers/Hour to Knots converter you can get answers to questions like the following:
• How many Knots are in 322 Kilometers/Hour?
• 322 Kilometers/Hour is equal to how many Knots?
• How to convert 322 Kilometers/Hour to Knots?
• How many is 322 Kilometers/Hour in Knots?
• What is 322 Kilometers/Hour in Knots?
• How much is 322 Kilometers/Hour in Knots?
• How many kt are in 322 km/h?
• 322 km/h is equal to how many kt?
• How to convert 322 km/h to kt?
• How many is 322 km/h in kt?
• What is 322 km/h in kt?
• How much is 322 km/h in kt? | 655 | 2,540 | {"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} | 3.46875 | 3 | CC-MAIN-2020-16 | latest | en | 0.882441 |
https://www.parkerslegacy.com/what-is-a-good-bat-speed-for-a-9-year-old/ | 1,725,768,103,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650958.30/warc/CC-MAIN-20240908020844-20240908050844-00058.warc.gz | 882,597,904 | 5,926 | What is a good bat speed for a 9 year old?
40-50 mph
Is it better to use a heavy or light bat?
If a player can maintain the same bat swing speed with a heavier bat, the heavier bat will produce higher batted ball velocity and an increase in distance. But, any player who has experimented swinging bats with widely different weights knows that it is easier to swing a light bat than a heavier bat.
How do you size a bat?
Bat length is measured in inches from knob to end cap. A longer bat gives you greater reach, allowing you to hit balls on the outside part of the plate. However, longer bats also tend to have more mass towards the end of the bat that requires more power to swing them.
How do I pick the right bat?
Position the bottom of the bat in the center of your chest, facing outward. If your arm can reach out and grab the barrel of the bat, then it is the correct length. Stand the bat up against the side of your leg. If the end of the bat reaches the center of your palm when you reach down, it's the appropriate length.
What is the average bat speed for a 10-year-old?
Swing Speed Depends on Bat Weight In a 1991 experiment published in “New Scientist,” researchers at the University of Arizona found that Little Leaguers ranging in age from 10 to 12 years bat an average of about 60 mph with a 10-ounce bat, 40 mph with a 20-ounce bat and 30 mph with a 30-ounce bat.Dec 5, 2018
Should I train with a heavier bat?
Practicing with a heavier bat significantly slows down the velocity of the bat head—depriving the batter of slugging power, exercise researchers at California State University, Fullerton, say. Both were far quicker than hacks with the heavy bat, which averaged just under 67.6 kilometers per hour.
Does a lighter bat hit farther?
A heavier bat will hit a ball farther than a lighter bat, when the speed of the bat swing, the pitch speed and the ball mass are kept constant. Increasing the mass of the bat gives the ball more momentum.
What is the average bat velocity by age?
The average bat speed for a 13-year-old is approximately 55-60 mph, while the average bat speed for 15-year-olds is 60-70 mph. Once you make the jump to college and the pros the minimum average speed is around 65-70 with an upper bat speed limit in the mid-80s for most players, with a few venturing into the low 90s.
Does the weight of a baseball bat affect how far the ball goes when it is hit?
The answer is yes the heavier the weight of the bat the farther the bat will go if everything above is also in line. Mass + Speed=Power its physics. However the heavier the bat the slower the speed the batter can swing it. So focus on swing mechanics and bat speed drills to improve your hitting.
What sizebat should 10-year-old use?
The right bat size for a 10-year-old, as determined by usage, is a 29 or 30-inch drop 10 or 11 bat. The drop is the numerical difference between the length of the bat in inches and the weight in ounces. 29 and 30-inch bats in the 18 to 20-ounce weight range make up more than 90% of total 10U bat usage.Oct 6, 2021
What is a good high school bat speed?
After learning and practicing the power hitting mechanics described in this book, most hitters will fall in a range from the low to mid-seventies. Elite high school hitters will eventually enjoy bat speeds 80 miles per hour and greater. Most college baseball hitters fall in a range between 70 and 80 mph.Dec 5, 2017
How do you know whatsizebat a kid needs?
- Children under 60 pounds should swing a bat between 26 and 29 inches long. - Children weighing more than 70 pounds should swing a bat ranging from 28 to 32 inches long.
What is good bat hand speed?
High school, mid-50s to mid 70s. College and pro, mid-60s to maybe mid-80s. Those are the ranges and averages, but there are a few factors, including the bat size and age, that would affect swing speed.”
What is the average bat speed of a MLB player?
Average MLB fastball speed is 91 mph out of the hand, and 83 mph at the plate. Example: MLB average exit speed is 103 mph, bat speed ranges roughly from 70-85 mph.
Is it better to have a heavy or light softball bat?
As a general rule, bigger, stronger players usually prefer a heavier bat for maximum power. Smaller players usually benefit from a lighter bat that allows greater bat speed.
What does a heavier bat do?
Bat Weight and Batted Ball Velocity The data shows that a heavier bat produces a faster batted ball speed. This makes intuitive sense since a heavier bat brings more momentum into the collision. Doubling the mass of the bat results in an increase of almost 12mph.
How do you size a batfor a kid?
Choosing the Correct Weight Youth Bat: Weigh Him/Her In general: Children under 60 pounds should swing a bat between 26 and 29 inches long. Children weighing more than 70 pounds should swing a bat ranging from 28 to 32 inches long.Apr 7, 2014
What is a good bat exit velocity?
For point of reference, the average major league Exit Velocity Per At Bat is about 68 mph.
What is a good catcher Velo?
High School Max-velocity on the low side is in the low 60's. Good for HS would be low-mid 70's; excellent would be mid to upper 70's. In the College group, the low side is 70-74; good would fall in 75-79, excellent would be 80-84, 85+ would be a serious prospect as far as velocity goes.
29 inch bat
Is it better to have a light or heavy cricket bat?
If a light bat was swung at the same speed as a heavy bat and both hit the same ball, the heavy bat would pack more power since it has more energy and more momentum. But light bats can be swung 10% faster. The end result is that heavy bats are about 1% more powerful than light bats. | 1,353 | 5,682 | {"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-2024-38 | latest | en | 0.931315 |
https://oeis.org/A108389 | 1,582,481,696,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875145818.81/warc/CC-MAIN-20200223154628-20200223184628-00051.warc.gz | 486,181,681 | 4,465 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A108389 Transmutable primes with four distinct digits. 3
133999337137, 139779933779, 173139331177, 173399913979, 177793993177, 179993739971, 391331737931, 771319973999, 917377131371, 933971311913, 997331911711, 1191777377177, 9311933973733, 9979333919939 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,1 COMMENTS This sequence is a subsequence of A108386 and of A108388. See the latter for the definition of transmutable primes and many more comments. Are any terms here doubly-transmutable also; i.e., terms of A108387? Palindromic too? Terms also of some other sequences cross-referenced below? a(7)=771319973999 is also a reversible prime (emirp). a(12)=9311933973733 also has the property that simultaneously removing all its 1's (93933973733), all its 3s (9119977) and all its 9s (3113373733) result in primes (but removing all 7s gives 93119339333=43*47*59*83*97^2, so a(12) is not also a term of A057876). Any additional terms have 14 or more digits. LINKS EXAMPLE a(0)=133999337137 is the smallest transmutable prime with four distinct digits (1,3,7,9): exchanging all 1's and 3's: 133999337137 ==> 311999117317 (prime), exchanging all 1's and 7's: 133999337137 ==> 733999331731 (prime), exchanging all 1's and 9's: 133999337137 ==> 933111337937 (prime), exchanging all 3's and 7's: 133999337137 ==> 177999773173 (prime), exchanging all 3's and 9's: 133999337137 ==> 199333997197 (prime) and exchanging all 7's and 9's: 133999337137 ==> 133777339139 (prime). No smaller prime with four distinct digits transmutes into six other primes. CROSSREFS Cf. A108386 (Primes p such that p's set of distinct digits is {1, 3, 7, 9}), A108388 (transmutable primes), A083983 (transmutable primes with two distinct digits), A108387 (doubly-transmutable primes), A006567 (reversible primes), A002385 (palindromic primes), A068652 (every cyclic permutation is prime), A107845 (transposable-digit primes), A003459 (absolute primes), A057876 (droppable-digit primes). Sequence in context: A072718 A034652 A015402 * A172561 A172600 A172704 Adjacent sequences: A108386 A108387 A108388 * A108390 A108391 A108392 KEYWORD base,hard,nonn AUTHOR Rick L. Shepherd, Jun 02 2005 STATUS approved
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Last modified February 23 13:14 EST 2020. Contains 332159 sequences. (Running on oeis4.) | 816 | 2,718 | {"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-2020-10 | latest | en | 0.816818 |
https://www.codingninjas.com/studio/problems/encode-the-message_699836 | 1,708,484,258,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473360.9/warc/CC-MAIN-20240221002544-20240221032544-00644.warc.gz | 783,744,639 | 42,412 | Topics
# Encode the Message
Easy
0/40
Average time to solve is 18m
## Problem statement
You have been given a text message. You have to return the Run-length Encoding of the given message.
Run-length encoding is a fast and simple method of encoding strings. The basic idea is to represent repeated successive characters as the character and a single count. For example, the string "aaaabbbccdaa" would be encoded as "a4b3c2d1a2".
Detailed explanation ( Input/output format, Notes, Images )
Constraints :
``````1 <= T <= 10
1 <= N <= 100000
Where 'N' is the length of the message string.
Time Limit: 1 sec
``````
Sample Input 1 :
``````3
aabbc
abcd
abbdcaas
``````
Sample Output 1 :
``````a2b2c1
a1b1c1d1
a1b2d1c1a2s1
``````
Explaination for Sample Input 1:
``````Test Case 1: As 2 consecutive 'a', 2 consecutive 'b', and 1 'c' are present in the given string so output is "a2b2c1".
Test Case 2: As 1 consecutive 'a', 1 consecutive 'b', 1 consecutive 'c' and 1 consecutive 'd' are present in the given string so output is "a1b1c1d1".
Test Case 3: As 1 consecutive 'a', 2 consecutive 'b', 1 consecutive 'd', 1consecutive 'c', 2 consecutive 'a' and 1 consecutive 's' are present in the given string so output is "a1b2d1c1a2s1".
``````
Sample Input 2:
``````2
``````s1a1d1a1s1d1 | 421 | 1,284 | {"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} | 2.6875 | 3 | CC-MAIN-2024-10 | latest | en | 0.872886 |
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Home » Countries » India » NFHS4: age at first intercourse, age at first cohabitation, age at first marriage (NFHS4: age at first intercourse, age at first cohabitation, age at first marriage)
NFHS4: age at first intercourse, age at first cohabitation, age at first marriage Mon, 27 July 2020 12:24
Rojin S. Messages: 2Registered: July 2020 Member
Dear all,
I am currently working on the timing of the following events with NFHS 4: age at first intercourse, age at first cohabitation, age at first marriage.
For age at first intercourse, I am using the variable v531.
But when it comes to age at first cohabitation/marriage, I'm not sure which variable I should use.
- variable v511 gives age at first cohabitation
- the variable s314c gives us the age at the first union (calculated) and its values are similar to those of the variable v511 (but not quite -> ex: 8 years)
- there is also variable s308y which gives us the year of first marriage and variable s309 which completes it (if I understood correctly) by giving the age of persons who did not know the year of their marriage
My questions are the following:
1) Do the variables v511 and s314c give us the same information? If so, are the first cohabitation and the first union considered to be exactly the same event? If so, how can we explain the differences between the two variables?
2) Are variables v511 and s314c related to variables s308y and s309? If not, which one is preferable to use for age at first marriage/union?
3) For a first sexual intercourse occurring in the same year as a first cohabitation/marriage, can one deduce which occurs first?
4) How much do you estimate the prevalence of an age at first intercourse lower than an age at first cohabitation?
Thank you very much for your help!
Re: NFHS4: age at first intercourse, age at first cohabitation, age at first marriage [message #19905 is a reply to message #19659] Wed, 26 August 2020 15:46
Trevor-DHS Messages: 721Registered: January 2013 Senior Member
1) V511 and s314c are the same except that in s314c there are cases included (incorrectly) for women for whom Gauna was not performed. Use v511.
2) V511 and S314 come from questions 313 and 314 in the questionnaire and are about the date and age of first union or cohabitation. S308Y and S309 are from questions 308 and 309 in the questionnaire and are about the date and age of first marriage. Formal marriage can come years before union/cohabitation, and may happen at very young ages of childhood. Thus we use the age and date of first union/cohabitation as the main variable that we use.
3) No, we can't tell which came first of first sexual intercourse occurring in the same year as a first cohabitation/marriage
4) You can compare the age at first sex in years with the age at first cohabitation, but that will only tell you if there was a difference in reported age, not which came first if both are reported at the same age.
Your questions highlighted an issue in the recode datasets. Variables V509, V511 and S314C should include all women ever in union including those that are now widowed, divorced, separated or divorced. However the variables currently do not include those formerly in union, and at present are limited only to those currently married . V509 and S314C do, though, incorrectly include cases reported as Married, Gauna not performed - these latter cases should be excluded from any analysis. This same issue exists in the men's recode dataset too.
Re: NFHS4: age at first intercourse, age at first cohabitation, age at first marriage [message #19957 is a reply to message #19905] Tue, 08 September 2020 03:33
Rojin S. Messages: 2Registered: July 2020 Member
Thank you very much for those answers! These data are much clearer now, especially for those that incorrectly include gauna. It's a pity that variables V509, V511 and S314C contain only currently married women. Will this be corrected in the updates? Thank you again.
Re: NFHS4: age at first intercourse, age at first cohabitation, age at first marriage [message #19960 is a reply to message #19957] Tue, 08 September 2020 09:49
Trevor-DHS Messages: 721Registered: January 2013 Senior Member
These will be corrected in an update in the future.
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Current Time: Wed Sep 23 19:45:48 Eastern Daylight Time 2020 | 1,085 | 4,470 | {"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-2020-40 | latest | en | 0.921649 |
https://www.physicsforums.com/threads/functional-derivative.412367/ | 1,508,421,021,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187823284.50/warc/CC-MAIN-20171019122155-20171019142155-00495.warc.gz | 980,583,128 | 13,809 | # Functional derivative
1. Jun 25, 2010
### alicexigao
For my current research, I need to prove the following:
$$\int_0^1 \frac{dC(q(x) + k'(q'(x) - q(x)))}{dk'}\,dk' = \int_0^1 \int_L^U p(q(x) + k(q'(x) - q(x)))(q'(x)-q(x)) dx dk$$
where $$C(q(x)) = \int_0^1 \int_L^U p(kq(x)) q(x)\,dx\,dk$$
Here's what I've tried using the definition of functional derivative:
$$\frac{\partial C(q(x))}{\partial q(x)}$$
$$= \lim_{\delta q(x) \rightarrow 0} \frac{C[q(x) + \delta q(x)] - C[q(x)]}{\delta q(x)}$$
$$= \int_L^U \int_0^1 \frac{\partial p(kq(x))}{\partial q(x)}q(x) + p(kq(x)) dk dx$$
My guess is that
$$\frac{dC(q(x) + k'(q'(x) - q(x)))}{dk'} = \frac{\partial C(q(x) + k'(q'(x) - q(x)))}{\partial (q(x) + k'(q'(x) - q(x)))} \frac{d(q(x) + k'(q'(x) - q(x)))}{dk'}$$
but I'm not sure what to do next. Any help will be greatly appreciated!
Thank you very much!
Alice
Last edited: Jun 25, 2010
2. Aug 17, 2010
### ross_tang
Hello, alicexigao. I can't really help you prove the identity. But I think it maybe able to evaluate to something more simple.
$$\int _0^1\frac{d C(q(x)+k'(q'(x)-q(x)))}{d k'}d k'$$
$$=\int _0^1d C(q(x)+k'(q'(x)-q(x)))$$
$$=[ C(q(x)+k'(q'(x)-q(x)))]_0^1$$
$$=C(q(x)+(q'(x)-q(x)))-C(q(x))$$
$$=C(q'(x))-C(q(x))$$
Now, you can do the substitution and continue. | 543 | 1,298 | {"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": 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} | 3.796875 | 4 | CC-MAIN-2017-43 | longest | en | 0.736902 |
https://astronomy.stackexchange.com/questions/38602/is-the-mean-orbit-centered-on-ellipse-center-or-ellipse-focus | 1,726,808,883,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700652130.6/warc/CC-MAIN-20240920022257-20240920052257-00834.warc.gz | 92,522,930 | 40,588 | # Is the "mean orbit" centered on ellipse center or ellipse focus?
Mean longitude and mean anomaly of a body are calculated as the angle covered by the body along a fictious circular orbit, and this is ok.
But where is this orbit centered? Five different sources say five different things:
Wikipedia: circle center = ellipse focus
University of Texas: who knows?!? No ellipse at all:
University of California: who knows? Mean anomaly mentioned but not drawn; only eclipse shown, no circle:
Britannica Enciclopedia: who knows? Both circle and ellipse shown, but mean anomaly not shown:
Max-Planck-Institut für Sonnensystemforschung: circle centered on ellipse center
Theory & formulas: longitudes and anomalies summed up together, no pictures at all.
I am very confused.
• Fitzpatrick@UTexas illustrates true and eccentric anomalies in a diagram here. Commented Aug 25, 2020 at 15:13
• The Wikipedia illustration has scale problems, and probably should have been removed from the Mean Anomaly article years ago. Commented Aug 25, 2020 at 15:18
• @notovny If the image on Wikipedia is wrong, anybody (also you) can remove/fix it. Commented Aug 26, 2020 at 7:24
But where is this orbit centered?
The true anomaly is the angle as measured from the central body between periapsis passage and the object's current location. The orbit is an ellipse with one of the two foci at the central body. This concept is central to Kepler's laws and Newtonian mechanics.
The eccentric anomaly is the angle as measured from the center of the ellipse between periapsis passage and the projection along the minor axis of the point in question onto the minimal bounding circle that encompasses the elliptical orbit. That minimal bounding circle is necessarily centered on the center of the orbit rather than the central body.
Mean anomaly is a fictitious angle. Unlike true anomaly and eccentric anomaly, mean anomaly does not indicate where the object is. Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body with the same semi-major axis length as the object in question would have passed through in the time since periapsis passage.
Aside: The image from wikipedia is flat-out wrong. The center is correct, but the size is incorrect. The circular orbit should intersect the ellipse, twice. It should be outside of the ellipse at periapsis and inside the ellipse at apoapsis.
• This answer does not answer. Question was: where is the fictitious orbit centered? I was not asking for definitions of the quantities. All sites say "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body", but none says where the circular orbit is centered. Commented Aug 26, 2020 at 7:19
• @jumpjack - I wrote "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body". I can update the answer if it isn't clear to you from what I already wrote that the central body is the center of the circular orbit. Commented Aug 26, 2020 at 7:24
• I'd write "circular orbit centered on ellipse focus". "Centered on central body" is not so helpful. Commented Aug 26, 2020 at 7:39
• @jumpjack - An ellipse has two foci. Writing "centered on the ellipse focus" is ambiguous because ellipses have two foci, so I am not changing my answer to your suggestion. The central body is in fact one of the two foci of a Keplerian ellipse. The other focus has no physical meaning. Commented Aug 26, 2020 at 7:44
Fortunately all diagrams in the question are consistent in these respects:
• the true anomaly $$\nu$$ or $$f$$ is measured around the focus at the central body
• the eccentric anomaly $$E$$ is measured around the center of the ellipse
• both are zero at the periapsis
The mean anomaly $$M$$ is a pseudo-angular quantity useful in computing $$E$$ and $$\nu$$, with the same zero and period but increasing linearly with time. $$M$$ is not a geometric angle around an actual point. There is no right place to draw it.
You could represent $$M$$ with a circle around the center of the ellipse, around the focus at the central body, or even around the other focus; any such choice is arbitrary. Better yet, leave it out of the diagram to avoid suggesting a geometric meaning where there isn't one.
• After looking at Kepler equation, I think we can say that M is centered on ellipse center: M = E - e sin(E) , being E centered on ellipse center. This means that the fictitious orbit is a circle centered on ellipse center. Given that the major semiaxis is the reference parameter of the ellipse, we could assume that this would be the radius of the fictitious orbit for e=0, hence such orbit is a circle tangent to the ellipse. Hence in the end the Max-Planck-Institut für Sonnensystemforschung picture is the right one. "Centered on ellipse center" has also geometrical sense. Commented Aug 26, 2020 at 16:51
• @jumpjack M = E - e sin E expresses a numerical relation, not a geometric one. The geometric center of M is undefined. If you must define one anyway, remember that you made it up. Commented Aug 26, 2020 at 18:46
• yes but what is confusing is that M = E - e sin E is a numerical relation, but for M is given a geometrical definition (the angle that IN THEORY a body would cover if it orbit was elliptic), and I have also formulas which sum up an angle centered somewhere to an angle centered (apparently) somewhere else ( aa.quae.nl/en/reken/zonpositie.html#5 ) , and to a further angle (capital $\pi$) centered on same center of both the ellipse and the circle... Commented Aug 27, 2020 at 10:37
• @jumpjack The quae.nl article's $\Pi$ is the heliocentric longitude of the perihelion, reckoned from the ellipse focus where the Sun is. Their equation (7) for $\lambda$ assumes that the orbital inclination $i$ is negligible. Commented Aug 27, 2020 at 13:48 | 1,374 | 5,916 | {"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": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2024-38 | latest | en | 0.881151 |
https://betterlesson.com/lesson/507211/towers-towers-towers-1-5?from=mtp_lesson | 1,544,882,688,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376826856.91/warc/CC-MAIN-20181215131038-20181215153038-00062.warc.gz | 536,084,631 | 22,481 | # Towers, Towers, Towers 1 - 5
7 teachers like this lesson
Print Lesson
## Objective
I can show I understand numbers 1 - 5 by building towers with that many blocks. I understand that when I count, each number is one more than the one before.
#### Big Idea
Children develop a true sense of number by working with real things. In this lesson, kindergarteners use blocks to build towers, learning what numbers 1-5 mean.
## Calendar & Daily Counting Practice
20 minutes
Each day we begin our math block with an interactive online calendar followed by counting songs and videos.
Calendar Time:
My class does calendar on Starfall.com. This website has free reading and math resources for primary teachers. It also has a “more” option that requires paying a yearly fee. The calendar use is free and the procedure is described in detail in the resources.
Counting with online sources:
We do daily counting practice to reinforce the counting skills. In the first two to three weeks of school, we watch two to three number recognition 0-10 videos (one to two minutes each) until all students can identify numbers correctly in random order. Depending on time, we may watch "Shawn the Train" and count objects with him. I may also choose to rotate songs, videos and counting depending on time and skill needs. As the students become more proficient at counting and number identification, I begin to add additional skills such as counting to 20 forward and back, counting by tens to 100 and counting to 100 by ones.
## Review
5 minutes
For this activity, we review our Bubble Maps of numbers 0-5.
We simply view them and read through each of them which leads us to thinking about counting, what numbers represent (quantity) and that gets us thinking about the visual representation of the numbers 1-5.
## Activity
20 minutes
One of the Common Core standards for kindergarten is understanding that when counting by ones, each number is one greater than the number before it and one less than the number after it. This activity helps develop the conceptual understanding of that concept by making it visual. This is what the activity looks like:
First I show the students the tower cards and explain how they will be used.
Next I demonstrate how to create the towers and I think out loud telling how I build each one. Each tower has one more cube than the one before.
Me: How many cubes are in my first tower?
Kids: One!
Me: I'm going to build a tower of two. (I build the tower) Look, this tower is one cube taller than the first tower I built. So that means 2 is one more than 1. That would make sense because that's how I count, 1, 2.
I continue with that pattern for all five towers. Then I put my towers in order and push them together into a staircase and say, "Wow! this looks like stairs. I can see that each tower is one greater than the tower before it." Then we count the towers together.
Me: This is how I check my work. If any of the "steps" were more or less than one more, I have to recount that tower.
Now it's your turn to build number towers 1-5.
**I roam the room while the kids build the towers. If I see someone with an incorrect set of towers, I have them push the towers together in order and I walk through it with them making sure each tower is one greater than the tower before it.
## Closure
5 minutes
I bring the students back together on the floor and have volunteers share their experience with this activity. I ask how they feel about it, if they feel successful and what they learned. We draw a picture of the steps (towers) touching to clarify the expectation that each tower they built should have been one more than the one before if they followed directions and built them in consecutive order as asked. The kids really grasp this concept and are very successful with this activity.
## Exit Ticket
5 minutes
The towers themselves are the exit ticket. I ask the students to save their towers and raise their hands as they finish. I walk around with my clipboard and provide feedback to the kids. I note on my observation notes who was successful and who struggled and in what way they struggled. I pull the strugglers to the floor the next day while the rest of the class attempt towers 6-10. | 917 | 4,248 | {"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} | 4.375 | 4 | CC-MAIN-2018-51 | longest | en | 0.942426 |
https://recording.org/threads/room-ratios-in-an-asymmetrical-room.34791/ | 1,516,284,976,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887414.4/warc/CC-MAIN-20180118131245-20180118151245-00625.warc.gz | 839,551,534 | 12,593 | # Room Ratios in an Asymmetrical Room
Discussion in 'Microphones (live or studio)' started by quiet, May 21, 2008.
Tags:
1. ### quietGuest
I'm new, here. And, yes, I've read the post with guidelines so as not to waste your time, which I am all in favor of. Can't seem to find anyone who's had exactly my question.
I live in Berkeley, Ca.
I am beginning a design to turn my 2-car garage into a rehearsal space & I am avidly reading Rod Gervais' book. My goal is a rehearsal space for jazz ensembles that will not annoy the neighbors in my residential neighborhood. Amplified music, but not quite rock 'n roll levels. I will start construction this summer.
I would give you more info on noise levels, proximity to neighbors, but it's not germane to my rather basic and preliminary questions i.e. shape of the room: I am building room-in-a-room, and I have some choices as to shape.
If I design symmetrically (which I don't believe I should) I have a space to carve up that is 16 feet by 20 feet, with an 8 foot ceiling. This is your classic, problematic small room, it seems to me.
Rod says (page 21) "Walls should be out of parallel by at least 1 in 10, or 6%." I plan to angle one of the walls at least this much.
My question is how to figure this in to the room ratios Rod offers on page 29 courtesy of Sepmeyer and Louden. The ratios are for H, W, and length. I will have four numbers if I cant one of walls. My plan is to cant one of the shorter walls by 6 to 10 percent. Should I just average the number for "Width?" They are pretty specific ratios that are being offered.
A related question: Rod actually says "Facing walls should be out of parallel ..." Is canting one wall - which I am proposing here - advisable? I'd rather not put both walls out of parallel for reasons I won't bore you with. But I could.
I can simply build for maximum square footage & deal with the outcome later, but I thought I'd try to apply some chapter 2 wisdom (Modes, Nodes, and other Terms of Confusion").
Just to - possibly - anticipate a response (or question), I don't have too many options within my small budget to tilt, vault, or recess the ceiling. I'm exploring skylights, but recessing them seems to me to complicate things in expensive ways from an engineering perspective if I'm to maintain the integrity of the room-within-a-room design.
Thanks !
Clark Suprynowicz
Berkeley, Ca.
Joined:
Mar 18, 2001
Location:
Sunny & warm NC
Clark,
I'd suggest you pop down a couple of floors to the and ask your questions. You'll likely get some good answers from Rod and several others who frequent the forum.
Just a couple of preliminary things...
Whatever you do, get your permits. Too many folks fail to, and if you go to sell the house, have an accident, or God forbid; a fire or anything where your insurance company or the city gets involved... you'll likely find yourself in serious piles of poo.
Next would be to ask how many rooms you are planning on. If it's at least two, e.g. tracking room and control room, then you're design considerations are kinda' misconceived. The control room should be as symmetrical as possible, and the tracking room as asymmetrical as possible. Thankfully, in normal practice this works out as a pretty natural occurrence by following the contours of the control room.
As far as figuring the ratio into distances... I used a std spreadsheet. Since you are dealing with an existing structure, you are pretty well limited to some maximum height. So, I would start with the ceiling height as a constant and calculate backwards to get the width and length.
About canting or angling walls. You are correct that you typically use an average when calculating some of these numbers.
Lastly, realize that to do this "right" (especially on a tight budget), you don't want to rush in and do this without a really thorough plan. I started to do that here, but I'm glad I actually held off and got a proper design and set of prints. However, I would also say that your process should certainly not take as long as mine has. (Oh brother, you just don't know) The one thing I can tell you is... whatever your budget is... unless you can buy all your materials at the same time and have room to store it... you're gonna have cost overruns... and without a good solid set of prints, it's liable to get UGLY in overruns.
Come on down to the acoustics forum!
Max
Corrected asymmetrical reference to tracking room[/edit]
3. ### RobakActive Member
Joined:
Mar 10, 2006
Location:
Poland
Joined:
Mar 18, 2001
Location:
Sunny & warm NC
Yeah... dat be me...
Oh man... Just noticed a SERIOUS typo error....
The control room should be as SYMMETRICAL as possible, and the tracking room as ASYMMETRICAL as possible.
(That's what I get for doing copy and paste with words that I commonly miss-spel)
Sorry for any confusion...
5. ### quietGuest
Thanks, guys. I've gone to the forum on acoustics, which you recommended. Appreciate the guidance
Clark | 1,172 | 4,976 | {"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-2018-05 | latest | en | 0.96757 |
http://roboblockly.org/curriculum/projects/spaceexploration/1.php | 1,653,551,927,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662604495.84/warc/CC-MAIN-20220526065603-20220526095603-00612.warc.gz | 43,996,064 | 33,571 | ### Learning Math and Coding with Robots
Grid: Tics Lines: Width px Hash Lines: Width px Labels: Font px Trace Lines: Robot 1: Width px Robot 2: Width px Robot 3: Width px Robot 4: Width px
Axes: x-axis y-axis Show Grid Grid: 24x24 inches 36x36 inches 72x72 inches 96x96 inches 192x192 inches Quad: 4 quadrants 1 quadrant Hardware Units: US Customary Metric
Background:
#### Robot 1
Initial Position: ( in, in) Initial Angle: deg Current Position: (0 in, 0 in) Current Angle: 90 deg Wheel Radius: 1.75 in1.625 in2.0 in Track Width: in
#### Robot 2
Initial Position: ( in, in) Initial Angle: deg Current Position: (6 in, 0 in) Current Angle: 90 deg Wheel Radius: 1.75 in1.625 in2.0 in Track Width: in
#### Robot 3
Initial Position: ( in, in) Initial Angle: deg Current Position: (12 in, 0 in) Current Angle: 90 deg Wheel Radius: 1.75 in1.625 in2.0 in Track Width: in
#### Robot 4
Initial Position: ( in, in) Initial Angle: deg Current Position: (18 in, 0 in) Current Angle: 90 deg Wheel Radius: 1.75 in1.625 in2.0 in Track Width: in
Draw the Moon
Problem Statement:
The pre-placed block will draw a star at any random x and y coordinate. Create a moon using the drawing blocks and edit the loop to draw 20 little stars.
```/* Code generated by RoboBlockly v2.0 */
#include <chplot.h>
double x;
double y;
CPlot plot;
int count;
plot.strokeColor("white");
plot.fillColor("white");
plot.circle(30, 28, 4);
count = 0;
while(count < 20) {
x = randint(1, 35);
y = randint(1, 35);
plot.regularPolygon(x, y, 3, 0.1, 90);
plot.regularPolygon(x, y, 3, 0.1, 270);
count = count + 1;
}
plot.grid(PLOT_OFF);
plot.label(PLOT_AXIS_XY, "");
plot.grid(PLOT_OFF);
plot.tics(PLOT_AXIS_XY, PLOT_OFF);
plot.axis(PLOT_AXIS_XY, PLOT_OFF);
plot.axisRange(PLOT_AXIS_XY, 0, 36);
plot.ticsRange(PLOT_AXIS_XY, 6);
plot.sizeRatio(1);
plot.plotting();
```
Blocks Save Blocks Load Blocks Show Ch Save Ch Workspace
Problem Statement:
The pre-placed block will draw a star at any random x and y coordinate. Create a moon using the drawing blocks and edit the loop to draw 20 little stars.
Time | 641 | 2,087 | {"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-2022-21 | latest | en | 0.536172 |
https://iot.stackexchange.com/questions/4809/how-can-i-transmit-this-information-from-my-device-using-the-least-data-possible | 1,719,070,003,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862404.32/warc/CC-MAIN-20240622144011-20240622174011-00098.warc.gz | 276,442,529 | 41,710 | # How can I transmit this information from my device using the least data possible?
I have got a Raspberry Pi setup to transmit things over socket from my project. It currently does this over cellular data on a pay as you go plan, so I would like to optimise it. All that needs to transmitted is three numbers, currently in the format of `int,int,int`. So, my question is which character uses the least data. I know this is nit picking, but it was more for theory.
If it were possible a list would be better as I would also need a character for negative numbers.
It will be from -100 to 100 but this could be any 200 range if reformatted at the other end
• A range of 200 certainly fits in one byte. You might even try to sneak them along in a protocol header field that doesn't count as data on your plan. Depends really on how accounting is working there.
– Helmar
Commented Jan 20, 2020 at 22:02
• Hi @Helmar thanks for the quick reply. I am an absolute beginner to python’s socket, could you point me in a direction of embedding in the protocol header please as a quick search only revealed how to access it. Thanks
– tejt
Commented Jan 22, 2020 at 6:56
## 2 Answers
What you want is to “pack” your 3 integers as such rather than express the in decimal as text.
If your 3 values were for instance 123, 1200 and 45678, then if you send them as three 16-bit integers it will take 6 bytes. If you send them as a human readable string made of the decimal representation, separated by commas, it would take 14 bytes.
The big difficulty with binary formats is to make sure each field has the appropriate number of bits/bytes for its range of values and precision. If any of your integers can have any value from 0 to 1 billion, you’ll need a 32-bit integer for that one. If it ranges from 0 to 100 then a single byte (8-bit integer) will be enough.
Integers can have either signed or unsigned representations. Here again, it’s a matter of matching the actual possible range of values:
• unsigned 8 bit: 0 to 255
• signed 8 bit: -128 to 127
• unsigned 16 bit: 0 to 65535
• signed 16 bit: -32768 to 32767
• unsigned 32 bit: 0 to 4 billion something
• signed 32 bit: -2 billion something to 2 billion something
Make sure you correctly define which format you use and that both ends use the same, and that it does indeed fit the whole range of values you need.
Of course, even if you need values from 0 to 10000, if all possible values are multiples of 100 (or of you don’t care about the last two digits), then a single byte will be enough (just divide by 100 on one end and multiply by 100 at the other end).
In some cases, using exponents may be more suitable. It really depends on the range and accuracy needed.
• There are fancier integer types like VARINT that take a variable number of bits and will be small for small values but expand for bigger ones. They are mostly used in data streams, since the exact location of an integer in a series can only be found by handling all of the integers that came before it. Commented Jan 22, 2020 at 3:01
• As @jcaron mentioned above, “packing” your 3 values will give you the smallest value possible. If you are sure your values always fit into -128 <= number <= 127, you can pack each integer into 1 byte and get the smallest value of 3 bytes total. In Python you can check the struct module, which in your case you can use like: `struct.pack('bbb', 100, 23, 100)` Commented Jan 27, 2020 at 20:27
Slightly too large for a comment, and does not answer your question as posted, but ...
Firstly, an upvote to @jcaron and, yes, it is a good idea to lean about data formats (integer/string/etc) and data compression (think .ZIP file).
However, an alternative would be to consider an IoT SIM card. A bunch of these have sprung up in recent years. They are data only, no voice , and beware that some might be 2G only, which your country may not support, or may drop.
The big advantage is that they charge per byte of data sent, and some don't even have a monthly fee.
I first became aware of them when buying an Onion Omega 3G expansion (*). Which recommended Hologram.
I will leave it to you to make your own decisions. Google for `IoT SIM` perhaps?
You can look at the Amazon listing for the Hologram SIM, which includes comparisons with others.
Here is the pricing for Hologram.
Here for ThingsMobile.
Here For Global M2M SIM.
Generally, you are talking about 5 cents for a mB, which is lot of data when you are only sending 3 integers.
• On the 3 pay as you go I get 1p a mB on 4g
– tejt
Commented Jan 23, 2020 at 16:20
• Thanks for that. You can send a shed-load of data for £1 then !
– Mawg
Commented Jan 24, 2020 at 8:56 | 1,188 | 4,689 | {"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-2024-26 | latest | en | 0.955964 |
https://essaypass.net/economics-126/ | 1,627,362,402,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046152236.64/warc/CC-MAIN-20210727041254-20210727071254-00107.warc.gz | 261,499,491 | 10,137 | # ECONOMICS
1. Joe quits his computer programming job, where he was earning a salary of \$50,000 per year, to start his own computer software business in a building that he owns and was previously renting out for \$24,000 per year. In his first year of business he has the following expenses: salary paid to himself, \$40,000; rent, \$0; other expenses, \$25,000. Find the accounting cost and the economic cost associated with Joe’s computer software business.
2. A firm has a fixed production cost of \$5000 and a constant marginal cost of production of \$500 per unit produced.
a. What is the firm’s total cost function? Average cost?
b. If the firm wanted to minimize the average total cost, would it choose to be very large or very small? Explain.
3. Suppose the economy takes a downturn, and that labor costs fall by 50% and are expected to stay at that level for a long time. Show graphically how this change in the relative price of labor and capital affects the firm’s expansion path.
4. You manage a plant that mass-produces engines by teams of workers using assembly machines. The technology is summarized by the production function 5 KL, where q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r \$10,000 per week, and each team costs w \$5000 per week. Engine costs are given by the cost of labor teams and machines, plus \$2000 per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design.
a. What is the cost function for your plant—namely, how much would it cost to produce q engines? What are average and marginal costs for producing q engines? How do average costs vary with output?
b. How many teams are required to produce 250 engines? What is the average cost per engine?
c. You are asked to make recommendations for the design of a new production facility. What capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total cost of producing at any level of output q?
5. The short-run cost function of a company is given by the equation TC 200 55q, where TC is the total cost and q is the total quantity of output, both measured in thousands.
a. What is the company’s fixed cost?
b. If the company produced 100,000 units of goods, what would be its average variable cost?
c. What would be its marginal cost of production?
d. What would be its average fixed cost?
e. Suppose the company borrows money and expands its factory. Its fixed cost rises by \$50,000, but its variable cost falls to \$45,000 per 1000 units. The cost of interest (i) also enters into the equation. Each 1-point increase in the interest rate raises costs by \$3000. Write the new cost equation.
6. Suppose that a firm’s production function is
11
22
10
qLK
=
. The cost of a unit of labor is \$20 and the cost of a unit of capital is \$80.a. The firm is currently producing 100 units of output and has determined that the cost-minimizing quantities of labor and capital are 20 and 5, respectively. Graphically illustrate this using isoquants and isocost lines.
b. The firm now wants to increase output to 140 units. If capital is fixed in the short run, how much labor will the firm require? Illustrate this graphically and find the firm’s new total cost.
c. Graphically identify the cost-minimizing level of capital and labor in the long run if the firm wants to produce 140 units.
d. If the marginal rate of technical substitution is
K
L
, find the optimal level of capital and labor required to produce the 140 units of output.Pindyck and Rubinfeld, Chapter 7 Appendix
1. The production function for a product is given by q 100KL. If the price of capital is \$120 per day and the price of labor \$30 per day, what is the minimum cost of producing 1000 units of output?
2. Suppose a production function is given by F(KL) KL2; the price of capital is \$10 and the price of labor \$15. What combination of labor and capital minimizes the cost of producing any given output? | 979 | 4,083 | {"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} | 2.921875 | 3 | CC-MAIN-2021-31 | latest | en | 0.955556 |
http://forums.wolfram.com/mathgroup/archive/2007/Aug/msg00130.html | 1,642,679,209,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00150.warc.gz | 28,763,094 | 8,389 | Re: Density Plot coloring issue
• To: mathgroup at smc.vnet.net
• Subject: [mg79742] Re: [mg79715] Density Plot coloring issue
• From: Brett Champion <brettc at wolfram.com>
• Date: Fri, 3 Aug 2007 06:27:39 -0400 (EDT)
• References: <200708020756.DAA01398@smc.vnet.net>
```On Aug 2, 2007, at 2:56 AM , Yvon wrote:
> I'm having an issue where my density plot will color all the cells
> EXCEPT the minimum above 0 and maximum values in my list. It seems to
> occur irregardless of whether I use the custom function I wrote or
> ColorFunction -> (Hue[2/5, 2/3, 1.2 - #] &)
>
Set ClippingStyle->Automatic in the ListDensityPlot.
Brett Champion
Wolfram Research
> This is driving me crazy, please tell me what I'm doing wrong. For
> the dataset given below, all the cells are colored except cell 3,4
> (value=3.93) and cell 6,1(value=7.41) They are white, along with the
> 0.0 values which is the first value in my color table.
>
> However, if I duplicate the min value by changing another value
> within the list (for example, replace (cell1,1) 6.75 with 3.93), then
> the color for both 3.93 cells change to what is expected. If I
> replace 6.75 with 7.41, both 7.41 cells are white, if I replace 6.75
> with 7.42, then cell 1,1 is white and the cell 6,1 changes to what is
> expected. Can someone tell me what is going on here?!!
>
> cList = {{6.75, 5.85, 0., 6.39, 7.41, 0.}, {6.4, 5.57, 0., 6.6, 6.68,
> 0.}, {5.99, 5.51, 0., 6.24, 6.37, 0.}, {6.1, 5.72, 3.93, 5.92,
> 5.89, 0.}, {5.71, 5.85, 6.05, 5.92, 6.06, 0.}, {5.62, 5.64, 5.61,
> 5.42, 5.14, 0.}, {0, 0, 0, 0, 0, 0}}
>
> numColors = 10
> minLWC = 0
> maxLWC = 10
>
> CellColor[index_] := Module[{cTable, s},
> (*White,purple,blue, light blue, light green, kahki, light yellow,
> light orange, pink, red *)
> cTable = {{1, 1, 1}, {0.5, 0, 0.5}, {0.4, .6,
> 0.9}, {0.7, .9, .96}, {.8, 1, .7}, {.6, 0.8, .2}, {1, 1,
> 0.5}, {1, .84, 0}, {1, .75, .8}, {.86, .08, .24}};
> s = Partition[Map[RGBColor, cTable], 1];
> RGBColor[cTable[[index]]]
> ]
>
> GetColor[x_] := (p =
> Round[Abs[(x - minLWC)*numColors/(maxLWC - minLWC)]] ;
> CellColor[p])
>
> ListDensityPlot[cList, InterpolationOrder -> 0,
> ColorFunction -> GetColor, ColorFunctionScaling -> {\!\(\*
> ButtonBox["False",
> ButtonData->"paclet:ref/True"]\)},
> Mesh -> {numCols - 1, numRows - 1}, PlotRangePadding -> None,
> DataRange -> {{0, numCols}, {0, numRows}},
> Epilog -> {Hue[0, 1, 1],
> MapIndexed[Text[#1, #2 - .5] &, Transpose[cList], {2}]}]
>
> Your help is greatly appreciated!!
>
> Yvon
>
```
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• Next by thread: Re: Density Plot coloring issue | 1,076 | 2,768 | {"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-2022-05 | latest | en | 0.782049 |
https://math.stackexchange.com/questions/4430515/what-is-the-smallest-3d-rotation-to-make-the-axes-line-up | 1,722,684,114,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640365107.3/warc/CC-MAIN-20240803091113-20240803121113-00428.warc.gz | 313,124,046 | 37,497 | # What is the smallest 3D rotation to make the axes line up
A 3x3 rotation matrix is considered axis-aligned if it consists of only 1, -1, and 0. Given an arbitrary rotation matrix, what is the smallest rotation required to make it axis-aligned?
For example, given
$$R=\begin{pmatrix} 0.0281568 & 0.8752862 & 0.4827849\\ 0.9936430 & 0.0281568 & -0.1089990\\ -0.1089990 & 0.4827849 & -0.8689292 \end{pmatrix}$$
The best I can think of is to try to get R closer to identity by permuting and negating the rows, and then compute the angle from identity (by converting to axis-angle form).
So in the example I would multiply R by
$$R'= \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} * R$$
and R' is about 30 degrees away from identity, so the answer is 30.
My method to compute the permutation is rather adhoc. Is there an better way?
It appears that the general solution will require searching. There are 24 possible 3x3 axis-aligned rotations. You could just compute the angle from R to each of these and choose the minimum.
For efficiency, you could reduce the set of possibilities from 24 down to 3 as follows. Add the columns of R together to get the vector m. The signs of the components of m tells you the octant of m and hence the octant that is nearest to R. Each octant has 3 axis-aligned matrices, so you only have to test the angle from R to each of these 3 and choose the minimum. I'm pretty sure the minimum solution will never cause the middle vector, m, to change to a different octant. But I can't prove it.
In the example, the sum of columns is approx $$[1.4, 0.9, -0.5]$$ so the octant is (+ + -). The three axis-aligned rotations in that octant are:
$$I_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$
$$I_2 = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$ $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}$$ Find the angle from R to each of these (using $$R' = I_i^T R$$), and choose the smallest. In this example, it is pretty obvious that $$I_1$$ will be the winner, but another example may be less obvious.
There is a unique matrix that will rotate the three axes of an arbitrary rotation matrix into standard position. And that matrix is just the transpose of the given matrix, because
$$R^T R = I$$ | 710 | 2,326 | {"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": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2024-33 | latest | en | 0.867248 |
http://mathforum.org/mathtools/discuss.html?context=recent&do=r&msg=164546 | 1,521,360,328,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257645550.13/warc/CC-MAIN-20180318071715-20180318091715-00582.warc.gz | 197,039,131 | 5,865 | You are not logged in.
Discussion: Recent Topics Topic: MathDash Feedback Request
Post a new topic to the Software Developers Discussion discussion
<< see all messages in this topic < previous message | next message >
Subject: RE: MathDash Feedback Request Author: fjl Date: Apr 10 2012
Hi Max and Everyone,
I think I just got a much better handle on the issue that Max was highlighting.
I just finished getting an update from my student Matt (MathDash Team Captain),
who recently came back from a very informative discussion with Suzanne and
others at the Math Forum.
So the issue is that the game might be reinforcing a common misconception that
kids might have with place values in that they treat them as literals. I
indicated in the earlier post that knowing place values might be a good thing
for learning abstraction, but issue is that the game is still treating it as a
literal one. This is true, and this is something that we need to address.
Suzanne and Ray offered some suggestions to Matt during the meeting, and we will
consider those. One idea that I've been playing with in my mind from Max's
initial feedback is something like this this:
1) When you combine two numbers, they result in three addends that sum up to
the two numbers. (e.g. 5,7 -> 2,4,6)
2) If there is more than one possible combination of 3 addends that sum up to
the two numbers, then one combination is randomly chosen.
Optional Rule
3) One of the addend must be 1. (This provides some consistency in that I'm
guaranteed to always get 1 from the combination)
This is an important point, certainly much more so than I initially thought. We
will fine some implementation that we are happy from gameplay point of view that
On Apr 6 2012, maxmathforum wrote:
> Hi Math Dashers,
I posted on your survey but also wanted to see
> if we could get a discussion started among all the smart Math and
> Tech folks here...
I'm wondering how best to handle the part of
> the game where two numbers are combined to make a new digit that's
> their sum.
If you combine 8 and 6 to get 14, the game currently
> gives you a 1 digit and a 4 digit. The 1 no longer represents a 1 in
> the 10s place with a value of 10 units.
To me, that seems like it
> might subtly reinforce student misconceptions of place value, and I
-What if combining 8 and 6 made a 4
> and ten 1's?
-What if the user got to choose, quickly, what three
> numbers they wanted to partition 14 into?
What else? What are the
> pluses and minuses of each plan?
Max | 610 | 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} | 3.234375 | 3 | CC-MAIN-2018-13 | latest | en | 0.95219 |
https://quant.stackexchange.com/questions/29523/interpreting-units-of-short-rate-parameters/29526 | 1,623,707,003,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487613453.9/warc/CC-MAIN-20210614201339-20210614231339-00219.warc.gz | 440,490,919 | 36,865 | # Interpreting Units of Short Rate Parameters
I've estimated the parameters for the Vasicek model $$dr(t) = a(b - r(t))dt + \sigma dW(t)$$ and the CIR model $$dr(t) = a(b - r(t))dt + \sigma\sqrt{r(t)} dW(t)$$ to one-year Treasury yield data from 1974 (which were around 8% then!). Let's say the estimates I got were $$Vasicek: a = 3.2, b = 8.1, \sigma = 6.0 \qquad (1)\\ CIR: a = 3.2, b = 8.1, \sigma = 2.3. \qquad (2)$$ N.b. these values correspond to $r(t)$ in percent, not decimal. So, my dimensions are short rate level measured in percent (%), and time, say, measured in seconds (s). The units of the parameters are then $$a = s^{-1}, b = \%, \sigma = \%/\sqrt{s}$$ My question is, how does one intuitively interpret the estimated values in (1) and (2)? I.e., I'm trying to think of the process as a physical process, and so what does a "mean reversion speed of $3.2 / s$" mean, e.g.? Actually, it seems that calling $a$ a "speed" is a misnomer, given the units.
Any insights welcome!
The processes revert towards their mean with the speed $E(dr(t)/dt) =a*(b-r(t))$ so $a$ is not the speed itself, only one factor of it. If $\sigma$ would vanish then $a$ would be $ln (2)$ times the inverse of the time - hence its unit $s^{-1}$- it would take for $b-r(t)$ to halve. Think of a as the decay-factor of the deviation from the mean!
• @M Lind, could you please be so kind to use the Latex formatting capabilities in your answer? Thanks. – Quantuple Aug 8 '16 at 12:58 | 459 | 1,473 | {"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} | 3.515625 | 4 | CC-MAIN-2021-25 | latest | en | 0.89702 |
https://brainmass.com/statistics/probability/credit-card-usage-289089 | 1,600,762,222,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400204410.37/warc/CC-MAIN-20200922063158-20200922093158-00599.warc.gz | 291,333,052 | 10,726 | Explore BrainMass
# Credit card usage
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
Each time that Ed charges his credit card, he omits the cents and records only the dollar value. If this month he has charged his credit card 20 times what can be said about the probability that the record shows at least \$15 less than the actual amount? | 88 | 404 | {"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-2020-40 | latest | en | 0.958064 |
https://www.eurocode.us/concrete-structures-eurocode-3/design-grid-line.html | 1,560,805,744,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998580.10/warc/CC-MAIN-20190617203228-20190617225228-00199.warc.gz | 751,339,671 | 8,141 | ## Design grid line
Effective depth, d d = 300 - 30 - 20 - 20 / 2 = 240 mm Flexure: column strip, sagging: MEd = 75.5 kNm / m
By inspection, z = 228 mm <Concise EC2 Table 15.5>
As = MEd / fydz = 75.5 x 106 / (228 x 500 / 1.15) = 761 mm2 / m
Deflection: column strip: By inspection, OK. Flexure: middle strip, sagging: MEd = 37.1 kNm / m By inspection, z = 228 mm As = MEd / fydz = 37.1 x 106 / (228 x 500 / 1.15) = 374 mm2 / m
By inspection, deflection OK. Try H10 @ 200 B2 (393 mm2 / m)
Flexure: column strip, hogging: MEd = 113.3 kNm / m
K = MEd / bd2fck = 113.3 x 106 / (1000 x 2402 x 30) = 0.065 <Concise EC2 Table 15.5>
As = MEd / fydz = 113.3 x 106 / (225 x 500 / 1.15) = 1158 mm2 / m
Flexure: middle strip, hogging: MEd = 18.5 kNm / m By inspection, z = 228 mm
As = MEd / fydz = 18.5 x 106 / (228 x 500 / 1.15) = 187 mm2 / m <Concise EC2 Table 15.5>
As before minimum area of reinforcement governs
As,min = 0.26 x 0.30 x 300666 x 1000 x 240 / 500 = 361 mm2 / m <9.3.1.1, 9.2.1.1>
Requirements:
Regarding the requirement to place 50% of At within a width equal to 0.125 of <9.4.1(2)>
ssss The hogging moment could have been considered at face of support to reduce the amount of reinforcement required.
the panel width on either side of the column. Area required = (3.0 x 1158 + 6.1 x 187) / 2 mm2 = 2307 mm2
Within = 2 x 0.125 x 6.0 m = 1500 mm centred on the column centreline.
i.e. require 2307 / 1.5 = 1538 mm2 / m for 750 mm either side of the column centreline.
Use H20 @ 200T2 (1570 mm2 / m) 750 mm either side of centre of support (p = 0.60%)
In column strip, outside middle 1500 mm, requirement is for 3.0 x 1158-15 x 1570 = 1119 mm2 in 1500 mm, i.e. 764 mm2 / m
Use H16 @ 250 T2 (804 mm2 / m) in remainder of column strip
In middle strip
## Greener Homes for You
Get All The Support And Guidance You Need To Be A Success At Living Green. This Book Is One Of The Most Valuable Resources In The World When It Comes To Great Tips on Buying, Designing and Building an Eco-friendly Home.
Get My Free Ebook | 762 | 2,040 | {"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} | 3.171875 | 3 | CC-MAIN-2019-26 | latest | en | 0.683484 |
https://www.mathbootcamps.com/area-circle-formula-examples/ | 1,721,046,165,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514696.4/warc/CC-MAIN-20240715102030-20240715132030-00335.warc.gz | 781,436,948 | 13,336 | # Area of a circle – formula and examples
The area of a circle can be thought of as the number of square units of space the circle occupies. This can be found using either the radius or the diameter, which we will cover in the examples below. We will also look at some examples of word problems involving area that you may come across in your studies.
## Examples of finding the area of a circle
We will use the following formula to find the area of any circle. Notice that this formula uses the radius, so we will have to convert when we are given the diameter instead. Let’s look at both cases.
Find the area of a circle with a radius of 5 meters.
### Solution
Apply the formula: $$A = \pi r^2$$ with radius $$r = 5$$. Remember that $$\pi$$ is about 3.14.
\begin{align}A &= \pi (5)^2 \\ &= 25\pi \\ &\approx \bbox[border: 1px solid black; padding: 2px]{75.5 \text{ m}^2}\end{align}
Since the units of the radius were in meters, the answer is in square meters. This can be written out in words, or as $$\text{m}^2$$. Also, the final answer can be written in terms of $$\pi$$ ($$25 \pi$$ square meters) or as a decimal approximation (75.5 square meters). Which one you use depends on the application and the problem you are working on.
### Example (given diameter)
Find the area of a circle with a diameter of 6 feet.
### Solution
The radius of any circle is always half the diameter. Since the diameter of the circle is 6 feet, the radius must be 3 feet (the radius is always half of the diameter). So, we can apply the formula using $$r = 3$$.
\begin{align}A &= \pi (3)^2 \\ &= 9\pi \\ &\approx \bbox[border: 1px solid black; padding: 2px]{28.3 \text{ ft}^2}\end{align}
As you can see, it is important to pay attention to whether or not you are given the radius or the diameter of the circle. In some word problems though, this may not always be as clear.
## Word problems involving the area of a circle
Not every problem you will encounter will simply say “find the area”. In the next two examples, you will see other types of questions you might be asked.
### Example
Jason is painting a large circle on one wall of his new apartment. The largest distance across the circle will be 8 feet. Approximately how many square feet of wall will the circle cover?
### Solution
Whenever you are asked to find the number of square feet covered by something, you are finding an area. To find the area of Jason’s circle, we first need to figure out if we have been given the radius or the diameter. By definition, the diameter of a circle is the longest distance across the circle, so we know here that the diameter is 8 feet. This means that the radius is 4 feet. Therefore:
\begin{align}A &= \pi (4)^2 \\ &= 16\pi \\ &\approx \bbox[border: 1px solid black; padding: 2px]{50.2 \text{ square feet}}\end{align}
So, Jason’s circle will cover about 50.2 square feet of his wall.
### Example
The area of a circle is $$81 \pi$$ square units. What is the radius of this circle?
### Solution
To answer this question, you will have to remember a little bit of algebra. Use the formula and substitute the values you know. Then, solve for the radius, $$r$$.
$$A = \pi r^2$$
Substitute in $$A = 81 \pi$$ since you know this is the area.
$$81\pi = \pi r^2$$
Divide both sides by $$\pi$$.
$$81 = r^2$$
This can also be written as:
$$r^2 = 81$$
Take the square root to find $$r$$. Since this is a radius, the value of $$r$$ must be positive.
\begin{align}r &= \sqrt{81} \\ &= \bbox[border: 1px solid black; padding: 2px]{9}\end{align}
Therefore, the radius must be 9. | 953 | 3,581 | {"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": 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": 4, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.875 | 5 | CC-MAIN-2024-30 | latest | en | 0.884465 |
https://se.tradingview.com/script/YHuROcQY-Machine-Learning-Multiple-Logistic-Regression/ | 1,716,818,109,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059040.32/warc/CC-MAIN-20240527113621-20240527143621-00088.warc.gz | 446,984,436 | 166,191 | # Machine Learning: Multiple Logistic Regression
Multiple Logistic Regression Indicator
The Logistic Regression Indicator for TradingView is a versatile tool that employs multiple logistic regression based on various technical indicators to generate potential buy and sell signals. By utilizing key indicators such as RSI, CCI, DMI, Aroon, EMA, and SuperTrend, the indicator aims to provide a systematic approach to decision-making in financial markets.
How It Works:
• Technical Indicators:
The script uses multiple technical indicators such as RSI, CCI, DMI, Aroon, EMA, and SuperTrend as input variables for the logistic regression model.
These indicators are normalized to create categorical variables, providing a consistent scale for the model.
• Logistic Regression:
The logistic regression function is applied to the normalized input variables (x1 to x6) with user-defined coefficients (b0 to b6).
The logistic regression model predicts the probability of a binary outcome, with values closer to 1 indicating a bullish signal and values closer to 0 indicating a bearish signal.
• Loss Function (Cross-Entropy Loss):
The cross-entropy loss function is calculated to quantify the difference between the predicted probability and the actual outcome.
The goal is to minimize this loss, which essentially measures the model's accuracy.
```// Error Function (cross-entropy loss)
loss(y, p) =>
-y * math.log(p) - (1 - y) * math.log(1 - p)
// y - depended variable
// p - multiple logistic regression```
Gradient descent is an optimization algorithm used to minimize the loss function by adjusting the weights of the logistic regression model.
The script iteratively updates the weights (b1 to b6) based on the negative gradient of the loss function with respect to each weight.
```// Adjusting model weights using gradient descent
b1 -= lr * (p + loss) * x1
b2 -= lr * (p + loss) * x2
b3 -= lr * (p + loss) * x3
b4 -= lr * (p + loss) * x4
b5 -= lr * (p + loss) * x5
b6 -= lr * (p + loss) * x6
// lr - learning rate or step of learning
// p - multiple logistic regression
// x_n - variables```
• Learning Rate:
The learning rate (lr) determines the step size in the weight adjustment process. It prevents the algorithm from overshooting the minimum of the loss function.
Users can set the learning rate to control the speed and stability of the optimization process.
• Visualization:
The script visualizes the output of the logistic regression model by coloring the SMA.
Arrows are plotted at crossover and crossunder points, indicating potential buy and sell signals.
Lables are showing logistic regression values from 1 to 0 above and below bars
• Table Display:
A table is displayed on the chart, providing real-time information about the input variables, their values, and the learned coefficients.
This allows traders to monitor the model's interpretation of the technical indicators and observe how the coefficients change over time.
How to Use:
Users can adjust the length of technical indicators (rsi_length, cci_length, etc.) and the Z score length based on their preference and market characteristics.
Set the initial values for the regression coefficients (b0 to b6) and the learning rate (lr) according to your trading strategy.
• Signal Interpretation:
Buy signals are indicated by an upward arrow (▲), and sell signals are indicated by a downward arrow (▼).
The color-coded SMA provides a visual representation of the logistic regression output by color.
• Table Information:
Monitor the table for real-time information on the input variables, their values, and the learned coefficients.
Keep an eye on the learning rate to ensure a balance between model adjustment speed and stability.
• Backtesting and Validation:
Before using the script in live trading, conduct thorough backtesting to evaluate its performance under different market conditions.
Validate the model against historical data to ensure its reliability.
Versionsinformation:
-
Versionsinformation:
Versionsinformation:
Alert UP: when logistic reg crossover 0.5 value
"Logistic Regression UP"
Alert DOWN: when logistic reg crossunder 0.5 value
"Logistic Regression DOWN"
"Logistic Regression: 1▲"
or
"Logistic Regression: 1▼"
Versionsinformation:
Skript med en öppen källkod
I sann TradingView-anda har författaren publicerat detta skript med öppen källkod så att andra handlare kan förstå och verifiera det. Hatten av för författaren! Du kan använda det gratis men återanvändning av den här koden i en publikation regleras av våra ordningsregler. Du kan ange den som favorit för att använda den i ett diagram.
Frånsägelse av ansvar
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Vill du använda det här skriptet i ett diagram? | 1,110 | 4,928 | {"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-2024-22 | latest | en | 0.818961 |
http://www.uwgb.edu/dutchs/STRUCTGE/SL12SDFromSC.HTM | 1,519,310,235,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891814124.25/warc/CC-MAIN-20180222140814-20180222160814-00562.warc.gz | 575,335,889 | 2,033 | # Find Strike and Dip of a Plane from Structure Contours
Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
First-time Visitors: Please visit Site Map and Disclaimer. Use "Back" to return here.
The strike is easy -- it's simply the azimuth of the structure contours.
In this case we have the spacing of the structure contours and need to find the dip. There are two ways to do this.
1. Draw a simple cross-section and measure the dip
2. Calculate the dip trigonometrically
H = S Tan dip, thus dip = Arctan (H/S)
You can use any vertical distance and the corresponding contour spacings in your diagram or calculation. In fact, it's probably best to measure across several contour intervals because your measurements will be more accurate.
## Example
1. The problem. Find the dip and strike of this layer from its structure contours. 2. The strike is simply the azimuth of the structure contours. 3. Determine the dip either by drawing a cross-section or by trigonometry. The fraction of a degree in the calculation is probably not significant. 4. Final result. | 239 | 1,098 | {"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} | 3.046875 | 3 | CC-MAIN-2018-09 | latest | en | 0.902383 |
https://goprep.co/ex-8.f-q6-in-how-many-ways-can-4-prizes-be-given-to-3-boys-i-1nlkel | 1,604,138,889,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00266.warc.gz | 348,197,251 | 46,350 | Q. 65.0( 1 Vote )
# In how many ways can 4 prizes be given to 3 boys when a boy is eligible for all prizes?
Let suppose 4 prizes be P1, P2, P3, P4 and 3 boys be B1, B2, B3
Now P1 can be distributed to 3 boys(B1, B2, B3) by 3 ways,
Similarly, P2 can be distributed to 3 boys(B1, B2, B3) by 3 ways,
Similarly, P3 can be distributed to 3 boys(B1, B2, B3) by 3 ways,
And P4 can be distributed to 3 boys(B1, B2, B3) by 3 ways
So total number of ways is 3 × 3 × 3 × 3 =81
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Shooting Fish in a Barrel
Earlier this month Graham Fletcher and Joshua Greene got together for a tag team take-down of a textbook lesson on finding the volume of rectangular prisms:
It never really had a chance.
Their analysis and make over suggestions are spot-on. And I hate to pile on, but what makes this especially egregious is that exploring volume of rectangular prisms, a major grade 5 content standard, can be, well, fun! Rich and I have had success with activities described here and here, and I'll offer up another one that we tried out with the fifth graders last year.
Andrew Stadel has written about the potential that starting arguments in math class has to reinforce both practice standards (especially numbers 1 and 3) and engage kids in learning, and I agree. For this project, I collected five boxes and asked the kids to guess their order from smallest to largest as measured by volume. I didn't want the boxes to be too similar, just similar enough that it wouldn't be too obvious.
I chose these five...
...and gave the kids time to get their hands on them.
Argument started!
With the intellectual need now built, it was time for the kids to resolve the dispute. They first had to estimate the volume of each box...
...then find the necessary measurements. We decided on nearest tenth of a centimeter to reinforce working with decimals.
We let them use a calculator because who wants to do all those calculations by hand?
The data was recorded.
And, due to faulty measuring, improper rounding, and incorrect number crunching, the argument continued to rage! In fact it took several days for the class to come to agreement on the volume of each box and the correct order. But they were days filled with engagement, collaboration, discussion, and multiple content and practice standards.
The activity isn't all that imaginative, and it's not that hard to prep for. All you need are some boxes, rulers, and calculators. And while it's not as easy as asking the kids to take out their books and do this...
. ...I guarantee you'll have more fun.
Tuesday, February 16, 2016
A Post About Counting Circles
"Children need repeated exposure to and practice with counting sequences in order to become fluent with counting. Counting sequences help children understand relationships between numbers and further develop their abilities to to apply these understandings to problem solving situations."
Jessica Shumway
Number Sense Routines, pg. 56
This year, our PLC is digging into Jessica Shumway's wonderful book:
Thanks to my principal for finding room in the budget to order multiple copies.
Comprised of teachers grades 1,2, and 3, we are on a mission to both critically examine our preexisting math routines and explore new possibilities. We have tackled the Calendar and Counting the Days in School routines, as well as the Quick Image/Dot Card routines that are now a part of our Everyday Math curriculum. Shumway's book has been an invaluable resource, providing background into learning trajectories and pedagogy as well as practical, how-to advice and student work examples.
Shumway devotes an entire chapter to counting routines. These routines have been a part of our curriculum for many years, commonly used as the warm-up to a lesson. But direction on implementation and usage is pretty vague, as illustrated by the following example from the grade 2 Teachers Manual:
As Everyday Math vets we are also used to seeing children engage in counting routines in their journal work:
Can you spot the error?
We asked ourselves some tough questions, using Shumway's book as a lens through which to view these routines with fresh eyes. Were we using the routine in a thoughtful way? Or were we just doing it because it was there? Were we in fact even doing it at all, or did the routine often end up on the cutting room floor due to lack of time? Were there ways we could make the routine more engaging? More meaningful?
Shumway offers a host of different ways to engage students in a counting routine. (See here for more detailed explanations as well as a video clip of a counting circle in action. Scroll down to #2, Counting Routines.) Counting circles where each class member adds to the count individually, choral counting, start and stop, whole class, small group, kinesthetic; Shumway explores many variations. She provides questions that help facilitate discussions about patterns, place value, number sense, and strategy.
After a lively discussion, and after engaging in a counting circle of our own, the teachers were ready to give it a try:
Shumway emphasizes the importance of planning. Kristin, after determining her second graders needed practice counting by 10s in three-digit numbers, sketched out her idea for recording the class count. Different patterns will emerge depending on how the count is recorded. She decided to record vertically, starting a new column when the digit in the hundreds place changed to highlight this particular trouble spot.
She recorded the count, and asked the class, "What do you notice?" The kids responded, and Kristin had the opportunity to reinforce concepts of place value in the changing numbers. There are also opportunities to explore the different strategies individual students employ to add 10.
Also from grade 2. Jane decided to have her class count backwards. Counting backwards is often neglected. Notice how she has elected to record the count horizontally.
Maggie, another grade 2 teacher, knows that her students will be asked to count by 25s on an upcoming progress check. She uses the routine to solidify the concept and also as a formative assessment.
Nicole uses an interactive hundreds grid with her first grade class. She often has her students do jumping jacks as they count.
Larissa, grade 4, wants in on the action:
She plans to have the class count by a fractional amount, and has some questions ready for her class to think about.
What do you notice? What are you wondering?
"Counting sequences," Shumway asserts, "Help children understand relationships among numbers and further develop their abilities to apply these understandings to problem-solving situations." Proof of that came last week:
Shannon's third graders were working on this task...
...and realized that skip counting would help them find their way to a solution. I felt this was a good example of a counting sequence application to a problem-solving situation.
Larissa's class collected data on the circumference of their heads...
...and then organized the data in a line plot.
Counting by 1/2s from 49 to 56. A perfect grade 4 counting circle task.
It's been fun to watch teachers experiment with this routine. The only way to learn how to do it is to, well, do it! It won't be perfect and it might get messy, but that's OK. Do what works best for you and your students. Here are some reflections based on my observations around the school
• The teacher needs to be alert to student participation and engagement. If the class is counting one at a time, once a child's turn is over, what will keep him paying attention to what's happening? Conversely, if the class is counting chorally, the teacher needs to be aware of any students trying to hide behind more able classmates. Those students would be good candidates for a small group counting circle.
• When conducting a counting circle, it is not uncommon for a student to get stuck. Shumway outlines moves a teacher can utilize in this instance, but unless a class culture of support and encouragement is in place (something Shumway also addresses in her book and Sadie Estrella emphasizes in her counting circle Ignite talk), the moment can turn into one of embarrassment and anxiety for the student in question.
• Planning is important. What will you count by? Where will you start? How will you record the count? What questions will you ask? The more thoughtful you are, the more you will be able to get out of the routine. And you will still be surprised by what students notice. So be ready!
• It's important to keep your eye on the clock, especially during the discussion time. Student interest can start to flag, and while there still may be ground to cover, if students are not engaged then nothing worthwhile can be accomplished. In other words, keep things moving!
• Counting circle routines provide opportunities for students to employ practice standards. Many students quickly realize that finding the emerging pattern that arises within the count will help them come up with the next number in the sequence, a good example of SMP 7 in action. SMP 8 comes into play when the teacher asks questions like, "What number will the last person say? How do you know?"
As I've learned from hard experience (see here and here), there's more to counting than meets the eye. "Students who struggle with mathematics," writes Shumway, "Often lack counting skills." Shaky understandings about place value, underdeveloped abilities to recognize patterns, poor estimation skills, misguided notions about the way our number system is organized, the inability to think in additive and multiplicative ways; well planned and executed counting circle routines target all these areas, as well as promote critical thinking and problem solving skills. What's not to like? | 2,034 | 9,406 | {"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} | 3.5 | 4 | CC-MAIN-2017-09 | longest | en | 0.970404 |
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Is there a way to understand Huygen's principle without running into overflow errors in the brain?
I get the gist, but if i extend the principle to literally treating every point as a source then I try to imagine infinite spheres emanating and infinite spheres emanating from them etc
just total chaos
3:15 AM
Yes, first of all if you're trying to use it in a classical context things are more subtle, see Feynman's comments here:
The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms a new wavefront. As such, the Huygens-Fresnel principle is a method of analysis applied to problems of luminous wave propagation both in the far-field limit and in near-field diffraction as well as reflection. == History == In 1678, Huygens proposed that every point reached...
In a QM context, you can think of a wave function at $x' = (t',\mathbf{x}')$ (i.e. a 'wave' at $x'$), $\psi(x')$, as a linear combination of waves that began at some earlier time $t$, $t'>t$, emanating from all points $\mathbf{x}$, where the strength/amplitude of the wave $\psi(x) = \psi(t,\mathbf{x})$ that arrives at $x'$ will be proportional to the original amplitude $\psi(x)$, where $K(x';x)$ is the proportionality.
Thus we are saying $\psi(x')$ is a sum of waves $K(x',x)\psi(x)$ i.e. $\psi(x') = \int d^3 \mathbf{x} K(x';x) \psi(x) = \int d^3 \mathbf{x} K(t',\mathbf{x}';t,\mathbf{x}) \psi(t,\mathbf{x})$, where $t' > t$ is assumed, which can be made explicit by writing this as $\theta(t' - t) \psi(x') = \int d^3 \mathbf{x} K(x';x) \psi(x)$
A non-trivial illustration of this arises when a wave function has stationary states, in which case it can be expanded as $\psi(x) = \sum_n c_n \psi_n^*(\mathbf{x}) e^{-iE_nt}$, where $c_n = \int d^3 \mathbf{x} e^{i E_n t} \psi(x) = c_n(t)$.
We can now insert $c_n(t_1)$ into $\psi(x_2)$ and get $\psi(x_2) = \int d^3 \mathbf{x}_1 \sum_n \psi_n(\mathbf{x}_2) \psi_n^*(\mathbf{x}_1) e^{iE_n (t_2 - t_1)} \psi(t_1,\mathbf{x}_1)$. Thus
$$\theta(t_2 - t_1) \psi(x_2) = \int d^3 \mathbf{x} [\theta(t_2 - t_1) \sum_n \psi_n(x_2) \psi_n^*(x_1)] \psi(x_1) = \int K(x_2;x_1) \psi(x_1)$$
Usually it's written as $K(x_2;x_1) = i G(x_2;x_1)$ because when you apply the Schrodinger equation to it, the $i$ in $i \partial_t$ cancels the $i$ in front of the $G$ when you apply the operator to $\theta(t_2 - t_1) \psi(x_2)$
Things get even more subtle when you apply this simple picture to relativity, where you now have 'negative energy' stationary states to deal with, which I wont go into
3:42 AM
(In Ket notation this is $\theta(t_2 - t_1) \psi(x_2) = \theta(t_2-t_1) <x_2|\psi> = \int d^3 \mathbf{x} \theta(t_2-t_1)<x_2|x_1><x_1|\psi> = \int d^3\mathbf{x} K(x_2;x_1) \psi(x_1)$, where $|x> = e^{i \hat{H} t}|\mathbf{x}>$)
@naturallyInconsistent Oh
@Slereah dirac cites a similar reason for pursuing what is now the dirac equation
3:58 AM
What is the physics sign convention for Thermo?
2 hours later…
5:36 AM
is there a formal way to think about a given a differential equation, e.g. $\frac{d}{dt}\psi(x,y,z,t) = \nabla^2\psi(x,y,z,t)$, and various "complexifications" of it? For instance, we might want to solve $i\frac{d}{dt}\psi(x,y,z,t) = \nabla^2\psi(x,y,z,t)$ instead using knowledge that we have of the original DE. Or vice versa.
1 hour later…
6:54 AM
hi
honk
7:15 AM
@SillyGoose are you trying to relate the heat equation and SE?
@Mr.Feynman this is the motivating example for the question
H O N K
@Mr.Feynman what sound is it that you make?
meow
the first 5 minutes of this talk talk about a quantum state as being a section youtube.com/watch?v=sshJyD0aWXg; i think this is the type of formulation of textbook quantum that i was trying to find earlier.
the related paper is here: arxiv.org/pdf/2302.10778.pdf
8:29 AM
Trying to figure out a bit what Kerr's singular curves are like and as an example I thought I'd look at the AdS null curves at infinity, but then I can't seem to find anything about whether or not these curves are singular or anything about them???
It's like people never did anything with finite affine length singular curves
Closest I can find is some Penrose paper on ideal points where he seems to discuss curves like that a bit, calling them null-finite $\infty$-TIP, but he doesn't really give much in the way of examples
8:54 AM
Time dilation experiment
Hafele–Keating experiment?
that's the one
Hafele on theleft, and that intelligent computer is Keating I assume
9:13 AM
@naturallyInconsistent the sound produced by waving hands
@SillyGoose well, if you consider an analytically extend functions of time $t$ to a complex variable $s$ and you restrict to imaginary $s=i\tau$ with $\tau\in\mathbb{R}$ (known as imaginary time) you have a correspondence between the two cases
Basically this turns imaginary exponentials into decreasing exponential and comparing the propagators of the respective equations you can see you're passing from one to the other
9:28 AM
@Mr.Feynman voom voom?
1 hour later…
10:31 AM
@SillyGoose i dont like this approach personally. it's too close to Bohmian mechanics
they've replaced a deterministic beable with a stochaistic beable
they also seem to be using decoherence to solve the measurement problem
but beables r not needed if u r using decoherence anyway
beables have problems with QFT, as there is no position basis. there is no preferred choice of what basis to use as a beable
3 hours later…
1:53 PM
> A magnitude if divisible one way is a line, if two ways a surface, and if three a body. Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all. For, as the Pythagoreans say, the world and all that is in it is determined by the number three, since beginning and middle and end give the number of an 'all', and the number they give is the triad.
And so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the Gods.
It's all connected
2:05 PM
Artistotle's On the heavens is a much nicer read than his physics book rly
not much metaphysical faffing about, he starts off like it is
All bodies are made of fire, air, etc
as we all know
2:19 PM
> Further, this circular motion is necessarily primary. For the perfect is naturally prior to the imperfect, and the circle is a perfect thing. This cannot be said of any straight line:-not of an infinite line; for, if it were perfect, it would have a limit and an end: nor of any finite line; for in every case there is something beyond it, since any finite line can be extended.
I'm not sold on his arguments
Proof : circles are great?
2:59 PM
lines are clearly gross
3:19 PM
I don't know why people think Aristotle is so great really
Bar was pretty low apparently
Aristotle is great for the other stuff, not for this part
I'd say something that stupid casts doubts on the rest
4:03 PM
it is not that stupid from the past's perspective
u can see urself writing this if u were living in the past
most children arent into empiricism. they just create a philosophy of the world in their heads
I'd argue it is v. stupid
when did the idea that we r sticking to the earth like it's a magnet come along?
was it the same time when we found that the earth is a sphere
Aristotle said that the natural motion of Earthly things is to go down
and i think he believed in geocentrism with a spherical earth
so he must have had this magnet idea
so he knew that down wasnt absolute
He very much did believe in geocentrism with a spherical earth
sorry the "didnt" was added by my subconscious
4:19 PM
He defines down in this very book
As towards the center
with a spherical earth, it's impossible to get this idea wrong
in flat earth models, there can be somehing like "everything falls down in the universe and the earth is falling down slower than the objects on earth"
did anyone belive in such notions of down in ancient history
spherical earth is extremely old knowledge, so unlikely
There was some notion of the earth falling down, although the more usual flat earth model was just that it was just floating on a big sea
Flat earth models were usually not amazingly complex
It was usually something like
The rarer half flat half round model
lol
i used to believe something similar to the first model in middle school
i read that it was a sphere but all i saw around me was flat
@Slereah so this picture is what i believed in. a dome with a flat surface
the dome being the sky
anyone have similar experiences?
4:35 PM
No I wasn't raised in Arkansas
lol
i also thought that rockets have to pierce through the sky
6 hours later…
10:18 PM
@ACuriousMind @naturallyInconsistent I think I have a concrete example that is physically relevant concerning the discussion earlier. In particular, consider a physical system with hamiltonian that is not equal to the total energy of the system $H$. What would it mean to treat this system quantum mechanically.
Consider a charged particle in an electromagnetic field. It seems like the classical Hamiltonian of a charged particle in an electromagnetic field is not identified with the total energy of the system.
@SillyGoose You just quantize it according to the rules of (constrained) quantization
why does it matter whether the Hamiltonian is "the energy"?
I guess when I say "treat this system quantum mechanically" I mean to include the usual identification of Hamiltonian eigenstates with states of definite energy.
And so how to interpret Hamiltonian eigenstates in this situation is what I am interested in knowing
depends on the system!
but saying that Hamiltonian eigenstates have to be identified with states of definite energy is definitely not what "treating a system quantum mechanically" means :P
let me revise to treating it at as systems are treated in an "introduction to quantum mechanics level textbook" :)
Sorry guys I wrote down in my notes that the ground state eigenfunction is always an even function. The problem is I know my professor said this so it must be true, nevertheless I can't think of a reason why this should be true and how I should show this. Anybody got any clues?
10:24 PM
@SillyGoose you will not meet a system whose Hamiltonian is not energy in an intro to QM textbook
well so then per the discussion we had before, "energy" ceases to be "useful" when we want to consider a charged particle in an electromagnetic field?
because typically these show up either in relativity (reparametrization-invariance forces zero Hamiltonian) or gauge theories (which require either ad-hoc methods or developing a theory of constrained quantization), neither of which you want to do in an intro context
what happens to statistical mechanics in these contexts :P...goodness
and the Hamiltonian for the EM field is energy, it's just not $T+V$ where the Lagrangian is $T-V$ (because the magnetic potential isn't a "real" potential associated with energy since the magnetic force does no work)
Let me see
symmetric potential
wrt to the x=0 axis
Oh I see maybe why you asked this
10:30 PM
this is kind of the inverse example you're looking for - the Hamiltonian is energy, but it doesn't fit into the T+V/T-V framework, see e.g. Qmechanic here
$\hat{p}^2$ is even since $\hat{p}$ is an odd operator
no wait the position operator is odd
@ClaudioMenchinelli i think theres an exercise in griffiths that proves this
Cohen said something about even hamiltonians
Oh I found it
@ClaudioMenchinelli why does it matter that the position operator is odd? There's no odd powers of the position operator in your Hamiltonian here (since by assumption it has a symmetric potential)
10:31 PM
it's the last paragraph
@ClaudioMenchinelli you can start with your definition of Schrödinger equation and consider the two cases: 1) suppose $\psi(x)$ is a solution, 2) then what is $\psi(-x)$?; the hypothesis that $V(x) = V(-x)$ must also be used
I thought the simmetry of $V(x)$ depended upon the position operator
@ClaudioMenchinelli So? No matter how it depends on it exactly, your assumption here is explicitly that $V(x)$ is symmetric/even
@ACuriousMind hm sorry what do you mean by is energy?
yep, I see now thanks, I was a bit distracted
10:34 PM
@SillyGoose it's the thing that will show up in the work-energy theorem
what do you mean by energy? :P
i'm not sure what energy is anymore...
now we're getting somewhere - if you can't even say what the definition is, why does it matter whether the Hamiltonian "is energy" or "is not energy" :P
I think it matters because I (~wrongly) took as definition that the Hamiltonian is the generator of time translations and that we call the quantity that is conserved under time translations energy. But this way of framing things seems inappropriate outside of an introductory textbook setting
what does it mean to say that a state is relativistically normalized? tong goes through this derivation to find the normalization term (i think) based on the measure being lorentz invariant, but i dont get the point of this? why cant we just normalize our states according to $\langle p \vert q \rangle = (2\pi)^3\delta(p - q)$ and be done with it?
@Relativisticcucumber you could but then you'd just get annoying factors of $2m$ all over the place
it's just the convention in relativistic contexts to not do that :P
10:43 PM
okay i see. so this is just how i should normalize the states in qft in general or is this a specific context?
I think my broad goal here is to tease out to what extent this Hamiltonian is useful. Previously, it seemed like quite a fundamental and ubiquitous tool in physics, but now it seems more like just another tool. I can adjust the way I see things in light of this to something along the lines of: we just want to find a language to write the physics in that is most convenient, i.e. most natural. it is a dream to imagine that one could find a single language in which all of physics is natural.
@SillyGoose a dream of Jaffe
and to clarify, I do not wish or mean to consider pathological cases when looking for these sorts of generalizations of concepts like energy and what not.
@SillyGoose What do you mean? The Hamiltonian still predicts the correct physics even when it's not energy!
if you hadn't been through years of intro physics where we drill this reified idea of energy into your head you would never have found this weird in the first place :P
@ACuriousMind I guess then I would like to know what this statement means. Moreover, if the Hamiltonian, whether it is energy or not, lets us make these accurate predictions, why even define such a thing as energy at all! Just call it the Hamiltonian
Another way of saying: If "being energy" is not what fundamentally makes it a valuable concept, then I would not like to call it energy as if to emphasize that energy is what is important :P
10:47 PM
@SillyGoose you want to explain Hamiltonian mechanics to students who first learn about collisions and how they obey energy and momentum conservation???
don't throw the baby out with the bath water - just because at some point you have to say goodbye to our naive idea of energy that doesn't retroactively invalidate all the cases where it was useful
I guess this suggests to me that Hamiltonian as energy is distinct from Hamiltonian in general. That is, the Hamiltonian is useful when it is identified with energy and it is useful when not but for utterly different reasons
but I would expect that the Hamiltonian is useful because of the same set of reasons for both cases. so the above is not intuitive to understand
well I guess I see it like this now and I am not sure if this is an accurate way to see things. Just like in say Tong's QFT lecture notes where the field and its conjugate field happen to be related by fourier transform, but this fourier relation is not generally true between a field and its conjugate; so is the relation between Hamiltonian and total energy of the system. It is convenient and it is useful because of its convenience, but strictly speaking the Hamiltonian has nothing
to do with total energy
I mean the problem is really that you have no abstract definition of "total energy"
it is definitely true that the notion of the Hamiltonian grew out of us realizing this conserved quantity called "energy" was kinda useful
but then the Hamiltonian formalism generalized to settings where suddenly the Hamiltonian wasn't anymore what we'd like "energy" to be (like identically zero), and the link between the Newtonian notion of "energy" and the Hamiltonian breaks
@Relativisticcucumber :-)
I don't think there's anything deep going on here - just one concept (the Hamiltonian) generalizing another (the energy)
well okay I can accept it being that way
11:06 PM
here, a and b are creation and annihilation operators, but i dont see why the solution to the KGE should include such operators?
i mean how do they appear in the general solution to that equation?
@Relativisticcucumber they're not "creation and annihilation operators" when you look at the solution to the classical KG equation, they're just coefficients for the two basic solutions $\mathrm{e}^{\mathrm{i}p_\mu x^\mu}$ and $\mathrm{e}^{-\mathrm{i}p_\mu x^\mu}$
that these turn out to be creation and annihilation operators after quantization isn't really relevant here
@ACuriousMind hm, so how is it proper to view these coefficients as the creation and annihilation operators?
@Relativisticcucumber what do you mean "proper"?
quantization turns classical observables into operators
i just mean purely in terms of looking at this as solving an ode
the classical observables of field theory are the fields, so your $\phi(x)$ becomes an operator, and so must the $a(k)$ and $b(k)$
11:16 PM
i dont see how the coefficients could be operators?
You can invert this Fourier transform to get an explicit formula for the $a(k)$ that looks like $a(k) = \phi(k) + \mathrm{i}\pi(k)$. Quantization turns the $\phi(k)$ and $\pi(k)$ into operators by definition, so the $a(k)$ become operators, too
what does it matter that this idea of decomposing the field into $a(k)$ and $b(k)$ was originally inspired by solving the KG equation?
the claim that $a(k)$ so defined is an operator is true no matter whether you even know about the KG equation or not
11:29 PM
@ACuriousMind so fields can act on states also, right?
what does being an operator mean if not that it's an operator on the space of states?
@ACuriousMind i thought a field is an operator valued function, so i more thought of it as a function but bleh so the fields are a kind of creation operator? conceptually i mean?
@Relativisticcucumber oh, that's just a language issue, we really don't keep saying "operator-valued function" all through QFT :P
both the $\phi(x)$ and $a(k)$ are "operator-valued functions", if you want to be pedantic
(if you want to be even more pedantic the $\phi(x)$ isn't even a function but a distribution but let's not get into that)
okay i see. so i am looking at two examples. it would seem that for free fields we have these two operators $b(k)$ and $a(k)$ but then when we do a complex scalar field, we have four of these operators. it's my understanding that this corresponds to having particles and antiparticles and the ability to create and annihilate them. im wondering how we show that these operators are unique? [...]
[...] like how we show that indeed the 4 operators in the complex scalar field case are four ind. operators that create and annihilate matter and antimatter?
...shouldn't the texts you're reading explain exactly that? :P
who's teaching you that these are creation and annihilation operators without showing why
11:37 PM
its possible that im just missing it. im referring to the complex scalar field example on page 13 here damtp.cam.ac.uk/user/tong/qft/two.pdf
13-14 i mean
and what exactly is the question here? You did the real scalar field before this, now you have a complex scalar field that's made out of two real scalar fields and hence has double the number of independent variables
i think i see how it workd for scalar field by looking at CRs but the antimatter and matter is confusing me as well as how we show these are two distinct types of matter
like how do i know the operators create a type of particle thats truly different than the matter counterpart?
because since the fields are just conjugates they arent even rigorously independent
so im skeptical
@Relativisticcucumber you're trying to run before you can walk - we're still just doing free field theory here
in a free scalar field theory there isn't really any meaning to saying something is "matter" or "antimatter" or whatever at all
i think my confusion is the statement that we have 4 independent operators. im only convinced that two are independent
because $\psi$ and $\psi^*$ dont introduce new info?
11:44 PM
note also that we do this in complex analysis all the time: $\frac{\partial z^\ast}{\partial z} = 0$
it's not really a physics trick, just how complex derivatives work
@ACuriousMind i think im still confused. so the answer by qmechanic says that we can solve this in 3 different ways and get the same physical result, right? and so of course we can rewrite it however we want, but in order to get 4 truly independent operators, we would need 4 degrees of freedom. what im seeing here is that regardless of how we write it, we still only have two degrees of freedom, as the first two methods state? maybe im missing the point?
well the difference between scalar and complex is the introduction of antimatter, right? so there must be a new DOF if we are now able to make and get rid of antimatter ? @ACuriousMind
@Relativisticcucumber where do you think the 2 operators for the real scalar field come from?
specifically a new DOF that comes from introducing "complexness"
ew why is it backwards
I'm confused why you accept the $a$ and $a^\dagger$ as two operators for the one real scalar field $\phi$ but then complain that we get $a,a^\dagger,b,b^\dagger$ for a complex scalar field, which is two real scalar fields
one set of c/a operators for a real scalar field, two sets of c/a operators for the complex one, which is just the complex sum of two real scalar fields
what's the problem?
11:59 PM
@ACuriousMind because its not two real scalar fields its the same field?
or i mean
by taking the complex conj of smth we get no new information, right?
@Relativisticcucumber $\phi = \phi_1 + \mathrm{i}\phi_2$ looks like two real scalar fields $\phi_1,\phi_2$ to me | 5,660 | 22,845 | {"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} | 3.3125 | 3 | CC-MAIN-2024-10 | latest | en | 0.870282 |
https://stats.stackexchange.com/questions/4717/what-is-the-difference-between-a-nested-and-a-non-nested-model/232452 | 1,560,927,501,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998923.96/warc/CC-MAIN-20190619063711-20190619085711-00123.warc.gz | 594,027,546 | 40,763 | # What is the difference between a “nested” and a “non-nested” model?
In the literature on hierarchical/multilevel models I have often read about "nested models" and "non-nested models", but what does this mean? Could anyone maybe give me some examples or tell me about the mathematical implications of this phrasing?
• This is a very overloaded term, depending on context. You have been warned. – fmark May 7 '12 at 2:00
## 6 Answers
Nested versus non-nested can mean a whole lot of things. You have nested designs versus crossed designs (see eg this explanation). You have nested models in model comparison. Nested means here that all terms of a smaller model occur in a larger model. This is a necessary condition for using most model comparison tests like likelihood ratio tests.
In the context of multilevel models I think it's better to speak of nested and non-nested factors. The difference is in how the different factors are related to one another. In a nested design, the levels of one factor only make sense within the levels of another factor.
Say you want to measure the oxygen production of leaves. You sample a number of tree species, and on every tree you sample some leaves on the bottom, in the middle and on top of the tree. This is a nested design. The difference for leaves in a different position only makes sense within one tree species. So comparing bottom leaves, middle leaves and top leaves over all trees is senseless. Or said differently: leaf position should not be modelled as a main effect.
Non-nested factors is a combination of two factors that are not related. Say you study patients, and are interested in the difference of age and gender. So you have a factor ageclass and a factor gender that are not related. You should model both age and gender as a main effect, and you can take a look at the interaction if necessary.
The difference is not always that clear. If in my first example the tree species are closely related in form and physiology, you could consider leaf position also as a valid main effect. In many cases, the choice for a nested design versus a non-nested design is more a decision of the researcher than a true fact.
Nested vs non-nested models come up in conjoint analysis and IIA. Consider the "red bus blue bus problem". You have a population where 50% of people take a car to work and the other 50% take the red bus. What happens if you add a blue bus which has the same specifications as the red bus to the equation? A multinomial logit model will predict 33% share for all three modes. We intuitively know this is not correct as the red bus and blue bus are more similar to one another than to the car and will thus take more share from one another before taking share from the car. That is where a nesting structure comes in, which is typically specified as a lambda coefficient on the similar alternatives.
Ben Akiva has put together a nice set of slides outlining the theory on this here. He begins talking about nested logit around slide 23.
One model is nested in another if you can always obtain the first model by constraining some of the parameters of the second model. For example, the linear model $y = a x + c$ is nested within the 2-degree polynomial $y = ax + bx^2 + c$, because by setting b = 0, the 2-deg. polynomial becomes identical to the linear form. In other words, a line is a special case of a polynomial, and so the two are nested.
The main implication if two models are nested is that it is relatively easy to compare them statistically. Simply put, with nested models you can consider the more complex one as being constructed by adding something to a more simple "null model". To select the best out of these two models, therefore, you simply have to find out whether that added something explains a significant amount of additional variance in the data. This scenario is actually equivalent to fitting the simple model first and removing its predicted variance from the data, and then fitting the additional component of the more complex model to the residuals from the first fit (at least with least squares estimation).
Non-nested models may explain entirely different portions of variance in the data. A complex model may even explain less variance than a simple one, if the complex one doesn't include the "right stuff" that the simple one does have. So in that case it is a bit more difficult to predict what would happen under the null hypothesis that both models explain the data equally well.
More to the point, under the null hypothesis (and given certain moderate assumptions), the difference in goodness-of-fit between two nested models follows a known distribution, the shape of which depends only on the difference in degrees of freedom between the two models. This is not true for non-nested models.
• great explanation. – mark Nov 10 '16 at 23:45
Two models are nonested or separate if one model cannot be obtained as limit of the other (or one model is not a particular case of the other)
• Can you clarify what you mean by 'limit of the other'? A nested model can be seen as one having some restriction on the parameters space compared to another, but I'm not sure if this what you intended to write. – chl Jul 20 '13 at 20:13
• I mean limit of the other for example the exponential distribution is a limit of the Gamma (as well a Weibull) distribution when the parameter of form Beta goes to 1 . – Basilio De Bragança Pereira Jun 24 '16 at 0:31
You asked about the difference between nested and nonnested models. See:
Where the subject of nonnested or separate models was treated for the first time or my forthcoming book: Choice of Separate or Nonnested Models.
• Welcome to the site, @BasilioDeBragancaPereira. It would be best to give a summary of what is in those papers so readers can decide if they want to track them down & read them. Also note that the OP specified "hierarchical/multilevel models" (students nested in classes nested in schools). Is that the context you are referring to here? – gung Jul 11 '16 at 23:28
See a simpler answer in this pdf. Essentially, a nested model is a model with less variables than a full model. One intention is to look for more parsimonious answers.
• Unfortunately, this is a simpler answer only because it is describing a different type of "nested model" than the type the OP is asking about. The OP asks instead about nested models in the context of hierarchical / multilevel models. That is, this answer, while correct in its own terms, is incorrect in the context of this thread. – gung Aug 24 '14 at 16:52
• Link is broken. – Waldir Leoncio Jun 15 '16 at 6:38
• The link says "forbidden" for me... and you don't explain much about what it says. – Glen_b Jun 15 '16 at 8:51 | 1,490 | 6,756 | {"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} | 2.734375 | 3 | CC-MAIN-2019-26 | latest | en | 0.963089 |
http://saravanamoorthy-physics.blogspot.com/2010/04/11.html | 1,484,695,854,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280128.70/warc/CC-MAIN-20170116095120-00456-ip-10-171-10-70.ec2.internal.warc.gz | 252,946,785 | 246,088 | ## 13 April 2010
### kinematics
Problem 1
A train covers 60 miles between 2 p.m. and 4 p.m. How fast was it going at 3 p.m.?
Solution:
The speed is traveled distance (60 miles) divided by traveled time (4pm – 2pm = 2hours):
Problem 2
Is it possible that the car could have accelerated to 55mph within 268 meters if the car can only accelerate from 0 to 60 mph in 15 seconds?
Solution:
Let us find the maximum acceleration of the car:
The car can accelerate from 0 to in 15 seconds. Then maximum acceleration is
If the car needs to accelerate to within 268 meters then its acceleration should be
This acceleration is less than the maximum possible acceleration, so the car can reach the speed 55 mph within 268 meters.
Problem 3
A car travels up a hill at a constant speed of 37 km/h and returns down the hill at a constant speed of 66 km/h. Calculate the average speed for the whole trip.
Solution:
By definition the average speed is the ration of the total traveled distance and the total traveled time. Let us introduce the total traveled distance of the car as L. Then the time of the travel up the hill is
The time of the travel down the hill is
The total traveled time is
Then the average velocity is
Problem 4
An archer shoots an arrow with a velocity of 30 m/s at an angle of 20 degrees with respect to the horizontal. An assistant standing on the level ground 30 m downrange from the launch point throws an apple straight up with the minimum initial speed necessary to meet the path of the arrow. What is the initial speed of the apple and at what time after the arrow is launched should the apple be thrown so that the arrow hits the apple?
Solution:
The motion of the arrow is a projectile motion. Then the motion of the arrow along horizontal direction is a motion with constant velocity:
Where the initial position is 0 and . Then
Motion along vertical axis is a motion with constant acceleration, then
Where , initial position is 0, and . Then
Then we need to find the time when the arrow will be exactly above the assistant. At this moment of time x(t) =100. Then we can find the time:
Then we can find the height of the arrow at this moment of time:
The assistant should through the apple with minimal velocity so it will reach point 5.4 m and at this height the velocity should be 0. From this condition we can find the minimal velocity:
The time of the motion of the apple to this point is
Then the apple should be thrown after
Problem 5
A box sits on a horizontal wooden board. The coefficient of static friction between the box and the board is 0.5. You grab one end of the board and lift it up, keeping the other end of the board on the ground. What is the angle between the board and the horizontal direction when the box begins to slide down the board?
Solution:
The critical angle is determined by the condition:
From this equation we can find an angle .
Problem 6
A 8 kg block is at rest on a horizontal floor. If you push horizontally on the 8 kg block with a force of 20 N, it just starts to move.
(a) What is the coefficient of static friction?
(b) A 10.0 kg block is stacked on top of the 8 kg block. What is the magnitude F of the force, acting horizontally on the 8 kg block as before, that is required to make the two blocks start to move?
Solution:
The magnitude of horizontal force should be equal to the magnitude of the maximal static friction force, which is equal to the product of the coefficient of static friction and the normal force (gravitation force in the present problem).
(a) The gravitation force is mg=8*9.8 = 78.4 N. Then the coefficient of static friction is
(b) Now we know the coefficient of static friction and we know the normal force: 18*9.8 = 176.4 N. Then we can find the magnitude of force F:
Problem 7
A car is accelerating at . Find its acceleration in .
Solution:
To find an acceleration in we need to use the relations:
,
Then we can write:
Problem 8.
We drive a distance of 1 km at 16 km/h. Then we drive an additional distance of 1 km at 32 km/h. What is our average speed?
Solution:
By definition the average speed is the ratio of traveled distance and traveled time.
The traveled distance is 2 km.
The traveled time is the sum of two contributions:
• time of the motion a distance 1 km with speed 16 km/h. It is
• time of the motion a distance 1 km with speed 32 km/h. It is
Then the average speed is
Problem 9.
An airliner reaches its takeoff speed of 163 mph in 36.2 s. What is the magnitude of its average acceleration.
Solution:
By definition the acceleration is the ratio of the change of velocity and the traveled time. In the present problem the change of velocity is 163 mph = 0.447*163 m/s = 72.9 m/s. The traveled time is 36.2 s.
Then the average acceleration is
Problem 10.
A car is initially traveling due north at 23 m/s.
(a) Find the velocity of the car after 4 s if its acceleration is due north.
(b) Find the velocity of the car after 4 s if its acceleration is instead due south.
Solution:
This is the motion with constant acceleration. The dependence of velocity on time is given by the equation:
Where .
(a) In this case the direction of acceleration is the same as the direction of initial velocity. Then and we have
(b) In this case the direction of acceleration is opposite to the direction of initial velocity. Then and we have
Problem 11.
From the dimensional analysis find the time t it takes for a ball to fall from a height h.
Solution:
We know that the time should depend on the height, h, and free fall acceleration, g. So we can write:
Where a is a dimensionless constant (we cannot find this constant) and x, y – are constant, which can be found from dimensional analysis.
The units of t is seconds. The units of h is meter, the unit of g is . Then we have:
From this equation we have:
Then
Then
Problem 12.
You are driving along the street at the speed limit (35mph) and 50 meters before reaching a traffic light you notice it becoming yellow. You accelerate to make the traffic light within the 3 seconds it takes for it to turn red. What is your speed as you cross the intersection? Assume that the acceleration is constant and that there is no air resistance.
Solution:
This is the motion with constant acceleration. If the acceleration of the car is a then we can write the expression for traveled distance:
We know the initial velocity , we also know that after 3 seconds the car travels distance 50 meters. Then we can find acceleration
Now we know acceleration, then we can find the final velocity:
Problem 13.
How high can a human throw a ball if he can throw it with initial velocity 90 mph.
Solution:
The height is given by the expression:
where . Then
Problem 14.
Mr. Letourneau is flying his broom stick parallel to the ground. He undergoes two consecutive displacements. The first is 100 km 10 degrees west of north, and the second is 120 km 50 degrees east of north. What is the magnitude of the broom stick's displacement?
Solution:
The displacement are shown schematically in the figure. From this figure we can see that the angle between the first and the second displacements is . Then from the triangle based on the first, second, and net displacements we can find the magnitude of the net displacement (cosine formula):
Problem 15.
In reaching her destination, a backpacker walks with an average velocity of 1 m/s, due west. This average velocity results, because she hikes for 6 km with an average velocity of 3 m/s due west, turns around, and hikes with an average velocity of 0.3 m/s due east.
How far east did she walk (in kilometers)?
Solution:
We will define the average velocity as the ratio of total traveled distance and total traveled time (the average velocity can be also defined as the ratio of displacement and traveled time).
The backpacker walks 6km=6000m due to west. The average velocity was 3 m/s. Then the traveled time for this motion is
Then she travels east. Let assume that her traveled time due to east is . Then the traveled distance due to east is
Then the total traveled distance is and the total traveled time is . Then the final average velocity is
From this equation we can find :
Then we can find :
Problem 16.
It takes you 9.5 minutes to walk with an average velocity of 1.2 m/s to the north from the bus stop to museum entrance. What is your displacement?
Solution:
If you travel along the straight line than the displacement is equal to the traveled distance.
Then the displacement is
where and .
Then
Problem 17.
What is 3.3 slug in kilograms?
Solution:
Since 1 slug = 14.5939 kg then we get:
Problem 18.
An athlete swims the length of a 50.0-m pool in 20.0s and makes the return trip to the starting position in 22.0s. Determine her average velocities in
(a) the first half of the swim,
(b) the second half of the swim, and
(c) the round trip.
Solution:
The average velocity is the ratio of traveled distance and traveled time.
Then
(a)
(b)
(c)
Problem 19.
A speedboat increases its speed uniformly from 20 m/s to 30 m/s in a distance of 200m. Find
(a) the magnitude of its acceleration and
(b) the time it takes the boat to travel the 200-m distance.
Solution:
(a) This is the motion with constant acceleration. We can use the following equation to find the magnitude of acceleration
where , , . Then
(b) We can find the time of the travel from the following equation:
Problem 20.
The car drives straight off the edge of a cliff that is 57 m high. The investigator at the scene of the accident notes that the point of impact is 130 m from the base of the cliff. How fast was the car traveling when it went over the cliff?
Solution:
Motion along axis x (horizontal axis) is the motion with constant velocity. So we can write down the dependence of x-coordinate as a function of time:
Where is the initial velocity (the initial velocity has only x-component, its direction is along axis x).
We know the final x-coordinate of the car – it is 130 m. But we do not know the traveled time and the initial velocity.
We can find the traveled time from the motion along axis y. This is the motion with constant acceleration (free fall acceleration). We know the height of the cliff – this is the traveled distance in y-direction. We know that initial velocity (in y-direction) is 0. Then we can write the following equation:
From this equation we can find the traveled time:
Then from the motion along axis x we have:
From this equation we can find initial velocity:
Problem 21.
An aircraft has a lift-off of 120km/h.
(a) What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff run of 240m?
(b) How long does it take the aircraft to become airborne?
Solution:
(a) This is the motion with constant acceleration. We know that the initial velocity is 0. And we know that after run of 240 m the velocity of aircraft becomes 120 km/h = 120 *1000/3600 m/s = 33.3 m/s.
Then we need to use the following equation:
From this equation we can find acceleration:
(b) Now we know acceleration and we know the final velocity so we can find the traveled time:
Problem 22.
A ball is thrown vertically upward with a speed of 25.0m/s.
(a) How high does it rise?
(b) How long does it take to reach its highest point?
(c) How long does the ball take to hit the ground after it reaches its highest point?
(d) What is its velocity when it returns to the level from which it started?
Solution:
This is the motion with constant acceleration (free fall acceleration).
(a) We need to use the following equation (the initial velocity is 25 m/s and the final velocity is 0):
(b) We need to use the following equation (the initial velocity is 25 m/s and the final velocity is 0):
(c) At this moment we do not know the final velocity, but we know the initial velocity (it is 0) and we know the height (the traveled distance) – it is 31.9 m. Then we can use the following equation:
From this equation we can find time:
It is the same time as in part (b).
(d) Now we can find the final velocity (the magnitude):
Problem 23.
A tortoise and a hare are in a road race to defend the honor of their breed. The tortoise crawls the entire 1000 meters at a speed of 0.2 m/s. The rabbit runs the first 200 meters at 2 m/s, stops to take a nap for 1.3 hours, and awakens to finish the last 800 meters with an average speed of 3 m/s. Who wins the race and by how much time?
Solution:
At first let us calculate the traveled time of tortoise. We know the speed and the distance, so we can easily find the time:
Now let us calculate the traveled time of rabbit. The traveled time consists of three parts:
1. He runs the first 200 m at 2 m/s. The time of this motion is
2. Then he take a nap for 1.3 hours:
3. Then he run the last 800 m with speed 3 m/s. The time of this motion is
Then the total traveled time is
Since then tortoise wins the race by 47 s.
Problem 24.
The slowest animal ever discovered is a crab found in the Red Sea that travels an average speed of 5.7 km/year. How long will it take this crab to travel 1 meter?
Solution:
The speed of the crab is
Then the time of the motion is
Problem 25.
A "moving sidewalk" in a busy airport terminal moves 1 m/s and is 200 m long. A passenger steps onto one end and walks, in the same direction as the sidewalk is moving, at a rate of 2.0 m/s relative to the moving sidewalk. How much time does it take the passenger to reach the opposite end of the walkway?
Solution:
The speed of the passenger relative to the ground is 2m/s+1m/s =3 m/s (since the he is working in the same direction as the direction of the motion of the sidewalk.
Then the passenger reaches the end of the sidewalk after
Problem 26.
Assume it takes 8 minutes to fill a 35.0 gal gasoline tank. (1 U.S. gal = 231 cubic inches)
(a) Calculate the rate at which the tank is filled in gallons per second.
(b) Calculate the rate at which the tank is filled in cubic meters per second.
(c) Determine the time interval, in hours, required to fill a volume at the same rate.
Solution:
The rate of filling is 35galons/8 minutes = 4.375 galons/minutes.
(a) Since 1 minute = 60 seconds then
(b) Since 1 minute = 60 seconds and then
(c) From part (b) - is filled within
Problem 27.
A plane flies 955 km due east, then turns due north and flies another 469 km. Draw an x-y axis at the starting point with the positive x-axis pointing east, and determine the polar coordinates of the plane's finale position.
Solution:
The positive x-axis is pointing east, while the positive y-axis is pointing north. Then the final position of the plane has coordinate:
x-coordinate is 955 km,
y-coordinate is 469 km.
Then the polar angle can be found from the equation:
Then .
Problem 28.
A car travels east at 89 km/h for 1 h. It then travels 26° east of north at 141 km/h for 1 h.
(a) What is the average speed for the trip?
(b) What is the average velocity for the trip?
Solution:
(a) The average speed is defined as the ratio of the total traveled distance and the traveled time. The total traveled distance is the sum of length AB (=89 km) and length BC (=141 km). The total traveled time is 2 hours. Then the average speed is
(b) The average velocity is defined as the ratio of the net displacement (the magnitude of the vector, connecting points A and C) and the traveled time (= 2 h)
We can find the length AC from the triangle ABC:
(cosine theorem).
Then the average velocity is
Problem 29.
(a) If a particle's position is given by (where t is in seconds, and x is in meters), what is it's velocity at t=1s?
(b) what is it's speed at t=1s?
(c) Is there ever an instant when the velocity is 0? If so, give the time.
Solution:
The velocity is the derivative of x(t) with respect to time. Then
(a) at t=1 s we get:
What we calculate here is an x-component of velocity. The negative sign means that the direction of velocity is opposite to the direction of axis x.
(b) the speed is the magnitude of velocity. Then at t=1 s the speed is 4 m/s
(c) to find time at which the velocity is 0 we just need to solve an equation:
From this equation we find time:
At this moment of time the velocity is 0.
Problem 30.
The captain of a plane wishes to proceed due west. The cruising speed of the plane is 245 m/s relative to the air. A weather report indicates that a 38-m/s wind is blowing from the south to the north. In what direction, measured to due west, should the pilot head the plane relative to the air?
Solution:
This is problem on the relative motion. The velocity of a plane is the vector sum of the velocity of an air and the relative velocity of the plane (relative to the air) as shown in the figure.
We know that the direction of the final velocity of the plane is from east to west (as shown in the figure).
Then from the triangle based shown in the figure we can find angle . This angle characterizes the direction of the relative velocity of the plane.
Then
Problem 31.
A missile is launched into the air at an initial velocity of 80 m/s. It is moving with constant velocity until it reaches 1000m, when the engine fails.
(a) How long does it take it to reach 1000m?
(b) How high does the missile go?
(c) How long does it take for it to fall back to the earth?
(d) How long does it stay in the air?
(e) How fast is it going when it hits the ground?
Solution:
(a) Since initial we have the motion with constant velocity we can easily find the time of the motion of the missile till it reaches the height 1000 m. The time is given by the expression:
After this point we have free fall motion – there is only one force acting on the object (it is gravitational force) – this force provide free fall acceleration.
The initial velocity is 80 m/s pointing upward. The acceleration is pointing downward. The initial height of the missile is 1000 m. Then the equations which describe this motion are the following:
(b) To find the maximum height of the missile we can use the last equation. The velocity at the maximum height is 0. Then
(c) To find the time when the missile hits the ground we need to use the first equation:
When the missile hits the ground h=0. Then
From this equation we can find time: 24.6 s.
(d) Then we can find the time when the missile is in the air: it is the sum of the time when it reaches 1000 m and the time when it hits the ground:
(e) To find the speed of the missile when it hits the ground we need to use the last equation:
When the missile hits the ground h=0. Then
Problem 32.
The highest barrier that a projectile can clear is 14 m, when the projectile is launched at an angle of 30.0 degrees above the horizontal. What is the projectile's launch speed?
The maximum height of the projectile is given by the equation:
where is the launch angle. Then we can find initial velocity:
Problem 33.
A rocket is fired vertically upwards with initial velocity 80 m/s at the ground level. Its engines then fire and it is accelerated at until it reaches an altitude of 1000 m. At that point the engines fail and the rocket goes into free-fall. Disregard air resistance.
(a) How long was the rocket above the ground?
(b) What is the maximum altitude?
(c) What is the velocity just before it collides with the ground?
Solution:
The first part of the motion is the motion with constant acceleration at . The initial velocity for this motion is 80 m/s. Then we can write the equation, which describe the dependence of height of the rocket on time:
From this equation we can find the time when the rocket reach the height 1000 m = h:
The solution of this equation is 10 s. So after 10 seconds the engine fails. The velocity at this moment of time is
After this moment of time we have free fall motion – there is only one force acting on the object (it is gravitational force) – this force provide free fall acceleration.
The initial velocity is 120 m/s pointing upward. The acceleration is pointing downward. The initial height of the rocket is 1000 m. Then the equations which describe this motion are the following:
To find the maximum height of the rocket we can use the last equation. The velocity at the maximum height is 0. Then
This is the answer to part (b).
To find the time when the rocket hits the ground we need to use the first equation:
When the rocket hits the ground h=0. Then
From this equation we can find time: 31 s.
Then we can find the time when the rocket is in the air: it is the sum of the time when it reaches 1000 m and the time when it hits the ground:
This is the answer to part (a).
To find the speed of the rocket when it hits the ground we need to use the last equation:
When the missile hits the ground h=0. Then
Problem 34.
An object is fired straight up at a speed of 9.8m/s. Compute its maximum altitude and the time it takes to reach that height. Ignore air resistance.
Solution:
This is the free fall motion. The initial velocity is 9.8 m/s pointing upward. The acceleration is pointing downward. The initial height of the object is 0 m. Then the equations which describe this motion are the following:
The maximum altitude corresponds to the condition that the velocity at this point is 0. Then from the second equation we can find the time:
Then
From the first equation we can find height:
Problem 35.
A rock is dropped from rest into a well.
(a) The sound of the splash is heard 4 s after the rock is released from rest.
How far below to top of the well is the surface of the water? (the speed of sound in air at ambient temperature is 336m/s).
(b) If the travel time for the sound is neglected, what % error is introduced when the depth of the well is calculated?
Solution:
(a) If h is the depth of the well then to find the time of the rock's fall we need to use the following equation:
From this equation we can find time:
After the rock hits the water the sound of splash will propagate the distance h with speed 336 m/s. After time the sound reaches the ground. Then total time is
This time is equal to 4 s. Then
From this equation we can find h:
(b) If we neglect the sound velocity then t=4 s and we can find the height from the equation:
If we compare this result with the result from part (a) then we can find an error:
Problem 36.
A ball is thrown upward with initial velocity v. What is the change in speed on the way up? What is the change in speed on the way down? Disregard air resistance.
Solution :
he speed of the ball is the magnitude of velocity.
On the way up: initial speed is v, the final speed is 0. Then the change of the speed is 0-v=-v (it is negative).
On the way down: initial speed is 0, the final speed is v. Then the change of the speed is v-0=v (it is positive).
Problem 37.
A displacement, s, of an object as a function of time, t, is given by
a) Find an expression for the acceleration of the object
b) Explain why this expression indicates that the acceleration is not constant
Solution :
(a) Acceleration is the second derivative of displacement with respect to time.
The first derivative of displacement with respect to time is a velocity:
The derivative of the velocity is an acceleration:
(b) From this expression we can see that the acceleration depends on time (at different moments of time we have different acceleration) – it means that acceleration is not constant.
Problem 38.
Snow is falling vertically at a constant speed of 3 m/s. At what angle from the vertical do the snowflakes appear to be falling as viewed by the driver of a car traveling on a straight, level road with a speed of 60 km/h?
Solution :
At first we need to have the same units, so we need to convert the speed of the car in m/s:
Then we need to find the direction of the velocity of the snowflakes relative to the car:
Then from the triangle shown in the picture we can find angle :
Then
Problem 39.
A dart is thrown horizontally with an initial speed of 10 m/s toward point P, the bull's-eye on a dart board. It hits at point Q on the rim, vertically below P, 0.2 s later. What is the distance PQ? How far away from the dart board is the dart released?
Solution :
The initial velocity is horizontal, the magnitude is 10 m/s. Then the equation of motion along horizontal axis has the following form:
After 0.2 seconds the dart hits the target. From this condition and from the above equation we can find the distance between the point where the dart is released and the board:
The equations which describe the motion along the vertical axis are the following:
If we substitute into the first equation the traveled time then we can find the distance between points P and Q:
The minus sign means that point Q is below point P.
Problem 40.
A boy walks to the store using the following path: 0.4 miles west, 0.2 miles north, 0.3 miles east. What is his total displacement? That is, what is the length and direction of the vector that points from her house directly to the store?
Solution :
The net displacement is the vector sum of three displacements as shown in the figure. It is more easier to find the net displacement in terms of components: x component (along east direction), y component (along north direction).
x component = x component of the first displacement + x component of the second displacement + x component of the third displacement = -0.4 + 0 + 0.3 = -0.1 miles.
y component = y component of the first displacement + y component of the second displacement + y component of the third displacement = 0 + 0.2 + 0 = 0.2 miles.
Then the magnitude of the net displacement is
The angle between the direction of the net displacement and west direction is
Then
Problem 41.
A motor scooter travels east at a speed of 13 m/s. The driver then reverses direction and heads west at 17 m/s. What was the change in velocity of the scooter?
Solution :
By definition the change in the velocity is equal to final velocity minus initial velocity.
The final velocity is 17 m/s pointing west. The initial velocity is 13 m/s pointing east. Since the velocity is a vector we need to describe the change of the velocity in terms of the components.
We choose the coordinate axis pointing to the west. Then the component of the final velocity is +17 m/s, and the component of the initial velocity is -13 m/s. Then the change in the velocity is
17 – (-13) = 30 m/s
Problem 42.
A bicycle travels 3.2 km due east in 0.1 h, the 3.2 km at 15.0 degrees east of north in 0.21 h, and finally another 3.2km due east in 0.1 h to reach its destination. The time lost in turning is negligible. What is the average velocity for the entire trip?
Solution :
By definition the average velocity is equal to the ratio of the magnitude of the net displacement and the traveled time. The net displacement is shown in the figure.
We can find the x and y components of the vector of net displacement.
The x-component is
The y-component is
Then the magnitude of the net displacement is
The traveled time is 0.1 h + 0.21 h + 0.1 h = 0.41 h. Then the average velocity is
Problem 43.
A swimmer dived off a cliff with a running horizontal leap. What must his minimum speed be just as he leaves the top of the cliff so that he will miss the ledge at the bottom which is 2 m wide and 9 m below the top of the cliff?
Solution:
This is the projectile motion with horizontal initial velocity. In this case the motion in horizontal direction is described by the following equation:
Where v is the initial velocity.
The motion along vertical axis y is described by the following equations (y-component of initial velocity is 0):
We know that the final point of the motion is the end of the ledge. The coordinates of the final point are: x=2 m and y = -9m.
We can substitute these numbers in the above equations and obtain:
From the seconds equation we can find time:
Then we substitute this time in the first equation and obtain the initial velocity:
Problem 44.
A ferries wheel with radius 20 m which rotates counterclockwise, is just starting up. At a given moment, a passenger on the rim of the wheel and passing through the lowest point of his circular motion is moving at 3.00m/s and in gaining speed at a rate of .
Find the magnitude of the passenger's acceleration at the instant.
Solution:
The acceleration at this point has two components: the tangential acceleration, which is equal to , and centripetal acceleration, which is given by the expression:
These components are orthogonal. Then the net acceleration is
Problem 45.
In an action film hero is supposed to throw a grenade from his car, which is going 90.km/h, to his enemy's car, which is going 110 km/h. The enemy's car is 15.8 m in front of the hero's when he lets go of the grenade. If the hero throws the grenade so its initial velocity relative to him is at an angle of 45(degree) above the horizontal, what should the magnitude of the initial velocity be? The cars are both traveling in the same direction on a level road. Ignore air resistance.
Find the magnitude of the velocity both relative to the hero and relative to the earth.
Solution:
At first we need to convert all the velocities into m/s:
We will describe the motion in the hero's reference frame. In this reference frame the velocity of the enemy's car is
And the grenade is released with the launch angle . We introduce the magnitude of initial velocity of grenade (relative to the hero's car) as v.
After the grenade is released it is moving according to the equations of projectile motion. Along axis x we have motion with constant velocity:
Along axis y we have motion with constant acceleration:
In these equations at the initial moment of time the grenade has zero coordinates.
We can also write down the equation of enemy's car motion:
Now we need to write down the condition that the grenade hits the enemy's car:
At some moment of time the x coordinate of grenade should be equation to the x-coordinate of enemy's car and at the same moment of time the y-coordinate of grenade should be equal 0.
Then we have:
Then we just need to solve the system of two equations with two unknown variables:
Then
And
Then
This is the velocity of the grenade relative to the hero.
We can also find the magnitude of velocity relative to the earth. The x-component of this velocity is
The y-component of the velocity is
Then the magnitude is
This is the magnitude of the velocity of grenade relative to the earth.
Problem 46.
A flowerpot falls from a windowsill 25.0 m above the sidewalk.
(a) how fast is the flowerpot moving when it strikes the ground?
(b) how much time does a passerby on the sidewalk below have to move out of the
way before the flowerpot hits the ground?
Solution :
This is the free fall motion with constant acceleration. The equations which describe this motion are the following:
where we introduced an axis y with upward direction. The origin of axis y is the initial position of the flowerpot. The initial velocity of flowerpot is 0.
We know the final position of the flowerpot: the coordinate of the final point is -25 m.
(a) We substitute this coordinate in the third equation an obtain the velocity:
The negative sign means that the direction of velocity is downward. The magnitude of velocity is 22 m/s.
(b) The traveled time can be found from the first equation in the above system. We substitute y(t)=-25 and obtain time:
Problem 47.
A tennis ball is thrown vertically upward with an initial velocity of +8.0 m/s.
(a) what will the ball\'s speed be when it returns to its starting point?
(b) how long will the ball take to reach its starting point?
Solution :
This is the free fall motion with constant acceleration. The equations which describe this motion are the following:
where we introduced an axis y with upward direction. The origin of axis y is the initial position of the ball. The initial velocity of the ball is 8 m/s.
(a) The final point is the starting point. It means that the coordinate of the final point is 0. We substitute this coordinate in the third equation and obtain:
It means that the magnitude of the final velocity is 8 m/s. The direction of the final velocity is downward. So we should write
Minus sign means that the direction is downward.
(b) To find the traveled time we substitute the final position (0 m) in the first equation of the system and obtain:
Then
Problem 48.
Problem 4.14 of Serway & Jewett: A firefighter, a distance d from a burning building,
directs a stream of water from a fire hose at angle above the horizontal. If the initial
speed of the stream is , at what height h does the water strike the building?
Solution :
This is the projectile motion. The equation which describe the motion along horizontal axis x is the following
where is the x-component of initial velocity.
Along axis y we have motion with constant acceleration:
In these equations at the initial moment of time the water has zero coordinate.
What do we know about the final point: we know only the x-coordinate of the final point – it is x=d. We can substitute this coordinate in the first equation (which describe the motion along axis x):
From this equation we can find the traveled time:
Then we substitute this time into the equation which describe the y-coordinate of the water:
This expression gives as the height where the water strikes the building.
Problem 49.
If it takes a player 3 seconds to run from the batter's box to the first base at an average speed of 6.5 m/s, what is the distance she covers in that time?
Solution :
In this problem we just need to use the relation between the traveled time, traveled distance, and the average velocity:
We know the traveled time and the average velocity, then the distance is
Problem 50.
A car goes down a certain road at an average speed of 40 km/h and returns along the same road at an average speed of 60 km/h. Calculate the average speed for the round trip.
Solution :
Solution:
By definition the average speed is the traveled distance divided by the traveled time. Let introduce L as the traveled distance of the car in one direction. Then the total traveled distance is 2L. Let us find the traveled time. It is
Then the average speed is
Problem 51.
(Inquiry into Physics-5th ed.,Ostdiek,Bord) On a day the wind is blowing toward the south at 3m/s, a runner jogs west at 4m/s. What is the velocity of the air relative to the runner?
Solution :
By definition the relative velocity is the vector difference of the velocity of the air minus velocity of the runner (see figure).
From the triangle we can find the magnitude of relative velocity:
Problem 52.
(Inquiry into Physics-5th ed.,Ostdiek,Bord) A runner has an average speed of 4 m/s during a race. How far does the runner travel in 20 minutes?
Solution :
By definition the traveled distance is equal to the product of traveled time and the average velocity. In this problem we need to convert time into seconds: 20 minutes = 20*60 seconds = 1200 seconds. Then
Problem 53.
A car starts from the rest and accelerates uniformly for t=5 seconds over a distance of 100m. Find the acceleration of the car.
Solution :
This is the motion with constant acceleration. The equations which describe this motion are the following:
We know the traveled time and we know the traveled distance, then from the first equation we can find acceleration:
Problem 54.
On the moon, an object is dropped from a height of 2 m. The acceleration of gravity on the moon is . Determine the time it takes for the object to fall to the surface of the moon.
Solution :
This is the motion with constant acceleration (free fall motion). The equations which describe this motion are the following:
We know the traveled distance (y=2 m), we know acceleration, then we can find time:
Problem 55.
A coin is dropped from a 400 m high tower. What is the coin's velocity when it hits the ground? How long does it take to get there?
Solution:
This is the free fall motion – motion with constant acceleration, . The equations, which describe this motion, are the following:
where the axis y has upward direction.
The final position of the coin is the ground. The coordinate of the final point is -400m. We can substitute this value in the first equation and obtain the traveled time:
Then
Then from the second equation we can find the velocity of the coin at the ground:
The minus sign means that the direction of velocity is downward.
Problem 56.
Does the odometer of a car measure a scalar or a vector quantity?
Solution :
The odometer of the car shows the magnitude of the vector quantity.
The odometer shows the speed of the car.
The speed of the car is the magnitude of the velocity (which is the vector) of the car.
So the odometer gives us only the number, it does not show us the direction of velocity. The odometer measure the scalar quantity – the speed of the car.
Problem 57.
The speed of light is about . Convert it into miles per hour (mph).
Solution :
Solution:
In this problem we need to know the following relations (see unit conversion):
1 mile = 1609 meter
1 hour = 3600 s
Then
Problem 58.
You are driving towards a traffic signal when it turns yellow. Your speed is 55 km/h, and your best deceleration has the magnitude of . Your best reaction time before breaking is . To avoid having the front of your car enter the intersection after the light turns red, should you break to a stop or continue to move at 55 km/h if the distance to the intersection and the duration of the yellow light are
(a) 40m and 2.8s, and
(b) 32m and 1.8s?
Solution :
The first step – we need to convert the speed to m/s:
We have two possible types of motion:
1. motion with constant speed till the driver reaches the intersection. Time of this motion is
2. motion with deceleration. The driver starts deceleration after . During this time the driver travels the distance:
The equations which describe these motion are the following:
where the initial position of the car has zero coordinate.
At the final point the velocity is 0. Then
During this time the position of the car is
Now we can analyze the problem:
(a) 40m and 2.8s. Then for motion 1 we have:
Since 2.6<2.8 then the driver can cross the intersection if he will move with constant speed.
(b) 32m and 1.8s. In this case:
Since 2.1>1,8 the driver cannot cross the intersection.
Therefore in this part we need to consider case 2. But even in this case the distance he will travel before he stops is 34.2 m, which is greater than 32 m. Therefore he cannot stop before the intersection.
Problem 59.
The tips of the blades in a food blender are moving with a speed of 20 m/s in a circle that has a radius of 0.06 m. How much time does it take for the blades to make one revolution?
Solution :
The traveled distance of a tip is
where R=0.06 m.
Then the time is equal to the ratio of the traveled time and the average velocity:
Problem 60.
While you are traveling in a car on a straight road at 90 km/hr, another car passes you in the same direction; its speedometer reads 120km/hr. What is your velocity relative to the other driver? What is the other car's velocity relative to you?
Solution :
The relative velocity is equal to the difference of velocities.
So if you need to find your relative velocity then you need to subtract from your velocity the velocity of another car:
The minus sign here gives us the direction of relative velocity: the direction is opposite to the directions of motions of the cars.
Similarly we can find:
The plus sign here gives us the direction of relative velocity: the direction is the same as the directions of motions of the cars.
Problem 61.
A ball is thrown from a point 1 m above the ground. The initial velocity is 20 m/s at an angle of 40 degrees above the horizontal.
(a) Find the maximum height of the ball above the ground.
(b) Calculate the speed of the ball at the highest point in the trajectory.
Solution :
This is the projectile motion. The equation which describe the motion along horizontal axis x is the following
where is the x-component of initial velocity.
Along axis y we have motion with constant acceleration:
(a) The point of the maximum height corresponds to the condition that . Then
and
Then
This is the maximum height.
(b) The y-component of velocity at maximum height is 0, then the speed is equal to x-component of velocity:
Problem 62.
A tortoise can run with a speed of 10.0 cm/s, and a hare can run exactly 10 times as fast. In a race, they both start at the same time, but the hare stops to rest for 3.00 min. The tortoise wins by 10 cm.
(a) How long does the race take?
(b) What is the length of the race?
Solution :
(a) If L is the length of the race then the tortoise run this race the time:
When the tortoise crosses the finish line the hare is 20 cm=0.2 m behind tortoise. It means that during time it traveled the distance L-0.2. Since he stopped for rest for 3 minutes =3*60 s=180 s and his speed is , then we can write the equation:
Taking into account the first equation we have:
Then
(b) Then we can find the length of the race:
Problem 63.
Emily takes a trip, driving with a constant velocity of 90 km/h to the north except for a 30 min rest stop. If Emily's average velocity is 75 km/h to the north, how long does the trip take?
Solution :
The average velocity is the ratio of traveled distance and the traveled time:
If the traveled time is t, then Emily traveled time (t-0.5)h with the speed 90 km/h. Then the traveled distance is
We substitute this relation in the first equation and obtain:
Then
Then we can find L (length of the trip):
Problem 64.
To qualify for the finals in a racing event, a race car must achieve an average speed of 250 km/h on a track with a total length of 2000 m. If a particular car covers the first half of the track at an average speed of 230 km/h, what minimum average speed must it have in the second half of the event to qualify?
Solution :
The average speed is the ratio of traveled distance and traveled time:
For the whole track we have: and . Then the traveled time is
The car travels the first half of the track (1 km) with speed 230 km/h. The time of this motion is
Since the total traveled time should be 0.008 h, then the second part of the track the car should travel 0.008-0.0043 = 0.0037 h. Since the distance of this motion is 1 km, then the average speed is
Problem 65.
A skier is accelerating down a 30 degree hill at .
(a) What is the vertical component of her acceleration?
(b) How long will it take her to reach the bottom of the hill, assuming she starts from rest and accelerates uniformly, if the elevation is 300 m?
Solution :
(a) If we assume that the direction of the vertical axis is downward then the vertical component is positive and is equal to
(b) This is the motion with constant acceleration then the equation which describe this motion is the following:
The traveled distance (L) is related to the elevation (h) by the equation:
Now we know acceleration and the traveled distance then we can find the traveled time:
Problem 66.
You are trying to cross a river that flows due south with a strong current. You start out in you motorboat on the west bank desiring to reach the east bank directly across from your starting point. Which direction should head your motorboat? Dram a picture of the river, the banks, and your motorboat, and include the relevant velocity vectors. What information would you need in order to determine the actual direction you need to head?
Solution :
To find the actual direction of the boat velocity we need to know the velocity of the river and the speed of the boat relative to the river (relative velocity). Then the actual velocity (this is the vector) is
The diagram is shown below.
Problem 67.
(a) One liter ( ) of oil is spilled onto a smooth lake. If the oil spreads out uniformly until it makes an oil slick just one molecule thick, with adjacent molecules just touching, estimate the diameter of the (roughly circular) oil slick. Assume the oil molecules have a diameter of .
(b) Recalculate for the Exxon Valdez oil spill (March 1989), in which 11 million gallons (42 million L) of crude oil coated Prince William Sound in Alaska.
Solution :
(a) To find the diameter of the oil slick we just need to write down the equation that the volume of the oil slick is equal to . The height of the oil slick is equal to the diameter of the molecule: .
Then the diameter of the oil slick can be found from the equation:
Then
(b) Now the volume is not but
Then using the same equations as in part (a) we obtain
Problem 68.
The acceleration of an object as a function of time is . Determine the
(a) velocity and
(b) the position of the object as a function of time
if it is located at x = 2 m and has a velocity of 3 m/s at time t = 0 s.
Solution :
(a) By definition the velocity of the object is the integral of acceleration with respect to time:
where is the initial velocity (at t=0).
(b) The position can found as an integral of velocity with respect to time:
where is the initial position (at t=0) of the object.
Problem 69.
An automobile traveling 90 km/h overtakes a 1.5-km-long train traveling in the same direction on a track parallel to the road. If the train's speed is 70 km/h,
(a) how long does it take the car to pass it, and
(b) how far will the car have traveled in the time?
Solution :
It is easier to solve this problem if we introduce the relative velocity of the car (relative to the train). The velocity of the car relative to the train is (90-70) km/h=20 km/h.
In the relative description the train is not moving and the car is moving with constant speed 20 km/h.
(a) Then we can easily find the time the car needs to pass the train. It is
(b) To find the actual traveled distance of the car we just need to multiply the traveled time (0.075 h) by the actual speed of the car:
Problem 70.
Two cannonballs, A and B, are fired from the ground with identical initial speeds, but with launch angle of cannonball A larger than the launch angle of cannonball B.
(a) Which cannonball reaches a higher elevation?
(b) Which cannonball stays longer in the air?
(c) Which cannonball travels farther?
Solution :
(a) The vertical component of the initial velocity of the projectile is proportional to sinus of the launch angle. It means that the vertical component of initial velocity of cannonball A is larger than the vertical component of cannonball B. Then cannonball A reaches a higher elevation.
(b) The time in the air depends only on the vertical component of initial velocity of projectile. The larger the vertical component of the velocity the larger the time in the air. Then the cannonball A will stay longer in the air.
(c) From the data of the problem we cannot tell which one will travel farther. The horizontal length is proportional to . It has maximum at angle . Therefore we can have both possibilities:
cannonball A travels further than cannonball B (for example, if the launch angle of cannonball A is and the launch angler of cannonball B is ;
cannonball B travels further than cannonball A (for example, if the launch angle of cannonball B is and launch angler of cannonball A is ;
Problem 71.
A stone falls freely from rest for 10 s. What is the stones displacement during this time.
Solution :
This is the free fall motion. The displacement (axis y is pointing downward) is given by the equation
Then at t=10 s we have:
Problem 72.
A typical atom has a diameter of . What is this in inches?
Solution :
In this problem we need to use the relation: 1m = 39.37 inch. Then
Problem 73.
A rock is shot up vertically upward from the edge of the top of the building. The rock reaches its maximum height 2 s after being shot. Them, after barely missing the edge of the building as it falls downward, the rock strikes the ground 8 s after it was launched. Find
(a) upward velocity the rock was shot at;
(b) the maximum height above the building the rock reaches; and
(c) how tall is the building?
Disregard air resistance.
Solution :
We have free fall motion of the rock. The equations, which describe the motion of the rock are the following:
Where – initial velocity, - initial height (height of the building).
(a) At the maximum height the velocity of the rock is zero. Then from the second equation we can find the initial velocity (since t= 2s)
(b) Then we can substitute the initial velocity, the traveled time (2 s) into the first equation and find the maximum height above the building:
(c) We know that after 8 s the rock hits the ground. It means that the position of the rock at this moment of time is 0. Then we substitute this time and this position into the first equation and find the height of the building:
Problem 74.
A car going at 10 m/s undergoes an acceleration of for 6 seconds. How far did it go when it was accelerating?
Solution :
In this problem we just need to use the equation:
Then at t = 6s we have:
Problem 75.
What is the displacement of a car accelerating from 5 m/s (right) to 10 m/s (right) in 2.0s?
Solution:
In this problem we need to use two equations which describe the motion with constant acceleration:
Where – acceleration, - initial velocity.
Then from the second equation we can find acceleration:
Then from the first equation we can find displacement
Problem 76.
A plane flies 50 degrees east of south for 100 km then 400 km north and then 20 degrees north of west for 250 km.
Solution:
This problem is easier to solve in terms of components.
The net displacement is the vector sum of three displacements: the first displacement (, the second displacement ( ), and the third displacement ( ):
Then we can find the x and y components of the displacement vectors and finally the net displacement:
Then the magnitude of the net displacement is
Problem 77.
Calculate the hang time of an athlete who jumps a vertical distance of .9 meters considering the acceleration of gravity is
Solution:
Without air resistance we have free fall motion. Then the maximum height can be related to the traveled distance (from the ground to the maximum height) by the equation:
where is the initial velocity.
At the maximum height the velocity is 0. Then
Then
And
The total hang time is the twice of this time: 2*0.42=0.84 s.
Problem 78.
A man is driving at the speed 40 mph when he see an obstacle at distance 300 ft ahead of his position. The driver applies the brakes and decelerates at How long does it take him to stop the vehicle? How far from the obstacle will the driver be when he finally stops?
Solution:
At first we need to convert all the variables into correct units (SI units):
Then we have motion with constant deceleration. Then
The car stops when then
And
The traveled distance is
Then the distance between the car and the obstacle is
Problem 79.
The highest barrier that a projectile can clear is 20 m, when the projectile is launched at an angle of 40.0 degrees above the horizontal. What is the projectile's launch speed?
Solution:
The meaning that the highest barrier the projectile can clear is 20 m is that the maximum height of the projectile is 20 m.
Since we are interested only in the height of the projectile then we can consider only the motion of the projectile along the vertical direction (axis y). Let us introduce the launch speed of the projectile as . Then the y component of the initial velocity of the projectile is
................................................(1)
The motion along axis y is a free fall motion and it is described by the following equations (only two equations are independent):
We know the initial height (y-coordinate) of the projectile: .
Then we introduce the final point – the point at which the projectile has the maximum height. We know this height: .
We also know that the y-component of final velocity (at the maximum height) is zero. We substitute these values in the third equation and obtain
From this equation we can find :
Then from equation (1) we can find the launch speed of the projectile:
Problem 80.
You travel on the highway at a rate of 60 mph for 1 hour and at 50 mph for 2 hours and 40 mph for 3 hours. What is the total distance you have traveled? What is your average speed during the trip?
Solution:
We divide the whole trip into three intervals:
(1) travel at a speed of 60 mph for 1 hour.
(2) travel at a speed of 50 mph for 2 hours,
(3) travel at a speed of 40 mph for 3 hours.
For each interval we know speed and time, then we can easily find the traveled distance (for each interval). The distance is equal to the product of time and speed (motion with constant velocity):
Then
(1) speed is 60 mph, time is 1 hour. Then
(2) speed is 50 mph, time is 2 hour. Then
(3) speed is 40 mph, time is 3 hour. Then
Then the total traveled distance is
The average speed is defined as the ratio of total traveled distance and the total traveled time. The total traveled time is
Now we can find the average speed:
Problem 81.
An object travels a distance of 5 km towards east, then 4 km towards north and finally 10 km towards east
1) what is total traveled distance?
2) what is resultant displacement?
Solution:
We show each displacement (travelled paths) in the figure.
(1) The total traveled distance is the sum of all traveled distances. It does not matter what the relative directions of displacements are, we just need to add the magnitude of displacements:
Problem 82.
Can two displacements (vectors) of different magnitudes be combined to give a zero displacement (resultant)?
Solution:
The sum of two vectors (displacements) is zero only if
1. vectors have opposite directions;
2. vectors have the same magnitude.
Indeed, the sum of two vectors is shown in the figure below for a fixed orientation of the first vector and different orientations of the second vector. In the picture we assume that the magnitude of the second vector is less than the magnitude of the first vector.
We can see that the sum of two vectors has the smallest magnitude when the vectors have opposite directions. In this case the sum is zero only if the vectors have the same magnitude.
Problem 83.
What is the speed (m/sec) needed for a stunt driver to launch from a 20 degree ramp to land 15 m away?
What is his maximum height?
Solution:
Initial speed.
This is the projectile motion. There are two sets of equations, which describe the motion of the projectile (stunt driver):
Set 1: motion along horizontal axis (axis x – see figure). This is the motion with constant velocity. There is only one equation, which describe this motion:
..............................(1)
Here and .
Set 2: motion along vertical axis (axisy – see figure). This is the motion with constant acceleration – free fall motion. There are three equations, which describe this motion. Only two equations are independent, but it is convenient to write all three equations:
................(2)
...........................(3)
...........................(4)
We know that the y-coordinate of the final point (point B) is 0 and the x-coordinate of the final point is 15 m. We substitute these values in equations (1) and (2) and obtain
Then from the first equation we can find :
Substitute this expression into the second equation:
From this equation we can easily find the initial velocity:
Maximum height.
The condition that the projectile is at the point with the maximum height is that the y-component of its velocity at this point is zero. It is easier to find the maximum height from equation (4). Indeed, we substitute in this equation and obtain:
Then
We know , then we can find the maximum height:
Problem 84.
A body moves 4 km towards East from a fixed point A and reaches point B. Then it covers 5 km towards North and arrives at point C. Find the distance and directions of the net displacement.
Solution:
We show two displacements (travelled paths) in the figure.
The net (resultant) displacement is the vector sum of two displacements: the first displacement () and the second displacement ():
The easiest way to find the net displacement is to introduce coordinate system (axes x and y) and then find the x and y components of the net vector-displacement. The x and y components of the displacement () and displacement () are the following:
Then the x and y components of the net displacement is
Then the magnitude of the net displacement is
The direction of the net displacement is characterized by angle (shown in the figure), which can be found from the known x and y components of the net displacement:
Problem 85.
A baseball player hits a homerun, and the ball lands in the left field seats, which is 120 m away from the point at which the ball was hit. The ball lands with a velocity of 20 m/s at an angle of 30 degrees below horizontal. Ignoring air resistance
(A) find the initial velocity and the angle above horizontal with which the ball leaves the bat;
(B) find the height of the ball relatively to the ground.
Solution:
(A) Initial velocity.
Without air resistance this is simple projectile motion. In the present problem we do not know initial velocity: we do not know the magnitude of the velocity (speed) and we do not know its direction.
There are two sets of equations, which describe the motion of the projectile (ball).
Set 1: motion along horizontal axis (axis x – see figure). This is the motion with constant velocity. There is only one equation, which describe this motion:
................................................(1)
Here .
Since the motion along the axis x is the motion with constant velocity then the x-component of the velocity is constant. We know the velocity at the final point. Then we can find the x-component of the velocity at the final point:
This x-component of the velocity is equal to the x-component of the initial velocity:
...........(2)
We also know the x-coordinate of the final point (point B): it is 120 m. We substitute this value in equation (1) and obtain
From this equation we can find the time of travel from point A to point B:
Now we need to analyze the second set of equations.
Set 2: motion along vertical axis (axis y – see figure). This is the motion with constant acceleration – free fall motion. There are three equations, which describe this motion. Only two equations are independent, but it is convenient to write all three equations:
.............(3)
................................................(4)
Since the initial y-coordinate is zero, then
.............................................(5)
We know the y component of the final velocity
This is the y-component of the velocity at the moment of time . We substitute these values in equation (4) and obtain
From this equation we can find the y-component of the initial velocity:
Finally we know the x- and y-components of the initial velocity:
From these expressions we can find the magnitude of the initial velocity and the direction (angle) of the initial velocity:
Now we know the initial velocity.
(B) Final height.
We need to find the final height of the ball (the final y-coordinate). To find the final height we can use equation (3). We just need to substitute the y-component of the initial velocity and the traveled time in this equation:
Problem 86.
A body covers 1/4 journey with a speed of 40 km/h, 1/2 of it with 50 km/h and remaining with the speed of 60 km/h. Calculate average speed for entire journey.
Solution:
In this problem we need to use the definition of the average speed. The average speed is equal to the ratio of the total travelled distance and the total traveled time:
We introduce the total traveled distance as and then calculate the total traveled time.
We know that 1/4 of the journey a body moves with a speed of 40 km/h. It means that the body moves a distance of with the speed of 40 km/h. Then we can find the time of this motion:
Then the body moves 1/2 of the journey with a speed of 50 km/h. It means that the body moves a distance of with the speed of 50 km/h. We can find the time of this motion:
Then the body moves the rest of the journey (which is 1/4 of the journey) with a speed of 60 km/h. It means that the body moves a distance of with the speed of 60 km/h. We can find the time of this motion:
Then the total traveled time is
Then the average speed is | 13,746 | 59,996 | {"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} | 4.5 | 4 | CC-MAIN-2017-04 | longest | en | 0.926962 |
https://istopdeath.com/simplify-x3-4x1-5/ | 1,675,008,775,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499744.74/warc/CC-MAIN-20230129144110-20230129174110-00779.warc.gz | 335,959,227 | 14,298 | # Simplify x^(3/4)*x^(1/5)
Use the power rule to combine exponents.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply by .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by . | 98 | 447 | {"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} | 3.234375 | 3 | CC-MAIN-2023-06 | latest | en | 0.933259 |
https://www.transentis.com/system-dynamics-simulation/en/ | 1,652,687,723,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00513.warc.gz | 1,193,018,339 | 99,646 | System Dynamics Simulation
An introduction
In this section, we will create a simulation model based on the stock and flow diagram we developed in the previous section on stock and flow diagrams. For ease of reference, we have included the diagram here:
As noted in the previous section, this stock and flow diagram is just a visual notation for a mathematical model of the system that can be used to simulate the system’s behavior using computer software. In these mathematical models, stocks and flows are represented by integral equations, converters are represented by functions of their input and/or time. In most cases, these integral equations cannot be solved analytically but instead are approximated using numerical integration.
Many techniques exist for numerical integration, the most basic of these is Euler integration, named after the Swiss mathematician Leonhard Euler (1707-1783): using this technique, we do not attempt to calculate the system’s state for every possible instance in time. Instead, we define a finite number of discrete time steps and make the basic assumption that the rates at which the system changes remain constant during these time steps. The smaller the time step becomes, the more precise our calculations will be. This approach is particularly reasonable in business because the accounting systems used in business are rarely updated more than once a day – it is also important to bear in mind here that our interest is not to precisely solve integral equations, but to make predictions about how your business will behave: mostly the simplifying assumptions you make about your business when creating your model have far more impact on the results than the size of the time step you choose.
Based on these assumptions, we can now build a simulation based on the stock and flow diagram:
1. System dynamics simulations run in simulated time. Simulation time starts at time t=0 and then runs for a predefined period of time. In business simulations, a typical time frame is 5 years of simulated time, but simulations may be shorter or longer. Thanks to modern computer power, simulating 5 years of time even in complex models only takes a few seconds of “real-time”.
2. As explained above, the calculations needed to perform the simulation are performed in discrete time steps – typical time steps in business simulations are days or weeks. This time step is referred to as dt.
3. The initial values for all stocks and constant converters need to be defined.
4. The exact relationships between a converter and its inputs (i.e., those converters that are connected to it) have to be defined using mathematical equations.
For each of the model elements, the following paragraphs give an overview of the initial values and equations for our simple project management simulation:
To make our model concrete, let us assume our project sets out to complete 100 tasks.
We assume that no tasks have been completed at the beginning of a simulation run.
Effort
This is the average planned effort of all tasks in the project. We make the following assumptions:
• Every task requires the same average effort to be completed. This assumption is fine for the purpose of this model. In a more realistic model, we could use different stocks for different kinds of tasks and assign individual efforts for each kind of task.
• To simplify calculations, we define this average effort to be 1 day/task.
Completion Rate
Completion Rate (t) = Productivity(t)*Overtime(t)/Effort
In an ideal situation, the actual time taken to complete a task is equal to the planned effort, and then the completion rate could be calculated as 1/Effort. But the discussion in the section on causal loop diagrams showed that the completion rate also depends on our productivity (how much of our time do we actually spend working on the task) and on overtime.
Schedule Pressure (t)
Schedule Pressure (t) = Open_Tasks(t)*Effort/Remaining_Time
The schedule pressure is just an indicator that compares the time that is needed to complete the remaining tasks to the time that is actually available. We, therefore, arrive at the formulation: Schedule_Pressure = Open_Tasks*Effort/Remaining_Time. Take a look at the above diagram – according to the diagram, the effort has no effect on schedule pressure! Clearly, at the time we had not thought of this connection – this is quite normal when creating system dynamics models: the initial causal loop diagrams and stock and flow diagrams are always less accurate than the final simulation models because our initial understanding of the problem situation is less mature.
Current Date(t)
Current Date(t)=t
The current date is always just equal to the simulated time.
To make calculations easy, we set our deadline to be equal to 100.
Remaining Time (t)
Remaining Time (t)=100 - Current Date(t) = 100-t
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Productivity
The productivity is calculated using a Look-up-Table depending on schedule pressure.
There seems to be no intuitive mathematical equation that defines the relationship between schedule pressure and productivity (or overtime, respectively). This happens quite often when creating simulation models, especially when it comes to modeling soft factors such as these ones. Experience shows that most people invest less time in projects when schedule pressure is low and work overtime when schedule pressure is high. The combined effect of schedule pressure on productivity and overtime can be sketched as in the following graph:
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Now that we have defined the initial values and equations, we can then set about simulating our model as follows:
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2. We then calculate the value of all flows at time t=0.
3. We can now calculate the value of our stock at time t=dt using the following formula: Stock(dt)=Stock(0)+ Sum of all Inflows(0)×dt − Sum of all Outflows(0)×dt
4. We now repeat this process for every time step using the general formula: Stock(t+td)=Stock(t)+ Sum of all Inflows(t)×dt − Sum of all Outflow(t)×dt
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01/02/2013 3:05 PM
We have a industrial office building in Ottawa Canada that had around 100 people in it and the associated computers. Presently our boilers are having issues in keeping the work area warm. My assumption is that the heat load of the workers is gone so the demand has gone up ?
What amount of heat does a average office worker generate ?
Does this make sense ?
Pathfinder Tags: boilers heat load heating
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#1
### Re: Heat Load in Office
01/02/2013 3:15 PM
your problem has little to do with how much heat people generate. BUT To answer that direction question, th average adult male "burns (uses up stored heat and energy of around 2000 calories. your problem isn't that cold people are absorbing excessive heat. either your heating equipment isn't working properly or the building design is allowing for a loss that exceeds your expectations.
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#12
### Re: Heat Load in Office
01/03/2013 8:25 AM
Your 2000 calorie value is based on FOOD CALORIES, not thermal calories.The actually thermal value is KILO CALORIES.2000000 calories.And this is based on 24 hours of average metabolism person.An 8 hour shift of course is obviously 1/3 of this value.
Based on the BTU's posted by another user, the heat load of 100 people could be several tons of cooling or heat.
Happy New Year.
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#13
### Re: Heat Load in Office
01/03/2013 8:34 AM
Nouncalorie
1. Either of two units of heat energy.
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#15
### Re: Heat Load in Office
01/03/2013 9:05 AM
Correct!.However, when using food calories, there is a factor of 1000 difference.Unless you are aware of this difference, it can result in large errors of load calculations.
Sometimes doctors refer to a 2000 calorie diet as a 2000 kilo calorie diet, which is more correct from a scientific standpoint.
The difference in symbology is upper case C for large calorie(The amount of energy required to raise the temperature of 1 kilogram of water 1degree C at 1 atmosphere of pressure.)
The lower case c(small calorie) is used for scientific purposes, and is the amount of energy required to raise the temperature of 1 gram of water by 1 degree C at atmospheric pressure.
The large Calorie is used normally in biological studies of metabolism.
Normally, in scientific or engineering calculations, the small calorie is used.Either will give a valid answer if the decimal point is positioned correctly.
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#2
### Re: Heat Load in Office
01/02/2013 3:24 PM
Wiki says 18.4 Btu/h·ft2.
I read somewhere that it is roughly equivalent to a single 100 watt incandescant light bulb.
Computers should be fairly simple to model.
And, Fredski, to borrow a phrase from "Oh Brother Where Art Thou" all the people's done "runn oft".
2
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#3
### Re: Heat Load in Office
01/02/2013 3:30 PM
Interesting,
Some examples from ASHRAE
Person seated at rest, 350 Btu/Hr
Seated, Light Office Work, 420 Btu/Hr
Standing Light Work, 640 Btu/Hr
Walking 3mph, 1040 Btu/Hr
Bowling, 960 Btu/Hr
Heavy Factory Work, 1600 Btu/Hr
Heavy Athletics, 1800 Btu/Hr
It also how much appliances are running, with the older CRT screen this puts out allot more heat, but current modern offices have LCD screens.
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#25
### Re: Heat Load in Office
01/03/2013 4:38 PM
Exactly!!
So in most office applications a person is factored in at 500 Btu/Hr and don't forget about the 1.2 CFM per person for the OSA (outside air in and office space) requirements.
Some people call it fresh air but after a 10k cost at a law firm we now call it OSA. They literally took it as fresh air, after they made us install the Hepa filter and ionizer racks the air met fresh air standards by their independent IAQ company they hired. So we now bring in OSA for air changes. Sorry for the tangent but if the people have left the space be sure that you have shut down the OSA to meet current demands, since that could also add to your Cold feelings.
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#27
### Re: Heat Load in Office
01/03/2013 5:18 PM
A small piece of history is in order. During one of the "energy crisis" around 1970s, it was noted that the OA requirements were around 20 CFM per person. Since this was deemed as "too much OA and a waste of energy", the codes were revised down to 5 CFM per person. Unfortunately, building envelops were improved to stop air leaks and infiltration. So in the 1980s and 1990s, we developed "sick building syndrome", and IAQ became part of our vocabulary. And codes were revised to increase the OA requirements to 15 CFM. 1.2 CFM is not enough air to keep CO2 levels at an acceptable level.
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#4
### Re: Heat Load in Office
01/02/2013 3:32 PM
To clarify the office was occupied and we recently moved the office staff to a new building so the office area is now vacant but we want to heat to a min of 68 degrees as we have a couple of staff members left here. I think I will need to add additional capacity to our boilers in order to replace the lost heat. Due to a expansion a couple of years back the boilers we operating at around 90% of capacity when the office area was full.
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#14
### Re: Heat Load in Office
01/03/2013 9:01 AM
It might be more efficient/economical if you moved those "couple" of staff members to a different location as well...or make them a smaller room with dedicated heating in it.
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#5
### Re: Heat Load in Office
01/02/2013 3:44 PM
In Toronto Canada when we re-purposed a office to a factory and had 44 people working there we had to add 40,000 BTU to maintain a suitable working temperature if this helps.
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#6
### Re: Heat Load in Office
01/02/2013 4:08 PM
100W per person is about right. What about the lighting as well?
If the 100 people, their computers, coffee machines and lighting have gone, why does the area have to be kept warm?
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#7
### Re: Heat Load in Office
01/02/2013 5:09 PM
If your boilers were at 90% capacity when the building was fully occupied then you are short on capacity bigtime now. The air returning to your AHU was probably ciculating thru the plenum space above the ceiling or if no ceiling was there then the top of the room. Either way the heat from people, lighting, equipment, etc. was raising the return air temp considerably. The heating coil is required to raise the air, now including outside air, to the supply air temp. The return air temp being lower creates more load on the coil/boiler. If the people are no longer there, your ventilating air requirements are lower and outside air quantity should be reduced. This will help your situation, hopefully enough.
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#8
### Re: Heat Load in Office
01/02/2013 5:19 PM
you're making an awful lot of assumptions, the guy never mentioned ductwork or any temps
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#11
### Re: Heat Load in Office
01/03/2013 12:42 AM
Yah, what is it? A 10,000 square foot room with a 30 foot ceiling or a 70,000 square foot multilevel office complex with 2 foot concrete walls?
What type of air interchange are we looking at? Every time the exterior door opens you get fresh air, or a complete change of air every 20 minutes using outside air? Is there any recirculation?
Do they have a heat recovery system?
Do they have a climate control system of any kind, shape, or form? Is it computerized (I hope)?
What temp is the boiler set to?
Do you use industrial control controls for heat/air-con (DDC's)?
Do they used simple VAV reheaters for each space? How many thermostatically controlled spaces are there? Are the thermostats in crowded/overutilized spaces and the VAV's they serve in areas which are cooler than they should be?
Holy crap, man! There is a lot going on here!!!
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#16
### Re: Heat Load in Office
01/03/2013 9:40 AM
The assumptions are resasonable for those who work with HVAC loads and understand how the systems work.
A building with 100 people should have about 15 CFM per person of outside air. At 100 people, that is 1500 CFM that needs to be heated. With 5 people, the outside air can be resit to 75 CFM that needs heating. Based on building codes, this is a simple assumption.
A person seated at work has a sensible heat load of 250 BTU/hr sensible, and 200-250 BTU/hr latent. 100 people will provide 45,000 BTU/hr heat easily. The lighting and office equipment will probably provide another 40-50,000 BTU/hr heat. Since only 5 people work here now, most of the lights and equipment is probably turned off to save money. Again this is a reasonable assumption.
The OP should have a commercial HVAC contractor survey the building and offer a better estimate (instead of our guesses and assumptions) of what can be done based on the long range plans for the building. I would guess short term, electric heaters in limited parts of the offices will be used this year while permanent HVAC modifications can be provided for next year.
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#17
### Re: Heat Load in Office
01/03/2013 9:44 AM
GA,
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#20
### Re: Heat Load in Office
01/03/2013 10:21 AM
you can make rough assumptions all day about how much heat each person is contributing to the total heat load and you'll never come close to coming up with a useful number. the question is silly to begin with. as usual there are a ton of variables and far too little data to make any calculations. your "cfm per person is nice but these days a VOC sensor or Co2 SENSOR is used to determine how much "make up" air is needed...it isn't a fixed number. we also have no idea how much heat is being lost to a poor design or a lack of insulation, etc. so he either needs "hotter people" or to get his equipment up to speed to math his loads....I suggest the latter
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#21
### Re: Heat Load in Office
01/03/2013 10:24 AM
The GA is to have it analysed to determine thermal losses.
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#22
### Re: Heat Load in Office
01/03/2013 11:19 AM
I suppose you're right. However, I'd like to see an empirical approach. I'd suggest pre-warming (at the start of each shift) the few people remaining to perhaps 180 degrees or so. Assuming the building has a good security system, this could be accomplished by cranking up the wattage on the personnel X ray system (or adding a simple walk-through microwave system.) Adjust body temperature to account for any thermal losses.
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#23
### Re: Heat Load in Office
01/03/2013 11:24 AM
Thats true, there was a post that had empirical info, and if its been idle for a while, there could be dampers that are stuck or frozen.
formulas only go so far.
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#28
### Re: Heat Load in Office
01/03/2013 5:36 PM
I disagree. My rough assumption was useful in that in allowed me to recommend getting an HVAC professional to do an on-site survey and develop a detailed plan to deal with the problem.
VOC sensing probably is not be appropriate for this situation. If desired, CO2 sensing could work, but it might be more money than the original poster wants or needs to spend.
You are correct that required OA is not a fixed number, but code authorities do require either measuring the IAQ to vary OA supplied or supplying 15 CFM per person. The owner can decide how he wants to have the system designed and operated.
Also, the original building was expanded at some point in the past, but there is no indication whether the boilers were expanded with the addition. Original boilers might have been sufficiently over sized that some one thought this was an opportunity to save money. This happens a lot with older buildings.
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#26
### Re: Heat Load in Office
01/03/2013 4:43 PM
Very GA!
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#18
### Re: Heat Load in Office
01/03/2013 9:48 AM
Good answer. In the old days, outside air dampers were arbitrarily set at 10 or 15% for fresh air. Depending on size of the system and the quality of the dampers, they might leak enough fully closed to satisfy the fresh air needed by the 4 or 5 people left behind. If you can afford it, a CO2 monitor could be used to control the damper to maintain 400 ppm CO2 so you only heat the outside air that you need.
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#9
### Re: Heat Load in Office
01/02/2013 10:45 PM
Maybe try leaving all the lights on?
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#29
### Re: Heat Load in Office
01/03/2013 5:39 PM
Switch to 150 watt incadescent bulbs?
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#10
### Re: Heat Load in Office
01/02/2013 10:51 PM
http://www.engineeringtoolbox.com/metabolic-heat-persons-d_706.html
This has a nice table (with metric units so it's easy to use).
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#19
### Re: Heat Load in Office
01/03/2013 10:19 AM
"Does this make sense ?"
No! To build a building and put heating in it depending on the number of people that will work there. Utilizing their body heat into the equation. Sounds a little off to me.
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#24
### Re: Heat Load in Office
01/03/2013 11:41 AM
I assume by implication that the space is no longer occupied. To answer your specific question the average sensible heat gain from an office occupant is approximately 250 BTU per hour of sensible heat. That is the portion of a person's emitted heat that will warm the air in the space. The rate of heat emitted from their various electronic devices, computers, printers, scanners, etc. can vary all over the map. A survey of the space will reveal this value. Don't forget the heat emitted by the office lighting fixtures and ballast.
If your objective is to determine the amount of heat required to maintain say 70 to 72 in the space during the winter months you should run a simple heat loss caLculation for the space or have it done by a reputable HVAC design/buid firm. The size and operating condition of the boiler is just one part of your concerns. The design and condition of the remaining parts of the heating system, I.E, pumps, fans, piping, ductwork, temperature controls, etc will determine the comfort of each part of the building.
A question hovering over all this is why is there any concern about the space temperature if the space is not occupied? Like the old saw "if a tree falls in the forest and no one is there to hear it is there any noise?" If the room temperature is 60 but no one is there, is it uncomfortable?
Good luck and Happy New Year!
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For the triangle shown, where A, B and C are all points on Go to page: 1, 2 Tags: Difficulty: 700-Level, Geometry, Source: Kaplan
enigma123
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11869
15 May 2017, 23:20
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If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?
Bunuel
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103
15 May 2017, 21:34
5
One number, n, is selected at random from a set of 10 integers. What
Bunuel
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516
15 May 2017, 20:12
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Are there more than five red chips on the table?
Bunuel
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182
15 May 2017, 19:47
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In the xy-coordinate plane, line l passes through the point (-3, 0).
Bunuel
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1823
15 May 2017, 19:31
If a marble is selected at random from a bag containing only red and
Bunuel
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85
15 May 2017, 19:22
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x+y+z is even, is x*y*z even?
Eden
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2633
15 May 2017, 15:34
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Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2,354 | 6,662 | {"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-2017-22 | latest | en | 0.867687 |
http://www.territorioscuola.com/wikipedia/en.wikipedia.php?title=Hosohedron | 1,398,313,627,000,000,000 | text/html | crawl-data/CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00213-ip-10-147-4-33.ec2.internal.warc.gz | 857,527,231 | 19,074 | # Hosohedron
Set of regular n-gonal hosohedra
Example hexagonal hosohedron on a sphere
Type Regular polyhedron or spherical tiling
Faces n digons
Edges n
Vertices 2
χ 2
Vertex configuration 2n
Schläfli symbol {2,n}
Wythoff symbol n | 2 2
Coxeter–Dynkin diagrams
Symmetry group Dnh, [2,n], (*22n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron dihedron
This beach ball shows a hosohedron with six lune faces, if the white circles on the ends are removed.
In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schläfli symbol {2, n}.
## Hosohedra as regular polyhedra
For a regular polyhedron whose Schläfli symbol is {mn}, the number of polygonal faces may be found by:
$N_2=\frac{4n}{2m+2n-mn}$
The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3. The restriction m ≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2, n} is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices.
A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere. A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.
Family of regular hosohedra
1 2 3 4 5 6 7 8 9 10 11 12 ...
{2,1}
{2,2}
{2,3}
{2,4}
{2,5}
{2,6}
{2,7}
{2,8}
{2,9}
{2,10}
{2,11}
{2,12}
## Kalidescopic symmetry
The digonal faces of a 2n-hosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: Cnv, [n], (*nn), order 2n. The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry Dnh, order 4n.
Symmetry C1v C2v C3v C4v C5v C6v
Hosohedron {2,2} {2,4} {2,6} {2,8} {2,10} {2,12}
Fundamental domains
## Relationship with the Steinmetz solid
The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.1
## Derivative polyhedra
The dual of the n-gonal hosohedron {2, n} is the n-gonal dihedron, {n, 2}. The polyhedron {2,2} is self-dual, and is both a hosohedron and a dihedron.
A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism.
## Hosotopes
Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol {2,p,...,q} has two vertices, each with a vertex figure {p,...,q}.
The two-dimensional hosotope {2} is a digon.
## Etymology
The term “hosohedron” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (osos/hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”. 2 | 999 | 3,205 | {"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": 1, "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-2014-15 | latest | en | 0.869407 |
http://www.xlstat.com/en/products-solutions/predict.html | 1,441,247,069,000,000,000 | text/html | crawl-data/CC-MAIN-2015-35/segments/1440645298065.68/warc/CC-MAIN-20150827031458-00303-ip-10-171-96-226.ec2.internal.warc.gz | 840,125,160 | 9,118 | # Predict Statistical solution for predicting and forecasting
Trial version
• Prices starting at*
• 50% off real value
• Company/Private : 995.00 USD
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The XLSTAT-Predict statistical solution has been designed for those who want to build explanatory or predictive models for risk, sale and stock forecasting. XLSTAT-Predict includes the Pro, 3DPlot, Time, PLS, Sim and Pivot XLSTAT modules. This combination of analytical software makes XLSTAT-Predict the best performing and affordably priced all-in-one statistical analysis package for Microsoft Excel available on the market.
## Predict includes the following XLSTAT modules:
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• Partial Least Squares (PLS) regression is a very powerful alternative to build linear models based on the latent structure of the data. With PLS regression you also get visual tools to explain relationships between observations and variables.
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## System configuration
• Windows:
• Versions: 9x/Me/NT/2000/XP/Vista/Win 7/Win 8
• Excel: 97 and later
• Processor: 32 or 64 bits
• Hard disk: 150 Mb
See all users' feedback | 521 | 2,408 | {"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-2015-35 | latest | en | 0.873353 |
https://de.maplesoft.com/support/help/Maple/view.aspx?path=StudyGuides/Calculus/Appendix/Examples/A-7/ExampleA-7-2 | 1,716,998,348,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059246.67/warc/CC-MAIN-20240529134803-20240529164803-00710.warc.gz | 160,141,761 | 23,144 | Example A-7-2 - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.
Appendix
A-7: Trigonometry
Example A-7.2
Two sides of a triangle have lengths a = 5 and b = 2 and the angle between them is $C=52°$. Use the Law of Sines and the Law of Cosines to find the other side c and the other two angles A and B opposite a and b, respectively.
Law of Sines: $\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$ Law of Cosines:
For more information on Maplesoft products and services, visit www.maplesoft.com | 188 | 634 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 41, "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} | 3.34375 | 3 | CC-MAIN-2024-22 | latest | en | 0.638966 |
http://studentsmerit.com/paper-detail/?paper_id=48594 | 1,481,436,042,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698544140.93/warc/CC-MAIN-20161202170904-00415-ip-10-31-129-80.ec2.internal.warc.gz | 265,372,385 | 7,141 | #### Description of this paper
##### Two Finance MCQs
Description
solution
Question
Question;Tomy Inc. has a 0.6 probability of a good year with operating cash flow of \$50,000, and 0.4 probability of a bad year with operating cash flow of \$30,000. The company has a debt of \$35,000 with 8% interest due next year. Assuming the company has no means of servicing its debt other than operations, and a 0% tax rate, which of the following is True:a) Shareholders expected claim is \$12,200b) Creditors expected claim is \$37,800c) Creditors expected claim is \$34680d) None of the above.2. Poto corporation has a net income of \$20,000 and tax rate of 35%. Its total debt is \$25,000 withPrincipal Payments of \$5000 due at the end of each year and an annual interest rate of 8%. What will be Poto Corporations?s interst tax shield in the upcoming year?a) \$8,750 b) \$700 c) \$9,450 d) \$2,450
Paper#48594 | Written in 18-Jul-2015
Price : \$22 | 262 | 950 | {"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-2016-50 | longest | en | 0.932274 |
https://ru.scribd.com/doc/16428233/rr100106-engineering-graphics | 1,563,238,714,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195524290.60/warc/CC-MAIN-20190715235156-20190716021156-00361.warc.gz | 535,514,078 | 59,542 | You are on page 1of 1
# Code No: RR100106 RR
## I B.Tech (RR) Supplementary Examinations, June 2009
ENGINEERING GRAPHICS
(Common to Civil Engineering and Mechanical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks
?????
1. Construct a diagonal scale of 1:5000 to show single metre and long enough to measure 300 meters.
Mark on the scale a distance of 285.5 meters. [16M]
2. Draw an involutes of a circle of 50 mm diameter. Also, draw a normal and tangent at any point on
the curve. [16M]
3. Three lines OA, OB and OC are respectively 25 mm, 45 mm 65 mm long, each making 120 degrees
angles with the other two and the shortest line being vertical. The figure is the top view of the three
rods OA, OB and OC whose ends A, B and C are on the ground, while O is 100 mm above it. Draw
the front view and determine the length of each rod and its inclination with the ground. [16M]
4. A pentagonal pyramid, base 30 mm side and axis 60 mm long, is lying on one of its triangular faces on
the H.P. with the axis parallel to the V.P. A vertical section plane, whose H.T. bisects the top view of
the axis and makes an angle of 30 degrees with the reference line, cuts the pyramid, removing its top
part. Draw the top view , sectional front view and true shape of the section. [16M]
5. A solid is in the form of a square prism of side of base 30 mm up to a height of 50 mm and thereafter
tapers into frustum of a square pyramid whose top surface is a square of 15 mm side. The total height
of the solid is 70 mm. Draw the development of the lateral surface of the solid. [16]
6. Draw the isometric projection of a Frustum of hexagonal pyramid, side of base 30 mm the side of top
face 15mm of height 50 mm. [16]
7. Convert the orthogonal projections shown in figure1 below into an isometric view of the actual picture.
[16M]
Figure 1:
8. A step block is on 15 mm high, 15 mm width and 15 mm long. The total length of the block is 30 mm.
The largest side makes an angle of 450 to PP. The observer is at a distance of 60 mm in front of the
edge and 10 mm to the left. The height of the observer is 50 mm. Draw the perspective projection of
the object. [16M]
????? | 596 | 2,206 | {"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} | 3.046875 | 3 | CC-MAIN-2019-30 | latest | en | 0.887864 |
https://metanumbers.com/58560 | 1,596,742,053,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439737019.4/warc/CC-MAIN-20200806180859-20200806210859-00261.warc.gz | 363,831,218 | 7,734 | ## 58560
58,560 (fifty-eight thousand five hundred sixty) is an even five-digits composite number following 58559 and preceding 58561. In scientific notation, it is written as 5.856 × 104. The sum of its digits is 24. It has a total of 9 prime factors and 56 positive divisors. There are 15,360 positive integers (up to 58560) that are relatively prime to 58560.
## Basic properties
• Is Prime? No
• Number parity Even
• Number length 5
• Sum of Digits 24
• Digital Root 6
## Name
Short name 58 thousand 560 fifty-eight thousand five hundred sixty
## Notation
Scientific notation 5.856 × 104 58.56 × 103
## Prime Factorization of 58560
Prime Factorization 26 × 3 × 5 × 61
Composite number
Distinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 9 Total number of prime factors rad(n) 1830 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 58,560 is 26 × 3 × 5 × 61. Since it has a total of 9 prime factors, 58,560 is a composite number.
## Divisors of 58560
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 61, 64, 80, 96, 120, 122, 160, 183, 192, 240, 244, 305, 320, 366, 480, 488, 610, 732, 915, 960, 976, 1220, 1464, 1830, 1952, 2440, 2928, 3660, 3904, 4880, 5856, 7320, 9760, 11712, 14640, 19520, 29280, 58560
56 divisors
Even divisors 48 8 4 4
Total Divisors Sum of Divisors Aliquot Sum τ(n) 56 Total number of the positive divisors of n σ(n) 188976 Sum of all the positive divisors of n s(n) 130416 Sum of the proper positive divisors of n A(n) 3374.57 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 241.992 Returns the nth root of the product of n divisors H(n) 17.3533 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 58,560 can be divided by 56 positive divisors (out of which 48 are even, and 8 are odd). The sum of these divisors (counting 58,560) is 188,976, the average is 33,74.,571.
## Other Arithmetic Functions (n = 58560)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 15360 Total number of positive integers not greater than n that are coprime to n λ(n) 480 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 5913 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares
There are 15,360 positive integers (less than 58,560) that are coprime with 58,560. And there are approximately 5,913 prime numbers less than or equal to 58,560.
## Divisibility of 58560
m n mod m 2 3 4 5 6 7 8 9 0 0 0 0 0 5 0 6
The number 58,560 is divisible by 2, 3, 4, 5, 6 and 8.
• Abundant
• Polite
• Practical
## Base conversion (58560)
Base System Value
2 Binary 1110010011000000
3 Ternary 2222022220
4 Quaternary 32103000
5 Quinary 3333220
6 Senary 1131040
8 Octal 162300
10 Decimal 58560
12 Duodecimal 29a80
20 Vigesimal 7680
36 Base36 196o
## Basic calculations (n = 58560)
### Multiplication
n×i
n×2 117120 175680 234240 292800
### Division
ni
n⁄2 29280 19520 14640 11712
### Exponentiation
ni
n2 3429273600 200818262016000 11759917423656960000 688660764329351577600000
### Nth Root
i√n
2√n 241.992 38.8329 15.5561 8.98504
## 58560 as geometric shapes
### Circle
Diameter 117120 367943 1.07734e+10
### Sphere
Volume 8.41186e+14 4.30935e+10 367943
### Square
Length = n
Perimeter 234240 3.42927e+09 82816.3
### Cube
Length = n
Surface area 2.05756e+10 2.00818e+14 101429
### Equilateral Triangle
Length = n
Perimeter 175680 1.48492e+09 50714.4
### Triangular Pyramid
Length = n
Surface area 5.93968e+09 2.36667e+13 47814
## Cryptographic Hash Functions
md5 ba44b9abb1a0331c5d134bafb71f993f 2a86cb8cde691f5b2f995ce605088319c1e10fc8 bc1ade9050c97875c25ddaa26e7c3d02c4d02272595aef5c228d7da4024bb963 8d4597ccd8de673d5358978f27a61d6af9e0df9e2b44ec3b0fceb1b3a106a666c77a1521ff58b2fb171c8e1ffbb195c857ec58bc6fcc93e613346b4a07c01c12 248a6956088b5505f39cf5f477e727fb21f7855a | 1,617 | 4,326 | {"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} | 3.671875 | 4 | CC-MAIN-2020-34 | latest | en | 0.753589 |
https://data.stackexchange.com/askubuntu/revision/38834/38834/testing-bentons-law | 1,606,530,824,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141194982.45/warc/CC-MAIN-20201128011115-20201128041115-00701.warc.gz | 264,502,710 | 4,622 | 0
## Please login or register to vote for this query.
(click on this box to dismiss)
Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale.
Q&A for Ubuntu users and developers
``````-- Testing Benton's Law
-- Benford's law, also called the first-digit law, states that in lists of
-- numbers from many (but not all) real-life sources of data, the leading digit
-- is distributed in a specific, non-uniform way. According to this law, the
-- first digit is 1 about 30% of the time, and larger digits occur as the
-- leading digit with lower and lower frequency, to the point where 9 as a
-- first digit occurs less than 5% of the time. This distribution of first
-- digits is the same as the widths of gridlines on the logarithmic scale.
-- Testing Benton's Law
-- Benford's law, also called the first-digit law, states that in lists of
-- numbers from many (but not all) real-life sources of data, the leading digit
-- is distributed in a specific, non-uniform way. According to this law, the
-- first digit is 1 about 30% of the time, and larger digits occur as the
-- leading digit with lower and lower frequency, to the point where 9 as a
-- first digit occurs less than 5% of the time. This distribution of first
-- digits is the same as the widths of gridlines on the logarithmic scale.
with data as (Select count(*) as total,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '1') as one,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '2') as two,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '3') as three,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '4') as four,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '5') as five,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '6') as six,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '7') as seven,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '8') as eight,
(Select cast(Count(*) as decimal) from Users where Reputation > 1 and SUBSTRING(CAST(Reputation AS VARCHAR), 1, 1) = '9') as nine
from Users where Reputation > 1) select
one/total as "1",
two/total as "2",
three/total as "3",
four/total as "4",
five/total as "5",
six/total as "6",
seven/total as "7",
eight/total as "8",
nine/total as "9"
from data;``````
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:records returned in :time ms:cached | 901 | 3,322 | {"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} | 3.265625 | 3 | CC-MAIN-2020-50 | latest | en | 0.944851 |
https://ecologyforthemasses.com/2022/06/08/who-is-simpson-and-what-does-his-paradox-mean-for-ecologists/ | 1,685,854,418,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649439.65/warc/CC-MAIN-20230604025306-20230604055306-00440.warc.gz | 260,754,404 | 31,513 | ## Who Is Simpson And What Does His Paradox Mean For Ecologists?
Edward H. Simpson was a codebreaker at Bletchley Park, the home of Allied code-breakers during the Second World War. While you’d think this would be his claim to fame, perhaps his most lasting contribution is his description of Simpson’s paradox. The paradox describes the phenomena whereby a relationship within a dataset dramatically changes if you look at the data by group or all together. More famous examples of the paradox stem from the medical world or the famous Berkeley admissions example. But what examples can we have in mind in ecological settings to guide us? Let’s consider the dimensions of penguins’ bills compiled from Palmer Station in Antarctica. If we are interested in the relationship between the bill depth and length we might do a preliminary analysis like the following linear regression.
There appears to be a negative relationship between bill depth and length. But what if we consider the fact that there are three different species of penguins at Palmer Station. Now we might want to let each species have its own relationship between bill depth and length. When we do this we get a surprise. Each penguin species now has a positive relationship between bill length and depth.
This is Simpson’s Paradox in full force. The relationship between two variables changed signs with and without considering subgroups in the data. You can also think of this situation as arising from a confounding variable. Without accounting for species, we misrepresent the relationship between penguin bill length and bill depth.
This situation can also occur when the confounder is quantitative. Just picture a quantitative variable split up into bins, where in each bin the relationship is the opposite of what you see overall, just like in this penguin example. It may not be anything fundamental about the species difference, it could simply be that one species is bigger than another. For instance, it might seem reasonable that larger penguins would have longer bills on average. Let’s have a look at the different species sizes.
We see that at least the Gentoo penguin species tends to have heavier individuals. What if we control for body mass in the regression between the bill depth and length instead of breaking the analysis into three different regressions?
Heavier penguins do tend to have longer bills, but we can still notice the different species’ effects. For example, the cluster above that we now know represents the Gentoo species has noticeably larger body mass values. We can also visualize this model fit to the data, going up a dimension, from line to plane, to account for the more complicated model. The relationship between bill depth and bill length in the presence of body mass is positive, so Simpson’s Paradox has been foiled by controlling for body mass (although it seems like accounting for species helps more).
Simpson’s Paradox can also creep up on us when analyses are performed on aggregated data (see Qian et al, 2019 for another example). This is related to the ecological fallacy that we have talked about before where extending findings made on aggregated data to individuals is dangerous. What if I didn’t have individual penguin data but only averages for each species?
Now I know that I have no business fitting a line to these three points, but for the sake of the example, let’s press on. We see the inappropriate sign return. Even though we see from previous figures that the variation within each species is positive, when collapsed into a summary statistic, we see a negative relationship.
How can we know to look for this paradox before it hurts us in our analysis? If there is a natural grouping, we can easily fit the model with and without accounting for groups to check that we do not see any dramatic changes. Searching for a potential confounder Z is a bit trickier. A confounder is related to both your response of interest Y and an explanatory variable X that you have in the model to try to explain your response. We can investigate what the relationship between X and Y is at many different slices of Z (again, think of binning a quantitative variable Z or looking at subgroups of a categorical variable Z) if we actually have information on Z.
If we didn’t collect information on that confounder, then we have our work cut out for us. We do know that a potential confounder is related to both Y and X. That means that without Z in the model, there is still a relationship between X and Y that is driven specifically by Z and unaccounted for in the model. Where does unaccounted for variation go? The residuals! If we plot the residuals of the model v. X and see a pattern, that might hint that there is a Z on the loose. (Thanks to a blog post by Jim Frost for this tip!)
I hope that with these concrete examples in mind (penguins seem pretty memorable, right?) and some guidance about what the pain points confounding variables can lead to will help you avoid falling into any Simpson’s Paradox traps in the wild.
Have a quantitative term or concept that mystifies you? Want it explained simply? Suggest a topic for next month → @sastoudt
Title Image Credit: Sybille H., Pixabay licence | 1,055 | 5,240 | {"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} | 2.59375 | 3 | CC-MAIN-2023-23 | latest | en | 0.952964 |
https://www.coursehero.com/file/6247908/Unit-1-Section-5/ | 1,526,971,145,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864626.37/warc/CC-MAIN-20180522053839-20180522073839-00615.warc.gz | 738,633,502 | 112,438 | {[ promptMessage ]}
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Unit 1 Section 5
# Unit 1 Section 5 - 1.5 The Field of Algebraic Expressions...
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1.5. The Field of Algebraic Expressions Our brief excursion into the world of polynomials led us to the following observations. Operations on real numbers carry over to operations on polynomials because polynomials are merely formulas for computing some real numbers. The parallelism, however, ends when we encounter division. Division of polynomials does not always yield a polynomial. The multiplicative inverse of a polynomial is not necessarily a polynomial. The most that we can say about the structure of polynomials is that it is a commutative ring with identity. There is a need for a structure where the multiplicative inverse of a polynomial is legitimate. Be ready for the field of algebraic expressions! The Field of Quotients Let P(x) be a polynomial of the form a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . . . + a n x n . Consider the set of algebraic expressions which can be written as quotients of polynomials, which will be called rational expressions . Thus, a rational expression is of the form P(x)/R(x), where R(x) ≠ 0. For example, 5 and , 1 x , x 1 , 1 x 1 x 2 x 3 2 are rational expressions, while the following are not: 2 3 2 x 1 x 2 x and x 1 x , x , x (Why?) We shall agree to write each rational expression in simplest form, i.e., each rational expression can be written as a quotient of polynomials which are relatively prime or with no common factor except 1. To this end, we shall use the algebraic fact that , b a c b c a provided b, c ≠ 0. Similar rational expressions are those that have the same denominators. Thus 2 2 x 1 x and x 2 are similar while 3 x 4 and x 3 are not. However, dissimilar rational expressions can be made similar by using, again, the algebraic fact stated above. Time to think! Express the following in simplest form. What conditions shall you impose? 2 2 2 2 3 2 2 y x y x ) 3 3 x 27 x ) 2 10 x 3 x 35 x 12 x ) 1 Time to think! Write the following as similar rational expressions: x 2 1 , 2 x 2 ) 4 ) 1 x ( x , 1 x 1 ) 3 3 x 1 , 2 x x ) 2 1 x 2 , 1 x 3 ) 1 3
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Unit1. Algebra as the Study of Structures Section 5 page 2 We are now ready to define addition of rational expressions. Definition 1.26 . Let P(x), Q(x), and R(x) be polynomials in x. Then ) x ( Q ) x ( R ) x ( P ) x ( Q ) x ( R ) x ( Q ) x ( P , provided Q(x) ≠ 0. Clearly, the sum of two rational expressions is again a rational expression. We define multiplication of two rational expressions in the following manner: Definition 1.27 . Let P(x), Q(x), R(x), and S(x) be polynomials in x. Then ) x ( S ) x ( Q ) x ( R ) x ( P ) x ( S ) x ( R ) x ( Q ) x ( P , provided Q(x), S(x) ≠ 0.
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Jill Tulane University ‘16, Course Hero Intern | 1,043 | 3,939 | {"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} | 4.59375 | 5 | CC-MAIN-2018-22 | latest | en | 0.905144 |
http://math.stackexchange.com/questions/108433/4-point-quadratic-curve?answertab=oldest | 1,462,451,821,000,000,000 | text/html | crawl-data/CC-MAIN-2016-18/segments/1461860127407.71/warc/CC-MAIN-20160428161527-00004-ip-10-239-7-51.ec2.internal.warc.gz | 176,398,303 | 17,275 | I can define a curve that passes through 3 points using a quadratic equation:
ax2 + bx + c = 0
I would like to know is it possible to define a curve that passes through 4 points using:
ax3 + bx2 + cx + d = 0
Cheers
-
## migrated from stackoverflow.comFeb 12 '12 at 8:10
This question came from our site for professional and enthusiast programmers.
Yes, it is possible. In general, for any n points in a plane, you can find an (n-1)th degree (or higher) polynomial that passes through all of them. Finding these polynomial involves solving matrix equations and can sometimes get a bit messy. – jpm Feb 10 '12 at 23:37
You can also use a Lagrange interpolating polynomial. – Anne Nonimus Feb 10 '12 at 23:40
@jpm is correct. For an easy formula (that works in most cases), see Lagrange polynomial - Wikipedia, the free encyclopedia. – Dennis Feb 10 '12 at 23:41
The answer was already in the comments upon migration: Use a Lagrange polynomial. The restriction "in most cases" is unnecessary; the Lagrange polynomial is completely general and yields a polynomial which interpolates the points as long as no two of them have the same $x$ coordinate; if they do, there can be no univariate function, polynomial or otherwise, that interpolates them. | 309 | 1,254 | {"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} | 3.265625 | 3 | CC-MAIN-2016-18 | latest | en | 0.928323 |
https://phpbuilder.com/skewness-kurtosis-and-average-statistical-function/ | 1,695,502,041,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506528.3/warc/CC-MAIN-20230923194908-20230923224908-00179.warc.gz | 506,893,074 | 10,546 | # Skewness, Kurtosis and average statistical function
By dhk
on April 19, 2001
Version: 1.0
Type: Function
Category: Math Functions
Description: Skewness, Kurtosis, and also average function by using array methodology
```<?php
/*
In this code fraction, there r 3 function that run on mean or average and skewness and kurtosis.
Program by : [email protected]
Date : Feb.18,2001
*/
function mean(&\$array) {
\$average = 0;
while (list(\$key, \$val)=each(\$array)) \$average += \$val;
reset(\$array);
\$average /= count(\$array);
return \$average;
}
function skewnessandkurtosis(\$array, &\$skew, &\$kurt) {
\$skew = "N/A";
\$kurt = "N/A";
\$amount = count(\$array);
if (\$amount > 2) {
for (\$i = 0, \$m2 =0,\$m3=0,\$m4=0; \$i < \$amount; \$i++) {
\$array [\$i] -= mean(\$array);
\$m2 += pow(\$array[\$i], 2);
\$m3 += pow(\$array[\$i], 3);
\$m4 += pow(\$array[\$i], 4);
}
\$m2 /= \$amount;
\$m3 /= \$amount;
\$m4 /= \$amount;
\$skew = \$m3 / pow(\$m2, 1.5);
\$skew *= sqrt(\$amount*(\$amount-1))/ (\$amount-2);
if (\$amount > 3) {
\$kurt = (\$m4/ pow(\$m2, 2))-3;
\$kurt = ((\$amount+1)*\$kurt)+6;
\$kurt *= (\$amount-1)/((\$amount-2)*(\$amount-3));
}
}
}
?>
<?
// and also example on how 2 use is here
\$samplearray = array(1.8,1.9,1.2,1.5,1.7);
skewnessandkurtosis(\$samplearray, \$skew, \$kurt);
echo "<p>Data average = ". mean(\$samplearray)."</p>";
echo "<p>skewness = \$skew </p>";
echo "<p>kurtosis=\$kurt</p>";
?> ```
| 529 | 1,436 | {"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-2023-40 | longest | en | 0.475592 |
https://kamus.sabda.org/dictionary/algorism | 1,701,888,168,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100602.36/warc/CC-MAIN-20231206162528-20231206192528-00508.warc.gz | 391,829,892 | 4,865 | POS
HYPHEN
WORDNET DICTIONARY
CIDE DICTIONARY
ROGET THESAURUS
# algorism
:
Noun
:
al=go=rism
## CIDE DICTIONARY
algorismn. [OE. algorism, algrim, augrim, OF. algorisme, F. algorithme (cf. Sp. algoritmo, OSp. alguarismo, LL. algorismus), fr. the Ar. al-Khowārezmī of Khowārezm, the modern Khiwa, surname of Abu Ja'far Mohammed ben Musā, author of a work on arithmetic early in the 9th century, which was translated into Latin, such books bearing the name algorismus. The spelling with th is due to a supposed connection with Gr. number.].
• The art of calculating by nine figures and zero; computation with Arabic figures. [1913 Webster]
• the Arabic system of numeration. [WordNet 1.5]
• The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc. [1913 Webster]
## ROGET THESAURUS
### Numeration
N numeration, numbering, pagination, tale, recension, enumeration, summation, reckoning, computation, supputation, calculation, calculus, algorithm, algorism, rhabdology, dactylonomy, measurement, statistics, arithmetic, analysis, algebra, geometry, analytical geometry, fluxions, differential calculus, integral calculus, infinitesimal calculus, calculus of differences, dead reckoning, muster, poll, census, capitation, roll call, recapitulation, account, notation, addition, subtraction, multiplication, division, rule of three, practice, equations, extraction of roots, reduction, involution, evolution, estimation, approximation, interpolation, differentiation, integration, abacus, logometer, slide rule, slipstick, tallies, Napier's bones, calculating machine, difference engine, suan- pan, adding machine, cash register, electronic calculator, calculator, computer, arithmetician, calculator, abacist, algebraist, mathematician, statistician, geometer, programmer, accountant, auditor, numeral, numerical, arithmetical, analytic, algebraic, statistical, numerable, computable, calculable, commensurable, commensurate, incommensurable, incommensurate, innumerable, unfathomable, infinite, quantitatively, arithmetically, measurably, in numbers.
copyright © 2012 Yayasan Lembaga SABDA (YLSA) | To report a problem/suggestion | 579 | 2,192 | {"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-2023-50 | latest | en | 0.760476 |
https://hoomd-blue.readthedocs.io/en/v4.6.0/tutorial/00-Introducing-HOOMD-blue/06-Equilibrating-the-System.html | 1,719,332,417,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198866143.18/warc/CC-MAIN-20240625135622-20240625165622-00406.warc.gz | 259,255,554 | 12,139 | # Equilibrating the System#
## Overview#
### Questions#
• What is equilibration?
• How do I save simulation results?
### Objectives#
• Explain the process of equilibration.
• Demonstrate using GSD to write the simulation trajectory to a file.
• Demonstrate best practices for move size tuning using Before and And Triggers.
## Boilerplate code#
[1]:
import math
import hoomd
The render function in the next (hidden) cell will render a snapshot using fresnel. Find the source in the hoomd-examples repository.
## Equilibration#
So far, this tutorial has placed N non-overlapping octahedra randomly in a box and then compressed it to a moderate volume fraction. The resulting configuration of particles is valid, but strongly dependent on the path taken to create it. There are many more equilibrium configurations in the set of possible configurations that do not depend on the path. Equilibrating the system is the process of taking an artificially prepared state and running a simulation. During the simulation run, the system will relax to equilibrium. Initialize the Simulation first:
[4]:
cpu = hoomd.device.CPU()
simulation = hoomd.Simulation(device=cpu, seed=20)
mc = hoomd.hpmc.integrate.ConvexPolyhedron()
mc.shape['octahedron'] = dict(
vertices=[
(-0.5, 0, 0),
(0.5, 0, 0),
(0, -0.5, 0),
(0, 0.5, 0),
(0, 0, -0.5),
(0, 0, 0.5),
]
)
simulation.operations.integrator = mc
The previous section of this tutorial wrote the compressed system to compressed.gsd. Initialize the system state from this file:
[5]:
simulation.create_state_from_gsd(filename='compressed.gsd')
## Writing simulation trajectories#
Save the system state to a file periodically so that you can observe the equilibration process. This tutorial previously used GSD files to store a single frame of the system state using either the GSD Python package or GSD.write. The GSD Writer (another operation) will create a GSD file with many frames in a trajectory.
We use the 'xb' mode to open the file to ensure that the file does not exist before opening it. If the file does exist an error will be raised rather than overwriting or appending to the file.
[6]:
gsd_writer = hoomd.write.GSD(
filename='trajectory.gsd', trigger=hoomd.trigger.Periodic(1000), mode='xb'
)
simulation.operations.writers.append(gsd_writer)
## Tuning the trial move size#
The previous section used the MoveSize tuner regularly during compression to adjust d and a to achieve a target acceptance ratio while the system density changed rapidly. Use it again during the equilibration run to ensure that HPMC is working optimally.
Move sizes should be tuned briefly at the beginning, then left constant for the duration of the run. Changing the move size throughout the simulation run violates detailed balance and can lead to incorrect results. Trigger the tuner every 100 steps but only for the first 5000 steps of the simulation by combining a Periodic and Before trigger with an And operation. Before returns True for all time steps t < value and the And trigger returns True when all of its child triggers also return True.
[7]:
tune = hoomd.hpmc.tune.MoveSize.scale_solver(
moves=['a', 'd'],
target=0.2,
trigger=hoomd.trigger.And(
[hoomd.trigger.Periodic(100), hoomd.trigger.Before(simulation.timestep + 5000)]
),
)
simulation.operations.tuners.append(tune)
[8]:
simulation.run(5000)
Check the acceptance ratios over the next 100 steps to verify that the tuner achieved the target acceptance ratios:
[9]:
simulation.run(100)
[10]:
rotate_moves = mc.rotate_moves
mc.rotate_moves[0] / sum(mc.rotate_moves)
[10]:
0.18988232534500957
[11]:
translate_moves = mc.translate_moves
mc.translate_moves[0] / sum(mc.translate_moves)
[11]:
0.1901174817532493
## Equilibrating the system#
To equilibrate the system, run the simulation. The length of the run needed is strongly dependent on the particular model, the system size, the density, and many other factors. Hard particle Monte Carlo self-assembly often takes tens of millions of time steps for systems with ~10,000 particles. This system is much smaller, so it takes fewer steps.
This cell may require several minutes to complete.
[12]:
simulation.run(1e5)
Here is the final state of the system after the run.
[13]:
render(simulation.state.get_snapshot())
[13]:
Is the final state an equilibrium state? The next section in this tutorial shows you how to analyze the trajectory and answer this question.
Flush the GSD write buffer. This is necessary if you run the notebook in the next section without shutting down this notebook first.
[14]:
gsd_writer.flush() | 1,078 | 4,615 | {"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} | 3.125 | 3 | CC-MAIN-2024-26 | latest | en | 0.782553 |
https://topic.alibabacloud.com/a/knights-in-fen-uva10422_1_31_32707922.html | 1,696,057,178,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510603.89/warc/CC-MAIN-20230930050118-20230930080118-00145.warc.gz | 610,828,515 | 16,835 | Knights in FEN UVA10422
Source: Internet
Author: User
Problem D
Knights in FEN
Input:Standard input
Output:Standard output
Time Limit:10 seconds
There are black and white knights on a 5 by 5 chessboard. there are twelve of each color, and there is one square that is empty. at any time, a knight can move into an empty square as long as it moves like a knight in normal chess (what else did you have CT ?).
Given an initial position of the board, the question is: what is the minimum number of moves in which we can reach the final position which is:
Input <喎?http: www.bkjia.com kf ware vc " target="_blank" class="keylink"> Vc3ryb25np1_vcd4kpha + ratio = "206" height = "207" src = "http://www.bkjia.com/uploadfile/Collfiles/20140107/20140107095121228.jpg" alt = "\">
Output
For each set your task is to find the minimum number of moves leading from the starting input configuration to the final one. If that number is bigger than 10, then output one line stating
`Unsolvable in less than 11 move(s).`
Otherwise output one line stating
Solvable inNMove (s ).
WhereN<= 10.
The output for each set is produced in a single line as shown in the sample output.
Sample Input
`2`
`01011`
`110 1`
`01110`
`01010`
`00100`
`10110`
`01 11`
`10111`
`01001`
`00000`
Sample Output
`Unsolvable in less than 11 move(s).`
`Solvable in 7 move(s).`
(Problem Setter: Piotr Rudnicki, University of Alberta, Canada)
"A man is as great as his dreams ."
A status chart is provided to move the server guard to the initial state.
Typical Implicit Graph Search problems: BFS search + hash. Hash is implemented using set. In addition, it should be noted that the server guard in chess uses Japanese characters.
```# Include
# Include
# Include
# Include
Using namespace std; typedef int state [25]; const int maxn = 5000000; const int dx [] = {,-1,-2,-2, -1}; const int dy [] = {-2,-, 1,-1,-2}; state st [maxn]; int dist [maxn]; int front, rear, s; set
Vis; state goal = {, 0, 0}; int try_to_insert (int s) // hash function {int v = 0; for (int I = 0; I <25; I ++) v = v * 2 + st [s] [I]; if (vis. count (v) return 0; vis. insert (v); return 1;} int bfs () {front = 1, rear = 2; vis. clear (); while (front
10) return-1; // pruning. if (memcmp (goal, p, sizeof (p) = 0) return front; int I, j, x, y; for (I = 0; I <25; I ++) if (st [front] [I] = 2) break; int z = I; x = I/5, y = I % 5; for (I = 0; I <8; I ++) {int newx = x + dx [I]; int newy = y + dy [I]; if (newx> = 0 & newx <5 & newy> = 0 & newy <5) {state & u = st [rear]; memcpy (& u, & p, sizeof (p )); u [x * 5 + y] = u [newx * 5 + newy]; u [newx * 5 + newy] = 2; dist [rear] = dist [front] + 1; if (try_to_insert (rear) rear ++;} fr Ont ++ }}int main () {cin> s; string str; getline (cin, str); while (s --) {memset (dist, 0, sizeof (dist); int I, j; for (I = 0; I <5; I ++) {getline (cin, str); for (j = 0; j <5; j ++) {if (str [j]! = '') St [1] [I * 5 + j] = str [j]-'0 '; else st [1] [I * 5 + j] = 2 ;}} int d = bfs (); if (d <= 0) cout <"Unsolvable in less than 11 move (s ). "<
```
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• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | 1,219 | 3,993 | {"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-2023-40 | latest | en | 0.853246 |
https://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/ | 1,591,150,451,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347428990.62/warc/CC-MAIN-20200603015534-20200603045534-00452.warc.gz | 570,846,336 | 21,355 | ## The Creation / Raising Operators for Appell Sequences
The Creation / Raising Operators for Appell Sequences is a pdf presenting reps of the raising operator $R$ and its exponentiation $\exp(tR)$ for normal and logarithmic Appell sequences of polynomials as differential and integral operators. The Riemann zeta and digamma, or Psi, function are connected to fractional calculus and associated Appell sequences for a characteristic genus discussed by Libgober and Lu.
Using the inverse Mellin transform rep of the Dirac delta function given in an earlier entry leads to the integral kernel $K(x,-m) = H(1-x) \frac{(1-x)^{-m-1}}{(-m-1)!}=(-1)^m \frac{d}{dx}^m \delta(1-x)=\delta^{(m)}(1-x)$ for $K(x,t)$ on page $10$.
(Added 9/8/2016) The post Bernoulli Appells contains yet another rep for an Appell polynomial raising operator:
$R = e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - x + e^{B.(0)D_x} \; x \; e^{\hat{B}(0)D_x} = x - e^{B.(0)D_x}[e^{\hat{B}(0)D_x},x]$,
which holds for any sequence of Appell polynomials $B_n(x)$ and its umbral inverse Appell sequence $\hat{B}(x)$. See OEIS A263634 for matrix reps of the raising op.
(Added 9/15/2016) For the convolution rep of the derivative op and its relation to the Mellin transform in Part III of the pdf, see the post Note on the Inverse Mellin Transform and the Dirac Delta Function on the inverse Mellin transform rep for the derivative of the Dirac delta :
$\displaystyle \delta^{'}(y-x) = \frac{d}{dy} \delta(y-x) = \frac{d}{dy} \int_{\sigma-i \infty }^{\sigma + \infty} y^{s-1}x^{-s}ds =\int_{\sigma-i \infty }^{\sigma + \infty} (s-1) y^{s-2}x^{-s}ds$,
so in this sense
$\displaystyle \delta^{'}(1-x) = \int_{\sigma-i \infty }^{\sigma + \infty} (s-1) x^{-s}ds = H(1-x) \frac{(1-x)^{t-1}}{(t-1)!}|_{t=-1}$.
Errata:
Pg. 2: $A(\phi.(:xD_x:+t)$ should be $A(\phi.(:xD_x:)+t)$.
Pg. 4: The lowering operator for the Bell partition and CIP polynomials gives $\frac{d}{dc_1} P_n = n \; P_{n-1}$, not the factor $(n-1)$.
Pg. 9: $e^{tR_x}x^s$ should be $e^{tR_x}$.
Related Stuff:
Two applications of elementary knot theory to Lie algebras and Vassiliev invariants” by Bar-Natan, Le, Thurston (note formulas containing sinh). See also the Thurston paper referenced in the post Bernoulli Appells. | 772 | 2,260 | {"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": 16, "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-2020-24 | latest | en | 0.646646 |
https://cupdf.com/document/computer-notes-huffman-encoding.html | 1,653,382,270,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662570051.62/warc/CC-MAIN-20220524075341-20220524105341-00592.warc.gz | 250,356,068 | 20,213 | # Computer notes - Huffman Encoding
Jun 26, 2015
## Education
Huffman code is method for the compression for standard text documents. It makes use of a binary tree to develop codes of varying lengths for the letters used in the original message. Huffman code is also part of the JPEG image compression scheme. The algorithm was introduced by David Huffman in 1952 as part of a course assignment at MIT.
#### nodes of lowest frequency
• 1. Class No.24Data Structures http://ecomputernotes.com
2. Huffman Encoding
• Huffman code is method for the compression for standard text documents.
• It makes use of a binary tree to develop codes of varying lengths for the letters used in the original message.
• Huffman code is also part of the JPEG image compression scheme.
• The algorithm was introduced by David Huffman in 1952 as part of a course assignment at MIT.
http://ecomputernotes.com 3. Huffman Encoding
• To understand Huffman encoding, it is best to use a simple example.
• Encoding the 32-character phrase: " traversing threaded binary trees ",
• If we send the phrase as a message in a network using standard 8-bit ASCII codes, we would have to send 8*32= 256 bits.
• Using the Huffman algorithm, we can send the message with only 116 bits.
http://ecomputernotes.com 4. Huffman Encoding
• List all the letters used, including the "space" character, along with the frequency with which they occur in the message.
• Consider each of these (character,frequency) pairs to be nodes; they are actually leaf nodes, as we will see.
• Pick the two nodes with the lowest frequency, and if there is a tie, pick randomly amongst those with equal frequencies.
http://ecomputernotes.com 5. Huffman Encoding
• Make a new node out of these two, and make the two nodes its children.
• This new node is assigned the sum of the frequencies of its children.
• Continue the process of combining the two nodes of lowest frequency until only one node, the root, remains.
http://ecomputernotes.com 6. Huffman Encoding
• Original text:traversing threaded binary trees
• size: 33 characters (space and newline)
• NL :1
• SP :3
• a :3
• b :1
• d :2
• e :5
• g :1
• h :1
• i :2
• n :2
• r :5
• s :2
• t :3
• v :1
• y :1
http://ecomputernotes.com 7. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 is equal to sumof the frequencies ofthe two children nodes. http://ecomputernotes.com a 3 8. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 There a number of ways to combine nodes. We have chosen just one such way. http://ecomputernotes.com a 3 9. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 http://ecomputernotes.com a 3 10. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 4 4 http://ecomputernotes.com a 3 11. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 6 http://ecomputernotes.com a 3 12. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 9 10 http://ecomputernotes.com a 3 13. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 http://ecomputernotes.com a 3 14. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 33 http://ecomputernotes.com a 3 15. Huffman Encoding
• List all the letters used, including the "space" character, along with the frequency with which they occur in the message.
• Consider each of these (character,frequency) pairs to be nodes; they are actually leaf nodes, as we will see.
• Pick the two nodes with the lowest frequency, and if there is a tie, pick randomly amongst those with equal frequencies.
http://ecomputernotes.com 16. Huffman Encoding
• Make a new node out of these two, and make the two nodes its children.
• This new node is assigned the sum of the frequencies of its children.
• Continue the process of combining the two nodes of lowest frequency until only one node, the root, remains.
http://ecomputernotes.com 17. Huffman Encoding
• Start at the root. Assign 0 to left branch and 1 to the right branch.
• Repeat the process down the left and right subtrees.
• To get the code for a character, traverse the tree from the root to the character leaf node and read off the 0 and 1 along the path.
http://ecomputernotes.com 18. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 33 1 0 http://ecomputernotes.com a 3 19. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 33 1 0 1 0 1 0 http://ecomputernotes.com a 3 20. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 33 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 http://ecomputernotes.com a 3 21. Huffman Encoding v 1 y 1 SP 3 r 5 h 1 e 5 g 1 b 1 NL 1 s 2 n 2 i 2 d 2 t 3 2 2 2 5 4 4 4 8 6 14 9 19 10 33 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 http://ecomputernotes.com a 3 22. Huffman Encoding
• Huffman character codes
• NL 10000
• SP 1111
• a 000
• b 10001
• d 0100
• e 101
• g 10010
• h 10011
• i 0101
• n 0110
• r 110
• s 0111
• t 001
• v 11100
• y 11101
• Notice that the code is variable length.
• Letters with higher frequencies have shorter codes.
• The tree could have been built in a number of ways; each would yielded different codes but the code would still be minimal.
http://ecomputernotes.com 23. Huffman Encoding
• Encoded:
• 001110000111001011100111010101101001011110011001111010100001001010100111110000101011000011011101111100111010110101110000
t r a v e http://ecomputernotes.com 24. Huffman Encoding
• With 8 bits per character, length is 264.
• Encoded:
• 001110000111001011100111010101101001011110011001111010100001001010100111110000101011000011011101111100111010110101110000
• Compressed into 122 bits, 54% reduction.
http://ecomputernotes.com 25. Mathematical Properties of Binary Trees http://ecomputernotes.com 26. Properties of Binary Tree
• Property: A binary tree with N internal nodes has N+1 external nodes.
http://ecomputernotes.com 27. Properties of Binary Tree
• A binary tree with N internal nodes has N+1 external nodes.
internal nodes: 9 external nodes: 10 external node internal node http://ecomputernotes.com D F B C G A E F E 28. Properties of Binary Tree
• Property:A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.
• Property:A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.
Internal links: 8 External links: 10 external link internal link http://ecomputernotes.com D F B C G A E F E 30. Properties of Binary Tree
• Property:A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.
• In every rooted tree, each node, except the root, has a unique parent.
• Every link connects a node to its parent, so there areN -1 links connecting internal nodes.
• Similarly, each of theN +1 external nodes has one link to its parent.
• ThusN -1+ N +1=2 Nlinks.
http://ecomputernotes.com
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https://www.taylorfrancis.com/chapters/mono/10.1201/b17127-6/introduction-simulation-susmita-bandyopadhyay-ranjan-bhattacharya?context=ubx&refId=0c9c3a2f-1cd7-43b0-a92b-3271395b2b94 | 1,722,781,004,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640404969.12/warc/CC-MAIN-20240804133418-20240804163418-00695.warc.gz | 793,689,870 | 41,640 | ## ABSTRACT
There are two types of systems-discrete and continuous. In a discrete system, the state variables’ values change at discrete points in time, whereas in a continuous system, the state variables’ values change continuously over time. The state variable of a system describes the state that reects the values of the variables. Experiments are conducted to nd the states of a dened system. Experimentation can be done on the actual system or a prototype of the system. If the actual system is large, it is difcult to observe the functioning of each of the components. On the contrary, developing a representative prototype for the system is a difcult task. If prototyping is not done properly, the purpose of simulation study will be in vain. There are two types of prototypes-physical and mathematical [1], also known as physical models and mathematical models, respectively. A physical model is a miniature of the actual system, whereas a mathematical model emphasizes only mathematics to represent the system on hand. Mathematical models can further be subdivided into two types: analytical systems, which use hardcore mathematical expressions to represent the working of a system, and simulation systems, which use mathematics but not in the form of explicit mathematical expressions. Simulation is entirely dependent on the uncertainties of a real system and thus uses probability distributions wherever
the condition of uncertainty is applicable. Figure 1.1 shows the type of experimentation that can be conducted to study a system. | 291 | 1,548 | {"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-2024-33 | latest | en | 0.887246 |
https://studysoup.com/tsg/159838/physics-principles-with-applications-6-edition-chapter-17-problem-34p | 1,596,895,900,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439737883.59/warc/CC-MAIN-20200808135620-20200808165620-00335.warc.gz | 511,031,113 | 11,378 | ×
×
# Solved: How much charge flows from each terminal of a
ISBN: 9780130606204 3
## Solution for problem 34P Chapter 17
Physics: Principles with Applications | 6th Edition
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Physics: Principles with Applications | 6th Edition
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Problem 34P
How much charge flows from each terminal of a 12.0-V battery when it is connected to a 7.00-μF capacitor?
Step-by-Step Solution:
Step 1 of 3
Weekly Assignment 13 https://session.masteringphysics.com/myct/assignmentPrintViewdispl... Weekly Assignment 13 Due: 9:00pm on Tuesday, April 26, 2016 To understand how points are awarded, read the Policy for this assignment. Problem 30.41 Part A What is the frequency of a photon that...
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##### ISBN: 9780130606204
Since the solution to 34P from 17 chapter was answered, more than 234 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 34P from chapter: 17 was answered by , our top Physics solution expert on 03/03/17, 03:53PM. Physics: Principles with Applications was written by and is associated to the ISBN: 9780130606204. The answer to “How much charge flows from each terminal of a 12.0-V battery when it is connected to a 7.00-?F capacitor?” is broken down into a number of easy to follow steps, and 19 words. This full solution covers the following key subjects: Battery, capacitor, Charge, connected, flows. This expansive textbook survival guide covers 35 chapters, and 3914 solutions. This textbook survival guide was created for the textbook: Physics: Principles with Applications, edition: 6.
#### Related chapters
Unlock Textbook Solution | 457 | 1,788 | {"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} | 2.71875 | 3 | CC-MAIN-2020-34 | latest | en | 0.913415 |
https://wiringsolver.com/what-size-generator-do-i-need-to-run-a-refrigerator/ | 1,723,134,104,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640736186.44/warc/CC-MAIN-20240808155812-20240808185812-00216.warc.gz | 492,253,303 | 31,104 | An increasing number of households are opting to buy a generator to ensure they have a reliable power source. But when it comes to powering larger appliances, such as a refrigerator, the size of the generator you choose is critical.
The answer is a bit critical. As there is no one singular answer for the question. The size of the generator will completely depend on the rating of the refrigerator that you wish to use. The greater the rating of the refrigerator, the greater the size of the generator.
In this article, we will be looking at what size generator you need to run a refrigerator, and the various factors to consider when making your choice.
## Generator Size Needed to Run a Refrigerator:
To understand what size generator you need for your fridge, you have to do a few things and keep some things in mind. So, let’s check them out below:
### Check the Wattage Rating
The very first thing you should do when you decide to buy a generator for your refrigerator is check the ratings on your refrigerator.
If you still have the sticker or label on the refrigerator, note what wattage rating it has, and if you have thrown it out, you could simply look up the model online to know the wattage ratings.
Another place where you are very likely to find the ratings is the user manual of the refrigerator
### Calculating the Wattage Rating
Oftentimes the wattage rating isn’t directly written on the sticker or label. Instead, they write the voltage and amperage ratings.
This means you will have to do simple math to know what the wattage rating of the refrigerator is. All you have to do is simply multiply the voltage rating with the amperage rating, and then there you have it, the wattage rating.
For example, your refrigerator at home is labeled with a 6.5 A current rating, and it plugs into a regular 120 V A.C. outlet. So, the average running wattage will be 6.5 A X 120 v, which is 780 Watts.
## What Size Generator is Needed to Run Window Unit and Fridge?
The 5000 BTU air conditioner is a very popular type of air conditioner, so let’s take it as an example. So, how big of a generator would you need to run a 5000 BTU air conditioner?
The starting wattage of this type of air conditioner is around 625 Watts, and then after a few seconds, it drops down to the continuous running watts, which are approximately 500 Watts.
Now, if you were wondering what size generator do I need to run my whole house, then the answer will be that it varies. The starting and running wattage of that generator must be greater than the sum of the starting and running wattages of all the appliances in your house.
Therefore, you will need a much larger generator to power your whole house than the generator you would need to just your refrigerator.
## How Can I Calculate the Size of Generator Needed?
To determine the size of generator you need to run a refrigerator, you’ll first need to know the power requirements of the appliance.
Most refrigerators list their power consumption in either watts or amps on the manufacturer’s label, usually located on the back of the unit or in the manual.
Once you have the power consumption information, you’ll want to consider the starting power requirements, also known as peak or surge watts, which are typically higher than the running watts.
The starting power is the amount of power required to start the compressor and get the refrigeration process going. This value is usually listed on the manufacturer’s label as well, and can be anywhere from 2 to 3 times the running watts.
Next, add up the total watts needed to run all the appliances you want to power with your generator, including the refrigerator. This number is known as the total load.
Additionally, consider the efficiency of the generator and factor in a 20% safety margin to account for any fluctuations in power consumption.
A generator’s rated power output should be slightly higher than your total load calculation to ensure that it can handle the appliances you want to power.
For example,
if your refrigerator uses 600 watts and has a starting power requirement of 1200 watts, and you want to power two lights (100 watts each) and a television (200 watts), your total load calculation would be:
600 + 100 + 100 + 200 + 20% = 1020 watts.
In this case, a generator with a rated power output of at least 1200 watts would be sufficient.
It’s important to choose a generator that is appropriately sized for your needs to ensure that it can power all the appliances you want to run, without overloading or damaging the generator or the appliances.
If you’re in doubt about what size generator you need, it’s always best to consult with a professional or the manufacturer of your appliances.
## Summary
Ultimately, selecting the correct size generator to power your refrigerator is not a simple task. You must consider a variety of factors, such as the size of your fridge, the wattage rating, and the type of fuel source. | 1,040 | 4,967 | {"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} | 3.125 | 3 | CC-MAIN-2024-33 | latest | en | 0.933919 |
https://www.theanswerbank.co.uk/Jobs-and-Education/Question247423.html | 1,631,820,220,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780053717.37/warc/CC-MAIN-20210916174455-20210916204455-00624.warc.gz | 1,065,708,509 | 14,371 | # maths - powers??
hattster | 14:19 Wed 14th Jun 2006 | Jobs & Education
Can someone please explain what powers are eg X tot he power of 5
1 to 2 of 2
No best answer has yet been selected by hattster. Once a best answer has been selected, it will be shown here.
'to the power of' refers to multiplication of a number by itself. Therefore, 4 (for example) 'to the power of' 2 (usually referred to as 'squared') means 4 x 4. Continuing from this, 4 'to the power of' 3 means 4 x 4 x 4, and so on.
yes, the prior answer is correct.
also, when the "power" is three, it is called "cubed".
setting numbers to the "power" of something is called exponents. the number is the "base" and the "power" is called the "exponent". the exponent tells you how many times to multiply the base by itself.
you do NOT multiply the base and the exponent together. the only time that works is with 2x2=4. (or any other time the base and the exponent are the same.)
you can have a negative base - the answer will be positive if the exponent is an even number (2, 4 6, etc.) or the answer will be negative if the exponent is an odd number (3, 5, 7, etc.).
you can have negative exponents - the answer may be a decimal or fraction (a number less than 1) - you would see this most commonly when you do scientific notation (really really big things or really really little things).
this may be more information than you were hoping for, but i am a 7th grade math teacher on summer break - i gotta use it or lose it! :D thanks for reading. good luck and happy math!
1 to 2 of 2
crankfwd
V1
V1
Peter Pedant
Crystal Tips | 432 | 1,607 | {"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-2021-39 | latest | en | 0.934022 |
https://nbviewer.jupyter.org/github/empet/Plotly-plots/blob/master/Dirichlet-Distribution.ipynb | 1,560,770,299,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998473.44/warc/CC-MAIN-20190617103006-20190617125006-00342.warc.gz | 529,379,002 | 9,366 | ## Dirichlet Distribution, $Dir(\alpha)$¶
In this Jupyter notebook we generate a plot for Wikipedia, illustrating the graph of a few probability density functions for the Dirichlet distribution, corresponding to different parameter vectors $\alpha$.
In [1]:
import numpy as np
import matplotlib.tri as tri
import scipy.stats as st
import cmocean #http://matplotlib.org/cmocean/
We deal with the Dirichlet distribution defined on the open simplex $\{(x_1, x_2, x_2)\:|\: x_1+x_2+x_3=1, x_k\in(0,1)\}$.
$(x_1, x_2, x_3)$ are interpreted as the baricentric coordinates of the points in a planar triangle.
We take an equilateral triangle and subdivide it uniformly and recursively, by a procedure of type 1-to-4 split:
In [2]:
def cartesian2baric(verts, point, dist=1.e-15):
#converts 2d cartesian coordinates to baricentric coordinates with respect
#to an equilateral triangle
midpts = [(vertices[(i + 1) % 3] + vertices[(i + 2) % 3]) / 2.0 for i in range(3)]
baric = [np.dot(verts[i] - midpts[i], point - midpts[i]) / 0.75 for i in range(3)]
return np.clip(baric, dist, 1.0 - dist)#clip coordinates to avoid points on the simplex boundary
def uniftriang(vertices, subdiv_level=7):
#define a uniform triangulation of the triangle of vertices vertices
triangle = tri.Triangulation(vertices[:, 0], vertices[:, 1])
refined_tri = tri.UniformTriRefiner(triangle)
finaltri = refined_tri.refine_triangulation(subdiv=subdiv_level)# final triangularization
#finaltri.triangles are the simplices of the triangulation
#finaltri.x, finaltri.y are the cartesian coordinates of the triangulation vertices
return finaltri
Define the vertices of an equilateral triangle, subdivide it, and compute the baricentric coordinates of the triangulation points:
In [3]:
vertices = np.array([[0, 0], [1, 0], [0.5, np.sqrt(3)/2]])
triangul=uniftriang(vertices, subdiv_level=7)
baric_coords=[cartesian2baric(vertices, point) for point in zip(triangul.x, triangul.y)]
We plot a surface representing the Dirichlet probability density function as a trisurf. Below are the functions that define the coloring method and the trisurf as a Plotly Mesh3d object:
In [4]:
import plotly.plotly as py
from plotly.graph_objs import *
from plotly import tools as tls
In [23]:
def map_z2color(zval, colormap, vmin, vmax):
#map the normalized value val to a corresponding color in the mpl colormap
if vmin>=vmax:
raise ValueError('incorrect relation between vmin and vmax')
t=(zval-vmin)/float((vmax-vmin))#normalize val
C=map(np.uint8, np.array(colormap(t)[:3])*255)
#convert to a Plotly color code:
return 'rgb'+str((C[0], C[1], C[2]))
In [24]:
def plotly_trisurf(x, y, z, simplices, colormap=cmocean.cm.bathy, scene='scene1'):
#x, y, z are lists of coordinates of the triangle vertices
#simplices are the simplices that define the triangulation;
#simplices is a numpy array of shape (no_triangles, 3)
#insert here the type check for input data
points3D=np.vstack((x,y,z)).T
tri_vertices= points3D[simplices]# vertices of the surface triangles
zmean=tri_vertices[:, :, 2].mean(-1)# mean values of z-coordinates of the
#triangle vertices
min_zmean=np.min(zmean)
max_zmean=np.max(zmean)
facecolor=[map_z2color(zz, colormap, min_zmean, max_zmean) for zz in zmean]
I,J,K=zip(*simplices)
triangles=Mesh3d(x=x,
y=y,
z=z,
facecolor=facecolor,
i=I,
j=J,
k=K,
name=''
)
return triangles
Define a list of parameters $\alpha$ for the Dirichlet distributions to be plotted:
In [16]:
alpha=[[(1.3, 1.3, 1.3), (3, 3, 3), (7, 7, 7)],
[ (2,6,11), (14, 9, 5), (6, 2, 6)]]
m=len(alpha)
n=len(alpha[0])
In [17]:
fig = tls.make_subplots(rows=m, cols=n, vertical_spacing=0.0075, horizontal_spacing=0.025,
specs=[ [{'is_3d': True}, {'is_3d': True}, {'is_3d': True}],
[{'is_3d': True}, {'is_3d': True}, {'is_3d': True}],
],
)
This is the format of your plot grid:
[ (1,1) scene1 ] [ (1,2) scene2 ] [ (1,3) scene3 ]
[ (2,1) scene4 ] [ (2,2) scene5 ] [ (2,3) scene6 ]
In [18]:
scenes=[['scene{}'.format(j+1+i*n) for j in range(n)] for i in range(m)]
In [19]:
axis = dict(
showbackground=True,
backgroundcolor="rgb(230, 230,230)",
gridcolor="rgb(255, 255, 255)",
zerolinecolor="rgb(255, 255, 255)",
tickfont=dict(size=11)
)
scene=Scene(xaxis=XAxis(axis),
yaxis=YAxis(axis),
zaxis=ZAxis(axis),
aspectratio=dict(x=1,
y=1,
z=0.25
)
)
In [20]:
cmap=cmocean.cm.bathy
for i in range(m):
for j in range(n):
X=st.dirichlet(np.array(alpha[i][j]))
C=[X.pdf(baric_coords[k]) for k in range(len(baric_coords)) ]
zmax=max(C)
trace=plotly_trisurf(triangul.x, triangul.y, C, triangul.triangles, cmap, scene=scenes[i][j])
fig.append_trace(trace, i+1, j+1)
fig['layout'][scenes[i][j]].update(scene)
fig['layout'][scenes[i][j]]['zaxis'].update(tickvals=[round(zmax/2,1), round(zmax,1)])
fig['layout'].update(title='Dirichlet distribution over an open 2-simplex'+
'<br> alpha=(1.3, 1.3, 1.3), (3, 3, 3), (7, 7, 7), '+
'<br>(2, 6, 11), (14, 9, 5), (6, 2, 6) ',
font=dict(family='Georgia, serif',
size=14),
margin=dict(t=135),
height=900,
width=1000,
showlegend=False,
)
In [ ]:
py.sign_in('empet', 'my_api_key')
py.plot(fig, filename='Dirichlet-distr')
In [12]:
from IPython.display import HTML
HTML('<iframe src=https://plot.ly/~empet/13886/ width=900 height=700></iframe>')
Out[12]:
In [13]:
from IPython.core.display import HTML
def css_styling(): | 1,717 | 5,315 | {"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} | 3.6875 | 4 | CC-MAIN-2019-26 | longest | en | 0.636777 |
http://fullhomework.com/downloads/qrb-501/ | 1,601,287,621,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600401598891.71/warc/CC-MAIN-20200928073028-20200928103028-00757.warc.gz | 50,158,570 | 10,789 | # QRB 501
Case 8-3 Template
INSTRUCTIONS: Read the case in the textbook. As a team, answer the questions in this spreadsheet, then save and submit the assignment as one Microsoft® Excel® attachment. Also, submit a 1-paragraph Microsoft® Word document explaining any issues or successes you had in answering these questions.
“Refer to and use the following abbreviations for the problems below:
• PP = Parma minimum supply purchase
• S1 = Supplier 1
• S2 = Supplier 2
• CDF = Chain Discount Factor
• NP = Net Price
• NDE = Net Decimal Equivalent
1. The Artist’s Palette purchases its inventory from a number of suppliers and each supplier offers different purchasing discounts. The manager of The Artist’s Palette, Marty Parma, is currently comparing two offers for purchasing modeling clay and supplies. The first company offers a chain discount of 20/10/5, and the second company offers a chain discount of 18/12/7 as long as the total purchases are \$300 or more. Assuming Parma purchases \$300 worth of supplies, a) what is the net price from supplier 1? And b) From supplier 2? And c) From which supplier would you recommend Parma purchase her modeling clay and supplies?
“(Use this cell to answer parts a, b, and c. Be sure to show your work.)
“Now use the marked cells below to fill in the appropriate numbers for the variables to check your answers.
2. What is the net decimal equivalent for supplier 1? For supplier 2?
“(Show your work in this cell to solve for the net decimal equivalent for the two suppliers.)
3. What is the trade discount from supplier 1? From supplier 2? “(Show your work in this cell to solve for the trade discount for the two suppliers.)
“Use the marked cells below to fill in the appropriate numbers for the variables to check your answers.
4. The Artist’s Palette recognizes that students may purchase supplies at the beginning of the term to cover all of their art class needs. Because this could represent a fairly substantial outlay, the Artist’s Palette offers discounts to those students who pay sooner than required. Assume that if students buy more than \$250 of art supplies in one visit, they may put it on a student account with terms of 2/10, n/30. If a student purchases \$250 of supplies on September 16, what amount is due by September 26? How much would the student save by paying early?
5. Assume that if students buy more than \$250 of art supplies in one visit, they may put the charge on a student account with terms of 2/10 EOM. If a student makes the purchase on September 16, on what day does the 2% discount expire? If the purchase is made on September 26, on what day does the 2% discount expire? If you were an art student, which method would you prefer: 2/10, n/30, or 2/10 EOM?
Case 9-1 Template
Karen is an acupuncturist with a busy practice. In addition to acupuncture services, Karen sells teas, herbal supplements, and rice- filled heating pads. Because Karen’s primary income is from acupuncture, she feels that she is providing the other items simply to fill a need and not as an important source of profits. As a matter of fact, the rice- filled heating pads are made by a patient who receives acupuncture for them instead of paying cash. The rice- filled pads cost Karen \$ 5.00, \$ 8.00, and \$ 12.00, respectively, for small, medium, and large sizes. The ginger tea, relaxing tea, cold & flu tea, and detox tea cost her \$ 2.59 per box plus \$ 5.00 shipping and handling for 24 boxes. Karen uses a cost plus markup method, whereby she adds the same set amount to each box of tea. She figures that each box costs \$ 2.59 plus \$ 0.21 shipping and handling, which totals \$ 2.80, then she adds \$ 0.70 profit to each box and sells it for \$ 3.50. Do you think this is a good pricing strategy? How would it compare to marking up by a percent-age of the cost?
INSTRUCTIONS: Read the case in the textbook. As a team, answer the questions in this spreadsheet, then save and submit the assignment as one Microsoft® Excel® attachment. Also, submit a 1-paragraph Microsoft® Word document explaining any issues or successes you had in answering these questions.
“Refer to and use the following abbreviations for the problems below:
• BT = Box of tea
• S&H = Shipping and handling
• P = Desired profit per box of tea | 1,005 | 4,289 | {"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} | 3.265625 | 3 | CC-MAIN-2020-40 | latest | en | 0.912123 |
https://www.transtutors.com/questions/b-find-vo-and-io-when-vs-10-v-c-what-are-vo-and-io-when-each-of-the-1-o-resistors-i-3925541.htm | 1,624,175,330,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487658814.62/warc/CC-MAIN-20210620054240-20210620084240-00529.warc.gz | 917,437,210 | 12,865 | # (b) Find vo and io when vs = 10 V. (c) What are vo and Io when each of the 1-O resistors is replaced 1 answer below »
(b) Find vo and io when vs = 10 V.
(c) What are vo and Io when each of the 1-Ω resistors is replaced by a 10-Ω resistor and vs = 10 V?
Figure 4.71
## Plagiarism Checker
Submit your documents and get free Plagiarism report
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Looking for Something Else? Ask a Similar Question | 127 | 423 | {"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-2021-25 | latest | en | 0.829358 |
https://groupprops.subwiki.org/wiki/LCS-Baer_Lie_group | 1,627,598,651,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153897.89/warc/CC-MAIN-20210729203133-20210729233133-00031.warc.gz | 309,386,581 | 11,328 | # LCS-Baer Lie group
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
## Definition
### Direct definition
A LCS-Baer Lie group or lower central series Baer Lie group is a group satisfying both the following properties:
1. It is a group of nilpotency class two, i.e., its nilpotency class is at most two.
2. Its derived subgroup is a 2-powered group, i.e., a uniquely 2-divisible group. Note that since the group has class at most two, the derived subgroup must also be abelian.
### Definition in terms of LCS-Lazard Lie group
A LCS-Baer Lie group is a LCS-Lazard Lie group that is also a group of nilpotency class two.
A LCS-Baer Lie group can serve on the group side of the LCS-Baer correspondence (the other side is the LCS-Baer Lie ring).
A finite group is a LCS-Baer Lie group if and only if it is a group of nilpotency class (at most) two and its 2-Sylow subgroup is abelian.
## Examples
### Finite examples
The finite LCS-Baer Lie groups are the groups of nilpotency class two whose 2-Sylow subgroup is abelian. In particular, when the 2-Sylow subgroup is nontrivial abelian, these examples are not Baer Lie groups.
### Infinite examples
Any infinite Baer Lie group gives an example. In addition, examples like direct product of UT(3,Q) and Z are examples of LCS-Baer Lie groups that are not Baer Lie groups. The reason it fails to be a Baer Lie group is that there is a separate part of the center (outside the derived subgroup) that is not 2-divisible.
## Relation with other properties
### Stronger properties
Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Baer Lie group class at most two, and whole group is uniquely 2-divisible cyclic group:Z2, or direct product of UT(3,Q) and Z |FULL LIST, MORE INFO
### Weaker properties
Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
CS-Baer Lie group class at most two, and some intermediate subgroup between derived subgroup and center where every element of derived subgroup has a unique half central product of UT(3,Z) and Q (an example that is in fact a UCS-Baer Lie group) |FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two CS-Baer Lie group|FULL LIST, MORE INFO
group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle CS-Baer Lie group|FULL LIST, MORE INFO
group whose derived subgroup is contained in the square of its center every element of the derived subgroup has a square root in the center CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two|FULL LIST, MORE INFO
group 1-isomorphic to an abelian group the group is 1-isomorphic to an abelian group CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle|FULL LIST, MORE INFO
group of nilpotency class two CS-Baer Lie group, Group of nilpotency class two whose commutator map is the double of a skew-symmetric cyclicity-preserving 2-cocycle, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, Group of nilpotency class two whose commutator map is the skew of a cyclicity-preserving 2-cocycle, Group whose derived subgroup is contained in the square of its center|FULL LIST, MORE INFO
### Incomparable properties
Property Meaning Proof that LCS-Baer Lie group may not have this property Proof that a group with this property may not be a LCS-Baer Lie group
UCS-Baer Lie group center is 2-powered any abelian group with 2-torsion, such as cyclic group:Z2 central product of UT(3,Z) and Q
LUCS-Baer Lie group derived subgroup has unique square roots in center any abelian group with 2-torsion, such as cyclic group:Z2 central product of UT(3,Z) and Q | 1,145 | 4,737 | {"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-2021-31 | latest | en | 0.87501 |
https://thecharlieblake.co.uk/1-introduction | 1,709,542,037,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947476432.11/warc/CC-MAIN-20240304065639-20240304095639-00159.warc.gz | 560,257,440 | 29,002 | 👋
# 1. Introduction
Below are the key ideas and equations in Chapter 1: Introduction of Probabilistic Machine Learning: An Introduction.
This is intended as a concise reference to accompany the text, and as the basis of an Anki flashcard deck
(see my explanation of the benefits of using Anki here:
📗
Probabilistic Machine Learning: An Introduction
)
📢
Press cmd/ctrl + option/alt + t to expand/close all the toggle lists at once (only on desktop)
## 1.2 Supervised Learning
### 1.2.1 Classification
#### 1.2.1.4 Empirical risk minimization
What is the definition of empirical risk? (in words)
The average loss calculated over the training set.
What is the definition of empirical risk? (equation)
where is our loss function.
#### 1.2.1.5 Uncertainty
Predictive error due to lack of knowledge is known as what? (two terms)
1. Epistemic uncertainty
1. Model uncertainty
Predictive error due to intrinsic stochasticity is known as what? (two terms)
1. Aleatoric uncertainty
1. Data uncertainty
What is the purpose of the softmax function?
It converts a vector of real values into a vector of probabilities.
What is the definition of the softmax function? (equation)
What is the definition of a logistic regression model? (equation)
where is the softmax function.
#### 1.2.1.6 Maximum likelihood estimation
What is the common choice of loss function for probabilistic models? (in words)
The negative log probability.
What is the common choice of loss function for probabilistic models? (equation)
What is the definition of negative log likelihood (NLL)? (in words)
The empirical risk, using negative log loss.
What is the definition of negative log likelihood (NLL)? (equation)
What is the definition of the maximum likelihood estimate (MLE)? (equation)
where is the negative log likelihood.
### 1.2.2 Regression
What is the difference between classification and regression?
For the former the class label is categorical, whereas for the latter its real-valued.
What is the definition of mean squared error (MSE)? (in words)
The empirical risk, using quadratic loss.
What is the definition of mean squared error (MSE)? (equation)
How are the NLL and MSE linked?
If we assume our predictions have Gaussian noise and compute the NLL, it is proportional to the MSE.
#### 1.2.2.2 Polynomial regression
What is the definition of a polynomial regression model, of degree ? (equation)
### 1.2.3 Overfitting and generalization
What is the definition of the population risk? (equation)
where is the true (but unknown) distribution used to generate the training set.
What is the definition of the generalization gap? (verbal)
The population risk minus the empirical risk.
How can we frame overfitting in terms of the generalisation gap?
It is present if a model has a large generalisation gap.
### 1.2.4 No free lunch theorem
What is the premise of the no free lunch theorem?
There is no single best model that works optimally for all kinds of problems.
## 1.3 Unsupervised learning
What distributions are supervised and unsupervised leaning trying to model?
These approaches model and respectively.
### 1.3.2 Discovering latent “factors of variation”
What is a latent factor?
A hidden low-dimensional variable, from which the observed high-dimensional variable is generated.
What is the definition of a factor analysis (FA) model? (equation)
Where are the latent factors.
### 1.3.3. Self-supervised learning
What is self-supervised learning?
A form of unsupervised learning that trains on 'proxy' supervised tasks, created from the unlabelled data.
## 1.4 Reinforcement learning
In reinforcement learning, what is a policy?
A model which specifies which action to take in response to each possible input.
## 1.5 Data
### 1.5.3 Preprocessing discrete input data
#### 1.5.3.1 One-hot encoding
What is the definition of a one-hot encoding for a variable that takes categorical values? (equation)
#### 1.5.3.1 Feature crosses
What is the definition of feature crosses? (verbal)
One-hot encoding over every possible combination of values, given multiple categorical values.
### 1.5.4 Preprocessing text data
#### 1.5.4.1 Bag of words model
What is a bag of words model?
A representation of a text document where we ignore word order.
What is stop word removal?
The dropping of common but uninformative words (e.g. "the", "and").
What is word stemming?
Replacing words with their base form (e.g. “running” → “run”)
What is the definition of a vector space model of text? (verbal)
Given a vocabulary of tokens, it encodes a document into a -dimensional vector where each element indicates the frequency of a word.
What is the definition of a term frequency (TF) matrix of a text dataset? (verbal)
A matrix where each entry is the frequency of term in document .
What is the definition of inverse document frequency (IDF)? (equation)
where is the number of documents with term .
#### 1.5.4.2 TF-IDF
What does TF-IDF stand for?
It stands for term frequency-inverse document frequency.
What is the definition of the term frequency-inverse document frequency (TF-IDF)? (equation)
where is the frequency of term in document , and is the inverse document frequency.
(we often normalise each row as well)
#### 1.5.4.3 Word embeddings
What is a word embedding?
A mapping from a high-dimensional one-hot vector , to a lower-dimensional dense vector , via multiplication by (i.e. indexing into) an embedding matrix :
What is the definition of a bag of word embeddings? (verbal)
The sum of the word embeddings of each token in a document
What is the definition of a bag of word embeddings? (equation)
where is the embedding matrix, and is the vector space model of the document.
### 1.5.5 Handling missing data
Name the three kinds of missing data.
1. Missing completely at random (MCAR)
1. Missing at random (MAR)
1. Not missing at random (NMAR)
What is the definition of missing completely at random (MCAR)? (verbal)
The 'missingness' of data does not depend on the hidden or observed features.
What is the definition of missing at random (MAR)? (verbal)
The 'missingness' of data does not depend on the hidden features, but may depend on the observed features.
What is the definition of not missing at random (NMAR)? (verbal)
The 'missingness' of data depends on the hidden features.
What must we do if we have not missing at random (NMAR) data?
If this is the case, we model the missing data mechanism, since the lack of information may be informative.
What is mean value imputation?
Replacing missing values by their empirical mean. | 1,536 | 6,549 | {"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} | 3.5 | 4 | CC-MAIN-2024-10 | longest | en | 0.895306 |
https://www.objectivebooks.com/2016/02/mechanical-engineering-hydraulics.html | 1,679,405,553,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296943698.79/warc/CC-MAIN-20230321131205-20230321161205-00422.warc.gz | 1,010,751,904 | 42,529 | Mechanical Engineering Hydraulics online Test - Set 22 - ObjectiveBooks
# Practice Test: Question Set - 22
1. When a cylindrical vessel containing liquid is revolved about its vertical axis at a constant angular velocity, the pressure
(A) Varies as the square of the radial distance
(B) Increases linearly as its radial distance
(C) Increases as the square of the radial distance
(D) Decreases as the square of the radial distance
2. Property of a fluid by which its own molecules are attracted is called
(B) Cohesion
(C) Viscosity
(D) Compressibility
3. According to Bazin's formula, the discharge over a rectangular weir is mL2g x H3/2where m is equal to
(A) 0.405 + (0.003/H)
(B) 0.003 + (0.405/H)
(C) 0.405 + (H/0.003)
(D) 0.003 + (H/0.405)
4. Which of the following is the unit of kinematic viscosity?
(A) Pascal
(B) Poise
(C) Stoke
5. The length AB of a pipe ABC in which the liquid is flowing has diameter (d1) and is suddenly contracted to diameter (d2) at B which is constant for the length BC. The loss of head due to sudden contraction, assuming coefficient of contraction as 0.62, is
(A) v₁²/2g
(B) v₂²/2g
(C) 0.5 v₁²/2g
(D) 0.375 v₂²/2g
6. Operation of McLeod gauge used for low pressure measurement is based on the principle of
(A) Gas law
(B) Boyle's law
(C) Charles law
(D) Pascal's law
7. Reynold's number is the ratio of the inertia force to the
(A) Surface tension force
(B) Viscous force
(C) Gravity force
(D) Elastic force
8. A piece weighing 3 kg in air was found to weigh 2.5 kg when submerged in water. Its specific gravity is
(A) 1
(B) 5
(C) 7
(D) 6
9. A differential manometer is used to measure
(A) Atmospheric pressure
(B) Pressure in pipes and channels
(C) Pressure in Venturimeter
(D) Difference of pressures between two points in a pipe
10. In a lock-gate, the reaction between two gates is (where P = Resultant pressure on the lock gate, and α = Inclination of the gate with the normal to the side of the lock)
(A) p/sinα
(B) 2p/sinα
(C) p/2sinα
(D) 2p/sin (α/2)
11. In a venturi-flume, the flow takes place at
(A) Atmospheric pressure
(B) Gauge pressure
(C) Absolute pressure
(D) None of these
12. The normal stress is same in all directions at a point in a fluid
(A) Only when the fluid is frictionless
(B) Only when the fluid is incompressible and has zero viscosity
(C) When there is no motion of one fluid layer relative to an adjacent layer
(D) Irrespective of the motion of one fluid layer relative to an adjacent layer
13. The highest efficiency is obtained with a channel of __________ section.
(A) Circular
(B) Square
(C) Rectangular
(D) Trapezoidal
14. A vertical wall is subjected to a pressure due to one kind of liquid, on one of its sides. The total pressure on the wall per unit length is (where w = Specific weight of liquid, and H = Height of liquid)
(A) wH
(B) wH/2
(C) wH2/2
(D) wH2/3
15. Which of the following manometer has highest sensitivity?
(A) U-tube with water
(B) Inclined U-tube
(C) U-tube with mercury
(D) Micro-manometer with water
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https://www.shaalaa.com/question-bank-solutions/heights-distances-a-vertical-tower-stands-horizontal-plane-surmounted-flagstaff-height-7m-point-plane-angle-elevation-bottom-flagstaff-30-that-top-flagstaff-45-find-height-tower_5761 | 1,521,914,224,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257650764.71/warc/CC-MAIN-20180324171404-20180324191404-00568.warc.gz | 859,640,693 | 16,201 | CBSE Class 10CBSE
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# Solution - A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30º and that of the top of the flagstaff is 45º. Find the height of the tower. - CBSE Class 10 - Mathematics
ConceptHeights and Distances
#### Question
A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30º and that of the top of the flagstaff is 45º. Find the height of the tower.
#### Solution
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#### APPEARS IN
NCERT Mathematics Textbook for Class 10
Chapter 9: Some Applications of Trigonometry
Q: 0 | Page no. 0
#### Reference Material
Solution for question: A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30º and that of the top of the flagstaff is 45º. Find the height of the tower. concept: Heights and Distances. For the course CBSE
S | 317 | 1,231 | {"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-2018-13 | latest | en | 0.865341 |
https://rd.springer.com/chapter/10.1007/978-0-8176-4791-9_2 | 1,534,594,361,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221213666.61/warc/CC-MAIN-20180818114957-20180818134957-00685.warc.gz | 797,660,148 | 14,392 | # Probability
• John R. Klauder
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)
## Abstract
Flip a coin, throw some dice, measure the voltage of a noisy circuit, and the outcomes are realizations of random variables. Let X denote the random variable in question, which we assume takes on strictly real values, and let J denote the set of values X can assume. The set J may be discrete (as for dice), continuous (as for noisy voltages), or the union of the two. Repeated and independent measurements yield a sequence of possible values x n , where$$n = 1, 2, \ldots , N, N < \infty,$$ that sample the possible outcomes of X in an unbiased way. Approximate averages involving the x n values can be obtained easily, such as
$$\overline{X} = \frac{1}{N} \sum^{N}_{n=1} x_n,$$
$$\overline{{X^2}} = \frac{1}{N} \sum^{N}_{n=1} x^{2}_n,$$
etc. Experience shows that the larger N becomes, the closer these values generally are to those values of the true averages for these quantities obtained ideally when $$N \rightarrow \infty$$
## Keywords
Probability Measure Characteristic Function Central Limit Theorem Divisible Distribution Poisson Variable
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
## Preview
Unable to display preview. Download preview PDF. | 342 | 1,396 | {"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": 2, "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-2018-34 | latest | en | 0.820071 |
https://us.metamath.org/mpeuni/scottss.html | 1,718,946,877,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862036.35/warc/CC-MAIN-20240621031127-20240621061127-00855.warc.gz | 526,951,727 | 3,591 | Mathbox for Rohan Ridenour < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > Mathboxes > scottss Structured version Visualization version GIF version
Theorem scottss 40889
Description: Scott's trick produces a subset of the input class. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Assertion
Ref Expression
scottss Scott 𝐴𝐴
Proof of Theorem scottss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-scott 40882 . 2 Scott 𝐴 = {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)}
21ssrab3 4032 1 Scott 𝐴𝐴
Colors of variables: wff setvar class Syntax hints: ∀wral 3130 ⊆ wss 3908 ‘cfv 6334 rankcrnk 9180 Scott cscott 40881 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2178 ax-ext 2794 This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2801 df-cleq 2815 df-clel 2894 df-nfc 2962 df-rab 3139 df-v 3471 df-in 3915 df-ss 3925 df-scott 40882 This theorem is referenced by: elscottab 40890
Copyright terms: Public domain W3C validator | 541 | 1,237 | {"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-2024-26 | latest | en | 0.284195 |
http://poker.stackexchange.com/questions/2863/how-much-must-hero-bet-for-villain-to-fold | 1,469,764,489,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257829970.64/warc/CC-MAIN-20160723071029-00182-ip-10-185-27-174.ec2.internal.warc.gz | 189,794,824 | 20,465 | How much must Hero bet for Villain to fold?
Game is \$1/\$3 NL
Hero (~315) - Been playing TAG; up \$115. Played villain in one earlier hand; villain folded on river (I filled a straight, which he may have discerned).
Villain (~140) - Seems like a good TAG player; plays few hands and plays them aggressively. Seems like a strong player; but not infallible. Looks a little like Gus from Breaking Bad FWIW.
Pre-Flop:
Hero is in early position; Villain is in late position.
Hero gets KQs and raises to \$10. There were a couple callers before reaching the villain, who raised to \$25. Action folds around to hero, who decides to call. Other callers fold so it's heads up to the flop.
Flop: (\$73 in the pot)
The flop is K 5 2 (rainbow). Hero is thrilled with top pair and good kicker. He can beat pocket 10s, Js, and Qs, although he's dead to pocket As, Ks, and AK. Hero decides to bet \$30 - doesn't want to get over-committed but enough that villain takes notice.
My thinking is that if villain missed, \$30 is enough for him to fold. Villain thinks for about 30 seconds and then calls. This makes me think he either has KQ, KJ (so debating whether his kicker is good enough) or is hollywooding (but he didn't seem that type of guy).
Turn: (\$133 in the pot)
The turn is a Q. Hero is doing mental gymnastics, having turned two pair. Bets \$70. Villain again pauses, noticing he has \$85 left. Waits about 10 seconds and goes all-in. Hero confidently calls the remaining \$15.
Result:
Turns out our villain made his 1-outer on the turn, making a set of Queens.
Questions:
1. Should Hero have called villain's pre-flop raise, re-raised, or folded?
2. If you were the villain with QQ on the flop, how much would hero have to bet to get you to fold?
-
Should Hero have called villain's pre-flop raise, re-raised, or folded?
The only thing you can do is call. You've labelled Villian as tight-aggressive and he's made a small reraise after you've opened under the gun. This is a fairly strong sign of strength. Let's look at your options:
Jamming: KQs is doing badly against the average TAG players range in this spot. Even if you give him a generous range of {99+, AJs+, AQo+, KQs}, KQs has 32.42% equity against this range. He isn't folding enough of that range to a jam to justify doing so.
Reraising Not-All-In: This is a bad idea. The majority of his range will jam if you reraise at which point you'll have to fold, giving up a significant pot. The rest of his range will likely call and go to a flop. Note that his calling portion isn't doing so bad against KQs either.
Calling: You flat his raise and go to a flop. The total pot is \$58 and you only need to call another \$15, giving you 3.87:1 odds; a very tempting proposition. Note that you'll have to play this hand out of position against a very strong range. You are giving excellent reverse implied odds in this situation. If your postflop skills are good you should be fine though.
Folding: This is definitely an option. It really depends on just how tight Villian is. If he's extremely tight I would fold this without a second thought.
So, the process of elimination only truly leaves you two options: call or fold. If you are fairly confident he's not a nit, see a flop.
Let's pretend Hero knew the villain had QQ on the flop. How much would Hero need to bet for villain to fold?
I'm sensing a flawed thought process here. You do not want Villian to fold QQ on the flop. You should be betting for value, not as an attempt to fold out his hand. You don't want to fold out the portion of his range you're beating. You want to extract chips from him.
I would personally never be betting this flop. You're out of position against an opponent that 3-bet your UTG open preflop and you've flopped TPGK which is still very vulnerable against Villian's range. He either has you crushed right now or has a hand that is scared due to the King.
Take a step back and think what leading out accomplishes. You simply fold out {AQ, AJs, QQ-99} a certain amount of the time (how often is dependent on how tight he is) and you valuetown yourself against {KK+, AK}. It's unlikely that Villian will fold to any flop bet with QQ, apart from a jam, and especially not against a \$30 bet into a \$73 pot. That means you will have to barrel the turn to push him off his hand which I just can't see happening with such short stacks. And that would be a terrible line to take as his range is very strong. You might as well just hand him your stack.
Consider checking. This way Villian will likely continuation bet his entire range, {KK+, AK} as well as the hands you're beating, {AQ, AJs, QQ-99}. And if he doesn't, you can be reasonably confident that he doesn't have {KK+, AK} and then you can bet the turn/river for value against {QQ-99}.
Your hand is relatively weak when compared to your opponents range but it is too strong to entirely give up on though. When you are in this kind of position, you should think of your hand as a bluffcatcher.
To answer your question entirely, if I for some reason decided to lead the flop here I would still bet \$30 as you did, but for a completely different reason: for value. \$30 bet into a \$74 pot on the flop and then jam the remaining effective \$85 on the turn when the pot is \$134. If I was going to try to push him off {QQ-99} I would just jam the flop (but I can think of literally no situation I would ever do that).
OP EDIT: If you were the villain with QQ on the flop, how much would hero have to bet to get you to fold?
~\$50+ when facing a tight-aggressive player that raised UTG, smooth called a small reraise, and then donk-bet 2/3rds pot on a super dry K52 rainbow flop. Tight-aggressive players just aren't doing this with worse than Kx. If he had {QQ-99}, he'd likely just check-call one or two streets. He also might be donk-betting with Ax hands that missed but it's such a small portion of his range you can usually ignore it, especially when you consider that most people won't continuation bet 2/3rds pot with air. They'll c-bet around half pot or slightly less.
If he donk-bet less, say \$30-\$40, I'd call one street and reevaluate on the turn. In this scenario, once I've hit my set I'm getting it in. Any non-Queen turn I would give up if he continued betting and if he didn't I'd check behind. Same on the river.
This changes drastically if you can't label your opponent as tight-aggressive. If he is even remotely loose, I'd be getting it in with QQ on K52 rainbow every time, every day. I'd never be folding.
-
I've reworded the second question to better describe my intention. I understand that I want villain to bet on the flop. I'm more curious about how much money it would take for someone to lay down that hand. – Craig Apr 7 '14 at 21:35
Edited my answer with a response to your new question. Hope that helps. – Brent Morrow Apr 7 '14 at 21:53
Hero's flop bet is terrible, IMO. You made only bad arguments for betting on the flop. Do you want villain to fold on the flop? Really? why? Which hands do you want him to fold?
pre-flop I'm almost always calling villain's 3bet (depends on table conditions, which you didn't specify). KQs is a good hand, which does well in this spot, both if more players call behind us and if we get folds behind us. If villain was deep stacked, say 150bb or more, then I think it'd be a much closer spot, and I might opt for a fold preflop.
Also, if open raises to 10\$ often get called in multiple spots, then my open raise sizing would be bigger.
-
I dont get your reasoning. The fact the villan is short is the exact reason I would fold preflop.I wouldnt want to go all in pre with KQ for 50bb and if we just call on the flop we have no room to manuver since the effective stack is so low and the opponent has the lead and position on us. The fact that he is short will also take our potential implied odds for any draws we might hit. Thus I would opt for the call when the oponent has a larger stack and we can apply our skills to outplay him postflop.Furthermore I wouldnt want to be involved in a lot of pots with Gus from Braking Badshivers:) – Daniel Apr 9 '14 at 9:09
@Daniel I agree with the general lines in your thought process, but I think you're wrong in this particular case. Everything you said is true about a hand like 98s. But KQs acts mostly as a high-card hand that also, as a bonus, plays better in multiway pots than KQo. It is not an implied-odds hand like 98s, since it makes less straights than 98s, it's less disguised, and it tends to make more dominated hands. In fact, KQs is a reverse implied odds (RIO) hand when the flop is seen heads-up in this spot. And with RIO hands, the deeper the stacks the less we want to play them. – mobius dumpling Apr 9 '14 at 11:21 | 2,196 | 8,780 | {"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-2016-30 | latest | en | 0.972728 |
https://www.extendoffice.com/documents/excel/7153-excel-restart-numbering-when-value-changes.html | 1,716,026,370,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971057379.11/warc/CC-MAIN-20240518085641-20240518115641-00690.warc.gz | 674,529,423 | 26,392 | How to restart numbering when value changes in Excel?
To number rows or columns in Excel can be quite quick and easy. But do you know how to automatically restart numbering when a different value is entered in a column, and continue when there are repeated values as shown below? Please follow the instructions to get the job done.
Separate serial numbers for different values in another column in Excel
Let's say you have a list of values, and there are serial numbers assigned to the values as shown below. To separate serial numbers for different values in the list, please do as follows.
1. Enter the following formula in the cell A2:
=COUNTIF(\$B\$2:\$B2,B2)
Note: In the formula, B2 is the top cell of the value list. You should leave the dollar signs (\$) the way they are, since the dollar sign makes the row letter and column number right behind it absolute, so the row and column won’t adjust when you copy the formula to below cells.
2. Select the cell A2, drag the fill handle (the small square in the lower-right corner of the selected cell) down to apply the formula to the below cells.
3. If you need to make the serial number and its corresponding value in one cell, you can use the formula below in the cell C2. After entering the formula, drag the fill handle down to the cells below.
=CONCATENATE(A2," ",B2)
Note: In the formula, A2 is the top cell of the serial number, and B2 is the top cell of the value list. " " is a space between A2 and B2. And you will get “1 A” as a result as shown below. You can change the snippet " " to other values such as ". " to get “1. A”.
4. Before you delete the column A and B, please convert the formulas in column C to text values so that you won’t get errors: Copy the column C2:C9, and then right click on a cell (say, cell C2) and select the Values button under Paste Options on the drop-down menu as shown below.
Alternatively, you can use Kutools for Excel’s To Actual feature to convert the formulas in C2:C9 to text directly by selecting the range and then click on the To Actual button on the Kutools tab.
Note: If you find the above two steps (step 3 and 4) are a bit annoying, please use Kutools for Excel's Combine rows, columns, or cells without losing data feature: Select the range A2:B9, then on Kutools tab, select Merge & Split > Combine rows, columns, or cells without losing data. See screenshot:
Related articles
How To Assign Serial Number To Duplicate Or Unique Values In Excel?
If you have a list of values which contains some duplicates, is it possible for us to assign sequential number to the duplicate or unique values? It means giving a sequential order for the duplicate values or unique values as following screenshot shown. This article, I will talk about some simple formulas to help you solving this task in Excel.
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• Increases your productivity by 50%, and reduces hundreds of mouse clicks for you every day! | 1,173 | 4,934 | {"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-2024-22 | longest | en | 0.860504 |
https://studen.com/mathematics/17887205 | 1,725,826,525,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651017.43/warc/CC-MAIN-20240908181815-20240908211815-00858.warc.gz | 525,768,187 | 45,421 | 11.05.2022
7. Washcloths were selling for $1.88 each. If 14 1/4 dozen cloths were sold, what was the retail value of the total sales 0 Step-by-step answer 24.06.2023, solved by verified expert Unlock the full answer The retail value of the total sales was$311.48
Step-by-step explanation:
14 1/4 dozen cloths were sold
This means that the total number of cloths sold was:
Each for $1.88 So the retail value of total sales was.$1.88*171 = $311.48 The retail value of the total sales was$311.48
Faq
Mathematics
The retail value of the total sales was $311.48 Step-by-step explanation: 14 1/4 dozen cloths were sold This means that the total number of cloths sold was: Each for$1.88
So the retail value of total sales was.
$1.88*171 =$311.48
The retail value of the total sales was $311.48 Mathematics Step-by-step answer P Answered by PhD Answer: 440 grams for 1.54 is the better value Explanation: Take the price and divide by the number of grams 1.54 / 440 =0.0035 per gram 1.26 / 340 =0.003705882 per gram 0.0035 per gram < 0.003705882 per gram Mathematics Step-by-step answer P Answered by PhD The answer is in the image Mathematics Step-by-step answer P Answered by PhD For every 8 cars there are 7 trucks Therefore, Cars:Truck=8:7 Answer is B)8:7 Mathematics Step-by-step answer P Answered by PhD The answer is in the image Mathematics Step-by-step answer P Answered by PhD F=ma where F=force m=mass a=acceleration Here, F=4300 a=3.3m/s2 m=F/a =4300/3.3 =1303.03kg Mathematics Step-by-step answer P Answered by PhD F=ma where F=force m=mass a=acceleration Here, F=4300 a=3.3m/s2 m=F/a =4300/3.3 =1303.03kg Approximately it is aqual to 1300kg Mathematics Step-by-step answer P Answered by PhD The answer is in the image Mathematics Step-by-step answer P Answered by PhD The solution is given in the image below Mathematics Step-by-step answer P Answered by PhD Here, tip=18%of$32
tip=(18/100)*32
=0.18*32
=$5.76 Total payment=32+5.76=$37.76
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FREE | 617 | 2,097 | {"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} | 4.21875 | 4 | CC-MAIN-2024-38 | latest | en | 0.924417 |
https://blancosilva.wordpress.com/teaching/ma142-summer-ii-2012/review-for-the-final-exam/ | 1,531,876,682,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676589980.6/warc/CC-MAIN-20180718002426-20180718022426-00084.warc.gz | 611,457,695 | 19,619 | ## Review for the Final Exam
Feel free to comment below if you need some guidance with any problem. As it is customary, I will provide with hints, but no solutions.
1. Evaluate the integral $\displaystyle{\int t^2 e^t\, dt}$
2. Evaluate the integral $\displaystyle{\int (x-3) \sqrt{ x^2-6x+5 }\, dx}$
3. Evaluate the integral $\displaystyle{\int \frac{1}{x^3 e^{1/x}}\, dx}$
4. Evaluate the integral $\displaystyle{\int \frac{x^3+1}{(x+1)^2(x^2+4)}\, dx}$
5. Evaluate the following integral, or indicate if it is divergent:
$\displaystyle{\int_0^\infty \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\, dx}$
6. Find the volume of the solid obtained by rotating the region bounded by $y=x^2$ and $y=2-x$ around the line $x=1.$
7. Find the volume of the solid obtained by rotating the region bounded by $y=e^{-x}, y=1/e,$ and $x=0$ around the line $y=0.$
8. Find the general term of the sequence $\big\{ 3,2, \frac{5}{3}, \frac{3}{2}, \frac{7}{5}, \frac{4}{3}, \dotsc \big\},$ and compute its limit.
9. Compute the limit of the sequence
$\bigg\{ \displaystyle{\frac{n^2+5n+2}{\sqrt{n^4+1}}} \bigg\}_{n=1}^\infty$
10. Compute the limit of the sequence $\big\{ \tan (\pi - 1/n) \big\}_{n=1}^\infty$
11. Compute $\displaystyle{ \lim_{n\to \infty} \bigg( 1- \frac{2}{n} \bigg)^n }$
12. Study the convergence of the series $\displaystyle{\sum_{n=2}^\infty \frac{3^n+4^n}{5^n}}.$ If convergent, evaluate the sum.
13. Classify the series $\displaystyle{\sum_{n=1}^\infty \frac{\cos (\pi n)}{n^{2/3}}}$ as absolutely convergent, conditionally convergent, or divergent.
14. Classify the series $\displaystyle{\sum_{n=1}^\infty \frac{(-1)^n n}{e^n}}$ as absolutely convergent, conditionally convergent, or divergent.
15. Find the interval of convergence of the power series $\displaystyle{ \sum_{n=0}^\infty \frac{(-3)^n x^n}{\sqrt{n+1}}}.$
16. Express the function $f(x) = \displaystyle{\frac{2x}{x^3+8}}$ as a power series. | 672 | 1,902 | {"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": 20, "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-2018-30 | latest | en | 0.619808 |
http://www.ccdconsultants.com/calculators/financial-ratios/gross-profit-margin-calculator-and-interpretation?tab=interpretation | 1,516,201,341,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886946.21/warc/CC-MAIN-20180117142113-20180117162113-00266.warc.gz | 406,955,758 | 9,047 | # Gross Profit Margin Interpretation
### What is Gross Profit Margin
Gross profit margin is a key financial indicator used to asses the profitability of a company's core activity, excluding fixed cost.
Gross profit margin formula is:
$Gross profit margin formula$
Gross profit margin measures company's manufacturing and distribution efficiency during the production process. It is a measurement of how much from each dollar of a company's revenue is available to cover overhead, other expenses and profits.
### Gross Profit Margin Analysis
The ideal level of gross profit margin depends on the industries, how long the business has been established and other factors.
High gross profit margin indicates that the company can make a reasonable profit, as long as it keeps the overhead cost in control.
Low gross profit margin indicates that the business is unable to control its production cost.
Gross profit margin can be used to compare a company with its competitors. More efficient firms will usually see a higher margin. Also, it provides clues about company's pricing, cost structure and production efficiency. Therefore, gross profit margin can be used to compare company's activity over time.
For most of the businesses, gross profit margin is affected by seasonality. Generally, in the good months, this margin is higher than in the slower months, when the company may use sales and other marketing tools to attract more customers.
Gross profit margin should be analyzed along with operating margin, which asses the profitability after including fixed cost and net profit margin, which asses the profitability after including fixed cost, interest expenses and taxes.
Input
Note: Gross Profit Margin calculator uses JavaScript, therefore you must have it enabled to use this calculator.
Results
Gross profit margin calculator measures company's manufacturing and distribution efficiency during the production process, the profitability of its core activity. Gross profit margin formula is:
$Gross profit margin formula$
Gross Profit Margin calculator is part of the Online financial ratios calculators, complements of our consulting team.
1. Complementarily, in order to calculate Gross Profit Margin for your business, we offer a calculator free of charge.
2. You may link to this calculator from your website as long as you give proper credit to C. C. D. Consultants Inc. and there exists a visible link to our website.
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### Definitions and terms used in Gross Profit Margin Calculator
Net Sales
The amount of revenue generated by a company after the deduction of returns, allowances for damaged or missing goods and any discounts allowed.
Net sales = Gross sales - Sales returns and allowances
Cost of Goods Sold (COGS)
The direct cost attributable to the production or purchasing of the goods sold by a company. It is also referred as Cost of sales.
Gross Profit
The difference between Net Sales and its Cost of Goods Sold, before deducting overhead, payroll, taxes, interest and other operating expenses.
Gross profit = Net sales - Cost of Goods Sold | 719 | 3,721 | {"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": 2, "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-2018-05 | latest | en | 0.923767 |
https://talkstats.com/threads/population-specific-odds-ratio-and-subject-specific-odds-ratio.67152/ | 1,596,953,664,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738425.43/warc/CC-MAIN-20200809043422-20200809073422-00507.warc.gz | 520,901,336 | 9,987 | # Population specific odds ratio and subject specific odds ratio
#### Iris Ilja
##### New Member
Hey guys,
For a school assignment I have to compute the population specific odds ratio and the subject specific odds ratio from the table below.
No GAD GAD Total
No MDD 350 (93%) 26 376 (62.7%)
MDD 153 71 224 (37.3%)
Total 503 (83.8%) 97 (16.2%) 600 (100%)
This is what I have on it so far:
The odds of no GAD in the no MDD group are 13.46 (350/26), and the odds of no GAD in the MDD group is 2.15 (153/71). The odds ratio in this case is 6.26 (13.46 / 2.15). So it can be said that the odds of having no GAD and no MDD is 6.12 times larger than the odds of having no GAD and having MDD.
Did I calculate the population or subject specific odds ratio there? And how to do the other one?
#### hlsmith
##### Less is more. Stay pure. Stay poor.
Not familiar with this, how exactly was the problem presented, verbatim?
#### Iris Ilja
##### New Member
I got this question on an assignment from school:
Consider major depression (MDD) and generalized anxiety disorder (GAD). The
interest is now in the question whether more people have GAD or MDD, therefore
look at the marginal distributions of MDD and GAD.
(a) Compute the population averaged odds ratio and interpret it.
(b) Compute the subject speci c odds ratio and interpret it
#### hlsmith
##### Less is more. Stay pure. Stay poor.
Marginal distributions are usually when you don't stratify the data by a group just base things on the totals. Though you are only presenting the data above as a single 2 x 2 table. Is that what they did in the problem?
#### Iris Ilja
##### New Member
No I created the table in order to calculate the odds ratio (I think population odds ratio). Do you need more info and if yes what do you need?
#### hlsmith
##### Less is more. Stay pure. Stay poor.
I hate to sound like a broken record, but what was the exact original question - presented with percentages and counts. So you reiterate the question, but how was data introduced? | 515 | 2,027 | {"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-2020-34 | latest | en | 0.926366 |
https://kalkinemedia.com/definition/f/frequency-distribution | 1,660,039,591,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00583.warc.gz | 336,469,034 | 31,436 | # Frequency Distribution
• Updated on
Definition – Frequency Distribution
A frequency distribution could be defined as a graphical or tabular display of data that summarise the data into a relatively small number of intervals. Ideally, a frequency distribution is a statistical tool that helps in analysing large amounts of data while working with all types of measurement scale.
## Steps to Construct a Frequency Distribution
As a tool to summarise a vast amount of data, frequency distribution follows some basic steps, which are as below:
• Arrange or sort the data in ascending order.
• Calculate range, which is defined as the difference between the maximum and minimum value of the underlying data.
• Decide the number of intervals.
• Determine the width of the interval.
• Determine successive intervals by adding the interval width to the minimum value.
• Terminate the above step after reaching the interval containing the maximum value.
• Identify the number of observations falling into each interval.
While each step sounds very easy to perform, except arranging the data in ascending order, other steps are usually more complicated than they sound and involves a lot of understanding and precautions.
## Terminology and Concepts Related to Frequency Distribution
• Interval
An interval is defined as a set of values within which each observation in a frequency distribution falls; thus, intervals group data in a frequency distribution. Also, as each observation of a data set falls into only one interval, the total number of interval covers all observations in a data set.
• Frequency or Absolute Frequency
The total number of observations falling into a given interval is described as frequency or absolute frequency.
• Relative Frequency
The relative frequency could be defined as the ratio of each observation relative to the total number of observations.
• Cumulative Relative Frequency
Cumulative relative frequency basically cumulates or adds up the relative frequency as we move from top to bottom of the frequency distribution table.
The cumulative relative frequency identifies what fraction of observation is less than the upper limit of the interval in frequency distribution and is a great summarisation tool.
## Constructing a Frequency Distribution
To construct a frequency distribution table, let us take a hypothetical portfolio of stocks with annual returns as mentioned in the below table:
To summarise the above-presented data in a frequency distribution, we need to follow the steps of constructing a frequency distribution table.
### Step 1
Arrange the annual returns in ascending order.
### Step 2
Determine the range by taking the difference between the maximum value and the minimum value.
So, the range for our data table would be
However, as the difference between the maximum value and the minimum value is ending into a rational number, we would convert the range into a positive integer by rounding. Furthermore, it is often advisable to round the range to the upper side rather than the lower side to capture the maximum value of the data into a frequency distribution.
Thus, let us take 5 as the number of range for constructing a frequency distribution of the above-presented portfolio.
### Step 3
Determine the number of intervals
For simplicity, let us try to summarise the annual returns of each stock present in the portfolio into 5 intervals.
### Step 4
Determine the interval width by dividing the range of the data with the number of intervals.
Thus, the interval width for our data set would be
### Step 5
Determine the interval by successively adding the interval width to the minimum value, leading to the interval table as below:
Once we have the interval table, we can move ahead with the construction of a frequency distribution by finding individual components of a frequency distribution such as frequency, relative frequency, cumulative frequency, and cumulative relative frequency.
## Interpreting the Frequency Distribution
In a frequency distribution table, an absolute frequency tells the number of observation falling into each interval and could address some questions like how many stocks delivered an annual return of greater than five per cent but less than 6 per cent and so on.
Likewise, relative frequency tells what fraction of observation falls into each interval and could address questions like what fractions of stock delivered a return higher than the expected range.
For example, if someone asks what fraction of the portfolio delivered a return between 8.0 to 9.0 per cent, the answer is just summarised in the face of absolute frequency, i.e., 31.58 per cent.
Apart from these two, cumulative relative frequency is also a great tool to summarise a data set, that tells what fraction of observation is less than the upper limit of the interval, and it can answer questions like percentage of stock delivering less than 9.0 per cent annual return in the above hypothetical portfolio.
I.e., 89.47 per cent.
Also, in the context of investment, frequency distribution could also serve as a first insight to determine the expected return on a portfolio or any individual stock by summarising the historical return.
For example, as the above-constructed frequency distribution table of our hypothetical portfolio reflects that nearly one-third of the observation falls between the interval of 7.0 to 8.0 per cent and nearly one-third of observation falls between the interval of 8.0 to 9.0 per cent; thus, a return between 7.0 to 9.0 per cent could be expected from the portfolio in the future, considering the historical performance.
However, in such a context, many other parameters need to be considered. For example, each stock in the portfolio should have an equal weightage to reach the determined expected return. Likewise, it should also be noticed that in the future context, frequency distribution only gives a vague idea as it is mainly a tool which summarises the actual past data. | 1,144 | 6,016 | {"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} | 4.34375 | 4 | CC-MAIN-2022-33 | longest | en | 0.884753 |
https://blog.finxter.com/5-best-ways-to-find-the-lexicographically-smallest-string-to-reach-a-destination-in-python/ | 1,718,552,102,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861665.97/warc/CC-MAIN-20240616141113-20240616171113-00047.warc.gz | 126,370,689 | 22,130 | # 5 Best Ways to Find the Lexicographically Smallest String to Reach a Destination in Python
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π‘ Problem Formulation: In certain computational problems, we are tasked with finding the lexicographically smallest string that represents the path from a starting point to a destination. Specifically, we are given a graph-like structure where each move corresponds to appending a character to a string. The challenge lies in determining the sequence of moves that results in the lexicographically smallest string. For example, given a start ‘A’ and a destination ‘C’ with possible moves ‘A to B (represented by ‘ab’)’ and ‘B to C (represented by ‘bc’)’, the desired output would be “abbc”.
## Method 1: Backtracking
Backtracking is a classic algorithmic approach where we attempt to build a solution incrementally, abandoning paths that do not lead to a valid solution. For finding the lexicographically smallest string, we perform a depth-first search on the graph of possible moves, backtracking as soon as a move does not lead us closer to the destination or if we’ve found a string that is not the smallest in lexicographical order.
Here’s an example:
```def smallest_string(start, destination, moves):
result = ['{']
def backtrack(path, last_node):
if last_node == destination:
if path < result[0]:
result[0] = path
return
for move in moves:
if move[0] == last_node and path + move[1] < result[0]:
backtrack(path + move[1], move[2])
backtrack('', start)
return result[0]
moves = [('A', 'ab', 'B'), ('B', 'bc', 'C')]
print(smallest_string('A', 'C', moves))
```
Output: `"abbc"`
In this code snippet, we define a function `smallest_string()` that takes the start, destination, and possible moves as input. It uses an inner function `backtrack()` to explore all possible paths using depth-first search and updates the result whenever a smaller string is found. The moves are tuples where the first element is the starting node, the second is the string to append, and the third is the ending node. The algorithm efficiently prunes the search space by checking if the current path is lexicographically larger than the already found smallest string.
## Method 2: Greedy Algorithm
A greedy algorithm builds a solution step by step, always choosing the next step that offers the most immediate benefit, with the hope of finding a global optimum. When searching for the lexicographically smallest string, a greedy algorithm would select the smallest viable character at each step. This method works well in structures where local optimal choices lead to a global optimum, but may not always result in the best solution in all scenarios.
Here’s an example:
```def smallest_string_greedy(start, destination, moves):
moves.sort(key=lambda x: x[1]) # Sort moves based on the lexicographical order of the appending string
path = ''
current = start
while current != destination:
for move in moves:
if move[0] == current:
current = move[2]
path += move[1]
break
return path
moves = [('A', 'ab', 'B'), ('B', 'bc', 'C')]
print(smallest_string_greedy('A', 'C', moves))
```
Output: `"abbc"`
The function `smallest_string_greedy()` takes the starting point, destination, and a list of possible moves. It initially sorts the moves so that we always use the lexicographically smallest option available. It then loops until the destination is reached, always choosing the smallest option at every step to append to the path. However, it’s important to note that this method can be suboptimal if the greedy choice at each step does not necessarily lead to the overall smallest solution.
## Method 3: Dynamic Programming
Dynamic Programming (DP) is an optimization over plain recursion. Where a recursive solution might solve the same subproblem multiple times, DP would remember the past results and reuse them to make the algorithm more efficient. For our lexicographically smallest string problem, we can use DP to store the smallest string found so far for each node, thus avoiding redundant calculations.
Here’s an example:
```def smallest_string_dp(start, destination, moves):
dp = {destination: ''}
moves.sort(key=lambda x: x[1], reverse=True)
for move in moves:
if move[2] in dp:
new_str = move[1] + dp[move[2]]
if move[0] in dp:
dp[move[0]] = min(dp[move[0]], new_str)
else:
dp[move[0]] = new_str
return dp[start]
moves = [('A', 'ab', 'B'), ('B', 'bc', 'C')]
print(smallest_string_dp('A', 'C', moves))
```
Output: `"abbc"`
The function `smallest_string_dp()` implements a bottom-up dynamic programming approach. It starts by sorting the moves in reverse lexicographical order and initializes a DP dictionary with the destination as the key and an empty string as the value. The algorithm then iteratively updates the DP table by combining the smallest strings leading to the destination. This method ensures that any lookup in the DP table gives us the smallest string to reach the destination from a given starting point.
## Method 4: Using Python Libraries
For many problems, there are existing Python libraries that can simplify the solution. In our case, we can leverage the heap data structure from the `heapq` library for efficiently finding the lexicographically smallest string. This approach is useful when dealing with a large number of moves or a complicated graph structure, as the library functions are typically optimized for performance.
Here’s an example:
```import heapq
def smallest_string_heapq(start, destination, moves):
heap = [(start, '')]
visited = set()
while heap:
current, path = heapq.heappop(heap)
if current == destination:
return path
if current not in visited:
for move in moves:
if move[0] == current:
heapq.heappush(heap, (move[2], path + move[1]))
moves = [('A', 'ab', 'B'), ('B', 'bc', 'C')]
print(smallest_string_heapq('A', 'C', moves))
```
Output: `"abbc"`
In this code snippet, we utilize the `heapq` module to prioritize moves based on their lexicographical order. A heap is a special tree-based data structure that satisfies the heap property. In a min-heap, the parent is always less than or equal to its children, which is perfect for our need to always choose the lexicographically smallest string. The function `smallest_string_heapq()` uses a priority queue to explore the graph and maintains a set of visited nodes to avoid cycles.
## Bonus One-Liner Method 5: Pythonic Way
Sometimes Python’s powerful one-liners allow us to express complex logic succinctly. We can use comprehensions, `min()` function, and conditional logic to find the lexicographically smallest string in a very concise way. This method is great for Python enthusiasts who love one-liners and want to leverage Python’s expressive syntax.
Here’s an example:
```moves = [('A', 'ab', 'B'), ('B', 'bc', 'C')]
print(min((a + b for a, b, c in sorted(moves) if c == 'C'), key=len))
```
Output: `"abbc"`
This one-liner uses list comprehension to create a generator of potential strings, each resulting from a move ending at the destination ‘C’. The `min()` function is then used to find the smallest string by length. However, this one-liner is very specific to our input structure and assumes all paths ending in ‘C’ are valid moves from ‘A’, which may not be the case in a more complex scenario.
## Summary/Discussion
• Method 1: Backtracking. Flexible. Can handle complex graphs. Potentially inefficient if the graph is large or complex due to the depth-first search approach.
• Method 2: Greedy Algorithm. Simple and fast. But, lacks optimization in the case of multiple paths. Not guaranteed to find the global minimum.
• Method 3: Dynamic Programming. Efficient for large graphs by avoiding redundant calculations. It requires careful thought to implement the DP table correctly.
• Method 4: Using Python Libraries. Leverages optimized library functions. Good for complex structures. Requires familiarity with Python’s standard library.
• Bonus Method 5: Pythonic One-Liner. Elegant and concise. However, limited to scenarios that match the one-liner’s assumptions. | 1,814 | 8,023 | {"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} | 3.21875 | 3 | CC-MAIN-2024-26 | latest | en | 0.821707 |
http://www.theinfolist.com/html/ALL/s/Function_of_several_variables.html | 1,652,810,694,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662519037.11/warc/CC-MAIN-20220517162558-20220517192558-00365.warc.gz | 117,225,951 | 19,778 | TheInfoList
In mathematics, a functionThe words map, mapping, transformation, correspondence, and operator are often used synonymously. . from a set (mathematics), set to a set assigns to each element of exactly one element of . The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept. A function is most often denoted by letters such as , and , and the value of a function at an element of its domain is denoted by . A function is uniquely represented by the set of all pair (mathematics), pairs , called the ''graph of a function, graph of the function''.This definition of "graph" refers to a ''set'' of pairs of objects. Graphs, in the sense of ''diagrams'', are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices). When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function. Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
# Definition
A function from a set to a set is an assignment of an element of to each element of . The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. A function, its domain, and its codomain, are declared by the notation , and the value of a function at an element of , denoted by , is called the ''image'' of under , or the ''value'' of applied to the ''argument'' . Functions are also called ''Map (mathematics), maps'' or ''mappings'', though some authors make some distinction between "maps" and "functions" (see ). Two functions and are equal if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, given and , we have if and only if for all .This follows from the axiom of extensionality, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example, The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set. Typically, this occurs in mathematical analysis, where "a function often refers to a function that may have a proper subsetcalled the ''domain of definition'' by some authors, notably computer science of as domain. For example, a "function from the reals to the reals" may refer to a real-valued function, real-valued function of a function of a real variable, real variable. However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval. Such a function is then called a partial function. For example, if is a function that has the real numbers as domain and codomain, then a function mapping the value to the value is a function from the reals to the reals, whose domain is the set of the reals , such that . The range of a function, range or Image (mathematics), image of a function is the set of the Image (mathematics), images of all elements in the domain.
## Relational approach
In the relational approach, a function is a binary relation between and that associates to each element of exactly one element of . That is, is defined by a set of ordered pairs with , such that every element of is the first component of exactly one ordered pair in . In other words, for every in , there is exactly one element such that the ordered pair belongs to the set of pairs defining the function . The set is called the graph of a function, graph of . Some authors identify it with the function; however, in common usage, the function is generally distinguished from its graph. In this approach, a function is defined as an ordered triple . In this notation, whether a function is surjective (see below) depends on the choice of . Any subset of the Cartesian product of two sets and defines a binary relation between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function given above. A binary relation is Functional relation, functional (also called right-unique) if :$\forall x\in X, \forall y\in Y, \forall z\in Y, \quad \left(\left(x,y\right)\in R \land \left(x,z\right)\in R\right)\implies y=z.$ A binary relation is Serial relation, serial (also called left-total) if :$\forall x\in X, \exists y\in Y, \quad\left(x,y\right)\in R.$ A partial function is a binary relation that is functional. A function is a binary relation that is functional and serial. Various properties of functions and function composition may be reformulated in the language of relations.Gunther Schmidt( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60, Cambridge University Press
CUP blurb for ''Relational Mathematics''
/ref> For example, a function is injective function, injective if the converse relation is functional, where the converse relation is defined as
## As an element of a Cartesian product over the domain
The set of all functions from some given domain to a codomain can be identified with the Cartesian product of copies of the codomain, index set, indexed by the domain. Namely, given sets and , any function is an element of the Cartesian product of copies of s over the index set : :$f\in Y^X:=\prod_Y.$ Viewing as tuple with coordinates, then for each , the th coordinate of this tuple is the value . This reflects the intuition that for each , the function ''picks'' some element , namely, . (This point of view is used for example in the discussion of a choice function.) Infinite Cartesian products are often simply "defined" as sets of functions.
# Notation
There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.
## Functional notation
In functional notation, the function is immediately given a name, such as , and its definition is given by what does to the explicit argument , using a formula in terms of . For example, the function which takes a real number as input and outputs that number plus 1 is denoted by :$f\left(x\right)=x+1$. If a function is defined in this notation, its domain and codomain are implicitly taken to both be $\R$, the set of real numbers. If the formula cannot be evaluated at all real numbers, then the domain is implicitly taken to be the maximal subset of $\R$ on which the formula can be evaluated; see Domain of a function. A more complicated example is the function :$f\left(x\right)=\sin\left(x^2+1\right)$. In this example, the function takes a real number as input, squares it, then adds 1 to the result, then takes the sine of the result, and returns the final result as the output. When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of . Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "" for the sine function, in contrast to italic font for single-letter symbols. When using this notation, one often encounters the abuse of notation whereby the notation can refer to the value of at , or to the function itself. If the variable was previously declared, then the notation unambiguously means the value of at . Otherwise, it is useful to understand the notation as being both simultaneously; this allows one to denote composition of two functions and in a succinct manner by the notation . However, distinguishing and can become important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a ''Functional (mathematics), functional''.) Other approaches of notating functions, detailed below, avoid this problem but are less commonly used.
## Arrow notation
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, $x\mapsto x+1$ is the function which takes a real number as input and outputs that number plus 1. Again a domain and codomain of $\R$ is implied. The domain and codomain can also be explicitly stated, for example: :$\begin \operatorname\colon \Z &\to \Z\\ x &\mapsto x^2.\end$ This defines a function from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose $f\colon X\times X\to Y;\;\left(x,t\right) \mapsto f\left(x,t\right)$ is a function in two variables, and we want to refer to a Partial application, partially applied function $X\to Y$ produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted $x\mapsto f\left(x,t_0\right)$ using the arrow notation. The expression $x\mapsto f\left(x,t_0\right)$ (read: "the map taking to ") represents this new function with just one argument, whereas the expression refers to the value of the function at the
## Index notation
Index notation is often used instead of functional notation. That is, instead of writing , one writes $f_x.$ This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence (mathematics), sequence, and, in this case the element $f_n$ is called the th element of sequence. The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map $x\mapsto f\left(x,t\right)$ (see above) would be denoted $f_t$ using index notation, if we define the collection of maps $f_t$ by the formula $f_t\left(x\right)=f\left(x,t\right)$ for all $x,t\in X$.
## Dot notation
In the notation $x\mapsto f\left(x\right),$ the symbol does not represent any value, it is simply a placeholder name, placeholder meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function from its value at . For example, $a\left(\cdot\right)^2$ may stand for the function $x\mapsto ax^2$, and $\int_a^ f(u)\,du$ may stand for a function defined by an integral with variable upper bound: $x\mapsto \int_a^x f(u)\,du$.
## Specialized notations
There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the Vector (mathematics and physics), vectors they act upon are denoted using a dual pair to show the underlying Duality (mathematics), duality. This is similar to the use of bra–ket notation in quantum mechanics. In Mathematical logic, logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function Abstraction (computer science), abstraction and Function application, application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions Commutative property, commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.
# Other terms
A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular ''map'' is often used in place of ''homomorphism'' for the sake of succinctness (e.g., linear map or ''map from to '' instead of ''group homomorphism from to ''). Some authors reserve the word ''mapping'' for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real number, real or complex number, complex numbers, and use the term ''mapping'' for more general functions. In the theory of dynamical systems, a map denotes an Discrete-time dynamical system, evolution function used to create Dynamical system#Maps, discrete dynamical systems. See also Poincaré map. Whichever definition of ''map'' is used, related terms like ''Domain of a function, domain'', ''codomain'', ''Injective function, injective'', ''Continuous function, continuous'' have the same meaning as for a function.
# Specifying a function
Given a function $f$, by definition, to each element $x$ of the domain of the function $f$, there is a unique element associated to it, the value $f\left(x\right)$ of $f$ at $x$. There are several ways to specify or describe how $x$ is related to $f\left(x\right)$, both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function $f$.
## By listing function values
On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if $A = \$, then one can define a function $f\colon A \to \mathbb$ by $f\left(1\right) = 2, f\left(2\right) = 3, f\left(3\right) = 4.$
## By a formula
Functions are often defined by a closed-form expression, formula that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, $f$ can be defined by the formula $f\left(n\right) = n+1$, for $n\in\$. When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zero of a function, zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from $\mathbb$ to $\mathbb,$ the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative. For example, $f\left(x\right)=\sqrt$ defines a function $f\colon \mathbb \to \mathbb$ whose domain is $\mathbb,$ because $1+x^2$ is always positive if is a real number. On the other hand, $f\left(x\right)=\sqrt$ defines a function from the reals to the reals whose domain is reduced to the interval . (In old texts, such a domain was called the ''domain of definition'' of the function.) Functions are often classified by the nature of formulas that define them: *A quadratic function is a function that may be written $f\left(x\right) = ax^2+bx+c,$ where are constant (mathematics), constants. *More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integers. For example, $f\left(x\right) = x^3-3x-1,$ and $f\left(x\right) = \left(x-1\right)\left(x^3+1\right) +2x^2 -1.$ *A rational function is the same, with divisions also allowed, such as $f\left(x\right) = \frac,$ and $f\left(x\right) = \frac 1+\frac 3x-\frac 2.$ *An algebraic function is the same, with nth root, th roots and zero of a function, roots of polynomials also allowed. *An elementary functionHere "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree. is the same, with logarithms and exponential functions allowed.
## Inverse and implicit functions
A function $f\colon X\to Y,$ with domain and codomain , is bijective, if for every in , there is one and only one element in such that . In this case, the inverse function of is the function $f^\colon Y \to X$ that maps $y\in Y$ to the element $x\in X$ such that . For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers onto the positive numbers. If a function $f\colon X\to Y$ is not bijective, it may occur that one can select subsets $E\subseteq X$ and $F\subseteq Y$ such that the restriction of a function, restriction of to is a bijection from to , and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval (mathematics), interval onto the interval , and its inverse function, called arccosine, maps onto . The other inverse trigonometric functions are defined similarly. More generally, given a binary relation between two sets and , let be a subset of such that, for every $x\in E,$ there is some $y\in Y$ such that . If one has a criterion allowing selecting such an for every $x\in E,$ this defines a function $f\colon E\to Y,$ called an implicit function, because it is implicitly defined by the relation . For example, the equation of the unit circle $x^2+y^2=1$ defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and . In this example, the equation can be solved in , giving $y=\pm \sqrt,$ but, in more complicated examples, this is impossible. For example, the relation $y^5+y+x=0$ defines as an implicit function of , called the Bring radical, which has $\mathbb R$ as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and nth root, th roots. The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.
## Using differential calculus
Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of that is 0 for . Another common example is the error function. More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for . Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by $e^x = \sum_^$. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential function, exponential and the trigonometric functions of a complex number.
## By recurrence
Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations. The factorial function on the nonnegative integers ($n\mapsto n!$) is a basic example, as it can be defined by the recurrence relation :$n!=n\left(n-1\right)!\quad\text\quad n>0,$ and the initial condition :$0!=1.$
# Representing a function
A Graph of a function, graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.
## Graphs and plots
Given a function $f\colon X\to Y,$ its ''graph'' is, formally, the set :$G=\.$ In the frequent case where and are subsets of the real numbers (or may be identified with such subsets, e.g. interval (mathematics), intervals), an element $\left(x,y\right)\in G$ may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a Plot (graphics), plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the ''graph of the function''. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function :$x\mapsto x^2,$ consisting of all points with coordinates $\left(x, x^2\right)$ for $x\in \R,$ yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function $x\mapsto x^2,$ with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates $\left(r,\theta\right) =\left(x,x^2\right),$ the plot obtained is Fermat's spiral.
## Tables
A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function $f\colon\^2 \to \mathbb$ defined as $f\left(x,y\right)=xy$ can be represented by the familiar multiplication table On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places: Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.
## Bar chart
Bar charts are often used for representing functions whose domain is a finite set, the natural numbers, or the integers. In this case, an element of the domain is represented by an interval (mathematics), interval of the -axis, and the corresponding value of the function, , is represented by a rectangle whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -axis).
# General properties
This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.
## Standard functions
There are a number of standard functions that occur frequently: * For every set , there is a unique function, called the from the empty set to . The graph of an empty function is the empty set.By definition, the graph of the empty function to is a subset of the Cartesian product , and this product is empty. The existence of the empty function is a convention that is needed for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. * For every set and every singleton set , there is a unique function from to , which maps every element of to . This is a surjection (see below) unless is the empty set. * Given a function $f\colon X\to Y,$ the ''canonical surjection'' of onto its image $f\left(X\right)=\$ is the function from to that maps to . * For every subset of a set , the inclusion map of into is the injective (see below) function that maps every element of to itself. * The identity function on a set , often denoted by , is the inclusion of into itself.
## Function composition
Given two functions $f\colon X\to Y$ and $g\colon Y\to Z$ such that the domain of is the codomain of , their ''composition'' is the function $g \circ f\colon X \rightarrow Z$ defined by :$\left(g \circ f\right)\left(x\right) = g\left(f\left(x\right)\right).$ That is, the value of $g \circ f$ is obtained by first applying to to obtain and then applying to the result to obtain . In the notation the function that is applied first is always written on the right. The composition $g\circ f$ is an operation (mathematics), operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both $g \circ f$ and $f \circ g$ satisfy these conditions, the composition is not necessarily commutative property, commutative, that is, the functions $g \circ f$ and $f \circ g$ need not be equal, but may deliver different values for the same argument. For example, let and , then $g\left(f\left(x\right)\right)=x^2+1$ and $f\left(g\left(x\right)\right) = \left(x+1\right)^2$ agree just for $x=0.$ The function composition is associative property, associative in the sense that, if one of $\left(h\circ g\right)\circ f$ and $h\circ \left(g\circ f\right)$ is defined, then the other is also defined, and they are equal. Thus, one writes :$h\circ g\circ f = \left(h\circ g\right)\circ f = h\circ \left(g\circ f\right).$ The identity functions $\operatorname_X$ and $\operatorname_Y$ are respectively a right identity and a left identity for functions from to . That is, if is a function with domain , and codomain , one has $f\circ \operatorname_X = \operatorname_Y \circ f = f.$ File:Function machine5.svg, A composite function ''g''(''f''(''x'')) can be visualized as the combination of two "machines". File:Example for a composition of two functions.svg, A simple example of a function composition File:Compfun.svg, Another composition. In this example, .
## Image and preimage
Let $f\colon X\to Y.$ The ''image'' under of an element of the domain is . If is any subset of , then the ''image'' of under , denoted , is the subset of the codomain consisting of all images of elements of , that is, :$f\left(A\right)=\.$ The ''image'' of is the image of the whole domain, that is, . It is also called the range of a function, range of , although the term ''range'' may also refer to the codomain.''Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology'', p. 15. ISO 80000-2 (ISO/IEC 2009-12-01) On the other hand, the ''inverse image'' or ''preimage'' under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by $f^\left(y\right)$ and is given by the equation :$f^\left(y\right) = \.$ Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to . It is denoted by $f^\left(B\right)$ and is given by the equation :$f^\left(B\right) = \.$ For example, the preimage of $\$ under the square function is the set $\$. By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage $f^\left(y\right)$ of an element of the codomain may be empty set, empty or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then $f^\left(0\right) = \mathbb$. If $f\colon X\to Y$ is a function, and are subsets of , and and are subsets of , then one has the following properties: * $A\subseteq B \Longrightarrow f\left(A\right)\subseteq f\left(B\right)$ * $C\subseteq D \Longrightarrow f^\left(C\right)\subseteq f^\left(D\right)$ * $A \subseteq f^\left(f\left(A\right)\right)$ * $C \supseteq f\left(f^\left(C\right)\right)$ * $f\left(f^\left(f\left(A\right)\right)\right)=f\left(A\right)$ * $f^\left(f\left(f^\left(C\right)\right)\right)=f^\left(C\right)$ The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber (mathematics), fiber of under . If a function has an inverse (see below), this inverse is denoted $f^.$ In this case $f^\left(C\right)$ may denote either the image by $f^$ or the preimage by of . This is not a problem, as these sets are equal. The notation $f\left(A\right)$ and $f^\left(C\right)$ may be ambiguous in the case of sets that contain some subsets as elements, such as $\.$ In this case, some care may be needed, for example, by using square brackets $f\left[A\right], f^\left[C\right]$ for images and preimages of subsets and ordinary parentheses for images and preimages of elements.
## Injective, surjective and bijective functions
Let $f\colon X\to Y$ be a function. The function is ''injective function, injective'' (or ''one-to-one'', or is an ''injection'') if for any two different elements and of . Equivalently, is injective if and only if, for any $y\in Y,$ the preimage $f^\left(y\right)$ contains at most one element. An empty function is always injective. If is not the empty set, then is injective if and only if there exists a function $g\colon Y\to X$ such that $g\circ f=\operatorname_X,$ that is, if has a left inverse function, left inverse. ''Proof'': If is injective, for defining , one chooses an element $x_0$ in (which exists as is supposed to be nonempty),The axiom of choice is not needed here, as the choice is done in a single set. and one defines by $g\left(y\right)=x$ if $y=f\left(x\right)$ and $g\left(y\right)=x_0$ if $y\not\in f\left(X\right).$ Conversely, if $g\circ f=\operatorname_X,$ and $y=f\left(x\right),$ then $x=g\left(y\right),$ and thus $f^\left(y\right)=\.$ The function is ''surjective'' (or ''onto'', or is a ''surjection'') if its range $f\left(X\right)$ equals its codomain $Y$, that is, if, for each element $y$ of the codomain, there exists some element $x$ of the domain such that $f\left(x\right) = y$ (in other words, the preimage $f^\left(y\right)$ of every $y\in Y$ is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then is surjective if and only if there exists a function $g\colon Y\to X$ such that $f\circ g=\operatorname_Y,$ that is, if has a right inverse function, right inverse. The axiom of choice is needed, because, if is surjective, one defines by $g\left(y\right)=x,$ where $x$ is an ''arbitrarily chosen'' element of $f^\left(y\right).$ The function is ''bijective'' (or is a ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective. That is, is bijective if, for any $y\in Y,$ the preimage $f^\left(y\right)$ contains exactly one element. The function is bijective if and only if it admits an inverse function, that is, a function $g\colon Y\to X$ such that $g\circ f=\operatorname_X$ and $f\circ g=\operatorname_Y.$ (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward). Every function $f\colon X\to Y$ may be factorization, factorized as the composition $i\circ s$ of a surjection followed by an injection, where is the canonical surjection of onto and is the canonical injection of into . This is the ''canonical factorization'' of . "One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Nicolas Bourbaki, Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps ''onto'' " differs from " maps ''into'' ", in that the former implies that is surjective, while the latter makes no assertion about the nature of . In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.
## Restriction and extension
If $f\colon X \to Y$ is a function and ''S'' is a subset of ''X'', then the ''restriction'' of $f$ to ''S'', denoted $f, _S$, is the function from ''S'' to ''Y'' defined by :$f, _S\left(x\right) = f\left(x\right)$ for all ''x'' in ''S''. Restrictions can be used to define partial inverse functions: if there is a subset ''S'' of the domain of a function $f$ such that $f, _S$ is injective, then the canonical surjection of $f, _S$ onto its image $f, _S\left(S\right) = f\left(S\right)$ is a bijection, and thus has an inverse function from $f\left(S\right)$ to ''S''. One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval (mathematics), interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called arccosine and is denoted . Function restriction may also be used for "gluing" functions together. Let $X=\bigcup_U_i$ be the decomposition of as a set union, union of subsets, and suppose that a function $f_i\colon U_i \to Y$ is defined on each $U_i$ such that for each pair $i, j$ of indices, the restrictions of $f_i$ and $f_j$ to $U_i \cap U_j$ are equal. Then this defines a unique function $f\colon X \to Y$ such that $f, _ = f_i$ for all . This is the way that functions on manifolds are defined. An ''extension'' of a function is a function such that is a restriction of . A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane. Here is another classical example of a function extension that is encountered when studying homography, homographies of the real line. A ''homography'' is a function $h\left(x\right)=\frac$ such that . Its domain is the set of all real numbers different from $-d/c,$ and its image is the set of all real numbers different from $a/c.$ If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting $h\left(\infty\right)=a/c$ and $h\left(-d/c\right)=\infty$.
# Multivariate function
A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. More formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. More generally, every mathematical operation is defined as a multivariate function. The Cartesian product $X_1\times\cdots\times X_n$ of sets $X_1, \ldots, X_n$ is the set of all -tuples $\left(x_1, \ldots, x_n\right)$ such that $x_i\in X_i$ for every with $1 \leq i \leq n$. Therefore, a function of variables is a function :$f\colon U\to Y,$ where the domain has the form :$U\subseteq X_1\times\cdots\times X_n.$ When using function notation, one usually omits the parentheses surrounding tuples, writing $f\left(x_1,x_2\right)$ instead of $f\left(\left(x_1,x_2\right)\right).$ In the case where all the $X_i$ are equal to the set $\R$ of real numbers, one has a function of several real variables. If the $X_i$ are equal to the set $\C$ of complex numbers, one has a function of several complex variables. It is common to also consider functions whose codomain is a product of sets. For example, Euclidean division maps every pair of integers with to a pair of integers called the ''quotient'' and the ''remainder'': :$\begin \text\colon\quad \Z\times \left(\Z\setminus \\right) &\to \Z\times\Z\\ \left(a,b\right) &\mapsto \left(\operatorname\left(a,b\right),\operatorname\left(a,b\right)\right). \end$ The codomain may also be a vector space. In this case, one talks of a vector-valued function. If the domain is contained in a Euclidean space, or more generally a manifold, a vector-valued function is often called a vector field.
# In calculus
The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued function, real-valued functions of a function of a real variable, real variable were considered, and all functions were assumed to be smooth function, smooth. But the definition was soon extended to #Multivariate function, functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined. Functions are now used throughout all areas of mathematics. In introductory calculus, when the word ''function'' is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.
## Real function
A ''real function'' is a real-valued function, real-valued function of a real variable, that is, a function whose codomain is the real number, field of real numbers and whose domain is a set of real numbers that contains an interval (mathematics), interval. In this section, these functions are simply called ''functions''. The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous function, continuous, differentiable function, differentiable, and even analytic function, analytic. This regularity insures that these functions can be visualized by their #Graph and plots, graphs. In this section, all functions are differentiable in some interval. Functions enjoy pointwise operations, that is, if and are functions, their sum, difference and product are functions defined by :$\begin \left(f+g\right)\left(x\right)&=f\left(x\right)+g\left(x\right)\\ \left(f-g\right)\left(x\right)&=f\left(x\right)-g\left(x\right)\\ \left(f\cdot g\right)\left(x\right)&=f\left(x\right)\cdot g\left(x\right)\\ \end.$ The domains of the resulting functions are the set intersection, intersection of the domains of and . The quotient of two functions is defined similarly by :$\frac fg\left(x\right)=\frac,$ but the domain of the resulting function is obtained by removing the zero of a function, zeros of from the intersection of the domains of and . The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function $x\mapsto \frac 1x,$ whose graph is a hyperbola, and whose domain is the whole real line except for 0. The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the function $x\mapsto\frac 1x$ is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for , is a differentiable function called the natural logarithm. A real function is monotonic function, monotonic in an interval if the sign of $\frac$ does not depend of the choice of and in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval , it has an inverse function, which is a real function with domain and image . This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function. Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation :$y\text{'}\text{'}+y=0$ such that :$\sin 0=0, \quad \cos 0=1, \quad\frac\left(0\right)=1, \quad\frac\left(0\right)=0.$
## Vector-valued function
When the elements of the codomain of a function are vector (mathematics and physics), vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function. Some vector-valued functions are defined on a subset of $\mathbb^n$ or other spaces that share geometric or topological properties of $\mathbb^n$, such as manifolds. These vector-valued functions are given the name ''vector fields''.
# Function space
In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued function, scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distribution (mathematics), distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topology, topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary differential equation, ordinary or partial differential equations result of the study of function spaces.
# Multi-valued functions
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood (mathematics), neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point $x_0,$ there are several possible starting values for the function. For example, in defining the square root as the inverse function of the square function, for any positive real number $x_0,$ there are two choices for the value of the square root, one of which is positive and denoted $\sqrt ,$ and another which is negative and denoted $-\sqrt .$ These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive , one value for 0 and no value for negative . In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps to a root of a function, root of $x^3-3x-y =0$ (see the figure on the right). For one may choose either $0, \sqrt 3,\text -\sqrt 3$ for . By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is . As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single ''multi-valued function'' of that has three values for , and only one value for and . Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets . There are generally two ways of solving the problem. One may define a function that is not continuous function, continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a ''multi-valued function'', which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.
# In the foundations of mathematics and set theory
The definition of a function that is given in this article requires the concept of set (mathematics), set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions. For example, the singleton set may be considered as a function $x\mapsto \.$ Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.; ; These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a Class (set theory), class. This theory includes the Von Neumann–Bernays–Gödel set theory#NBG's axiom of replacement, replacement axiom, which may be stated as: If is a set and is a function, then is a set.
# In computer science
In computer programming, a Function (programming), function is, in general, a piece of a computer program, which implementation, implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many programming languages every subroutine is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the computer memory. Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else is a function that takes three functions as arguments, and, depending on the result of the first function (''true'' or ''false''), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below). Except for computer-language terminology, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computable function, computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being μ-recursive function, general recursive functions, lambda calculus and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a ''computable function'' defines also the same functions. General recursive functions are partial functions from integers to integers that can be defined from * constant functions, * successor function, successor, and * projection function, projection functions via the operators * #Function composition, composition, * primitive recursion, and * μ operator, minimization. Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties: * a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...), * every sequence of symbols may be coded as a sequence of bits, * a bit sequence can be interpreted as the binary representation of an integer. Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of ''terms'' that are either variables, function definitions ('-terms), or applications of functions to terms. Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation. In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of ''type'' in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.
## Subpages
* List of types of functions * List of functions * Function fitting * Implicit function
## Generalizations
* Higher-order function * Homomorphism * Morphism * Microfunction * Distribution (mathematics), Distribution * Functor
## Related topics
* Associative array * Closed-form expression * Elementary function * Functional (mathematics), Functional * Functional decomposition * Functional predicate * Functional programming * Parametric equation * Set function * Simple function
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December 13, 2005, 13:24 Inlets - Outles activation time #1 Chucho Guest Posts: n/a Is there any way to define the time at which the inlets and outlets should be opened? Is there any way of defining a kind of algorithm that rules the opening and closing activity of inlets? any help would help, tx
December 14, 2005, 03:51 Re: Inlets - Outles activation time #2 TB Guest Posts: n/a why do you need that? Is it a transient simulation? Are you talking about "opening" boundary condition?
December 14, 2005, 15:07 Re: Inlets - Outles activation time #3 Chucho Guest Posts: n/a Yes it is a transient simulation, i have inlet and oulet boundary conditions that works as fans or grilles, and what i need is to simulate the activation of this fans and the opening and closing of grilles. So is there any way to set this times?
December 15, 2005, 07:09 Re: Inlets - Outles activation time #4 Martin Guest Posts: n/a that's pretty easy If you want to set e.g. inletvelocity to 10 m/s at 5 s you can write to the velocity at your inlet step(5-Time/1[s])*0[m/s] + step(Time/1[s]-5)*10 [m/s] step return 0 if the expression is <=0 and 1 if >0 you have to work without dimension within step, so you need Time/1[s]. bye martin
December 15, 2005, 08:31 Re: Inlets - Outles activation time #5 TB Guest Posts: n/a Another method....Use profile boundary condition if you have some experimental data at inlet/outlet...
December 20, 2005, 12:24 Re: Inlets - Outles activation time #6 Chucho Guest Posts: n/a When I look into the inlet properties the Profile Boundary option is disabled. How can I activate this option so I can enter experimental inlet data?
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